Fil-C on Nix

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the formal campaign

rocq/theories/Machine.v

Campaign 3's machine: Semantics.v's steps run over a Runtime (Runtime.v), and the first invariant of every run, that the runtime stays ready to allocate (formal.md, Campaign 3; rocq.md).

Semantics.v's step_rel takes any fresh base for an allocation and has no other thread. The machine of a runtime R is what one thread's program sees run under R: R chooses each allocation's base, R's free runs each call of libc's free (Run.v's Driver makes the genv so), and R may act before an instruction where R.at_safepoint holds and before a poll edge is taken, and nowhere else. That is what lets the validator keep evidence across every other instruction and edge.

Decisions

DecisionWhy
the runtime acts *before* an at_safepoint instruction or a poll edgethe instruction then runs on what the runtime left, as native code does after a pollcheck
an allocation with a negative extent has no machine stepfresh_allocation covers extents from 0; only a local's constant size can be negative, and valid refuses it (sizes 0 to 2^31-1), while heap sizes are read at l and so never negative
ending a local is not lifetime_step's freeRuntime.v's region_step lets any live object end, of any kind; the axioms' lifetime_step speaks of the runtime's free, which empties only heap objects
From Stdlib Require Import String ZArith List Bool Lia Relations
From Nanofil Require Import Term Syntax Values Memory Ops Semantics Runtime
Import ListNotations
Open Scope Z_scope

Case analysis

A tactic: split on every match in the hypothesis H whose scrutinee names no bound variable, outermost first, dropping the cases where H's computation gives no result or another one.

33Ltac cases H ≔
repeat match type of H withcontext [match ?x with _ ⇒ _ end]⇒destruct x eqn :? ; cbv beta iota zeta in H ; try discriminate Hend

What the program does to memory

Leaving memory as it is is kept.

42Lemma kept_refl
∀ m
keptm m
Explore proof 2 tactics

Initial goal

⊢
∀m : mem, keptm m

intros m

m :
mem
⊢
keptm m

split ; intros b reg′ E ; exists reg' ; auto | auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros m
split ; intros b reg′ E ; exists reg' ; auto | auto

Two kept changes in a row, from m1 to m2 and m2 to m3, are one kept change.

47Lemma kept_trans
∀ m1 m2 m3
keptm1 m2
keptm2 m3
keptm1 m3
Explore proof 20 tactics

Initial goal

⊢
∀m1 m2 m3 : mem, keptm1 m2 → keptm2 m3 → keptm1 m3

intros m1 m2 m3 R12 & O12 R23 & O23

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
keptm1 m3

split

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
region_stepm1 m3
focused goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
∀a : Z, ¬in_objectm1 a → m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

–intros b reg3 E3

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
b :
Z
reg3 :
region
E3 :
object_regionb m_regionsm3 = Somereg3
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg3 = reg ∨ r_lifereg = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
∀a : Z, ¬in_objectm1 a → m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

destruct R23_ _ E3 as reg2 & E2 & H23

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
b :
Z
reg3 :
region
E3 :
object_regionb m_regionsm3 = Somereg3
reg2 :
region
E2 :
object_regionb m_regionsm2 = Somereg2
H23 :
reg3 = reg2 ∨ r_lifereg2 = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg2; r_life ≔ Freed
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg3 = reg ∨ r_lifereg = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
∀a : Z, ¬in_objectm1 a → m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

destruct R12_ _ E2 as reg1 & E1 & H12

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
b :
Z
reg3 :
region
E3 :
object_regionb m_regionsm3 = Somereg3
reg2 :
region
E2 :
object_regionb m_regionsm2 = Somereg2
H23 :
reg3 = reg2 ∨ r_lifereg2 = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg2; r_life ≔ Freed
reg1 :
region
E1 :
object_regionb m_regionsm1 = Somereg1
H12 :
reg2 = reg1 ∨ r_lifereg1 = Live ∧ reg2 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg3 = reg ∨ r_lifereg = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
∀a : Z, ¬in_objectm1 a → m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

exists reg1

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
b :
Z
reg3 :
region
E3 :
object_regionb m_regionsm3 = Somereg3
reg2 :
region
E2 :
object_regionb m_regionsm2 = Somereg2
H23 :
reg3 = reg2 ∨ r_lifereg2 = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg2; r_life ≔ Freed
reg1 :
region
E1 :
object_regionb m_regionsm1 = Somereg1
H12 :
reg2 = reg1 ∨ r_lifereg1 = Live ∧ reg2 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
⊢
object_regionb m_regionsm1 = Somereg1 ∧ reg3 = reg1 ∨ r_lifereg1 = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
∀a : Z, ¬in_objectm1 a → m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

split ; auto

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
b :
Z
reg3 :
region
E3 :
object_regionb m_regionsm3 = Somereg3
reg2 :
region
E2 :
object_regionb m_regionsm2 = Somereg2
H23 :
reg3 = reg2 ∨ r_lifereg2 = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg2; r_life ≔ Freed
reg1 :
region
E1 :
object_regionb m_regionsm1 = Somereg1
H12 :
reg2 = reg1 ∨ r_lifereg1 = Live ∧ reg2 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
⊢
reg3 = reg1 ∨ r_lifereg1 = Live ∧ reg3 = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
∀a : Z, ¬in_objectm1 a → m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

destruct H23 as → | L2 & → ; destruct H12 as → | L1 & → ; auto

deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
⊢
∀a : Z, ¬in_objectm1 a → m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

–intros a Hn

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

assert Hn2 : ¬in_objectm2 a

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
⊢
¬in_objectm2 a
focused goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

{intros b & reg2 & E2 & Hr

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg2 :
region
E2 :
object_regionb m_regionsm2 = Somereg2
Hr :
b ≤ a < r_endreg2
⊢
False
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

destruct R12_ _ E2 as reg1 & E1 & → | _ & →

3 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg1 :
region
Hr :
b ≤ a < r_endreg1
E2 :
object_regionb m_regionsm2 = Somereg1
E1 :
object_regionb m_regionsm1 = Somereg1
⊢
False
focused goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg1 :
region
Hr :
b ≤ a < r_endr_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E2 :
object_regionb m_regionsm2 = Somer_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E1 :
object_regionb m_regionsm1 = Somereg1
⊢
False
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

–apply Hn

3 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg1 :
region
Hr :
b ≤ a < r_endreg1
E2 :
object_regionb m_regionsm2 = Somereg1
E1 :
object_regionb m_regionsm1 = Somereg1
⊢
in_objectm1 a
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg1 :
region
Hr :
b ≤ a < r_endr_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E2 :
object_regionb m_regionsm2 = Somer_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E1 :
object_regionb m_regionsm1 = Somereg1
⊢
False
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

exists b, reg1

3 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg1 :
region
Hr :
b ≤ a < r_endreg1
E2 :
object_regionb m_regionsm2 = Somereg1
E1 :
object_regionb m_regionsm1 = Somereg1
⊢
object_regionb m_regionsm1 = Somereg1 ∧ b ≤ a < r_endreg1
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg1 :
region
Hr :
b ≤ a < r_endr_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E2 :
object_regionb m_regionsm2 = Somer_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E1 :
object_regionb m_regionsm1 = Somereg1
⊢
False
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

auto

2 open

deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg1 :
region
Hr :
b ≤ a < r_endr_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E2 :
object_regionb m_regionsm2 = Somer_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E1 :
object_regionb m_regionsm1 = Somereg1
⊢
False
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

–simpl in Hr

2 open

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
b :
Z
reg1 :
region
Hr :
b ≤ a < b
E2 :
object_regionb m_regionsm2 = Somer_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg1; r_life ≔ Freed
E1 :
object_regionb m_regionsm1 = Somereg1
⊢
False
deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

lia

deferred goal
m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

destruct O12a Hn as B1 A1

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
B1 :
m_bytesm2 a = m_bytesm1 a
A1 :
m_auxm2 a = m_auxm1 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

destruct O23a Hn2 as B2 A2

m1, m2, m3 :
mem
R12 :
region_stepm1 m2
O12 :
∀a : Z, ¬in_objectm1 a → m_bytesm2 a = m_bytesm1 a ∧ m_auxm2 a = m_auxm1 a
R23 :
region_stepm2 m3
O23 :
∀a : Z, ¬in_objectm2 a → m_bytesm3 a = m_bytesm2 a ∧ m_auxm3 a = m_auxm2 a
a :
Z
Hn :
¬in_objectm1 a
Hn2 :
¬in_objectm2 a
B1 :
m_bytesm2 a = m_bytesm1 a
A1 :
m_auxm2 a = m_auxm1 a
B2 :
m_bytesm3 a = m_bytesm2 a
A2 :
m_auxm3 a = m_auxm2 a
⊢
m_bytesm3 a = m_bytesm1 a ∧ m_auxm3 a = m_auxm1 a

split ; congruence

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros m1 m2 m3 R12 & O12 R23 & O23
split
–intros b reg3 E3
destruct R23_ _ E3 as reg2 & E2 & H23
destruct R12_ _ E2 as reg1 & E1 & H12
exists reg1
split ; auto
destruct H23 as → | L2 & → ; destruct H12 as → | L1 & → ; auto
–intros a Hn
assert Hn2 : ¬in_objectm2 a
{intros b & reg2 & E2 & Hr
destruct R12_ _ E2 as reg1 & E1 & → | _ & →
–apply Hn
exists b, reg1
auto
–simpl in Hr
lia
destruct O12a Hn as B1 A1
destruct O23a Hn2 as B2 A2
split ; congruence

A change the runtime's free may make (lifetime_step) is one the program may make (region_step): it ends only live objects.

62Lemma lifetime_region_step
∀ m m′
lifetime_stepm m′
region_stepm m′
Explore proof 5 tactics

Initial goal

⊢
∀m m′ : mem, lifetime_stepm m′ → region_stepm m′

intros m m′ H b reg′ E

m, m′ :
mem
H :
lifetime_stepm m′
b :
Z
reg′ :
region
E :
object_regionb m_regionsm′ = Somereg′
⊢
∃reg : region, object_regionb m_regionsm = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

destruct H_ _ E as reg & E0 & → | Hk & Hl & → ; exists reg ; split ; auto

m, m′ :
mem
H :
lifetime_stepm m′
b :
Z
E :
object_regionb m_regionsm′ = Somer_base ≔ b; r_end ≔ b; r_kind ≔ Heap; r_life ≔ Freed
reg :
region
E0 :
object_regionb m_regionsm = Somereg
Hk :
r_kindreg = Heap
Hl :
r_lifereg = Live
⊢
r_base ≔ b; r_end ≔ b; r_kind ≔ Heap; r_life ≔ Freed = reg ∨ r_lifereg = Live ∧ r_base ≔ b; r_end ≔ b; r_kind ≔ Heap; r_life ≔ Freed = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

right

m, m′ :
mem
H :
lifetime_stepm m′
b :
Z
E :
object_regionb m_regionsm′ = Somer_base ≔ b; r_end ≔ b; r_kind ≔ Heap; r_life ≔ Freed
reg :
region
E0 :
object_regionb m_regionsm = Somereg
Hk :
r_kindreg = Heap
Hl :
r_lifereg = Live
⊢
r_lifereg = Live ∧ r_base ≔ b; r_end ≔ b; r_kind ≔ Heap; r_life ≔ Freed = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

rewrite Hk

m, m′ :
mem
H :
lifetime_stepm m′
b :
Z
E :
object_regionb m_regionsm′ = Somer_base ≔ b; r_end ≔ b; r_kind ≔ Heap; r_life ≔ Freed
reg :
region
E0 :
object_regionb m_regionsm = Somereg
Hk :
r_kindreg = Heap
Hl :
r_lifereg = Live
⊢
r_lifereg = Live ∧ r_base ≔ b; r_end ≔ b; r_kind ≔ Heap; r_life ≔ Freed = r_base ≔ b; r_end ≔ b; r_kind ≔ Heap; r_life ≔ Freed

auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros m m′ H b reg′ E
destruct H_ _ E as reg & E0 & → | Hk & Hl & → ; exists reg ; split ; auto
right
rewrite Hk
auto

Every memory change the function free makes is kept: what a free that meets the lifetime and retention axioms does.

71Definition free_keepsfree : word → mem → optionresunit × mem : Prop ≔
∀x m r m′, freex m = Somer, m′ → keptm m′

A successful access check covers its bytes with one object's bounds: if the check of size bytes at a passes, every address from a up to a + size is inside an object.

77Lemma access_in
∀ m a c size r align u b
access_resm a c size r align = SomeRValu
a ≤ b < a + size
in_objectm b
Explore proof 3 tactics

Initial goal

⊢
∀m : mem a : Z c : cap size : Z r : rights align : Z u : unit, access_resm a c size r align = SomeRValu → ∀b : Z, a ≤ b < a + size → in_objectm b

unfold access_res , cap_access_res

⊢
∀ m : mem a : Z c : cap size : Z r : rights align : Z u : unit  ,
match c with
CNone
⇒
SomeRTrapNullCapability
Objbase
⇒
let* reg ≔ object_regionbase m_regionsm in
Some
base ≤? a && a + size ≤? r_endreg
→
aligned_resa align
else
RTrapOutOfBounds
Robase
⇒
let* reg ≔ object_regionbase m_regionsm in
Some
base ≤? a && a + size ≤? r_endreg
→
readonly_access_resr a align
else
RTrapOutOfBounds
Fn_
⇒
SomeRTrapOutOfBounds
External_
⇒
None
=
SomeRValu
→
∀b : Z, a ≤ b < a + size → in_objectm b

intros m a c size r align u H b Hb

m :
mem
a :
Z
c :
cap
size :
Z
r :
rights
align :
Z
u :
unit
H :
match c with
CNone
⇒
SomeRTrapNullCapability
Objbase
⇒
let* reg ≔ object_regionbase m_regionsm in
Some
base ≤? a && a + size ≤? r_endreg
→
aligned_resa align
else
RTrapOutOfBounds
Robase
⇒
let* reg ≔ object_regionbase m_regionsm in
Some
base ≤? a && a + size ≤? r_endreg
→
readonly_access_resr a align
else
RTrapOutOfBounds
Fn_
⇒
SomeRTrapOutOfBounds
External_
⇒
None
=
SomeRValu
b :
Z
Hb :
a ≤ b < a + size
⊢
in_objectm b

destruct c as | base | base | f | n ; try discriminate ; destruct object_regionbase m_regionsm as reg | eqn: E ; try discriminate ; destruct base ≤? a && a + size ≤? r_endreg eqn: Eb ; try discriminate ; apply andb_true_iff in Eb as E1 E2 ; apply Z.leb_le in E1 ; apply Z.leb_le in E2 ; exists base, reg ; split ; auto ; lia

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold access_res , cap_access_res
intros m a c size r align u H b Hb
destruct c as | base | base | f | n ; try discriminate ; destruct object_regionbase m_regionsm as reg | eqn: E ; try discriminate ; destruct base ≤? a && a + size ≤? r_endreg eqn: Eb ; try discriminate ; apply andb_true_iff in Eb as E1 E2 ; apply Z.leb_le in E1 ; apply Z.leb_le in E2 ; exists base, reg ; split ; auto ; lia

Writing size bytes at a is kept if they are all inside objects.

90Lemma store_bytes_kept
∀ m a size x
∀b, a ≤ b < a + size → in_objectm b
keptm store_bytesm a size x
Explore proof 15 tactics

Initial goal

⊢
∀m : mem a size x : Z, ∀b : Z, a ≤ b < a + size → in_objectm b → keptm store_bytesm a size x

intros m a size x Hin

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
⊢
keptm store_bytesm a size x

split

2 open

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
⊢
region_stepm store_bytesm a size x
focused goal
m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
⊢
∀a0 : Z, ¬in_objectm a0 → m_bytesstore_bytesm a size x a0 = m_bytesm a0 ∧ m_auxstore_bytesm a size x a0 = m_auxm a0

–intros b reg′ E

2 open

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
reg′ :
region
E :
object_regionb m_regionsstore_bytesm a size x = Somereg′
⊢
∃reg : region, object_regionb m_regionsm = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed
deferred goal
m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
⊢
∀a0 : Z, ¬in_objectm a0 → m_bytesstore_bytesm a size x a0 = m_bytesm a0 ∧ m_auxstore_bytesm a size x a0 = m_auxm a0

exists reg′

2 open

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
reg′ :
region
E :
object_regionb m_regionsstore_bytesm a size x = Somereg′
⊢
object_regionb m_regionsm = Somereg′ ∧ reg′ = reg′ ∨ r_lifereg′ = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg′; r_life ≔ Freed
deferred goal
m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
⊢
∀a0 : Z, ¬in_objectm a0 → m_bytesstore_bytesm a size x a0 = m_bytesm a0 ∧ m_auxstore_bytesm a size x a0 = m_auxm a0

split ; exact E | auto

deferred goal
m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
⊢
∀a0 : Z, ¬in_objectm a0 → m_bytesstore_bytesm a size x a0 = m_bytesm a0 ∧ m_auxstore_bytesm a size x a0 = m_auxm a0

–intros b Hn

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
⊢
m_bytesstore_bytesm a size x b = m_bytesm b ∧ m_auxstore_bytesm a size x b = m_auxm b

simpl

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
⊢
a ≤? b && b <? a + size
→
byte_ofx b − a
else
m_bytesm b
=
m_bytesm b
∧
m_auxm b = m_auxm b

split ; auto

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
⊢
a ≤? b && b <? a + size
→
byte_ofx b − a
else
m_bytesm b
=
m_bytesm b

destruct a ≤? b && b <? a + size eqn: E ; auto

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
E :
a ≤? b && b <? a + size = true
⊢
byte_ofx b − a = m_bytesm b

exfalso

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
E :
a ≤? b && b <? a + size = true
⊢
False

apply Hn , Hin

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
E :
a ≤? b && b <? a + size = true
⊢
a ≤ b < a + size

apply andb_true_iff in E as E1 E2

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
E1 :
a ≤? b = true
E2 :
b <? a + size = true
⊢
a ≤ b < a + size

apply Z.leb_le in E1

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
E1 :
a ≤ b
E2 :
b <? a + size = true
⊢
a ≤ b < a + size

apply Z.ltb_lt in E2

m :
mem
a, size, x :
Z
Hin :
∀b : Z, a ≤ b < a + size → in_objectm b
b :
Z
Hn :
¬in_objectm b
E1 :
a ≤ b
E2 :
b < a + size
⊢
a ≤ b < a + size

lia

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros m a size x Hin
split
–intros b reg′ E
exists reg′
split ; exact E | auto
–intros b Hn
simpl
split ; auto
destruct a ≤? b && b <? a + size eqn: E ; auto
exfalso
apply Hn , Hin
apply andb_true_iff in E as E1 E2
apply Z.leb_le in E1
apply Z.ltb_lt in E2
lia

Storing a capability in the aux at a is kept if a is inside an object.

