Implement most operators on assertions

This commit is contained in:
Théophile Bastian 2017-11-16 15:22:35 +01:00
parent 98286d1ad8
commit 90ca9bc5bb

93
wp.v
View file

@ -66,21 +66,12 @@ Arguments find_lub_correct {T} {chain} _.
Definition subst: Mem -> Var -> Z -> Mem :=
fun (m: Mem) (v: Var) (z: Z) (v2: Var) =>
if nat_eq v v2 then z else m v2.
Notation "m [ x <- z ]" := (subst m x z) (at level 50, left associativity).
Definition MemCpo := cpo Mem.
Definition MemError := CpoError Mem.
Definition MemElem := CpoElem Mem.
Fixpoint repeat_n (n: nat) (inst: Instr) :=
let aux := (fix aux (n: nat) (inst: Instr) (acc: Instr) :=
match n with
| 0 => skip
| S n => aux n inst (seq acc inst) end) in
aux n inst (skip).
Definition while_sem_n: Expr -> Instr -> nat -> Instr :=
fun guard inst n =>
seq (repeat_n n (ifonly guard inst)) (ifonly guard abort).
Fixpoint interp (inst: Instr) (mem: MemCpo) : MemCpo :=
match mem with
| CpoError _ => MemError
@ -88,17 +79,77 @@ Fixpoint interp (inst: Instr) (mem: MemCpo) : MemCpo :=
match inst with
| skip => MemElem mem0
| abort => MemError
| assign v e => (MemElem (subst mem0 v (e mem0)))
| assign v e => (MemElem (mem0 [v <- (e mem0)]))
| seq instr1 instr2 => interp instr2 (interp instr1 (MemElem mem0))
| ifelse exp instrIf instrElse =>
if ((exp mem0) =? 0) % Z
then interp instrIf (MemElem mem0)
else interp instrElse (MemElem mem0)
| while guard instr =>
let fix while_chain (guard: Expr) (inst: Instr)
(mem: MemCpo) : (Sequence MemCpo) :=
fun n => interp (while_sem_n guard inst n) mem
in find_lub (while_chain guard instr (MemElem mem0))
| ifelse guard instrIf instrElse =>
if ((guard mem0) =? 0) % Z
then interp instrIf mem
else interp instrElse mem
| while guard body =>
let fix while_chain (mem: MemCpo) (n: nat): MemCpo :=
match n with
| 0 => mem
| S m =>
match while_chain (MemElem mem0) m with
| CpoError _ => MemError
| CpoElem _ submem =>
if ((guard submem) =? 0) % Z
then interp body (MemElem submem)
else mem
end
end
in find_lub (fun n =>
match while_chain (MemElem mem0) n with
| CpoError _ => MemError
| CpoElem _ submem =>
if ((guard submem) =? 0) % Z
then MemError
else MemElem submem
end)
end
end.
(***** Validite, prouvabilite pour Hoare *************************************)
Definition Assert := Mem -> Prop.
Delimit Scope assert with assert.
Definition assertTop : Assert := fun _ => True.
Definition assertBot : Assert := fun _ => False.
Definition assertNot : Assert -> Assert :=
fun orig mem => ~ (orig mem).
Notation "~ x" := (assertNot x) (at level 75, right associativity) : assert.
Definition assertAnd : Assert -> Assert -> Assert :=
fun x1 x2 mem => (x1 mem) /\ (x2 mem).
Infix "/\" := assertAnd : assert.
Definition assertOr : Assert -> Assert -> Assert :=
fun x1 x2 mem => (x1 mem) \/ (x2 mem).
Infix "\/" := assertOr : assert.
Definition assertImpl : Assert -> Assert -> Assert :=
fun x1 x2 => (~x1 \/ x2) % assert.
Infix "->" := assertImpl : assert.
Definition assertForall : Var -> Assert -> Assert :=
fun ident asser mem => forall (z: Z), asser (mem [ident <- z]).
Notation "\-/ x" := (assertForall x) (at level 90, no associativity) : assert.
Definition existsForall : Var -> Assert -> Assert :=
fun ident asser mem => exists (z: Z), asser (mem [ident <- z]).
Notation "'exists' x" := (existsForall x)
(at level 87, no associativity): assert.
Definition assertMemForall : Assert -> Assert :=
fun asser mem => forall (mem: Mem), asser mem.
Notation "'\-/m' x" := (assertMemForall x)
(at level 90, no associativity): assert.
Definition existsMemForall : Assert -> Assert :=
fun asser mem => exists (mem: Mem), asser mem.
Notation "'exists_m' x" := (existsMemForall x)
(at level 87, no associativity): assert.
Definition substAssert : Assert -> Var -> Z -> Assert :=
fun asser ident val mem => asser (mem [ident <- val]).
Notation "a < x <- z >" := (substAssert a x z)
(at level 50, left associativity).
Definition substAssertExpr : Assert -> Var -> Expr -> Assert :=
fun asser ident expr mem => asser (mem [ident <- (expr mem)]).
Notation "a < x <- 'expr' z >" := (substAssertExpr a x z)
(at level 50, left associativity).