103Lemma store_aux_kept
∀ m a c
in_objectm a
keptm store_auxm a c
Explore proof 12 tactics

Initial goal

⊢
∀m : mem a : Z c : cap, in_objectm a → keptm store_auxm a c

intros m a c Hin

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
⊢
keptm store_auxm a c

split

2 open

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
⊢
region_stepm store_auxm a c
focused goal
m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
⊢
∀a0 : Z, ¬in_objectm a0 → m_bytesstore_auxm a c a0 = m_bytesm a0 ∧ m_auxstore_auxm a c a0 = m_auxm a0

–intros b reg′ E

2 open

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
b :
Z
reg′ :
region
E :
object_regionb m_regionsstore_auxm a c = Somereg′
⊢
∃reg : region, object_regionb m_regionsm = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed
deferred goal
m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
⊢
∀a0 : Z, ¬in_objectm a0 → m_bytesstore_auxm a c a0 = m_bytesm a0 ∧ m_auxstore_auxm a c a0 = m_auxm a0

exists reg′

2 open

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
b :
Z
reg′ :
region
E :
object_regionb m_regionsstore_auxm a c = Somereg′
⊢
object_regionb m_regionsm = Somereg′ ∧ reg′ = reg′ ∨ r_lifereg′ = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg′; r_life ≔ Freed
deferred goal
m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
⊢
∀a0 : Z, ¬in_objectm a0 → m_bytesstore_auxm a c a0 = m_bytesm a0 ∧ m_auxstore_auxm a c a0 = m_auxm a0

split ; exact E | auto

deferred goal
m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
⊢
∀a0 : Z, ¬in_objectm a0 → m_bytesstore_auxm a c a0 = m_bytesm a0 ∧ m_auxstore_auxm a c a0 = m_auxm a0

–intros b Hn

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
b :
Z
Hn :
¬in_objectm b
⊢
m_bytesstore_auxm a c b = m_bytesm b ∧ m_auxstore_auxm a c b = m_auxm b

simpl

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
b :
Z
Hn :
¬in_objectm b
⊢
m_bytesm b = m_bytesm b
∧
b =? a
→
c
else
m_auxm b
=
m_auxm b

split ; auto

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
b :
Z
Hn :
¬in_objectm b
⊢
b =? a
→
c
else
m_auxm b
=
m_auxm b

destruct Z.eqbb a eqn: E ; auto

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
b :
Z
Hn :
¬in_objectm b
E :
b =? a = true
⊢
c = m_auxm b

apply Z.eqb_eq in E

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
b :
Z
Hn :
¬in_objectm b
E :
b = a
⊢
c = m_auxm b

subst

m :
mem
a :
Z
c :
cap
Hin :
in_objectm a
Hn :
¬in_objectm a
⊢
c = m_auxm a

contradiction

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros m a c Hin
split
–intros b reg′ E
exists reg′
split ; exact E | auto
–intros b Hn
simpl
split ; auto
destruct Z.eqbb a eqn: E ; auto
apply Z.eqb_eq in E
subst
contradiction

Writes leave the objects where they were: an address inside an object before a byte write is inside one after it.

114Lemma in_object_store_bytes
∀ m a size x b
in_objectm b
in_objectstore_bytesm a size x b
Explore proof 1 tactics

Initial goal

⊢
∀m : mem a size x b : Z, in_objectm b → in_objectstore_bytesm a size x b

auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
auto

The same for a capability stored in the aux.

119Lemma in_object_store_aux
∀ m a c b
in_objectm b
in_objectstore_auxm a c b
Explore proof 1 tactics

Initial goal

⊢
∀m : mem a : Z c : cap b : Z, in_objectm b → in_objectstore_auxm a c b

auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
auto

A store as store_res/7 does it is kept: it writes the bytes, and a pointer's capability, only after its access check passed.

124Lemma store_res_kept
∀ w m a c x r m′
store_resw m a c x = Somer, m′
keptm m′
Explore proof 3 tactics

Initial goal

⊢
∀w : width m : mem a : Z c : cap x : word r : resunit m′ : mem, store_resw m a c x = Somer, m′ → keptm m′

unfold store_res

⊢
∀ w : width m : mem a : Z c : cap x : word r : resunit m′ : mem  ,
match w with
Wp
⇒
match x with
WPtrv
⇒
let* chk ≔ access_resm a c width_bytesw Write 8 in
Some
chk
match chk with
RVal_
⇒
store_auxstore_bytesm a width_bytesw ptr_addrv a ptr_capv
_
⇒
m
_
⇒
None
_
⇒
match x with
WIntv
⇒
let* chk ≔
access_resm a c width_bytesw Write
match w with
Wx
⇒
8
_
⇒
1
in
Some
chk
match chk with
RVal_
⇒
store_bytesm a width_bytesw v
_
⇒
m
_
⇒
None
=
Somer, m′
→
keptm m′

intros w m a c x r m′ H

w :
width
m :
mem
a :
Z
c :
cap
x :
word
r :
resunit
m′ :
mem
H :
match w with
Wp
⇒
match x with
WPtrv
⇒
let* chk ≔ access_resm a c width_bytesw Write 8 in
Some
chk
match chk with
RVal_
⇒
store_auxstore_bytesm a width_bytesw ptr_addrv a ptr_capv
_
⇒
m
_
⇒
None
_
⇒
match x with
WIntv
⇒
let* chk ≔
access_resm a c width_bytesw Write
match w with
Wx
⇒
8
_
⇒
1
in
Some
chk
match chk with
RVal_
⇒
store_bytesm a width_bytesw v
_
⇒
m
_
⇒
None
=
Somer, m′
⊢
keptm m′

destruct w , x ; try discriminate ; cases H ; inversion H ; subst ; try apply kept_refl;match goal withE : access_res_ _ _ _ _ _ = SomeRVal_ |- kept_ store_aux_ _ _⇒eapply kept_trans ; | apply store_aux_kept ; apply store_bytes_kept ; intros b Hb ; eapply access_in ; exact E | exact Hb | apply in_object_store_bytes ; eapply access_in ; exact E | simpl ; liaE : access_res_ _ _ _ _ _ = SomeRVal_ |- _⇒apply store_bytes_kept ; intros b Hb ; eapply access_in ; exact E | exact Hbend

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold store_res
intros w m a c x r m′ H
destruct w , x ; try discriminate ; cases H ; inversion H ; subst ; try apply kept_refl;match goal withE : access_res_ _ _ _ _ _ = SomeRVal_ |- kept_ store_aux_ _ _⇒eapply kept_trans ; | apply store_aux_kept ; apply store_bytes_kept ; intros b Hb ; eapply access_in ; exact E | exact Hb | apply in_object_store_bytes ; eapply access_in ; exact E | simpl ; liaE : access_res_ _ _ _ _ _ = SomeRVal_ |- _⇒apply store_bytes_kept ; intros b Hb ; eapply access_in ; exact E | exact Hbend

A fold of memory changes that starts from no memory ends with none.

140Lemma fold_left_none
∀ A : Type f : mem → A → optionmem xs
fold_left
λ acc x  ⇒
let* m1 ≔ acc in
fm1 x
xs
None
=
None
Explore proof 2 tactics

Initial goal

⊢
∀ A : Type f : mem → A → optionmem xs : listA  ,
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
None
=
None

intros A f xs

A :
Type
f :
mem → A → optionmem
xs :
listA
⊢
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
None
=
None

induction xs ; simpl ; auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros A f xs
induction xs ; simpl ; auto

A fold of memory changes f over xs, from m0, is kept if each change is kept and keeps P, where P holds of m0.

146Lemma fold_kept
∀ A : Type P : mem → Prop f : mem → A → optionmem xs m0 m′
Pm0
∀m1 x m2, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
fold_left
λ acc x  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
keptm0 m′
Explore proof 10 tactics

Initial goal

⊢
∀ A : Type P : mem → Prop f : mem → A → optionmem xs : listA m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′

intros A P f xs

A :
Type
P :
mem → Prop
f :
mem → A → optionmem
xs :
listA
⊢
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′

induction xs as | x xs IH ; simpl ; intros m0 m′ H0 Hf H

2 open

A :
Type
P :
mem → Prop
f :
mem → A → optionmem
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
H :
Somem0 = Somem′
⊢
keptm0 m′
focused goal
A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
fm0 x
=
Somem′
⊢
keptm0 m′

–inversion H ; subst

2 open

A :
Type
P :
mem → Prop
f :
mem → A → optionmem
m′ :
mem
H0 :
Pm′
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
H :
Somem′ = Somem′
⊢
keptm′ m′
deferred goal
A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
fm0 x
=
Somem′
⊢
keptm0 m′

apply kept_refl

deferred goal
A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
fm0 x
=
Somem′
⊢
keptm0 m′

–destruct fm0 x as m1 | eqn: E

2 open

A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
m1 :
mem
E :
fm0 x = Somem1
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem1
=
Somem′
⊢
keptm0 m′
focused goal
A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
E :
fm0 x = None
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
None
=
Somem′
⊢
keptm0 m′

+destruct Hf_ _ _ H0 E as H1 K1

2 open

A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
m1 :
mem
E :
fm0 x = Somem1
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem1
=
Somem′
H1 :
Pm1
K1 :
keptm0 m1
⊢
keptm0 m′
deferred goal
A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
E :
fm0 x = None
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
None
=
Somem′
⊢
keptm0 m′

eapply kept_trans ; exact K1 |

2 open

A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
m1 :
mem
E :
fm0 x = Somem1
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem1
=
Somem′
H1 :
Pm1
K1 :
keptm0 m1
⊢
keptm1 m′
deferred goal
A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
E :
fm0 x = None
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
None
=
Somem′
⊢
keptm0 m′

eapply IH ; eauto

deferred goal
A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
E :
fm0 x = None
H :
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
None
=
Somem′
⊢
keptm0 m′

+rewrite fold_left_none in H

A :
Type
P :
mem → Prop
f :
mem → A → optionmem
x :
A
xs :
listA
IH :
∀ m0 m′ : mem  ,
Pm0
→
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
→
fold_left
λ acc : optionmem x : A  ⇒
let* m1 ≔ acc in
fm1 x
xs
Somem0
=
Somem′
→
keptm0 m′
m0, m′ :
mem
H0 :
Pm0
Hf :
∀m1 : mem x : A m2 : mem, Pm1 → fm1 x = Somem2 → Pm2 ∧ keptm1 m2
E :
fm0 x = None
H :
None = Somem′
⊢
keptm0 m′

discriminate

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros A P f xs
induction xs as | x xs IH ; simpl ; intros m0 m′ H0 Hf H
–inversion H ; subst
apply kept_refl
–destruct fm0 x as m1 | eqn: E
+destruct Hf_ _ _ H0 E as H1 K1
eapply kept_trans ; exact K1 |
eapply IH ; eauto
+rewrite fold_left_none in H
discriminate

A pointer store's effects es (pointer_store_effects/4) at the address a are kept, if the access check of its 8 bytes passed.

161Lemma pointer_store_effects_kept
∀ a x m es m′
∀b, a ≤ b < a + 8 → in_objectm b
pointer_store_effectsa x m es = Somem′
keptm m′
Explore proof 20 tactics

Initial goal

⊢
∀a : Z x : ptr m : mem es : listeffect m′ : mem, ∀b : Z, a ≤ b < a + 8 → in_objectm b → pointer_store_effectsa x m es = Somem′ → keptm m′

intros a x m es m′ Hin H

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
⊢
keptm m′

apply fold_kepteffect λm1 ⇒ m_regionsm1 = m_regionsm pointer_store_effecta x es m m′ ; auto

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
⊢
∀m1 : mem x0 : effect m2 : mem, m_regionsm1 = m_regionsm → pointer_store_effecta x m1 x0 = Somem2 → m_regionsm2 = m_regionsm ∧ keptm1 m2

intros m1 e m2 Hr E

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
⊢
m_regionsm2 = m_regionsm ∧ keptm1 m2

assert Hin1 : ∀b, a ≤ b < a + 8 → in_objectm1 b

2 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
⊢
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
focused goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
m_regionsm2 = m_regionsm ∧ keptm1 m2

{intros b Hb

2 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
b :
Z
Hb :
a ≤ b < a + 8
⊢
in_objectm1 b
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
m_regionsm2 = m_regionsm ∧ keptm1 m2

unfold in_object

2 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
b :
Z
Hb :
a ≤ b < a + 8
⊢
∃b0 : Z reg : region, object_regionb0 m_regionsm1 = Somereg ∧ b0 ≤ b < r_endreg
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
m_regionsm2 = m_regionsm ∧ keptm1 m2

rewrite Hr

2 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
b :
Z
Hb :
a ≤ b < a + 8
⊢
∃b0 : Z reg : region, object_regionb0 m_regionsm = Somereg ∧ b0 ≤ b < r_endreg
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
m_regionsm2 = m_regionsm ∧ keptm1 m2

apply Hin

2 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
b :
Z
Hb :
a ≤ b < a + 8
⊢
a ≤ b < a + 8
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
m_regionsm2 = m_regionsm ∧ keptm1 m2

auto

deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
e :
effect
m2 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
pointer_store_effecta x m1 e = Somem2
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
m_regionsm2 = m_regionsm ∧ keptm1 m2

destruct e ; simpl in E ; try discriminate ; inversion E ; subst ; split ; auto using kept_refl

3 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_auxm1 a ptr_capx = Somestore_auxm1 a ptr_capx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_auxm1 a ptr_capx
focused goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesm1 a 8 ptr_addrx = Somestore_bytesm1 a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_bytesm1 a 8 ptr_addrx
focused goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx

–apply store_aux_kept , Hin1

3 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_auxm1 a ptr_capx = Somestore_auxm1 a ptr_capx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
a ≤ a < a + 8
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesm1 a 8 ptr_addrx = Somestore_bytesm1 a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_bytesm1 a 8 ptr_addrx
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx

lia

2 open

deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesm1 a 8 ptr_addrx = Somestore_bytesm1 a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_bytesm1 a 8 ptr_addrx
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx

–apply store_bytes_kept

2 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesm1 a 8 ptr_addrx = Somestore_bytesm1 a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx

exact Hin1

deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx

–eapply kept_trans ; | apply store_bytes_kept

2 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
keptm1 store_auxm1 a ptr_capx
focused goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
∀b : Z, a ≤ b < a + 8 → in_objectstore_auxm1 a ptr_capx b

+apply store_aux_kept , Hin1

2 open

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
a ≤ a < a + 8
deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
∀b : Z, a ≤ b < a + 8 → in_objectstore_auxm1 a ptr_capx b

lia

deferred goal
a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
⊢
∀b : Z, a ≤ b < a + 8 → in_objectstore_auxm1 a ptr_capx b

+intros b Hb

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
b :
Z
Hb :
a ≤ b < a + 8
⊢
in_objectstore_auxm1 a ptr_capx b

apply in_object_store_aux , Hin1

a :
Z
x :
ptr
m :
mem
es :
listeffect
m′ :
mem
Hin :
∀b : Z, a ≤ b < a + 8 → in_objectm b
H :
pointer_store_effectsa x m es = Somem′
m1 :
mem
Hr :
m_regionsm1 = m_regionsm
E :
Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx = Somestore_bytesstore_auxm1 a ptr_capx a 8 ptr_addrx
Hin1 :
∀b : Z, a ≤ b < a + 8 → in_objectm1 b
b :
Z
Hb :
a ≤ b < a + 8
⊢
a ≤ b < a + 8

auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros a x m es m′ Hin H
apply fold_kepteffect λm1 ⇒ m_regionsm1 = m_regionsm pointer_store_effecta x es m m′ ; auto
intros m1 e m2 Hr E
assert Hin1 : ∀b, a ≤ b < a + 8 → in_objectm1 b
{intros b Hb
unfold in_object
rewrite Hr
apply Hin
auto
destruct e ; simpl in E ; try discriminate ; inversion E ; subst ; split ; auto using kept_refl
–apply store_aux_kept , Hin1
lia
–apply store_bytes_kept
exact Hin1
–eapply kept_trans ; | apply store_bytes_kept
+apply store_aux_kept , Hin1
lia
+intros b Hb
apply in_object_store_aux , Hin1
auto

Reading a variadic argument (va_arg_res/4) is kept: its one write is a checked store.

181Lemma va_arg_res_kept
∀ k p m r m′
va_arg_resk p m = Somer, m′
keptm m′
Explore proof 3 tactics

Initial goal

⊢
∀k : class p : ptr m : mem r : resword m′ : mem, va_arg_resk p m = Somer, m′ → keptm m′

unfold va_arg_res

⊢
∀ k : class p : ptr m : mem r : resword m′ : mem  ,
match access_resm ptr_addrp ptr_capp 8 Read 8 with
SomeRVal_
⇒
match va_cellk with
Somestep, align
⇒
let* w ≔ class_widthk in
match access_resm ptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align ptr_capptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp step Read 8 with
SomeRVal_
⇒
match store_resWl m ptr_addrp ptr_capp WIntptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align + step with
SomeRVal_, m1
⇒
match k with
P
⇒
SomeRValWPtrptr_addr ≔ load_bytesm1 ptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align 8; ptr_cap ≔ aux_atm1 ptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align, m1
_
⇒
SomeRValWIntload_bytesm1 ptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align width_bytesw, m1
SomeRTrapwhy, m1
⇒
SomeRTrapwhy, m1
_
⇒
None
SomeRTrapwhy
⇒
SomeRTrapwhy, m
_
⇒
None
None
⇒
None
SomeRTrapwhy
⇒
SomeRTrapwhy, m
_
⇒
None
=
Somer, m′
→
keptm m′

intros k p m r m′ H

k :
class
p :
ptr
m :
mem
r :
resword
m′ :
mem
H :
match access_resm ptr_addrp ptr_capp 8 Read 8 with
SomeRVal_
⇒
match va_cellk with
Somestep, align
⇒
let* w ≔ class_widthk in
match access_resm ptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align ptr_capptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp step Read 8 with
SomeRVal_
⇒
match store_resWl m ptr_addrp ptr_capp WIntptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align + step with
SomeRVal_, m1
⇒
match k with
P
⇒
SomeRValWPtrptr_addr ≔ load_bytesm1 ptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align 8; ptr_cap ≔ aux_atm1 ptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align, m1
_
⇒
SomeRValWIntload_bytesm1 ptr_addrptr_addr ≔ load_bytesm ptr_addrp 8; ptr_cap ≔ aux_atm ptr_addrp + align − 1 / align × align width_bytesw, m1
SomeRTrapwhy, m1
⇒
SomeRTrapwhy, m1
_
⇒
None
SomeRTrapwhy
⇒
SomeRTrapwhy, m
_
⇒
None
None
⇒
None
SomeRTrapwhy
⇒
SomeRTrapwhy, m
_
⇒
None
=
Somer, m′
⊢
keptm m′

cases H;match type of H withSome_, ?x = Some_, ?y⇒assert x = y as ← by congruenceend;first|apply kept_refl|match goal withE : store_res_ _ _ _ _ = Some_ |- _⇒exact store_res_kept_ _ _ _ _ _ _ Eend

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold va_arg_res
intros k p m r m′ H
cases H;match type of H withSome_, ?x = Some_, ?y⇒assert x = y as ← by congruenceend;first|apply kept_refl|match goal withE : store_res_ _ _ _ _ = Some_ |- _⇒exact store_res_kept_ _ _ _ _ _ _ Eend

congruence, not inversion: inversion takes seconds over the loaded words' arithmetic

An op changes memory only by va_arg: an integer op leaves it as it is.

193Lemma op_res_int_mem
∀ o k xs m r m′
op_res_into k xs m = SomeOpResr m′
m′ = m
Explore proof 3 tactics

Initial goal

⊢
∀o : op k : class xs : listword m : mem r : resword m′ : mem, op_res_into k xs m = SomeOpResr m′ → m′ = m

unfold op_res_int

⊢
∀ o : op k : class xs : listword m : mem r : resword m′ : mem  ,
match o with
Add | Sub
⇒
let* w ≔ additive_resk o xs in
SomeOpResRValw m
Allocalign
⇒
match xs with
WIntsize
⇒
max_allocation_size <? size
→
SomeOpResRTrapTooBig m
else
SomeOpAllocheap_reqalign size λbase : Z ⇒ WPtrptr_addr ≔ base; ptr_cap ≔ Objbase, allocatedm heap_reqalign size base
WIntsize :: _ :: _
⇒
None
_
⇒
None
Checklo hi r0 al
⇒
match k with
A
⇒
match xs with
WPtrp
⇒
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r0 al in
SomeOpReschecked_reschk WPtrp m
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
CheckAheadlo hi r0 al ahi
⇒
match k with
A
⇒
match xs with
WPtrp
⇒
match access_resm raw_addressptr_addrp lo ptr_capp ahi − lo r0 al with
SomeRTrapwhy
⇒
SomeOpResRTrapwhy m
None
⇒
None
_
⇒
match access_resm raw_addressptr_addrp lo ptr_capp hi − lo r0 al with
SomeRTrapwhy
⇒
SomeOpResRTrapEarlywhy m
None
⇒
None
_
⇒
SomeOpResRValWPtrp m
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
Renewlo hi r0 al
⇒
match k with
A
⇒
match xs with
[ ]
⇒
None
q1
⇒
None
q1; x
⇒
match q1 with
WPtrp1
⇒
match x with
WPtrp
⇒
cap_eqbptr_capp1 ptr_capp
→
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r0 al in
let* r′ ≔ renew_reschk p m in
SomeOpResr′ m
else
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
q1 :: x :: _ :: _
⇒
None
_
⇒
None
RenewHulllo hi r0 al
⇒
match k with
A
⇒
match xs with
[ ]
⇒
None
ql
⇒
None
ql; qh
⇒
None
ql; qh; x
⇒
match ql with
WPtrpl
⇒
match qh with
WPtrph
⇒
match x with
WPtrp
⇒
cap_eqbptr_cappl ptr_capp && cap_eqbptr_capph ptr_capp
→
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r0 al in
let* r′ ≔ renew_reschk p m in
SomeOpResr′ m
else
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
ql :: qh :: x :: _ :: _
⇒
None
_
⇒
None
HalfCheckbd lo hi r0 al
⇒
match k with
A
⇒
match xs with
[ ]
⇒
None
q1
⇒
None
q1; x
⇒
match q1 with
WPtrp1
⇒
match x with
WPtrp
⇒
cap_eqbptr_capp1 ptr_capp
→
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r0 al in
match half_passesbd m raw_addressptr_addrp lo hi − lo ptr_capp r0 al with
SomeSomeok′
⇒
SomeOpReshalf_reschk ok′ WPtrp m
SomeNone
⇒
SomeOpResRUncheckedHalf m
None
⇒
None
else
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
q1 :: x :: _ :: _
⇒
None
_
⇒
None
SpanTestlo hi
⇒
match k with
W
⇒
match xs with
WPtrp
⇒
match ptr_capp with
Objbase | Robase
⇒
let* reg ≔ object_regionbase m_regionsm in
Some
OpRes
RVal
WInt
base ≤? raw_addressptr_addrp lo && raw_addressptr_addrp lo <? raw_addressptr_addrp hi && raw_addressptr_addrp hi ≤? r_endreg
→
1
else
0
m
_
⇒
SomeOpResRValWInt0 m
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
RenewIflo hi r0 al
⇒
match k with
A
⇒
match xs with
WPtrp; WIntg
⇒
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r0 al in
g =? 0
→
SomeOpReschecked_reschk WPtrp m
else
match chk with
RTrapMisaligned
⇒
SomeOpResRTrapMisaligned m
_
⇒
let* r′ ≔ renew_reschk p m in
SomeOpResr′ m
WPtrp :: WIntg :: _ :: _
⇒
None
WPtrp | WPtrp :: WPtr_ :: _ | WPtrp :: WLimit_ _ :: _
⇒
None
_
⇒
None
_
⇒
None
Limitsize
⇒
match k with
L
⇒
match xs with
WPtrp
⇒
SomeOpResRValWLimitptr_capp size m
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
CheckLimitlo hi r0 al
⇒
match k with
A
⇒
match xs with
WPtrp
⇒
None
WPtrp; u
⇒
match u with
WLimitc size
⇒
cap_eqbc ptr_capp && size =? hi − lo
→
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r0 al in
SomeOpReschecked_reschk WPtrp m
else
SomeOpResRUncheckedWhyLimit m
_
⇒
SomeOpResRUncheckedWhyLimit m
WPtrp :: u :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
RawLoadt off
⇒
match xs with
WPtrp
⇒
let* r0 ≔ int_load_rest k m raw_addressptr_addrp off ptr_capp in
SomeOpResraw_resr0 m
WPtrp :: _ :: _
⇒
None
_
⇒
None
RawLoadPointeroff es
⇒
match k with
P
⇒
match xs with
WPtrp
⇒
match access_resm raw_addressptr_addrp off ptr_capp 8 Read 8 with
SomeRVal_
⇒
let* x0 ≔ pointer_load_effectsm raw_addressptr_addrp off es in
SomeOpResRValWPtrx0 m
SomeRTrapwhy | SomeRUncheckedwhy
⇒
SomeOpResRUncheckedwhy m
None
⇒
None
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
ExtF_ | Truncd | Truncx | FloatToInt_ _ | IntToFloat_ _ | Cast | CheckLocal_ _ _ _ _ | CheckFunction | VaArgs | VaArg
⇒
None
_
⇒
let* w ≔ aluo k xs in
SomeOpResRValw m
=
SomeOpResr m′
→
m′ = m

intros o k xs m r m′ H

o :
op
k :
class
xs :
listword
m :
mem
r :
resword
m′ :
mem
H :
match o with
Add | Sub
⇒
let* w ≔ additive_resk o xs in
SomeOpResRValw m
Allocalign
⇒
match xs with
WIntsize
⇒
max_allocation_size <? size
→
SomeOpResRTrapTooBig m
else
SomeOpAllocheap_reqalign size λbase : Z ⇒ WPtrptr_addr ≔ base; ptr_cap ≔ Objbase, allocatedm heap_reqalign size base
WIntsize :: _ :: _
⇒
None
_
⇒
None
Checklo hi r al
⇒
match k with
A
⇒
match xs with
WPtrp
⇒
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r al in
SomeOpReschecked_reschk WPtrp m
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
CheckAheadlo hi r al ahi
⇒
match k with
A
⇒
match xs with
WPtrp
⇒
match access_resm raw_addressptr_addrp lo ptr_capp ahi − lo r al with
SomeRTrapwhy
⇒
SomeOpResRTrapwhy m
None
⇒
None
_
⇒
match access_resm raw_addressptr_addrp lo ptr_capp hi − lo r al with
SomeRTrapwhy
⇒
SomeOpResRTrapEarlywhy m
None
⇒
None
_
⇒
SomeOpResRValWPtrp m
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
Renewlo hi r al
⇒
match k with
A
⇒
match xs with
[ ]
⇒
None
q1
⇒
None
q1; x
⇒
match q1 with
WPtrp1
⇒
match x with
WPtrp
⇒
cap_eqbptr_capp1 ptr_capp
→
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r al in
let* r′ ≔ renew_reschk p m in
SomeOpResr′ m
else
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
q1 :: x :: _ :: _
⇒
None
_
⇒
None
RenewHulllo hi r al
⇒
match k with
A
⇒
match xs with
[ ]
⇒
None
ql
⇒
None
ql; qh
⇒
None
ql; qh; x
⇒
match ql with
WPtrpl
⇒
match qh with
WPtrph
⇒
match x with
WPtrp
⇒
cap_eqbptr_cappl ptr_capp && cap_eqbptr_capph ptr_capp
→
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r al in
let* r′ ≔ renew_reschk p m in
SomeOpResr′ m
else
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
ql :: qh :: x :: _ :: _
⇒
None
_
⇒
None
HalfCheckbd lo hi r al
⇒
match k with
A
⇒
match xs with
[ ]
⇒
None
q1
⇒
None
q1; x
⇒
match q1 with
WPtrp1
⇒
match x with
WPtrp
⇒
cap_eqbptr_capp1 ptr_capp
→
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r al in
match half_passesbd m raw_addressptr_addrp lo hi − lo ptr_capp r al with
SomeSomeok′
⇒
SomeOpReshalf_reschk ok′ WPtrp m
SomeNone
⇒
SomeOpResRUncheckedHalf m
None
⇒
None
else
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
_
⇒
SomeOpResRUncheckedAlias m
q1 :: x :: _ :: _
⇒
None
_
⇒
None
SpanTestlo hi
⇒
match k with
W
⇒
match xs with
WPtrp
⇒
match ptr_capp with
Objbase | Robase
⇒
let* reg ≔ object_regionbase m_regionsm in
Some
OpRes
RVal
WInt
base ≤? raw_addressptr_addrp lo && raw_addressptr_addrp lo <? raw_addressptr_addrp hi && raw_addressptr_addrp hi ≤? r_endreg
→
1
else
0
m
_
⇒
SomeOpResRValWInt0 m
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
RenewIflo hi r al
⇒
match k with
A
⇒
match xs with
WPtrp; WIntg
⇒
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r al in
g =? 0
→
SomeOpReschecked_reschk WPtrp m
else
match chk with
RTrapMisaligned
⇒
SomeOpResRTrapMisaligned m
_
⇒
let* r′ ≔ renew_reschk p m in
SomeOpResr′ m
WPtrp :: WIntg :: _ :: _
⇒
None
WPtrp | WPtrp :: WPtr_ :: _ | WPtrp :: WLimit_ _ :: _
⇒
None
_
⇒
None
_
⇒
None
Limitsize
⇒
match k with
L
⇒
match xs with
WPtrp
⇒
SomeOpResRValWLimitptr_capp size m
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
CheckLimitlo hi r al
⇒
match k with
A
⇒
match xs with
WPtrp
⇒
None
WPtrp; u
⇒
match u with
WLimitc size
⇒
cap_eqbc ptr_capp && size =? hi − lo
→
let* chk ≔ access_resm raw_addressptr_addrp lo ptr_capp hi − lo r al in
SomeOpReschecked_reschk WPtrp m
else
SomeOpResRUncheckedWhyLimit m
_
⇒
SomeOpResRUncheckedWhyLimit m
WPtrp :: u :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
RawLoadt off
⇒
match xs with
WPtrp
⇒
let* r ≔ int_load_rest k m raw_addressptr_addrp off ptr_capp in
SomeOpResraw_resr m
WPtrp :: _ :: _
⇒
None
_
⇒
None
RawLoadPointeroff es
⇒
match k with
P
⇒
match xs with
WPtrp
⇒
match access_resm raw_addressptr_addrp off ptr_capp 8 Read 8 with
SomeRVal_
⇒
let* x0 ≔ pointer_load_effectsm raw_addressptr_addrp off es in
SomeOpResRValWPtrx0 m
SomeRTrapwhy | SomeRUncheckedwhy
⇒
SomeOpResRUncheckedwhy m
None
⇒
None
WPtrp :: _ :: _
⇒
None
_
⇒
None
_
⇒
None
ExtF_ | Truncd | Truncx | FloatToInt_ _ | IntToFloat_ _ | Cast | CheckLocal_ _ _ _ _ | CheckFunction | VaArgs | VaArg
⇒
None
_
⇒
let* w ≔ aluo k xs in
SomeOpResRValw m
=
SomeOpResr m′
⊢
m′ = m

destruct o ; cases H ; inversion H ; auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold op_res_int
intros o k xs m r m′ H
destruct o ; cases H ; inversion H ; auto

An op that gives a result changes memory as kept allows.

199Lemma op_res_kept
∀ o k xs m r m′
op_reso k xs m = SomeOpResr m′
keptm m′
Explore proof 3 tactics

Initial goal

⊢
∀o : op k : class xs : listword m : mem r : resword m′ : mem, op_reso k xs m = SomeOpResr m′ → keptm m′

unfold op_res , float_integer

⊢
∀ o : op k : class xs : listword m : mem r : resword m′ : mem  ,
match o with
Add | Sub | Mul | Div | Neg | Copy
⇒
match fmt_ofk with
Somef
⇒
let* r0 ≔ float_arithmetico f xs in
SomeOpResRValWIntr0 m
None
⇒
op_res_into k xs m
Cmpc ck
⇒
match xs with
WInta; WIntb
⇒
match fmt_ofck with
Somef
⇒
SomeOpResRValWIntfloat_comparec f a b m
None
⇒
op_res_into k xs m
WInta :: WIntb :: _ :: _
⇒
match fmt_ofck with
Some_
⇒
None
None
⇒
op_res_into k xs m
WInta | WInta :: WPtr_ :: _ | WInta :: WLimit_ _ :: _
⇒
match fmt_ofck with
Some_
⇒
None
None
⇒
op_res_into k xs m
_
⇒
match fmt_ofck with
Some_
⇒
None
None
⇒
op_res_into k xs m
ExtF_ | Truncd | Truncx
⇒
match xs with
WIntx
⇒
let* r0 ≔ float_converto k x in
SomeOpResRValWIntr0 m
WIntx :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
FloatToIntfk sg
⇒
match xs with
WIntx
⇒
let* from ≔ fmt_offk in
let* m0 ≔ modulusk in
let 'lo, hi ≔
match sg with
Signed
⇒
−m0 / 2, m0 / 2 − 1
Unsigned
⇒
0, m0 − 1
in
match Ieee.ieee_to_intfrom x with
Somen
⇒
lo ≤? n && n ≤? hi
→
SomeOpResRValWIntn mod m0 m
else
SomeOpUBTAtom"float_conversion"
None
⇒
SomeOpUBTAtom"float_conversion"
WIntx :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
IntToFloatik sg
⇒
match fmt_ofk with
Somef
⇒
match xs with
WIntx
⇒
let* n ≔ integer_sourceik sg x in
SomeOpResRValWIntIeee.ieee_from_intf n m
WIntx :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
None
⇒
op_res_into k xs m
Cast
⇒
match xs with
WIntx
⇒
SomeOpResRValWIntx m
WIntx :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
VaArg
⇒
match xs with
WPtrp
⇒
let* rm ≔ va_arg_resk p m in
SomeOpResfstrm sndrm
WPtrp :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
=
SomeOpResr m′
→
keptm m′

intros o k xs m r m′ H

o :
op
k :
class
xs :
listword
m :
mem
r :
resword
m′ :
mem
H :
match o with
Add | Sub | Mul | Div | Neg | Copy
⇒
match fmt_ofk with
Somef
⇒
let* r ≔ float_arithmetico f xs in
SomeOpResRValWIntr m
None
⇒
op_res_into k xs m
Cmpc ck
⇒
match xs with
WInta; WIntb
⇒
match fmt_ofck with
Somef
⇒
SomeOpResRValWIntfloat_comparec f a b m
None
⇒
op_res_into k xs m
WInta :: WIntb :: _ :: _
⇒
match fmt_ofck with
Some_
⇒
None
None
⇒
op_res_into k xs m
WInta | WInta :: WPtr_ :: _ | WInta :: WLimit_ _ :: _
⇒
match fmt_ofck with
Some_
⇒
None
None
⇒
op_res_into k xs m
_
⇒
match fmt_ofck with
Some_
⇒
None
None
⇒
op_res_into k xs m
ExtF_ | Truncd | Truncx
⇒
match xs with
WIntx
⇒
let* r ≔ float_converto k x in
SomeOpResRValWIntr m
WIntx :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
FloatToIntfk sg
⇒
match xs with
WIntx
⇒
let* from ≔ fmt_offk in
let* m0 ≔ modulusk in
let 'lo, hi ≔
match sg with
Signed
⇒
−m0 / 2, m0 / 2 − 1
Unsigned
⇒
0, m0 − 1
in
match Ieee.ieee_to_intfrom x with
Somen
⇒
lo ≤? n && n ≤? hi
→
SomeOpResRValWIntn mod m0 m
else
SomeOpUBTAtom"float_conversion"
None
⇒
SomeOpUBTAtom"float_conversion"
WIntx :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
IntToFloatik sg
⇒
match fmt_ofk with
Somef
⇒
match xs with
WIntx
⇒
let* n ≔ integer_sourceik sg x in
SomeOpResRValWIntIeee.ieee_from_intf n m
WIntx :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
None
⇒
op_res_into k xs m
Cast
⇒
match xs with
WIntx
⇒
SomeOpResRValWIntx m
WIntx :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
VaArg
⇒
match xs with
WPtrp
⇒
let* rm ≔ va_arg_resk p m in
SomeOpResfstrm sndrm
WPtrp :: _ :: _
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
_
⇒
op_res_into k xs m
=
SomeOpResr m′
⊢
keptm m′

destruct o ; cases H ; inversion H ; subst;first|apply kept_refl|match goal withE : op_res_int_ _ _ _ = Some_ |- _⇒rewrite op_res_int_mem_ _ _ _ _ _ E ; apply kept_reflend|match goal withE : va_arg_res_ _ _ = Some?rm |- _⇒destruct rm ; exact va_arg_res_kept_ _ _ _ _ Eend

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold op_res , float_integer
intros o k xs m r m′ H
destruct o ; cases H ; inversion H ; subst;first|apply kept_refl|match goal withE : op_res_int_ _ _ _ = Some_ |- _⇒rewrite op_res_int_mem_ _ _ _ _ _ E ; apply kept_reflend|match goal withE : va_arg_res_ _ _ = Some?rm |- _⇒destruct rm ; exact va_arg_res_kept_ _ _ _ _ Eend

Only the op alloc allocates: a heap object with a positive alignment, and it writes nothing into the new memory.

212Lemma op_res_alloc
∀ o k xs m q kont
op_reso k xs m = SomeOpAllocq kont
0 < req_alignq ∧ ∀base, keptallocatedm q base sndkontbase
Proof 4 steps Qed
unfold op_res , op_res_int , float_integer
intros o k xs m q kont H
destruct o ; cases H ; inversion H ; subst
split ; simpl ; lia | intros base ; apply kept_refl

Ending an invocation's locals (invalidate_locals/3) is kept: it ends live stack objects, and writes nothing.

223Lemma invalidate_locals_kept
∀ locals m m′
invalidate_localslocals m = Somem′
keptm m′
Explore proof 20 tactics

Initial goal

⊢
∀locals : listname × optionZ m m′ : mem, invalidate_localslocals m = Somem′ → keptm m′

unfold invalidate_locals

⊢
∀ locals : listname × optionZ m m′ : mem  ,
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m0 ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm0 in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm0 in
Somem_bytes ≔ m_bytesm0; m_regions ≔ rs; m_aux ≔ m_auxm0
Freed
⇒
None
_
⇒
None
None
⇒
Somem0
locals
Somem
=
Somem′
→
keptm m′

intros locals m m′ H

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
⊢
keptm m′

eapply fold_keptname × optionZ λ_ ⇒ True ; exact I | | exact H

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
⊢
∀ m1 : mem x : name × optionZ m2 : mem  ,
True
→
λ m0 : mem l : name × optionZ  ⇒
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm0 in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm0 in
Somem_bytes ≔ m_bytesm0; m_regions ≔ rs; m_aux ≔ m_auxm0
Freed
⇒
None
_
⇒
None
None
⇒
Somem0
m1 x
=
Somem2
→
True ∧ keptm1 m2

intros m1 x slot m2 _ E

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
slot :
optionZ
m2 :
mem
E :
match sndx, slot with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm1 in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 in
Somem_bytes ≔ m_bytesm1; m_regions ≔ rs; m_aux ≔ m_auxm1
Freed
⇒
None
_
⇒
None
None
⇒
Somem1
=
Somem2
⊢
True ∧ keptm1 m2

split ; auto

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
slot :
optionZ
m2 :
mem
E :
match sndx, slot with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm1 in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 in
Somem_bytes ≔ m_bytesm1; m_regions ≔ rs; m_aux ≔ m_auxm1
Freed
⇒
None
_
⇒
None
None
⇒
Somem1
=
Somem2
⊢
keptm1 m2

simpl in E

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
slot :
optionZ
m2 :
mem
E :
match slot with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm1 in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 in
Somem_bytes ≔ m_bytesm1; m_regions ≔ rs; m_aux ≔ m_auxm1
Freed
⇒
None
_
⇒
None
None
⇒
Somem1
=
Somem2
⊢
keptm1 m2

cases E ; inversion E ; subst ; try apply kept_refl

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
⊢
keptm1 m_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1

match goal withEr : replace_region?base _ _ = Some_ |- _⇒split ; intros b reg′ Eb ; simpl in Eb ; rewrite object_region_replace_ _ _ _ b Er eq_refl in Eb ; destruct Z.eqbb base eqn: Ez | intros a _ ; simpl ; autoend

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = true
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed
focused goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

–apply Z.eqb_eq in Ez

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b = z
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

subst

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
reg′ :
region
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somereg′
⊢
∃reg : region, object_regionz m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ z; r_end ≔ z; r_kind ≔ r_kindreg; r_life ≔ Freed
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

inversion Eb ; subst

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed
⊢
∃reg : region, object_regionz m_regionsm1 = Somereg ∧ r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = reg ∨ r_lifereg = Live ∧ r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = r_base ≔ z; r_end ≔ z; r_kind ≔ r_kindreg; r_life ≔ Freed
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

eexists

3 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed
⊢
object_regionz m_regionsm1 = Some?reg ∧ r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = ?reg ∨ r_life?reg = Live ∧ r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = r_base ≔ z; r_end ≔ z; r_kind ≔ r_kind?reg; r_life ≔ Freed
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed
shelved goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed
⊢
region

split ; eassumption |

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed
⊢
r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = r ∨ r_lifer = Live ∧ r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = r_base ≔ z; r_end ≔ z; r_kind ≔ r_kindr; r_life ≔ Freed
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

right

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed
⊢
r_lifer = Live ∧ r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = r_base ≔ z; r_end ≔ z; r_kind ≔ r_kindr; r_life ≔ Freed
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

split ; assumption |

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed
⊢
r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = r_base ≔ z; r_end ≔ z; r_kind ≔ r_kindr; r_life ≔ Freed
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

f_equal

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed
⊢
Stack = r_kindr
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

symmetry

2 open

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
Eb :
Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed = Somer_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed
⊢
r_kindr = Stack
deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

assumption

deferred goal
locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
∃reg : region, object_regionb m_regionsm1 = Somereg ∧ reg′ = reg ∨ r_lifereg = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg; r_life ≔ Freed

–exists reg′

locals :
listname × optionZ
m, m′ :
mem
H :
fold_left
λ acc : optionmem l : name × optionZ  ⇒
let* m ≔ acc in
match sndl with
Somebase
⇒
let* reg ≔ object_regionbase m_regionsm in
match r_kindreg with
Stack
⇒
match r_lifereg with
Live
⇒
let* rs ≔ replace_regionbase r_base ≔ base; r_end ≔ base; r_kind ≔ Stack; r_life ≔ Freed m_regionsm in
Somem_bytes ≔ m_bytesm; m_regions ≔ rs; m_aux ≔ m_auxm
Freed
⇒
None
_
⇒
None
None
⇒
Somem
locals
Somem
=
Somem′
m1 :
mem
x :
name
z :
Z
r :
region
Heqo0 :
object_regionz m_regionsm1 = Somer
Heqk :
r_kindr = Stack
Heql :
r_lifer = Live
l :
listregion
Heqo1 :
replace_regionz r_base ≔ z; r_end ≔ z; r_kind ≔ Stack; r_life ≔ Freed m_regionsm1 = Somel
E :
Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1 = Somem_bytes ≔ m_bytesm1; m_regions ≔ l; m_aux ≔ m_auxm1
b :
Z
reg′ :
region
Ez :
b =? z = false
Eb :
object_regionb m_regionsm1 = Somereg′
⊢
object_regionb m_regionsm1 = Somereg′ ∧ reg′ = reg′ ∨ r_lifereg′ = Live ∧ reg′ = r_base ≔ b; r_end ≔ b; r_kind ≔ r_kindreg′; r_life ≔ Freed

auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold invalidate_locals
intros locals m m′ H
eapply fold_keptname × optionZ λ_ ⇒ True ; exact I | | exact H
intros m1 x slot m2 _ E
split ; auto
simpl in E
cases E ; inversion E ; subst ; try apply kept_refl
match goal withEr : replace_region?base _ _ = Some_ |- _⇒split ; intros b reg′ Eb ; simpl in Eb ; rewrite object_region_replace_ _ _ _ b Er eq_refl in Eb ; destruct Z.eqbb base eqn: Ez | intros a _ ; simpl ; autoend
–apply Z.eqb_eq in Ez
subst
inversion Eb ; subst
eexists
split ; eassumption |
right
split ; assumption |
f_equal
symmetry
assumption
–exists reg′
auto

What a step does to memory

The argument snapshot a variadic callee takes writes its words into the snapshot object: if the 8 bytes per word from a on are inside objects, snapshot_values is kept.

242Lemma snapshot_kept
∀ xs a m
∀b, a ≤ b < a + 8 × Z.of_natlengthxs → in_objectm b
keptm snapshot_valuesxs a m
Explore proof 26 tactics

Initial goal

⊢
∀xs : listword a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m

induction xs as | x xs IH ; simpl ; intros a m Hin

2 open

a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + 0 → in_objectm b
⊢
keptm m
focused goal
x :
word
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm
match x with
WIntn
⇒
snapshot_valuesxs a + 8 store_bytesm a 8 n
WPtrp
⇒
snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
WLimit_ _
⇒
snapshot_valuesxs a + 8 m

–apply kept_refl

deferred goal
x :
word
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm
match x with
WIntn
⇒
snapshot_valuesxs a + 8 store_bytesm a 8 n
WPtrp
⇒
snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
WLimit_ _
⇒
snapshot_valuesxs a + 8 m

–destruct x as n | p | c size

3 open

n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_bytesm a 8 n
focused goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
focused goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

+eapply kept_trans ; | apply IH

4 open

n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm store_bytesm a 8 n
focused goal
n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_bytesm a 8 n b
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

∗apply store_bytes_kept

4 open

n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a ≤ b < a + 8 → in_objectm b
deferred goal
n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_bytesm a 8 n b
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

intros b Hb

4 open

n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a ≤ b < a + 8
⊢
in_objectm b
deferred goal
n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_bytesm a 8 n b
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

apply Hin

4 open

n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a ≤ b < a + 8
⊢
a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0
deferred goal
n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_bytesm a 8 n b
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

lia

3 open

deferred goal
n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_bytesm a 8 n b
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

∗intros b Hb

3 open

n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs
⊢
in_objectstore_bytesm a 8 n b
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

apply in_object_store_bytes , Hin

3 open

n :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs
⊢
a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

lia

2 open

deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

+eapply kept_trans ; | apply IH

3 open

p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm store_auxstore_bytesm a 8 ptr_addrp a ptr_capp
focused goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

∗eapply kept_trans ; | apply store_aux_kept

4 open

p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm store_bytesm a 8 ptr_addrp
focused goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
in_objectstore_bytesm a 8 ptr_addrp a
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

--apply store_bytes_kept

4 open

p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a ≤ b < a + 8 → in_objectm b
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
in_objectstore_bytesm a 8 ptr_addrp a
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

intros b Hb

4 open

p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a ≤ b < a + 8
⊢
in_objectm b
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
in_objectstore_bytesm a 8 ptr_addrp a
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

apply Hin

4 open

p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a ≤ b < a + 8
⊢
a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
in_objectstore_bytesm a 8 ptr_addrp a
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

lia

3 open

deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
in_objectstore_bytesm a 8 ptr_addrp a
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

--apply in_object_store_bytes , Hin

3 open

p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
a ≤ a < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0
deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

lia

2 open

deferred goal
p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

∗intros b Hb

2 open

p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs
⊢
in_objectstore_auxstore_bytesm a 8 ptr_addrp a ptr_capp b
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

apply in_object_store_aux , in_object_store_bytes , Hin

2 open

p :
ptr
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs
⊢
a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0
deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

lia

deferred goal
c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
keptm snapshot_valuesxs a + 8 m

+apply IH

c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
⊢
∀b : Z, a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs → in_objectm b

intros b Hb

c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs
⊢
in_objectm b

apply Hin

c :
cap
size :
Z
xs :
listword
IH :
∀a : Z m : mem, ∀b : Z, a ≤ b < a + 8 × Z.of_natDatatypes.lengthxs → in_objectm b → keptm snapshot_valuesxs a m
a :
Z
m :
mem
Hin :
∀b : Z, a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0 → in_objectm b
b :
Z
Hb :
a + 8 ≤ b < a + 8 + 8 × Z.of_natDatatypes.lengthxs
⊢
a ≤ b < a + Z.posPosDef.Pos.of_succ_natDatatypes.lengthxs~0~0~0

lia

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
induction xs as | x xs IH ; simpl ; intros a m Hin
–apply kept_refl
–destruct x as n | p | c size
+eapply kept_trans ; | apply IH
∗apply store_bytes_kept
intros b Hb
apply Hin
lia
∗intros b Hb
apply in_object_store_bytes , Hin
lia
+eapply kept_trans ; | apply IH
∗eapply kept_trans ; | apply store_aux_kept
--apply store_bytes_kept
intros b Hb
apply Hin
lia
--apply in_object_store_bytes , Hin
lia
∗intros b Hb
apply in_object_store_aux , in_object_store_bytes , Hin
lia
+apply IH
intros b Hb
apply Hin
lia

A heap request's extent covers its size: the runtime is asked for at least as many bytes as the object needs.

262Lemma heap_req_extent
∀ align size
size ≤ req_extentheap_reqalign size
Explore proof 5 tactics

Initial goal

⊢
∀align size : Z, size ≤ req_extentheap_reqalign size

intros align size

align, size :
Z
⊢
size ≤ req_extentheap_reqalign size

simpl

align, size :
Z
⊢
size ≤ size + 15 / 16 × 16

pose proof Z.div_modsize + 15 16 ltac:lia

align, size :
Z
H :
size + 15 = 16 × size + 15 / 16 + size + 15 mod 16
⊢
size ≤ size + 15 / 16 × 16

pose proof Z.mod_pos_boundsize + 15 16 ltac:lia

align, size :
Z
H :
size + 15 = 16 × size + 15 / 16 + size + 15 mod 16
H0 :
0 ≤ size + 15 mod 16 < 16
⊢
size ≤ size + 15 / 16 × 16

lia

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros align size
simpl
pose proof Z.div_modsize + 15 16 ltac:lia
pose proof Z.mod_pos_boundsize + 15 16 ltac:lia
lia

The new object's bytes are inside it: after allocating q at base, every address from base up to base plus the extent is inside an object.

272Lemma in_allocated
∀ m q base b
base ≤ b < base + req_extentq
in_objectallocatedm q base b
Explore proof 5 tactics

Initial goal

⊢
∀m : mem q : alloc_req base b : Z, base ≤ b < base + req_extentq → in_objectallocatedm q base b

intros m q base b Hb

m :
mem
q :
alloc_req
base, b :
Z
Hb :
base ≤ b < base + req_extentq
⊢
in_objectallocatedm q base b

exists base, mkRegionbase base + req_extentq req_kindq Live

m :
mem
q :
alloc_req
base, b :
Z
Hb :
base ≤ b < base + req_extentq
⊢
object_regionbase m_regionsallocatedm q base = Somer_base ≔ base; r_end ≔ base + req_extentq; r_kind ≔ req_kindq; r_life ≔ Live ∧ base ≤ b < r_endr_base ≔ base; r_end ≔ base + req_extentq; r_kind ≔ req_kindq; r_life ≔ Live

simpl

m :
mem
q :
alloc_req
base, b :
Z
Hb :
base ≤ b < base + req_extentq
⊢
base =? base
→
Somer_base ≔ base; r_end ≔ base + req_extentq; r_kind ≔ req_kindq; r_life ≔ Live
else
object_regionbase m_regionsm
=
Somer_base ≔ base; r_end ≔ base + req_extentq; r_kind ≔ req_kindq; r_life ≔ Live
∧
base ≤ b < base + req_extentq

rewrite Z.eqb_refl

m :
mem
q :
alloc_req
base, b :
Z
Hb :
base ≤ b < base + req_extentq
⊢
Somer_base ≔ base; r_end ≔ base + req_extentq; r_kind ≔ req_kindq; r_life ≔ Live = Somer_base ≔ base; r_end ≔ base + req_extentq; r_kind ≔ req_kindq; r_life ≔ Live ∧ base ≤ b < base + req_extentq

auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros m q base b Hb
exists base, mkRegionbase base + req_extentq req_kindq Live
simpl
rewrite Z.eqb_refl
auto

A variadic callee's snapshot is written into its new object.

280Lemma snapshot_alloc_kept
∀ xs m base
let q ≔ heap_req16 8 × Z.of_natlengthxs in
keptallocatedm q base snapshot_valuesxs base allocatedm q base
Explore proof 7 tactics

Initial goal

⊢
∀ xs : listword m : mem base : Z  ,
let q ≔ heap_req16 8 × Z.of_natDatatypes.lengthxs in
keptallocatedm q base snapshot_valuesxs base allocatedm q base

intros xs m base q

xs :
listword
m :
mem
base :
Z
q :=
heap_req16 8 × Z.of_natDatatypes.lengthxs : alloc_req
⊢
keptallocatedm q base snapshot_valuesxs base allocatedm q base

apply snapshot_kept

xs :
listword
m :
mem
base :
Z
q :=
heap_req16 8 × Z.of_natDatatypes.lengthxs : alloc_req
⊢
∀b : Z, base ≤ b < base + 8 × Z.of_natDatatypes.lengthxs → in_objectallocatedm q base b

intros b Hb

xs :
listword
m :
mem
base :
Z
q :=
heap_req16 8 × Z.of_natDatatypes.lengthxs : alloc_req
b :
Z
Hb :
base ≤ b < base + 8 × Z.of_natDatatypes.lengthxs
⊢
in_objectallocatedm q base b

apply in_allocated

xs :
listword
m :
mem
base :
Z
q :=
heap_req16 8 × Z.of_natDatatypes.lengthxs : alloc_req
b :
Z
Hb :
base ≤ b < base + 8 × Z.of_natDatatypes.lengthxs
⊢
base ≤ b < base + req_extentq

pose proof heap_req_extent16 8 × Z.of_natlengthxs

xs :
listword
m :
mem
base :
Z
q :=
heap_req16 8 × Z.of_natDatatypes.lengthxs : alloc_req
b :
Z
Hb :
base ≤ b < base + 8 × Z.of_natDatatypes.lengthxs
H :
8 × Z.of_natDatatypes.lengthxs ≤ req_extentheap_req16 8 × Z.of_natDatatypes.lengthxs
⊢
base ≤ b < base + req_extentq

unfold q

xs :
listword
m :
mem
base :
Z
q :=
heap_req16 8 × Z.of_natDatatypes.lengthxs : alloc_req
b :
Z
Hb :
base ≤ b < base + 8 × Z.of_natDatatypes.lengthxs
H :
8 × Z.of_natDatatypes.lengthxs ≤ req_extentheap_req16 8 × Z.of_natDatatypes.lengthxs
⊢
base ≤ b < base + req_extentheap_req16 8 × Z.of_natDatatypes.lengthxs

lia

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros xs m base q
apply snapshot_kept
intros b Hb
apply in_allocated
pose proof heap_req_extent16 8 × Z.of_natlengthxs
unfold q
lia

The instruction i is the one doomed_result/5 reads: a check_ahead.

290Definition ahead_insti : inst : bool ≔
match i with
Let_ A CheckAhead_ _ _ _ _ _
⇒
true
_
⇒
false

For every instruction but a check_ahead, doomed_result changes only an unchecked outcome: in a doomed frame it becomes the early trap, unless it is an alias or a wrong limit.

296Lemma doomed_result_other
∀ ge fr m i out
ahead_insti = false
doomed_resultge fr m i out
=
match out with
IStopUncheckedw
⇒
match fr_doomedfr, w with
Somewhy, Alias | WhyLimit
⇒
Someout, fr_doomedfr
Somewhy, _
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None, _
⇒
Someout, None
_
⇒
Someout, fr_doomedfr
Explore proof 6 tactics

Initial goal

⊢
∀ ge : genv fr : frame m : mem i : inst out : inst_out  ,
ahead_insti = false
→
doomed_resultge fr m i out
=
match out with
IStopUncheckedw
⇒
match fr_doomedfr with
Somewhy
⇒
match w with
Alias | WhyLimit
⇒
Someout, fr_doomedfr
_
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None
⇒
Someout, None
_
⇒
Someout, fr_doomedfr

intros ge fr m i out E

ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
E :
ahead_insti = false
⊢
doomed_resultge fr m i out
=
match out with
IStopUncheckedw
⇒
match fr_doomedfr with
Somewhy
⇒
match w with
Alias | WhyLimit
⇒
Someout, fr_doomedfr
_
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None
⇒
Someout, None
_
⇒
Someout, fr_doomedfr

destruct i ; try reflexivity

ge :
genv
fr :
frame
m :
mem
d :
name
k :
class
o :
op
args :
listvalue
out :
inst_out
E :
ahead_instLetd k o args = false
⊢
doomed_resultge fr m Letd k o args out
=
match out with
IStopUncheckedw
⇒
match fr_doomedfr with
Somewhy
⇒
match w with
Alias | WhyLimit
⇒
Someout, fr_doomedfr
_
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None
⇒
Someout, None
_
⇒
Someout, fr_doomedfr

destruct k ; try reflexivity

ge :
genv
fr :
frame
m :
mem
d :
name
o :
op
args :
listvalue
out :
inst_out
E :
ahead_instLetd A o args = false
⊢
doomed_resultge fr m Letd A o args out
=
match out with
IStopUncheckedw
⇒
match fr_doomedfr with
Somewhy
⇒
match w with
Alias | WhyLimit
⇒
Someout, fr_doomedfr
_
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None
⇒
Someout, None
_
⇒
Someout, fr_doomedfr

destruct o ; try reflexivity

ge :
genv
fr :
frame
m :
mem
d :
name
lo, hi :
Z
r :
rights
align, ahi :
Z
args :
listvalue
out :
inst_out
E :
ahead_instLetd A CheckAheadlo hi r align ahi args = false
⊢
doomed_resultge fr m Letd A CheckAheadlo hi r align ahi args out
=
match out with
IStopUncheckedw
⇒
match fr_doomedfr with
Somewhy
⇒
match w with
Alias | WhyLimit
⇒
Someout, fr_doomedfr
_
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None
⇒
Someout, None
_
⇒
Someout, fr_doomedfr

destruct args as | v | v′ args ; try reflexivity

ge :
genv
fr :
frame
m :
mem
d :
name
lo, hi :
Z
r :
rights
align, ahi :
Z
v :
value
out :
inst_out
E :
ahead_instLetd A CheckAheadlo hi r align ahi v = false
⊢
doomed_resultge fr m Letd A CheckAheadlo hi r align ahi v out
=
match out with
IStopUncheckedw
⇒
match fr_doomedfr with
Somewhy
⇒
match w with
Alias | WhyLimit
⇒
Someout, fr_doomedfr
_
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None
⇒
Someout, None
_
⇒
Someout, fr_doomedfr

discriminate E

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros ge fr m i out E
destruct i ; try reflexivity
destruct k ; try reflexivity
destruct o ; try reflexivity
destruct args as | v | v′ args ; try reflexivity
discriminate E

doomed_result (doomed_result/5) passes the instruction's result on, or continues with the memory unchanged, or stops.

316Lemma doomed_result_out
∀ ge fr m i out od
doomed_resultge fr m i out = Someod
fstod = out ∨ ∃e l, fstod = IContinuee l m ∨ ∃o, fstod = IStopo
Explore proof 10 tactics

Initial goal

⊢
∀ge : genv fr : frame m : mem i : inst out : inst_out od : inst_out × optionreason, doomed_resultge fr m i out = Someod → fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

intros ge fr m i out od H

ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

destruct ahead_insti eqn: Ea

2 open

ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
Ea :
ahead_insti = true
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo
focused goal
ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
Ea :
ahead_insti = false
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

–destruct i ; try discriminate Ea

2 open

ge :
genv
fr :
frame
m :
mem
d :
name
k :
class
o :
op
args :
listvalue
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m Letd k o args out = Someod
Ea :
ahead_instLetd k o args = true
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o0 : outcome, fstod = IStopo0
deferred goal
ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
Ea :
ahead_insti = false
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

destruct k ; try discriminate Ea

2 open

ge :
genv
fr :
frame
m :
mem
d :
name
o :
op
args :
listvalue
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m Letd A o args out = Someod
Ea :
ahead_instLetd A o args = true
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o0 : outcome, fstod = IStopo0
deferred goal
ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
Ea :
ahead_insti = false
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

destruct o ; try discriminate Ea

2 open

ge :
genv
fr :
frame
m :
mem
d :
name
lo, hi :
Z
r :
rights
align, ahi :
Z
args :
listvalue
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m Letd A CheckAheadlo hi r align ahi args out = Someod
Ea :
ahead_instLetd A CheckAheadlo hi r align ahi args = true
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo
deferred goal
ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
Ea :
ahead_insti = false
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

destruct args as | v | v′ args ; try discriminate Ea

2 open

ge :
genv
fr :
frame
m :
mem
d :
name
lo, hi :
Z
r :
rights
align, ahi :
Z
v :
value
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m Letd A CheckAheadlo hi r align ahi v out = Someod
Ea :
ahead_instLetd A CheckAheadlo hi r align ahi v = true
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo
deferred goal
ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
Ea :
ahead_insti = false
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

unfold doomed_result in H

2 open

ge :
genv
fr :
frame
m :
mem
d :
name
lo, hi :
Z
r :
rights
align, ahi :
Z
v :
value
out :
inst_out
od :
inst_out × optionreason
H :
match out with
IStopTrapEarlywhy
⇒
let* p ≔ value_xge fr_envfr P v in
Some
IContinueputd p fr_envfr fr_localsfr m
match fr_doomedfr with
Somew
⇒
Somew
None
⇒
Somewhy
IStopUncheckedw
⇒
match fr_doomedfr with
Somewhy
⇒
match w with
Alias | WhyLimit
⇒
Someout, fr_doomedfr
_
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None
⇒
Someout, None
_
⇒
Someout, fr_doomedfr
=
Someod
Ea :
ahead_instLetd A CheckAheadlo hi r align ahi v = true
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo
deferred goal
ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
Ea :
ahead_insti = false
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

cases H ; inversion H ; subst ; simpl ; eauto

deferred goal
ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
doomed_resultge fr m i out = Someod
Ea :
ahead_insti = false
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

–rewrite doomed_result_other_ _ _ _ _ Ea in H

ge :
genv
fr :
frame
m :
mem
i :
inst
out :
inst_out
od :
inst_out × optionreason
H :
match out with
IStopUncheckedw
⇒
match fr_doomedfr with
Somewhy
⇒
match w with
Alias | WhyLimit
⇒
Someout, fr_doomedfr
_
⇒
SomeIStopTrapEarlywhy, fr_doomedfr
None
⇒
Someout, None
_
⇒
Someout, fr_doomedfr
=
Someod
Ea :
ahead_insti = false
⊢
fstod = out ∨ ∃e : env l : listname × optionZ, fstod = IContinuee l m ∨ ∃o : outcome, fstod = IStopo

cases H ; inversion H ; subst ; simpl ; eauto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros ge fr m i out od H
destruct ahead_insti eqn: Ea
–destruct i ; try discriminate Ea
destruct k ; try discriminate Ea
destruct o ; try discriminate Ea
destruct args as | v | v′ args ; try discriminate Ea
unfold doomed_result in H
cases H ; inversion H ; subst ; simpl ; eauto
–rewrite doomed_result_other_ _ _ _ _ Ea in H
cases H ; inversion H ; subst ; simpl ; eauto

The ops inst_step reads in a let of its own (a local's check, a function check, the variadic area); every other op's let is the generic one.

330Definition special_leto : op : bool ≔
match o with
CheckLocal_ _ _ _ _ | CheckFunction | VaArgs
⇒
true
_
⇒
false

For any other op o, inst_step of a let is the generic case: read the arguments at their classes, run the op, and bind its result, stop, or allocate.

336Lemma inst_step_let
∀ ge fr m d k o args
special_leto = false
inst_stepge fr m Letd k o args
=
let e ≔ fr_envfr in
let locals ≔ fr_localsfr in
let* classes ≔ op_arg_classeso k in
let* xs ≔ map2_optionvalue_xge e classes args in
let* out ≔ op_reso k xs m in
match out with
OpResr m′
⇒
Somebind_resd r e locals m′
OpUBwhy
⇒
SomeIStopUBwhy
OpAllocq kont
⇒
Some
IFreshq
λ base  ⇒
let x, m′ ≔ kontbase in
putd x e, locals, m′
Explore proof 2 tactics

Initial goal

⊢
∀ ge : genv fr : frame m : mem d : name k : class o : op args : listvalue  ,
special_leto = false
→
inst_stepge fr m Letd k o args
=
let e ≔ fr_envfr in
let locals ≔ fr_localsfr in
let* classes ≔ op_arg_classeso k in
let* xs ≔ map2_optionvalue_xge e classes args in
match op_reso k xs m with
SomeOpResr m′
⇒
Somebind_resd r e locals m′
SomeOpUBwhy
⇒
SomeIStopUBwhy
SomeOpAllocq kont
⇒
Some
IFreshq
λ base : Z  ⇒
let x, m′ ≔ kontbase in
putd x e, locals, m′
None
⇒
None

intros ge fr m d k o args Es

ge :
genv
fr :
frame
m :
mem
d :
name
k :
class
o :
op
args :
listvalue
Es :
special_leto = false
⊢
inst_stepge fr m Letd k o args
=
let e ≔ fr_envfr in
let locals ≔ fr_localsfr in
let* classes ≔ op_arg_classeso k in
let* xs ≔ map2_optionvalue_xge e classes args in
match op_reso k xs m with
SomeOpResr m′
⇒
Somebind_resd r e locals m′
SomeOpUBwhy
⇒
SomeIStopUBwhy
SomeOpAllocq kont
⇒
Some
IFreshq
λ base : Z  ⇒
let x, m′ ≔ kontbase in
putd x e, locals, m′
None
⇒
None

destruct o ; try discriminate Es ; destruct k ; reflexivity

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros ge fr m d k o args Es
destruct o ; try discriminate Es ; destruct k ; reflexivity

An instruction that continues changes memory as kept allows, where the genv's free does (free_keeps).

354Lemma inst_step_kept
∀ ge fr m i e l m′
free_keepsge_freege
inst_stepge fr m i = SomeIContinuee l m′
keptm m′
Explore proof 4 tactics

Initial goal

⊢
∀ge : genv fr : frame m : mem i : inst e : env l : listname × optionZ m′ : mem, free_keepsge_freege → inst_stepge fr m i = SomeIContinuee l m′ → keptm m′

intros ge fr m i e l m′ Hfree H

ge :
genv
fr :
frame
m :
mem
i :
inst
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m i = SomeIContinuee l m′
⊢
keptm m′

destruct i

8 open

ge :
genv
fr :
frame
m :
mem
d :
name
k :
class
o :
op
args :
listvalue
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Letd k o args = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
o :
op
args :
listflight_arg
lower :
flight_lower
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Flightd o args lower = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
w :
width
off :
Z
v, a :
value
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m RawStorew off v a = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
off :
Z
v, a :
value
es :
listeffect
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m RawStorePointeroff v a es = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
r :
call_result
callee :
value
args :
listarg
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Callr callee args = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
r :
call_result
callee :
value
args :
listarg
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m RawCallr callee args = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
x :
name
size :
Z
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m LocalAllocx size = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
file :
term
line :
Z
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Locfile line = SomeIContinuee l m′
⊢
keptm m′

1: destruct special_leto eqn: Es ; destruct o ; try discriminate Es | rewrite inst_step_let_ _ _ _ _ _ _ Es in H

11 open

ge :
genv
fr :
frame
m :
mem
d :
name
k :
class
x :
name
off, bytes :
Z
r :
rights
align :
Z
args :
listvalue
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Letd k CheckLocalx off bytes r align args = SomeIContinuee l m′
Es :
special_letCheckLocalx off bytes r align = true
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
k :
class
args :
listvalue
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Letd k CheckFunction args = SomeIContinuee l m′
Es :
special_letCheckFunction = true
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
k :
class
args :
listvalue
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Letd k VaArgs args = SomeIContinuee l m′
Es :
special_letVaArgs = true
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
k :
class
o :
op
args :
listvalue
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
let e ≔ fr_envfr in
let locals ≔ fr_localsfr in
let* classes ≔ op_arg_classeso k in
let* xs ≔ map2_optionvalue_xge e classes args in
match op_reso k xs m with
SomeOpResr m′
⇒
Somebind_resd r e locals m′
SomeOpUBwhy
⇒
SomeIStopUBwhy
SomeOpAllocq kont
⇒
Some
IFreshq
λ base : Z  ⇒
let x, m′ ≔ kontbase in
putd x e, locals, m′
None
⇒
None
=
SomeIContinuee l m′
Es :
special_leto = false
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
o :
op
args :
listflight_arg
lower :
flight_lower
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Flightd o args lower = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
w :
width
off :
Z
v, a :
value
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m RawStorew off v a = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
off :
Z
v, a :
value
es :
listeffect
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m RawStorePointeroff v a es = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
r :
call_result
callee :
value
args :
listarg
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Callr callee args = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
r :
call_result
callee :
value
args :
listarg
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m RawCallr callee args = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
x :
name
size :
Z
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m LocalAllocx size = SomeIContinuee l m′
⊢
keptm m′
focused goal
ge :
genv
fr :
frame
m :
mem
file :
term
line :
Z
e :
env
l :
listname × optionZ
m′ :
mem
Hfree :
free_keepsge_freege
H :
inst_stepge fr m Locfile line = SomeIContinuee l m′
⊢
keptm m′

all: unfold inst_step , external_call , bind_res , store_out in H ; cases H ; inversion H ; subst;first|apply kept_refl|match goal withE : op_res_ _ _ _ = SomeOpRes_ _ |- _⇒exact op_res_kept_ _ _ _ _ _ Eend|match goal withE : ge_free_ _ _ = Some_, _ |- _⇒exact Hfree_ _ _ _ Eend|match goal withE : store_res_ _ _ _ _ = Some_, _ |- _⇒exact store_res_kept_ _ _ _ _ _ _ Eend|match goal withA : access_res_ _ _ _ _ _ = SomeRVal_ , E : pointer_store_effects_ _ _ _ = Some_ |- _⇒apply pointer_store_effects_kept_ _ _ _ _ access_in_ _ _ _ _ _ _ A Eend

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros ge fr m i e l m′ Hfree H
destruct i
1: destruct special_leto eqn: Es ; destruct o ; try discriminate Es | rewrite inst_step_let_ _ _ _ _ _ _ Es in H
all: unfold inst_step , external_call , bind_res , store_out in H ; cases H ; inversion H ; subst;first|apply kept_refl|match goal withE : op_res_ _ _ _ = SomeOpRes_ _ |- _⇒exact op_res_kept_ _ _ _ _ _ Eend|match goal withE : ge_free_ _ _ = Some_, _ |- _⇒exact Hfree_ _ _ _ Eend|match goal withE : store_res_ _ _ _ _ = Some_, _ |- _⇒exact store_res_kept_ _ _ _ _ _ _ Eend|match goal withA : access_res_ _ _ _ _ _ = SomeRVal_ , E : pointer_store_effects_ _ _ _ = Some_ |- _⇒apply pointer_store_effects_kept_ _ _ _ _ access_in_ _ _ _ _ _ _ A Eend

An instruction that allocates asks for a positive alignment, and writes nothing yet into the memory with the new object.

374Lemma inst_step_fresh
∀ ge fr m i q k
inst_stepge fr m i = SomeIFreshq k
0 < req_alignq ∧ ∀base, keptallocatedm q base sndkbase
Explore proof 5 tactics

Initial goal

⊢
∀ge : genv fr : frame m : mem i : inst q : alloc_req k : Z → env × listname × optionZ × mem, inst_stepge fr m i = SomeIFreshq k → 0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase

intros ge fr m i q k H

ge :
genv
fr :
frame
m :
mem
i :
inst
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m i = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase

destruct i

8 open

ge :
genv
fr :
frame
m :
mem
d :
name
k0 :
class
o :
op
args :
listvalue
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Letd k0 o args = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
o :
op
args :
listflight_arg
lower :
flight_lower
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Flightd o args lower = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
w :
width
off :
Z
v, a :
value
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m RawStorew off v a = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
off :
Z
v, a :
value
es :
listeffect
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m RawStorePointeroff v a es = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
r :
call_result
callee :
value
args :
listarg
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Callr callee args = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
r :
call_result
callee :
value
args :
listarg
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m RawCallr callee args = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
x :
name
size :
Z
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m LocalAllocx size = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
file :
term
line :
Z
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Locfile line = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase

1: destruct special_leto eqn: Es ; destruct o ; try discriminate Es | rewrite inst_step_let_ _ _ _ _ _ _ Es in H

11 open

ge :
genv
fr :
frame
m :
mem
d :
name
k0 :
class
x :
name
off, bytes :
Z
r :
rights
align :
Z
args :
listvalue
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Letd k0 CheckLocalx off bytes r align args = SomeIFreshq k
Es :
special_letCheckLocalx off bytes r align = true
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
k0 :
class
args :
listvalue
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Letd k0 CheckFunction args = SomeIFreshq k
Es :
special_letCheckFunction = true
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
k0 :
class
args :
listvalue
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Letd k0 VaArgs args = SomeIFreshq k
Es :
special_letVaArgs = true
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
k0 :
class
o :
op
args :
listvalue
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
let e ≔ fr_envfr in
let locals ≔ fr_localsfr in
let* classes ≔ op_arg_classeso k0 in
let* xs ≔ map2_optionvalue_xge e classes args in
match op_reso k0 xs m with
SomeOpResr m′
⇒
Somebind_resd r e locals m′
SomeOpUBwhy
⇒
SomeIStopUBwhy
SomeOpAllocq kont
⇒
Some
IFreshq
λ base : Z  ⇒
let x, m′ ≔ kontbase in
putd x e, locals, m′
None
⇒
None
=
SomeIFreshq k
Es :
special_leto = false
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
d :
name
o :
op
args :
listflight_arg
lower :
flight_lower
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Flightd o args lower = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
w :
width
off :
Z
v, a :
value
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m RawStorew off v a = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
off :
Z
v, a :
value
es :
listeffect
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m RawStorePointeroff v a es = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
r :
call_result
callee :
value
args :
listarg
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Callr callee args = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
r :
call_result
callee :
value
args :
listarg
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m RawCallr callee args = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
x :
name
size :
Z
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m LocalAllocx size = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase
focused goal
ge :
genv
fr :
frame
m :
mem
file :
term
line :
Z
q :
alloc_req
k :
Z → env × listname × optionZ × mem
H :
inst_stepge fr m Locfile line = SomeIFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedm q base sndkbase

all: unfold inst_step , external_call , bind_res , store_out in H ; cases H ; inversion H ; subst

2 open

ge :
genv
fr :
frame
m :
mem
d :
name
k0 :
class
o :
op
args :
listvalue
q :
alloc_req
l :
listclass
Heqo0 :
op_arg_classeso k0 = Somel
l0 :
listword
Heqo1 :
map2_optionvalue_xge fr_envfr l args = Somel0
k1 :
Z → word × mem
H :
Some
IFreshq
λ base : Z  ⇒
let x, m′ ≔ k1base in
putd x fr_envfr, fr_localsfr, m′
=
Some
IFreshq
λ base : Z  ⇒
let x, m′ ≔ k1base in
putd x fr_envfr, fr_localsfr, m′
Heqo2 :
op_reso k0 l0 m = SomeOpAllocq k1
Es :
special_leto = false
⊢
0 < req_alignq
∧
∀ base : Z  ,
keptallocatedm q base
snd
let x, m′ ≔ k1base in
putd x fr_envfr, fr_localsfr, m′
focused goal
ge :
genv
fr :
frame
m :
mem
x :
name
size :
Z
Heqo :
lookupx fr_localsfr = SomeNone
H :
SomeIFreshlocal_reqsize λbase : Z ⇒ fr_envfr, putx Somebase fr_localsfr, allocatedm local_reqsize base = SomeIFreshlocal_reqsize λbase : Z ⇒ fr_envfr, putx Somebase fr_localsfr, allocatedm local_reqsize base
⊢
0 < req_alignlocal_reqsize ∧ ∀base : Z, keptallocatedm local_reqsize base sndfr_envfr, putx Somebase fr_localsfr, allocatedm local_reqsize base

all: first|split ; simpl ; lia | intros base ; simpl ; apply kept_refl|match goal withE : op_res_ _ _ _ = SomeOpAlloc_ _ |- _⇒destruct op_res_alloc_ _ _ _ _ _ E as Ha Hkend;split;|exact Ha|intros base ; specialize Hkbase ; simpl;match goal withHk : kept_ snd?t |- _⇒destruct tend;exact Hk

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros ge fr m i q k H
destruct i
1: destruct special_leto eqn: Es ; destruct o ; try discriminate Es | rewrite inst_step_let_ _ _ _ _ _ _ Es in H
all: unfold inst_step , external_call , bind_res , store_out in H ; cases H ; inversion H ; subst
all: first|split ; simpl ; lia | intros base ; simpl ; apply kept_refl|match goal withE : op_res_ _ _ _ = SomeOpAlloc_ _ |- _⇒destruct op_res_alloc_ _ _ _ _ _ E as Ha Hkend;split;|exact Ha|intros base ; specialize Hkbase ; simpl;match goal withHk : kept_ snd?t |- _⇒destruct tend;exact Hk

or_stuck gives a next configuration only if it was given one.

392Lemma or_stuck_next
∀ why r c
or_stuckwhy r = Nextc
r = SomeNextc
Explore proof 2 tactics

Initial goal

⊢
∀why : string r : optionstep_result c : config, or_stuckwhy r = Nextc → r = SomeNextc

intros why r c H

why :
string
r :
optionstep_result
c :
config
H :
or_stuckwhy r = Nextc
⊢
r = SomeNextc

destruct r ; simpl in H ; congruence

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros why r c H
destruct r ; simpl in H ; congruence

or_stuck gives an allocation only if it was given one.

396Lemma or_stuck_fresh
∀ why r q k
or_stuckwhy r = Freshq k
r = SomeFreshq k
Explore proof 2 tactics

Initial goal

⊢
∀why : string r : optionstep_result q : alloc_req k : Z → config, or_stuckwhy r = Freshq k → r = SomeFreshq k

intros why r q k H

why :
string
r :
optionstep_result
q :
alloc_req
k :
Z → config
H :
or_stuckwhy r = Freshq k
⊢
r = SomeFreshq k

destruct r ; simpl in H ; congruence

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros why r q k H
destruct r ; simpl in H ; congruence

A tactic: where the hypothesis H says an or_stuck gave a step, split on the result it was given.

401Ltac unstuck H ≔
lazymatch type of H withor_stuck_ _ = Next_⇒apply or_stuck_next in H ; cases Hor_stuck_ _ = Fresh_ _⇒apply or_stuck_fresh in H ; cases H_⇒idtacend

A return changes memory as kept allows: it only ends the invocation's locals.

410Lemma return_step_kept
∀ c o c′
return_stepc o = Nextc′
keptc_memc c_memc′
Explore proof 5 tactics

Initial goal

⊢
∀c : config o : outcome c′ : config, return_stepc o = Nextc′ → keptc_memc c_memc′

unfold return_step

⊢
∀ c : config o : outcome c′ : config  ,
or_stuck"return"
let* m ≔ invalidate_localsfr_localsc_framec c_memc in
match c_callersc with
[ ]
⇒
SomeDoneo
caller, pd :: callers
⇒
let* o′ ≔
match pd_convpd with
Somecw, kw
⇒
converted_outcomecw kw o
None
⇒
Someo
in
let* e ≔ returned_envpd_respd o′ fr_envcaller in
SomeNextc_frame ≔ fr_fn ≔ fr_fncaller; fr_env ≔ e; fr_locals ≔ fr_localscaller; fr_doomed ≔ fr_doomedcaller; fr_at ≔ fr_atcaller; c_callers ≔ callers; c_mem ≔ m; c_fuel ≔ c_fuelc
=
Nextc′
→
keptc_memc c_memc′

intros c o c′ H

c :
config
o :
outcome
c′ :
config
H :
or_stuck"return"
let* m ≔ invalidate_localsfr_localsc_framec c_memc in
match c_callersc with
[ ]
⇒
SomeDoneo
caller, pd :: callers
⇒
let* o′ ≔
match pd_convpd with
Somecw, kw
⇒
converted_outcomecw kw o
None
⇒
Someo
in
let* e ≔ returned_envpd_respd o′ fr_envcaller in
SomeNextc_frame ≔ fr_fn ≔ fr_fncaller; fr_env ≔ e; fr_locals ≔ fr_localscaller; fr_doomed ≔ fr_doomedcaller; fr_at ≔ fr_atcaller; c_callers ≔ callers; c_mem ≔ m; c_fuel ≔ c_fuelc
=
Nextc′
⊢
keptc_memc c_memc′

apply or_stuck_next in H

c :
config
o :
outcome
c′ :
config
H :
let* m ≔ invalidate_localsfr_localsc_framec c_memc in
match c_callersc with
[ ]
⇒
SomeDoneo
caller, pd :: callers
⇒
let* o′ ≔
match pd_convpd with
Somecw, kw
⇒
converted_outcomecw kw o
None
⇒
Someo
in
let* e ≔ returned_envpd_respd o′ fr_envcaller in
SomeNextc_frame ≔ fr_fn ≔ fr_fncaller; fr_env ≔ e; fr_locals ≔ fr_localscaller; fr_doomed ≔ fr_doomedcaller; fr_at ≔ fr_atcaller; c_callers ≔ callers; c_mem ≔ m; c_fuel ≔ c_fuelc
=
SomeNextc′
⊢
keptc_memc c_memc′

cases H ; inversion H ; subst ; simpl

c :
config
o :
outcome
m :
mem
Heqo0 :
invalidate_localsfr_localsc_framec c_memc = Somem
l :
listframe × pending
f :
frame
p0 :
pending
Heql :
c_callersc = f, p0 :: l
o0 :
outcome
Heqo1 :
match pd_convp0 with
Somecw, kw
⇒
converted_outcomecw kw o
None
⇒
Someo
=
Someo0
e :
env
Heqo2 :
returned_envpd_resp0 o0 fr_envf = Somee
H :
SomeNextc_frame ≔ fr_fn ≔ fr_fnf; fr_env ≔ e; fr_locals ≔ fr_localsf; fr_doomed ≔ fr_doomedf; fr_at ≔ fr_atf; c_callers ≔ l; c_mem ≔ m; c_fuel ≔ c_fuelc = SomeNextc_frame ≔ fr_fn ≔ fr_fnf; fr_env ≔ e; fr_locals ≔ fr_localsf; fr_doomed ≔ fr_doomedf; fr_at ≔ fr_atf; c_callers ≔ l; c_mem ≔ m; c_fuel ≔ c_fuelc
⊢
keptc_memc m

match goal withE : invalidate_locals_ _ = Some_ |- _⇒exact invalidate_locals_kept_ _ _ Eend

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold return_step
intros c o c′ H
apply or_stuck_next in H
cases H ; inversion H ; subst ; simpl
match goal withE : invalidate_locals_ _ = Some_ |- _⇒exact invalidate_locals_kept_ _ _ Eend

A jump changes memory as kept allows: only a return changes it at all.

419Lemma jump_step_kept
∀ ge c l j c′
jump_stepge c l j = Nextc′
keptc_memc c_memc′
Explore proof 4 tactics

Initial goal

⊢
∀ge : genv c : config l : name j : jump c′ : config, jump_stepge c l j = Nextc′ → keptc_memc c_memc′

unfold jump_step , edge_step

⊢
∀ ge : genv c : config l : name j : jump c′ : config  ,
match fr_doomedc_framec with
Somewhy
⇒
match j with
JGotoTo_ as t
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JGotoPoll_ as t
⇒
match fr_doomedc_framec with
Somewhy0
⇒
DoneUncheckedAheadwhy0
None
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JBranchv t1 t2
⇒
or_stuck"branch"
match value_xge fr_envc_framec W v with
SomeWIntn
⇒
Some
match if n =? 0 then t2 else t1 with
To_
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
Poll_
⇒
match fr_doomedc_framec with
Somewhy0
⇒
DoneUncheckedAheadwhy0
None
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
_
⇒
None
_
⇒
DoneUncheckedAheadwhy
None
⇒
match j with
JGotoTo_ as t
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JGotoPoll_ as t
⇒
match fr_doomedc_framec with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JBranchv t1 t2
⇒
or_stuck"branch"
match value_xge fr_envc_framec W v with
SomeWIntn
⇒
Some
match if n =? 0 then t2 else t1 with
To_
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
Poll_
⇒
match fr_doomedc_framec with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
_
⇒
None
JRetv
⇒
or_stuck"ret"
match fn_retfr_fnc_framec with
RClassk
⇒
let* x ≔ value_xge fr_envc_framec k v in
Somereturn_stepc Retx
_
⇒
None
JRetsvs
⇒
or_stuck"rets"
match fn_retfr_fnc_framec with
RTupleks
⇒
let* xs ≔ map2_optionvalue_xge fr_envc_framec ks vs in
Somereturn_stepc Retsxs
_
⇒
None
JRetVoid
⇒
return_stepc RetVoid
JUnreachable
⇒
DoneUnreachable
=
Nextc′
→
keptc_memc c_memc′

intros ge c l j c′ H

ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
H :
match fr_doomedc_framec with
Somewhy
⇒
match j with
JGotoTo_ as t
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JGotoPoll_ as t
⇒
match fr_doomedc_framec with
Somewhy0
⇒
DoneUncheckedAheadwhy0
None
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JBranchv t1 t2
⇒
or_stuck"branch"
match value_xge fr_envc_framec W v with
SomeWIntn
⇒
Some
match if n =? 0 then t2 else t1 with
To_
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
Poll_
⇒
match fr_doomedc_framec with
Somewhy0
⇒
DoneUncheckedAheadwhy0
None
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
_
⇒
None
_
⇒
DoneUncheckedAheadwhy
None
⇒
match j with
JGotoTo_ as t
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JGotoPoll_ as t
⇒
match fr_doomedc_framec with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JBranchv t1 t2
⇒
or_stuck"branch"
match value_xge fr_envc_framec W v with
SomeWIntn
⇒
Some
match if n =? 0 then t2 else t1 with
To_
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
Poll_
⇒
match fr_doomedc_framec with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
_
⇒
None
JRetv
⇒
or_stuck"ret"
match fn_retfr_fnc_framec with
RClassk
⇒
let* x ≔ value_xge fr_envc_framec k v in
Somereturn_stepc Retx
_
⇒
None
JRetsvs
⇒
or_stuck"rets"
match fn_retfr_fnc_framec with
RTupleks
⇒
let* xs ≔ map2_optionvalue_xge fr_envc_framec ks vs in
Somereturn_stepc Retsxs
_
⇒
None
JRetVoid
⇒
return_stepc RetVoid
JUnreachable
⇒
DoneUnreachable
=
Nextc′
⊢
keptc_memc c_memc′

cases H ; unstuck H

9 open

ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
r :
reason
Heqo :
fr_doomedc_framec = Somer
t :
target
l0 :
name
Heqt0 :
t = Tol0
Heqj0 :
j = JGotoTol0
H :
Nextcontinue_atc with_pointc_framec Enteredge_labelTol0 Somel c_memc c_fuelc = Nextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
r :
reason
Heqo :
fr_doomedc_framec = Somer
v :
value
t1, t2 :
target
Heqj0 :
j = JBranchv t1 t2
w :
word
n :
Z
Heqw0 :
w = WIntn
Heqo0 :
value_xge fr_envc_framec W v = SomeWIntn
l0 :
name
Heqt :
n =? 0
→
t2
else
t1
=
Tol0
H :
SomeNextcontinue_atc with_pointc_framec Enteredge_labelTol0 Somel c_memc c_fuelc = SomeNextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
Heqo :
fr_doomedc_framec = None
t :
target
l0 :
name
Heqt0 :
t = Tol0
Heqj0 :
j = JGotoTol0
H :
Nextcontinue_atc with_pointc_framec Enteredge_labelTol0 Somel c_memc c_fuelc = Nextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
Heqo :
fr_doomedc_framec = None
t :
target
l0 :
name
Heqt0 :
t = Polll0
Heqj0 :
j = JGotoPolll0
H :
Nextcontinue_atc with_pointc_framec Enteredge_labelPolll0 Somel c_memc c_fuelc = Nextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
Heqo :
fr_doomedc_framec = None
v :
value
t1, t2 :
target
Heqj0 :
j = JBranchv t1 t2
w :
word
n :
Z
Heqw0 :
w = WIntn
Heqo0 :
value_xge fr_envc_framec W v = SomeWIntn
l0 :
name
Heqt :
n =? 0
→
t2
else
t1
=
Tol0
H :
SomeNextcontinue_atc with_pointc_framec Enteredge_labelTol0 Somel c_memc c_fuelc = SomeNextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
Heqo :
fr_doomedc_framec = None
v :
value
t1, t2 :
target
Heqj0 :
j = JBranchv t1 t2
w :
word
n :
Z
Heqw0 :
w = WIntn
Heqo0 :
value_xge fr_envc_framec W v = SomeWIntn
l0 :
name
Heqt :
n =? 0
→
t2
else
t1
=
Polll0
H :
SomeNextcontinue_atc with_pointc_framec Enteredge_labelPolll0 Somel c_memc c_fuelc = SomeNextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
Heqo :
fr_doomedc_framec = None
v :
value
Heqj0 :
j = JRetv
k :
class
Heqr :
fn_retfr_fnc_framec = RClassk
w :
word
Heqo0 :
value_xge fr_envc_framec k v = Somew
H :
Somereturn_stepc Retw = SomeNextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
Heqo :
fr_doomedc_framec = None
vs :
listvalue
Heqj0 :
j = JRetsvs
ks :
listclass
Heqr :
fn_retfr_fnc_framec = RTupleks
l0 :
listword
Heqo0 :
map2_optionvalue_xge fr_envc_framec ks vs = Somel0
H :
Somereturn_stepc Retsl0 = SomeNextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c :
config
l :
name
j :
jump
c′ :
config
Heqo :
fr_doomedc_framec = None
Heqj0 :
j = JRetVoid
H :
return_stepc RetVoid = Nextc′
⊢
keptc_memc c_memc′

all: first|match goal withE : return_step_ _ = Next_ |- _⇒exact return_step_kept_ _ _ Eend|inversion H ; subst;first|exact kept_refl_|match goal withE : return_step_ _ = Next_ |- _⇒exact return_step_kept_ _ _ Eend

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold jump_step , edge_step
intros ge c l j c′ H
cases H ; unstuck H
all: first|match goal withE : return_step_ _ = Next_ |- _⇒exact return_step_kept_ _ _ Eend|inversion H ; subst;first|exact kept_refl_|match goal withE : return_step_ _ = Next_ |- _⇒exact return_step_kept_ _ _ Eend

A return never allocates.

431Lemma return_step_not_fresh
∀ c o q k
return_stepc o ≠ Freshq k
Explore proof 4 tactics

Initial goal

⊢
∀c : config o : outcome q : alloc_req k : Z → config, return_stepc o ≠ Freshq k

unfold return_step

⊢
∀ c : config o : outcome q : alloc_req k : Z → config  ,
or_stuck"return"
let* m ≔ invalidate_localsfr_localsc_framec c_memc in
match c_callersc with
[ ]
⇒
SomeDoneo
caller, pd :: callers
⇒
let* o′ ≔
match pd_convpd with
Somecw, kw
⇒
converted_outcomecw kw o
None
⇒
Someo
in
let* e ≔ returned_envpd_respd o′ fr_envcaller in
SomeNextc_frame ≔ fr_fn ≔ fr_fncaller; fr_env ≔ e; fr_locals ≔ fr_localscaller; fr_doomed ≔ fr_doomedcaller; fr_at ≔ fr_atcaller; c_callers ≔ callers; c_mem ≔ m; c_fuel ≔ c_fuelc
≠
Freshq k

intros c o q k H

c :
config
o :
outcome
q :
alloc_req
k :
Z → config
H :
or_stuck"return"
let* m ≔ invalidate_localsfr_localsc_framec c_memc in
match c_callersc with
[ ]
⇒
SomeDoneo
caller, pd :: callers
⇒
let* o′ ≔
match pd_convpd with
Somecw, kw
⇒
converted_outcomecw kw o
None
⇒
Someo
in
let* e ≔ returned_envpd_respd o′ fr_envcaller in
SomeNextc_frame ≔ fr_fn ≔ fr_fncaller; fr_env ≔ e; fr_locals ≔ fr_localscaller; fr_doomed ≔ fr_doomedcaller; fr_at ≔ fr_atcaller; c_callers ≔ callers; c_mem ≔ m; c_fuel ≔ c_fuelc
=
Freshq k
⊢
False

apply or_stuck_fresh in H

c :
config
o :
outcome
q :
alloc_req
k :
Z → config
H :
let* m ≔ invalidate_localsfr_localsc_framec c_memc in
match c_callersc with
[ ]
⇒
SomeDoneo
caller, pd :: callers
⇒
let* o′ ≔
match pd_convpd with
Somecw, kw
⇒
converted_outcomecw kw o
None
⇒
Someo
in
let* e ≔ returned_envpd_respd o′ fr_envcaller in
SomeNextc_frame ≔ fr_fn ≔ fr_fncaller; fr_env ≔ e; fr_locals ≔ fr_localscaller; fr_doomed ≔ fr_doomedcaller; fr_at ≔ fr_atcaller; c_callers ≔ callers; c_mem ≔ m; c_fuel ≔ c_fuelc
=
SomeFreshq k
⊢
False

cases H

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold return_step
intros c o q k H
apply or_stuck_fresh in H
cases H

A jump never allocates.

435Lemma jump_step_not_fresh
∀ ge c l j q k
jump_stepge c l j ≠ Freshq k
Explore proof 4 tactics

Initial goal

⊢
∀ge : genv c : config l : name j : jump q : alloc_req k : Z → config, jump_stepge c l j ≠ Freshq k

unfold jump_step , edge_step

⊢
∀ ge : genv c : config l : name j : jump q : alloc_req k : Z → config  ,
match fr_doomedc_framec with
Somewhy
⇒
match j with
JGotoTo_ as t
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JGotoPoll_ as t
⇒
match fr_doomedc_framec with
Somewhy0
⇒
DoneUncheckedAheadwhy0
None
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JBranchv t1 t2
⇒
or_stuck"branch"
match value_xge fr_envc_framec W v with
SomeWIntn
⇒
Some
match if n =? 0 then t2 else t1 with
To_
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
Poll_
⇒
match fr_doomedc_framec with
Somewhy0
⇒
DoneUncheckedAheadwhy0
None
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
_
⇒
None
_
⇒
DoneUncheckedAheadwhy
None
⇒
match j with
JGotoTo_ as t
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JGotoPoll_ as t
⇒
match fr_doomedc_framec with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JBranchv t1 t2
⇒
or_stuck"branch"
match value_xge fr_envc_framec W v with
SomeWIntn
⇒
Some
match if n =? 0 then t2 else t1 with
To_
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
Poll_
⇒
match fr_doomedc_framec with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
_
⇒
None
JRetv
⇒
or_stuck"ret"
match fn_retfr_fnc_framec with
RClassk0
⇒
let* x ≔ value_xge fr_envc_framec k0 v in
Somereturn_stepc Retx
_
⇒
None
JRetsvs
⇒
or_stuck"rets"
match fn_retfr_fnc_framec with
RTupleks
⇒
let* xs ≔ map2_optionvalue_xge fr_envc_framec ks vs in
Somereturn_stepc Retsxs
_
⇒
None
JRetVoid
⇒
return_stepc RetVoid
JUnreachable
⇒
DoneUnreachable
≠
Freshq k

intros ge c l j q k H

ge :
genv
c :
config
l :
name
j :
jump
q :
alloc_req
k :
Z → config
H :
match fr_doomedc_framec with
Somewhy
⇒
match j with
JGotoTo_ as t
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JGotoPoll_ as t
⇒
match fr_doomedc_framec with
Somewhy0
⇒
DoneUncheckedAheadwhy0
None
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JBranchv t1 t2
⇒
or_stuck"branch"
match value_xge fr_envc_framec W v with
SomeWIntn
⇒
Some
match if n =? 0 then t2 else t1 with
To_
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
Poll_
⇒
match fr_doomedc_framec with
Somewhy0
⇒
DoneUncheckedAheadwhy0
None
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
_
⇒
None
_
⇒
DoneUncheckedAheadwhy
None
⇒
match j with
JGotoTo_ as t
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JGotoPoll_ as t
⇒
match fr_doomedc_framec with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
Nextcontinue_atc with_pointc_framec Enteredge_labelt Somel c_memc c_fuelc
JBranchv t1 t2
⇒
or_stuck"branch"
match value_xge fr_envc_framec W v with
SomeWIntn
⇒
Some
match if n =? 0 then t2 else t1 with
To_
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
Poll_
⇒
match fr_doomedc_framec with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
Next
continue_atc
with_pointc_framec
Enter
edge_label
n =? 0
→
t2
else
t1
Somel
c_memc
c_fuelc
_
⇒
None
JRetv
⇒
or_stuck"ret"
match fn_retfr_fnc_framec with
RClassk
⇒
let* x ≔ value_xge fr_envc_framec k v in
Somereturn_stepc Retx
_
⇒
None
JRetsvs
⇒
or_stuck"rets"
match fn_retfr_fnc_framec with
RTupleks
⇒
let* xs ≔ map2_optionvalue_xge fr_envc_framec ks vs in
Somereturn_stepc Retsxs
_
⇒
None
JRetVoid
⇒
return_stepc RetVoid
JUnreachable
⇒
DoneUnreachable
=
Freshq k
⊢
False

cases H ; unstuck H ; first exact return_step_not_fresh_ _ _ _ H | discriminate H | idtac

2 open

ge :
genv
c :
config
l :
name
j :
jump
q :
alloc_req
k :
Z → config
Heqo :
fr_doomedc_framec = None
v :
value
Heqj0 :
j = JRetv
k0 :
class
Heqr :
fn_retfr_fnc_framec = RClassk0
w :
word
Heqo0 :
value_xge fr_envc_framec k0 v = Somew
H :
Somereturn_stepc Retw = SomeFreshq k
⊢
False
focused goal
ge :
genv
c :
config
l :
name
j :
jump
q :
alloc_req
k :
Z → config
Heqo :
fr_doomedc_framec = None
vs :
listvalue
Heqj0 :
j = JRetsvs
ks :
listclass
Heqr :
fn_retfr_fnc_framec = RTupleks
l0 :
listword
Heqo0 :
map2_optionvalue_xge fr_envc_framec ks vs = Somel0
H :
Somereturn_stepc Retsl0 = SomeFreshq k
⊢
False

all: injection H as H ; exact return_step_not_fresh_ _ _ _ H

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold jump_step , edge_step
intros ge c l j q k H
cases H ; unstuck H ; first exact return_step_not_fresh_ _ _ _ H | discriminate H | idtac
all: injection H as H ; exact return_step_not_fresh_ _ _ _ H

A tactic for the goal that memory is kept: an instruction's result after doomed_result, if it continues, continues as the instruction did, or with the memory unchanged. Hfree says the genv's free is kept.

446Ltac doomed_continue Hfree ≔
match goal withE : doomed_result_ _ _ _ _ = Some?od , Hc : fst?od = IContinue_ _ _ |- _⇒destruct doomed_result_out_ _ _ _ _ _ E as Ho | e′ & l′ & Ho | o′ & Ho ; rewrite Ho in Hc ; try discriminate Hc;|subst;match goal withE2 : inst_step_ _ _ _ = SomeIContinue_ _ _ |- _⇒exact inst_step_kept_ _ _ _ _ _ _ Hfree E2end|inversion Hc ; subst ; apply kept_reflend

A step that allocates nothing changes memory as kept allows, where the genv's free does: its writes pass an access check, so they are inside an object, and it ends objects (free, and an invocation's locals) without moving one.

461Lemma step_next_kept
∀ ge c c′
free_keepsge_freege
stepge c = Nextc′
keptc_memc c_memc′
Explore proof 4 tactics

Initial goal

⊢
∀ge : genv c c′ : config, free_keepsge_freege → stepge c = Nextc′ → keptc_memc c_memc′

unfold step , inst_next

⊢
∀ ge : genv c c′ : config  ,
free_keepsge_freege
→
match fr_atc_framec with
Snapshot[ ] l
⇒
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ 0; ptr_cap ≔ CNone fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc
Snapshot_ :: _ as xs l
⇒
negbforallbsnapshot_word xs
→
Stuck"snapshot"
else
Freshheap_req16 8 × Z.of_natDatatypes.lengthxs λbase : Z ⇒ c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ base; ptr_cap ≔ Robase fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ snapshot_valuesxs base allocatedc_memc heap_req16 8 × Z.of_natDatatypes.lengthxs base; c_fuel ≔ c_fuelc
Enterl from
⇒
c_fuelc =? 0
→
DoneOutOfFuel
else
or_stuck"block"
let* b ≔ find_blockl fn_blocksfr_fnc_framec in
let* e ≔ phis_envge b_phisb from fr_envc_framec in
SomeNextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Execl b_instsb b_jumpb; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc − 1
Execl [ ] j
⇒
jump_stepge c l j
Execl i :: rest j
⇒
match let* why ≔ fr_doomedc_framec in if ahead_transparenti then None else Somewhy with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
or_stuck"instruction"
let* out0 ≔ inst_stepge c_framec c_memc i in
let* od ≔ doomed_resultge c_framec c_memc i out0 in
Some
match fstod with
IContinuee locals m′
⇒
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndod; fr_at ≔ Execl rest j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
IStopo
⇒
Doneo
ICallg xs pd
⇒
match call_frameg xs with
Somecallee
⇒
Nextc_frame ≔ callee; c_callers ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ sndod; fr_at ≔ Execl rest j, pd :: c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc
None
⇒
Stuck"call"
IFreshq k
⇒
Freshq
λ base : Z  ⇒
let p, m′ ≔ kbase in
let e, locals ≔ p in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndod; fr_at ≔ Execl rest j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
ICallOutside_
⇒
Stuck"outside the core subset"
=
Nextc′
→
keptc_memc c_memc′

intros ge c c′ Hfree H

ge :
genv
c, c′ :
config
Hfree :
free_keepsge_freege
H :
match fr_atc_framec with
Snapshot[ ] l
⇒
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ 0; ptr_cap ≔ CNone fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc
Snapshot_ :: _ as xs l
⇒
negbforallbsnapshot_word xs
→
Stuck"snapshot"
else
Freshheap_req16 8 × Z.of_natDatatypes.lengthxs λbase : Z ⇒ c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ base; ptr_cap ≔ Robase fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ snapshot_valuesxs base allocatedc_memc heap_req16 8 × Z.of_natDatatypes.lengthxs base; c_fuel ≔ c_fuelc
Enterl from
⇒
c_fuelc =? 0
→
DoneOutOfFuel
else
or_stuck"block"
let* b ≔ find_blockl fn_blocksfr_fnc_framec in
let* e ≔ phis_envge b_phisb from fr_envc_framec in
SomeNextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Execl b_instsb b_jumpb; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc − 1
Execl [ ] j
⇒
jump_stepge c l j
Execl i :: rest j
⇒
match let* why ≔ fr_doomedc_framec in if ahead_transparenti then None else Somewhy with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
or_stuck"instruction"
let* out0 ≔ inst_stepge c_framec c_memc i in
let* od ≔ doomed_resultge c_framec c_memc i out0 in
Some
match fstod with
IContinuee locals m′
⇒
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndod; fr_at ≔ Execl rest j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
IStopo
⇒
Doneo
ICallg xs pd
⇒
match call_frameg xs with
Somecallee
⇒
Nextc_frame ≔ callee; c_callers ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ sndod; fr_at ≔ Execl rest j, pd :: c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc
None
⇒
Stuck"call"
IFreshq k
⇒
Freshq
λ base : Z  ⇒
let p, m′ ≔ kbase in
let e, locals ≔ p in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndod; fr_at ≔ Execl rest j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
ICallOutside_
⇒
Stuck"outside the core subset"
=
Nextc′
⊢
keptc_memc c_memc′

cases H ; unstuck H

5 open

ge :
genv
c, c′ :
config
Hfree :
free_keepsge_freege
xs :
listword
l :
name
Heql0 :
xs = [ ]
Heqp :
fr_atc_framec = Snapshot[ ] l
H :
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ 0; ptr_cap ≔ CNone fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc = Nextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c, c′ :
config
Hfree :
free_keepsge_freege
l :
name
from :
optionname
Heqp :
fr_atc_framec = Enterl from
Heqb :
c_fuelc =? 0 = false
b :
block
Heqo :
find_blockl fn_blocksfr_fnc_framec = Someb
e :
env
Heqo0 :
phis_envge b_phisb from fr_envc_framec = Somee
H :
SomeNextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Execl b_instsb b_jumpb; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc − 1 = SomeNextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c, c′ :
config
Hfree :
free_keepsge_freege
l :
name
insts :
listinst
j :
jump
Heql0 :
insts = [ ]
Heqp :
fr_atc_framec = Execl [ ] j
H :
jump_stepge c l j = Nextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c, c′ :
config
Hfree :
free_keepsge_freege
l :
name
insts :
listinst
j :
jump
i :
inst
l0 :
listinst
Heql0 :
insts = i :: l0
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
i0 :
inst_out
Heqo0 :
inst_stepge c_framec c_memc i = Somei0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i i0 = Somep
e :
env
locals :
listname × optionZ
m :
mem
Heqi1 :
fstp = IContinuee locals m
H :
SomeNextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m; c_fuel ≔ c_fuelc = SomeNextc′
⊢
keptc_memc c_memc′
focused goal
ge :
genv
c, c′ :
config
Hfree :
free_keepsge_freege
l :
name
insts :
listinst
j :
jump
i :
inst
l0 :
listinst
Heql0 :
insts = i :: l0
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
i0 :
inst_out
Heqo0 :
inst_stepge c_framec c_memc i = Somei0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i i0 = Somep
f :
func
xs :
listword
pd :
pending
Heqi1 :
fstp = ICallf xs pd
f0 :
frame
Heqo2 :
call_framef xs = Somef0
H :
SomeNextc_frame ≔ f0; c_callers ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j, pd :: c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc = SomeNextc′
⊢
keptc_memc c_memc′

all: first|match goal withE : jump_step_ _ _ _ = Next_ |- _⇒exact jump_step_kept_ _ _ _ _ Eend|inversion H ; subst ; simpl ; first exact kept_refl_ | doomed_continue Hfree

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold step , inst_next
intros ge c c′ Hfree H
cases H ; unstuck H
all: first|match goal withE : jump_step_ _ _ _ = Next_ |- _⇒exact jump_step_kept_ _ _ _ _ Eend|inversion H ; subst ; simpl ; first exact kept_refl_ | doomed_continue Hfree

An allocation step: its request has a positive alignment, and what the step then writes, such as a variadic callee's argument snapshot, is inside the new object.

474Lemma step_fresh_kept
∀ ge c q k
stepge c = Freshq k
0 < req_alignq ∧ ∀base, keptallocatedc_memc q base c_memkbase
Explore proof 16 tactics

Initial goal

⊢
∀ge : genv c : config q : alloc_req k : Z → config, stepge c = Freshq k → 0 < req_alignq ∧ ∀base : Z, keptallocatedc_memc q base c_memkbase

unfold step , inst_next

⊢
∀ ge : genv c : config q : alloc_req k : Z → config  ,
match fr_atc_framec with
Snapshot[ ] l
⇒
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ 0; ptr_cap ≔ CNone fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc
Snapshot_ :: _ as xs l
⇒
negbforallbsnapshot_word xs
→
Stuck"snapshot"
else
Freshheap_req16 8 × Z.of_natDatatypes.lengthxs λbase : Z ⇒ c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ base; ptr_cap ≔ Robase fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ snapshot_valuesxs base allocatedc_memc heap_req16 8 × Z.of_natDatatypes.lengthxs base; c_fuel ≔ c_fuelc
Enterl from
⇒
c_fuelc =? 0
→
DoneOutOfFuel
else
or_stuck"block"
let* b ≔ find_blockl fn_blocksfr_fnc_framec in
let* e ≔ phis_envge b_phisb from fr_envc_framec in
SomeNextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Execl b_instsb b_jumpb; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc − 1
Execl [ ] j
⇒
jump_stepge c l j
Execl i :: rest j
⇒
match let* why ≔ fr_doomedc_framec in if ahead_transparenti then None else Somewhy with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
or_stuck"instruction"
let* out0 ≔ inst_stepge c_framec c_memc i in
let* od ≔ doomed_resultge c_framec c_memc i out0 in
Some
match fstod with
IContinuee locals m′
⇒
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndod; fr_at ≔ Execl rest j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
IStopo
⇒
Doneo
ICallg xs pd
⇒
match call_frameg xs with
Somecallee
⇒
Nextc_frame ≔ callee; c_callers ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ sndod; fr_at ≔ Execl rest j, pd :: c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc
None
⇒
Stuck"call"
IFreshq0 k0
⇒
Freshq0
λ base : Z  ⇒
let p, m′ ≔ k0base in
let e, locals ≔ p in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndod; fr_at ≔ Execl rest j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
ICallOutside_
⇒
Stuck"outside the core subset"
=
Freshq k
→
0 < req_alignq ∧ ∀base : Z, keptallocatedc_memc q base c_memkbase

intros ge c q k H

ge :
genv
c :
config
q :
alloc_req
k :
Z → config
H :
match fr_atc_framec with
Snapshot[ ] l
⇒
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ 0; ptr_cap ≔ CNone fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc
Snapshot_ :: _ as xs l
⇒
negbforallbsnapshot_word xs
→
Stuck"snapshot"
else
Freshheap_req16 8 × Z.of_natDatatypes.lengthxs λbase : Z ⇒ c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ base; ptr_cap ≔ Robase fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ snapshot_valuesxs base allocatedc_memc heap_req16 8 × Z.of_natDatatypes.lengthxs base; c_fuel ≔ c_fuelc
Enterl from
⇒
c_fuelc =? 0
→
DoneOutOfFuel
else
or_stuck"block"
let* b ≔ find_blockl fn_blocksfr_fnc_framec in
let* e ≔ phis_envge b_phisb from fr_envc_framec in
SomeNextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Execl b_instsb b_jumpb; c_callers ≔ c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc − 1
Execl [ ] j
⇒
jump_stepge c l j
Execl i :: rest j
⇒
match let* why ≔ fr_doomedc_framec in if ahead_transparenti then None else Somewhy with
Somewhy
⇒
DoneUncheckedAheadwhy
None
⇒
or_stuck"instruction"
let* out0 ≔ inst_stepge c_framec c_memc i in
let* od ≔ doomed_resultge c_framec c_memc i out0 in
Some
match fstod with
IContinuee locals m′
⇒
Nextc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndod; fr_at ≔ Execl rest j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
IStopo
⇒
Doneo
ICallg xs pd
⇒
match call_frameg xs with
Somecallee
⇒
Nextc_frame ≔ callee; c_callers ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ sndod; fr_at ≔ Execl rest j, pd :: c_callersc; c_mem ≔ c_memc; c_fuel ≔ c_fuelc
None
⇒
Stuck"call"
IFreshq k
⇒
Freshq
λ base : Z  ⇒
let p, m′ ≔ kbase in
let e, locals ≔ p in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndod; fr_at ≔ Execl rest j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
ICallOutside_
⇒
Stuck"outside the core subset"
=
Freshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedc_memc q base c_memkbase

cases H ; unstuck H ; try exfalso ; exact jump_step_not_fresh_ _ _ _ _ _ H

2 open

ge :
genv
c :
config
q :
alloc_req
k :
Z → config
xs :
listword
l :
name
w :
word
l0 :
listword
Heql0 :
xs = w :: l0
Heqp :
fr_atc_framec = Snapshotw :: l0 l
Heqb :
negbforallbsnapshot_word w :: l0 = false
H :
Freshheap_req16 8 × Z.of_natDatatypes.lengthw :: l0 λbase : Z ⇒ c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ base; ptr_cap ≔ Robase fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ snapshot_valuesw :: l0 base allocatedc_memc heap_req16 8 × Z.of_natDatatypes.lengthw :: l0 base; c_fuel ≔ c_fuelc = Freshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedc_memc q base c_memkbase
focused goal
ge :
genv
c :
config
q :
alloc_req
k :
Z → config
l :
name
insts :
listinst
j :
jump
i :
inst
l0 :
listinst
Heql0 :
insts = i :: l0
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
i0 :
inst_out
Heqo0 :
inst_stepge c_framec c_memc i = Somei0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i i0 = Somep
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqi1 :
fstp = IFreshq0 k0
H :
Some
Freshq0
λ base : Z  ⇒
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc
=
SomeFreshq k
⊢
0 < req_alignq ∧ ∀base : Z, keptallocatedc_memc q base c_memkbase

all: injection H as Hq Hk ; subst q k

2 open

ge :
genv
c :
config
xs :
listword
l :
name
w :
word
l0 :
listword
Heql0 :
xs = w :: l0
Heqp :
fr_atc_framec = Snapshotw :: l0 l
Heqb :
negbforallbsnapshot_word w :: l0 = false
⊢
0 < req_alignheap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 ∧ ∀base : Z, keptallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base c_memc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ base; ptr_cap ≔ Robase fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ match w with WIntn ⇒ snapshot_valuesl0 base + 8 store_bytesallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base base 8 n | WPtrp ⇒ snapshot_valuesl0 base + 8 store_auxstore_bytesallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base base 8 ptr_addrp base ptr_capp | WLimit_ _ ⇒ snapshot_valuesl0 base + 8 allocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base end; c_fuel ≔ c_fuelc
focused goal
ge :
genv
c :
config
l :
name
insts :
listinst
j :
jump
i :
inst
l0 :
listinst
Heql0 :
insts = i :: l0
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
i0 :
inst_out
Heqo0 :
inst_stepge c_framec c_memc i = Somei0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i i0 = Somep
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqi1 :
fstp = IFreshq0 k0
⊢
0 < req_alignq0
∧
∀ base : Z  ,
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

–split ; simpl ; lia |

2 open

ge :
genv
c :
config
xs :
listword
l :
name
w :
word
l0 :
listword
Heql0 :
xs = w :: l0
Heqp :
fr_atc_framec = Snapshotw :: l0 l
Heqb :
negbforallbsnapshot_word w :: l0 = false
⊢
∀base : Z, keptallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base c_memc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ base; ptr_cap ≔ Robase fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ match w with WIntn ⇒ snapshot_valuesl0 base + 8 store_bytesallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base base 8 n | WPtrp ⇒ snapshot_valuesl0 base + 8 store_auxstore_bytesallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base base 8 ptr_addrp base ptr_capp | WLimit_ _ ⇒ snapshot_valuesl0 base + 8 allocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base end; c_fuel ≔ c_fuelc
deferred goal
ge :
genv
c :
config
l :
name
insts :
listinst
j :
jump
i :
inst
l0 :
listinst
Heql0 :
insts = i :: l0
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
i0 :
inst_out
Heqo0 :
inst_stepge c_framec c_memc i = Somei0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i i0 = Somep
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqi1 :
fstp = IFreshq0 k0
⊢
0 < req_alignq0
∧
∀ base : Z  ,
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

intros base

2 open

ge :
genv
c :
config
xs :
listword
l :
name
w :
word
l0 :
listword
Heql0 :
xs = w :: l0
Heqp :
fr_atc_framec = Snapshotw :: l0 l
Heqb :
negbforallbsnapshot_word w :: l0 = false
base :
Z
⊢
keptallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base c_memc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ putva_snapshot WPtrptr_addr ≔ base; ptr_cap ≔ Robase fr_envc_framec; fr_locals ≔ fr_localsc_framec; fr_doomed ≔ fr_doomedc_framec; fr_at ≔ Enterl None; c_callers ≔ c_callersc; c_mem ≔ match w with WIntn ⇒ snapshot_valuesl0 base + 8 store_bytesallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base base 8 n | WPtrp ⇒ snapshot_valuesl0 base + 8 store_auxstore_bytesallocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base base 8 ptr_addrp base ptr_capp | WLimit_ _ ⇒ snapshot_valuesl0 base + 8 allocatedc_memc heap_req16 Z.posPosDef.Pos.of_succ_natDatatypes.lengthl0~0~0~0 base end; c_fuel ≔ c_fuelc
deferred goal
ge :
genv
c :
config
l :
name
insts :
listinst
j :
jump
i :
inst
l0 :
listinst
Heql0 :
insts = i :: l0
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
i0 :
inst_out
Heqo0 :
inst_stepge c_framec c_memc i = Somei0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i i0 = Somep
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqi1 :
fstp = IFreshq0 k0
⊢
0 < req_alignq0
∧
∀ base : Z  ,
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

match goal withHp : fr_at_ = Snapshot?xs _ |- _⇒exact snapshot_alloc_keptxs c_memc baseend

deferred goal
ge :
genv
c :
config
l :
name
insts :
listinst
j :
jump
i :
inst
l0 :
listinst
Heql0 :
insts = i :: l0
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
i0 :
inst_out
Heqo0 :
inst_stepge c_framec c_memc i = Somei0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i i0 = Somep
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqi1 :
fstp = IFreshq0 k0
⊢
0 < req_alignq0
∧
∀ base : Z  ,
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

–match goal withE : doomed_result_ _ _ _ _ = Some?od , Hc : fst?od = IFresh_ _ |- _⇒destruct doomed_result_out_ _ _ _ _ _ E as Ho | e′ & l′ & Ho | o′ & Ho ; rewrite Ho in Hc ; try discriminate Hcend

ge :
genv
c :
config
l :
name
insts :
listinst
j :
jump
i :
inst
l0 :
listinst
Heql0 :
insts = i :: l0
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
i0 :
inst_out
Heqo0 :
inst_stepge c_framec c_memc i = Somei0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i i0 = Somep
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqi1 :
i0 = IFreshq0 k0
Ho :
fstp = i0
⊢
0 < req_alignq0
∧
∀ base : Z  ,
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

subst

ge :
genv
c :
config
l :
name
j :
jump
i :
inst
l0 :
listinst
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqo0 :
inst_stepge c_framec c_memc i = SomeIFreshq0 k0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i IFreshq0 k0 = Somep
Ho :
fstp = IFreshq0 k0
⊢
0 < req_alignq0
∧
∀ base : Z  ,
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

match goal withE2 : inst_step_ _ _ _ = SomeIFresh_ _ |- _⇒destruct inst_step_fresh_ _ _ _ _ _ E2 as Ha Hkend

ge :
genv
c :
config
l :
name
j :
jump
i :
inst
l0 :
listinst
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqo0 :
inst_stepge c_framec c_memc i = SomeIFreshq0 k0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i IFreshq0 k0 = Somep
Ho :
fstp = IFreshq0 k0
Ha :
0 < req_alignq0
Hk :
∀base : Z, keptallocatedc_memc q0 base sndk0base
⊢
0 < req_alignq0
∧
∀ base : Z  ,
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

split ; auto

ge :
genv
c :
config
l :
name
j :
jump
i :
inst
l0 :
listinst
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqo0 :
inst_stepge c_framec c_memc i = SomeIFreshq0 k0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i IFreshq0 k0 = Somep
Ho :
fstp = IFreshq0 k0
Ha :
0 < req_alignq0
Hk :
∀base : Z, keptallocatedc_memc q0 base sndk0base
⊢
∀ base : Z  ,
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

intros base

ge :
genv
c :
config
l :
name
j :
jump
i :
inst
l0 :
listinst
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqo0 :
inst_stepge c_framec c_memc i = SomeIFreshq0 k0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i IFreshq0 k0 = Somep
Ho :
fstp = IFreshq0 k0
Ha :
0 < req_alignq0
Hk :
∀base : Z, keptallocatedc_memc q0 base sndk0base
base :
Z
⊢
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

specialize Hkbase

ge :
genv
c :
config
l :
name
j :
jump
i :
inst
l0 :
listinst
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqo0 :
inst_stepge c_framec c_memc i = SomeIFreshq0 k0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i IFreshq0 k0 = Somep
Ho :
fstp = IFreshq0 k0
Ha :
0 < req_alignq0
base :
Z
Hk :
keptallocatedc_memc q0 base sndk0base
⊢
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

simpl

ge :
genv
c :
config
l :
name
j :
jump
i :
inst
l0 :
listinst
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqo0 :
inst_stepge c_framec c_memc i = SomeIFreshq0 k0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i IFreshq0 k0 = Somep
Ho :
fstp = IFreshq0 k0
Ha :
0 < req_alignq0
base :
Z
Hk :
keptallocatedc_memc q0 base sndk0base
⊢
keptallocatedc_memc q0 base
c_mem
let p0, m′ ≔ k0base in
let e, locals ≔ p0 in
c_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e; fr_locals ≔ locals; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m′; c_fuel ≔ c_fuelc

match goal withHk : kept_ snd?t |- _⇒destruct t as e1 l1 m1end

ge :
genv
c :
config
l :
name
j :
jump
i :
inst
l0 :
listinst
Heqp :
fr_atc_framec = Execl i :: l0 j
Heqo :
let* why ≔ fr_doomedc_framec in
ahead_transparenti
→
None
else
Somewhy
=
None
q0 :
alloc_req
k0 :
Z → env × listname × optionZ × mem
Heqo0 :
inst_stepge c_framec c_memc i = SomeIFreshq0 k0
p :
inst_out × optionreason
Heqo1 :
doomed_resultge c_framec c_memc i IFreshq0 k0 = Somep
Ho :
fstp = IFreshq0 k0
Ha :
0 < req_alignq0
base :
Z
e1 :
env
l1 :
listname × optionZ
m1 :
mem
Hk :
keptallocatedc_memc q0 base snde1, l1, m1
⊢
keptallocatedc_memc q0 base c_memc_frame ≔ fr_fn ≔ fr_fnc_framec; fr_env ≔ e1; fr_locals ≔ l1; fr_doomed ≔ sndp; fr_at ≔ Execl l0 j; c_callers ≔ c_callersc; c_mem ≔ m1; c_fuel ≔ c_fuelc

exact Hk

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold step , inst_next
intros ge c q k H
cases H ; unstuck H ; try exfalso ; exact jump_step_not_fresh_ _ _ _ _ _ H
all: injection H as Hq Hk ; subst q k
–split ; simpl ; lia |
intros base
match goal withHp : fr_at_ = Snapshot?xs _ |- _⇒exact snapshot_alloc_keptxs c_memc baseend
–match goal withE : doomed_result_ _ _ _ _ = Some?od , Hc : fst?od = IFresh_ _ |- _⇒destruct doomed_result_out_ _ _ _ _ _ E as Ho | e′ & l′ & Ho | o′ & Ho ; rewrite Ho in Hc ; try discriminate Hcend
subst
match goal withE2 : inst_step_ _ _ _ = SomeIFresh_ _ |- _⇒destruct inst_step_fresh_ _ _ _ _ _ E2 as Ha Hkend
split ; auto
intros base
specialize Hkbase
simpl
match goal withHk : kept_ snd?t |- _⇒destruct t as e1 l1 m1end
exact Hk

The machine of a runtime

The machine of a runtime R: one thread's program run under R, and the proof that R stays ready in every state of a run.

501Module Machine R : Runtime

A machine state: the configuration and the runtime's state.

504Definition mstate: Type ≔
config × R.state%type

The configuration c with its memory replaced by m.

507Definition with_memc : config m : mem : config ≔
mkConfigc_framec c_callersc m c_fuelc

The edge the jump j of the frame fr takes, read as jump_step reads it; None for a return or unreachable.

512Definition taken_targetge : genv fr : frame j : jump : optiontarget ≔
match j with
JGotot
⇒
Somet
JBranchv t1 t2
⇒
match value_xge fr_envfr W v with
SomeWIntn
⇒
Some
n =? 0
→
t2
else
t1
_
⇒
None
_
⇒
None

The runtime may act now, in the configuration c: before c's next instruction where R.at_safepoint holds, or before a poll edge is taken.

526Definition safepoint_duege : genv c : config : bool ≔
match fr_atc_framec with
Exec_ i :: _ _
⇒
R.at_safepointi
Exec_ [ ] j
⇒
match taken_targetge c_framec j with
SomePoll_
⇒
true
_
⇒
false
_
⇒
false

A machine step:

  • mstep_next: a step that allocates nothing;
  • mstep_alloc: an allocation at R's alloc, taking its base and R's next state; where R refuses (out of memory) there is no step, and the run traps;
  • mstep_safepoint: R's action at a safepoint, where one is due: R.safepoint may change memory and R's state.
541Inductive mstepge : genv : mstate → mstate → Prop ≔
∀ c c′ s
stepge c = Nextc′
mstep_next
mstepge c, s c′, s
∀ c q k s base s′
stepge c = Freshq k
0 ≤ req_extentq
R.allocs q = Somebase, s′
mstep_alloc
mstepge c, s kbase, s′
∀ c s s′ m′
safepoint_duege c = true
R.safepoints c_memc s′ m′
mstep_safepoint
mstepge c, s with_memc m′, s′

The machine state st' follows st after any number of machine steps.

557Definition mreachablege : genv : mstate → mstate → Prop ≔
clos_refl_transmstate mstepge

The runtime is ready to allocate in the memory: R.ready of the runtime's state and the configuration's memory.

562Definition ready_statest : mstate : Prop ≔
R.readysndst c_memfstst

R's free changes memory as kept allows, by R's free_lifetime and retention.

566Lemma free_kept
free_keepsR.free
Explore proof 7 tactics

Initial goal

⊢
free_keepsR.free

intros x m r m′ H

x :
word
m :
mem
r :
resunit
m′ :
mem
H :
R.freex m = Somer, m′
⊢
keptm m′

split

2 open

x :
word
m :
mem
r :
resunit
m′ :
mem
H :
R.freex m = Somer, m′
⊢
region_stepm m′
focused goal
x :
word
m :
mem
r :
resunit
m′ :
mem
H :
R.freex m = Somer, m′
⊢
∀a : Z, ¬in_objectm a → m_bytesm′ a = m_bytesm a ∧ m_auxm′ a = m_auxm a

–apply lifetime_region_step

2 open

x :
word
m :
mem
r :
resunit
m′ :
mem
H :
R.freex m = Somer, m′
⊢
lifetime_stepm m′
deferred goal
x :
word
m :
mem
r :
resunit
m′ :
mem
H :
R.freex m = Somer, m′
⊢
∀a : Z, ¬in_objectm a → m_bytesm′ a = m_bytesm a ∧ m_auxm′ a = m_auxm a

eapply R.free_lifetime ; eauto

deferred goal
x :
word
m :
mem
r :
resunit
m′ :
mem
H :
R.freex m = Somer, m′
⊢
∀a : Z, ¬in_objectm a → m_bytesm′ a = m_bytesm a ∧ m_auxm′ a = m_auxm a

–intros a _

x :
word
m :
mem
r :
resunit
m′ :
mem
H :
R.freex m = Somer, m′
a :
Z
⊢
m_bytesm′ a = m_bytesm a ∧ m_auxm′ a = m_auxm a

destruct R.retention_ _ _ _ H as → →

x :
word
m :
mem
r :
resunit
m′ :
mem
H :
R.freex m = Somer, m′
a :
Z
⊢
m_bytesm a = m_bytesm a ∧ m_auxm a = m_auxm a

auto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros x m r m′ H
split
–apply lifetime_region_step
eapply R.free_lifetime ; eauto
–intros a _
destruct R.retention_ _ _ _ H as → →
auto

Every machine step keeps the runtime ready, where the genv's free is R's:

  • a step that allocates nothing changes memory only as kept allows (step_next_kept), so R's ready_kept applies;
  • an allocation is R's fresh_allocation, and what the step then writes is inside the new object (step_fresh_kept);
  • a safepoint is R's safepoint_ready.
580Theorem mstep_ready
∀ ge st st′
ge_freege = R.free
mstepge st st′
ready_statest
ready_statest′
Explore proof 14 tactics

Initial goal

⊢
∀ge : genv st st′ : mstate, ge_freege = R.free → mstepge st st′ → ready_statest → ready_statest′

unfold ready_state

⊢
∀ge : genv st st′ : mstate, ge_freege = R.free → mstepge st st′ → R.readysndst c_memfstst → R.readysndst′ c_memfstst′

intros ge st st′ Hge Hs Hr

ge :
genv
st, st′ :
mstate
Hge :
ge_freege = R.free
Hs :
mstepge st st′
Hr :
R.readysndst c_memfstst
⊢
R.readysndst′ c_memfstst′

destruct Hs as c c′ s E | c q k s base s′ E He Hal | c s s′ m′ _ Hsp

3 open

ge :
genv
Hge :
ge_freege = R.free
c, c′ :
config
s :
R.state
E :
stepge c = Nextc′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndc′, s c_memfstc′, s
focused goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndkbase, s′ c_memfstkbase, s′
focused goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

–simpl in *

3 open

ge :
genv
Hge :
ge_freege = R.free
c, c′ :
config
s :
R.state
E :
stepge c = Nextc′
Hr :
R.readys c_memc
⊢
R.readys c_memc′
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndkbase, s′ c_memfstkbase, s′
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

eapply R.ready_kept ; exact Hr |

3 open

ge :
genv
Hge :
ge_freege = R.free
c, c′ :
config
s :
R.state
E :
stepge c = Nextc′
Hr :
R.readys c_memc
⊢
keptc_memc c_memc′
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndkbase, s′ c_memfstkbase, s′
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

eapply step_next_kept ; | exact E

3 open

ge :
genv
Hge :
ge_freege = R.free
c, c′ :
config
s :
R.state
E :
stepge c = Nextc′
Hr :
R.readys c_memc
⊢
free_keepsge_freege
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndkbase, s′ c_memfstkbase, s′
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

rewrite Hge

3 open

ge :
genv
Hge :
ge_freege = R.free
c, c′ :
config
s :
R.state
E :
stepge c = Nextc′
Hr :
R.readys c_memc
⊢
free_keepsR.free
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndkbase, s′ c_memfstkbase, s′
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

exact free_kept

2 open

deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndkbase, s′ c_memfstkbase, s′
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

–simpl in *

2 open

ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readys c_memc
⊢
R.readys′ c_memkbase
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

destruct step_fresh_kept_ _ _ _ E as Ha Hk

2 open

ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readys c_memc
Ha :
0 < req_alignq
Hk :
∀base : Z, keptallocatedc_memc q base c_memkbase
⊢
R.readys′ c_memkbase
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

destruct R.fresh_allocation_ _ _ _ _ Hr Ha He Hal as _ Hr′

2 open

ge :
genv
Hge :
ge_freege = R.free
c :
config
q :
alloc_req
k :
Z → config
s :
R.state
base :
Z
s′ :
R.state
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Hr :
R.readys c_memc
Ha :
0 < req_alignq
Hk :
∀base : Z, keptallocatedc_memc q base c_memkbase
Hr′ :
R.readys′ allocatedc_memc q base
⊢
R.readys′ c_memkbase
deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

eapply R.ready_kept ; exact Hr′ | apply Hk

deferred goal
ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readysndc, s c_memfstc, s
⊢
R.readysndwith_memc m′, s′ c_memfstwith_memc m′, s′

–simpl in *

ge :
genv
Hge :
ge_freege = R.free
c :
config
s, s′ :
R.state
m′ :
mem
Hsp :
R.safepoints c_memc s′ m′
Hr :
R.readys c_memc
⊢
R.readys′ m′

eapply R.safepoint_ready ; eauto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold ready_state
intros ge st st′ Hge Hs Hr
destruct Hs as c c′ s E | c q k s base s′ E He Hal | c s s′ m′ _ Hsp
–simpl in *
eapply R.ready_kept ; exact Hr |
eapply step_next_kept ; | exact E
rewrite Hge
exact free_kept
–simpl in *
destruct step_fresh_kept_ _ _ _ E as Ha Hk
destruct R.fresh_allocation_ _ _ _ _ Hr Ha He Hal as _ Hr′
eapply R.ready_kept ; exact Hr′ | apply Hk
–simpl in *
eapply R.safepoint_ready ; eauto

Ready initially, ready ever after: if the runtime is ready in st and the genv's free is R's, it is ready in every state a run from st reaches.

595Theorem mreachable_ready
∀ ge st st′
ge_freege = R.free
mreachablege st st′
ready_statest
ready_statest′
Explore proof 3 tactics

Initial goal

⊢
∀ge : genv st st′ : mstate, ge_freege = R.free → mreachablege st st′ → ready_statest → ready_statest′

intros ge st st′ Hge H

ge :
genv
st, st′ :
mstate
Hge :
ge_freege = R.free
H :
mreachablege st st′
⊢
ready_statest → ready_statest′

induction H as st st′ Hs | st | st1 st2 st3 _ IH1 _ IH2 ; auto

ge :
genv
Hge :
ge_freege = R.free
st, st′ :
mstate
Hs :
mstepge st st′
⊢
ready_statest → ready_statest′

eapply mstep_ready ; eauto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros ge st st′ Hge H
induction H as st st′ Hs | st | st1 st2 st3 _ IH1 _ IH2 ; auto
eapply mstep_ready ; eauto

Ready initially, no object of any state a run reaches ends past 2^48 (R's address_space): the coverage lemma's objects_fit hypothesis (Coverage.v) holds in every state of a run that starts ready.

606Corollary mreachable_objects_fit
∀ ge st st′
ge_freege = R.free
mreachablege st st′
ready_statest
objects_fitm_regionsc_memfstst′
Explore proof 3 tactics

Initial goal

⊢
∀ge : genv st st′ : mstate, ge_freege = R.free → mreachablege st st′ → ready_statest → objects_fitm_regionsc_memfstst′

intros ge st st′ Hge H Hr

ge :
genv
st, st′ :
mstate
Hge :
ge_freege = R.free
H :
mreachablege st st′
Hr :
ready_statest
⊢
objects_fitm_regionsc_memfstst′

apply R.address_spacesndst′

ge :
genv
st, st′ :
mstate
Hge :
ge_freege = R.free
H :
mreachablege st st′
Hr :
ready_statest
⊢
R.readysndst′ c_memfstst′

exact mreachable_ready_ _ _ Hge H Hr

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
intros ge st st′ Hge H Hr
apply R.address_spacesndst′
exact mreachable_ready_ _ _ Hge H Hr

The machine refines the step relation: from a ready state, a machine step is a step of step_rel, its allocation's base fresh, or the runtime's action at a safepoint that was due. So whatever the soundness theorem proves of step_rel and the safepoint relation holds of every run of every runtime.

619Theorem mstep_refines
∀ ge c s c′ s′
ready_statec, s
mstepge c, s c′, s′
step_relge c c′ ∨ safepoint_duege c = true ∧ ∃m′, c′ = with_memc m′ ∧ R.safepoints c_memc s′ m′
Explore proof 13 tactics

Initial goal

⊢
∀ge : genv c : config s : R.state c′ : config s′ : R.state, ready_statec, s → mstepge c, s c′, s′ → step_relge c c′ ∨ safepoint_duege c = true ∧ ∃m′ : mem, c′ = with_memc m′ ∧ R.safepoints c_memc s′ m′

unfold ready_state

⊢
∀ge : genv c : config s : R.state c′ : config s′ : R.state, R.readysndc, s c_memfstc, s → mstepge c, s c′, s′ → step_relge c c′ ∨ safepoint_duege c = true ∧ ∃m′ : mem, c′ = with_memc m′ ∧ R.safepoints c_memc s′ m′

simpl

⊢
∀ge : genv c : config s : R.state c′ : config s′ : R.state, R.readys c_memc → mstepge c, s c′, s′ → step_relge c c′ ∨ safepoint_duege c = true ∧ ∃m′ : mem, c′ = with_memc m′ ∧ R.safepoints c_memc s′ m′

intros ge c s c′ s′ Hr Hs

ge :
genv
c :
config
s :
R.state
c′ :
config
s′ :
R.state
Hr :
R.readys c_memc
Hs :
mstepge c, s c′, s′
⊢
step_relge c c′ ∨ safepoint_duege c = true ∧ ∃m′ : mem, c′ = with_memc m′ ∧ R.safepoints c_memc s′ m′

inversion Hs as c0 c1 s0 E | c0 q k s0 base s1 E He Hal | c0 s0 s1 m′ Hd Hsp ; subst

3 open

ge :
genv
c, c′ :
config
s′ :
R.state
Hs :
mstepge c, s′ c′, s′
Hr :
R.readys′ c_memc
E :
stepge c = Nextc′
⊢
step_relge c c′ ∨ safepoint_duege c = true ∧ ∃m′ : mem, c′ = with_memc m′ ∧ R.safepoints′ c_memc s′ m′
focused goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
k :
Z → config
base :
Z
Hs :
mstepge c, s kbase, s′
q :
alloc_req
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
⊢
step_relge c kbase ∨ safepoint_duege c = true ∧ ∃m′ : mem, kbase = with_memc m′ ∧ R.safepoints c_memc s′ m′
focused goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
step_relge c with_memc m′ ∨ safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

–left

3 open

ge :
genv
c, c′ :
config
s′ :
R.state
Hs :
mstepge c, s′ c′, s′
Hr :
R.readys′ c_memc
E :
stepge c = Nextc′
⊢
step_relge c c′
deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
k :
Z → config
base :
Z
Hs :
mstepge c, s kbase, s′
q :
alloc_req
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
⊢
step_relge c kbase ∨ safepoint_duege c = true ∧ ∃m′ : mem, kbase = with_memc m′ ∧ R.safepoints c_memc s′ m′
deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
step_relge c with_memc m′ ∨ safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

apply step_next

3 open

ge :
genv
c, c′ :
config
s′ :
R.state
Hs :
mstepge c, s′ c′, s′
Hr :
R.readys′ c_memc
E :
stepge c = Nextc′
⊢
stepge c = Nextc′
deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
k :
Z → config
base :
Z
Hs :
mstepge c, s kbase, s′
q :
alloc_req
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
⊢
step_relge c kbase ∨ safepoint_duege c = true ∧ ∃m′ : mem, kbase = with_memc m′ ∧ R.safepoints c_memc s′ m′
deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
step_relge c with_memc m′ ∨ safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

exact E

2 open

deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
k :
Z → config
base :
Z
Hs :
mstepge c, s kbase, s′
q :
alloc_req
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
⊢
step_relge c kbase ∨ safepoint_duege c = true ∧ ∃m′ : mem, kbase = with_memc m′ ∧ R.safepoints c_memc s′ m′
deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
step_relge c with_memc m′ ∨ safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

–left

2 open

ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
k :
Z → config
base :
Z
Hs :
mstepge c, s kbase, s′
q :
alloc_req
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
⊢
step_relge c kbase
deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
step_relge c with_memc m′ ∨ safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

destruct step_fresh_kept_ _ _ _ E as Ha _

2 open

ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
k :
Z → config
base :
Z
Hs :
mstepge c, s kbase, s′
q :
alloc_req
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Ha :
0 < req_alignq
⊢
step_relge c kbase
deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
step_relge c with_memc m′ ∨ safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

destruct R.fresh_allocation_ _ _ _ _ Hr Ha He Hal as Hf _

2 open

ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
k :
Z → config
base :
Z
Hs :
mstepge c, s kbase, s′
q :
alloc_req
E :
stepge c = Freshq k
He :
0 ≤ req_extentq
Hal :
R.allocs q = Somebase, s′
Ha :
0 < req_alignq
Hf :
freshc_memc q base
⊢
step_relge c kbase
deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
step_relge c with_memc m′ ∨ safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

eapply step_fresh ; eauto

deferred goal
ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
step_relge c with_memc m′ ∨ safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

–right

ge :
genv
c :
config
s, s′ :
R.state
Hr :
R.readys c_memc
m′ :
mem
Hs :
mstepge c, s with_memc m′, s′
Hd :
safepoint_duege c = true
Hsp :
R.safepoints c_memc s′ m′
⊢
safepoint_duege c = true ∧ ∃m'0 : mem, with_memc m′ = with_memc m'0 ∧ R.safepoints c_memc s′ m'0

eauto

No remaining goals.

Qed

✓ Proof checked and closed by Rocq.

Source script
unfold ready_state
simpl
intros ge c s c′ s′ Hr Hs
inversion Hs as c0 c1 s0 E | c0 q k s0 base s1 E He Hal | c0 s0 s1 m′ Hd Hsp ; subst
–left
apply step_next
exact E
–left
destruct step_fresh_kept_ _ _ _ E as Ha _
destruct R.fresh_allocation_ _ _ _ _ Hr Ha He Hal as Hf _
eapply step_fresh ; eauto
–right
eauto