315 lines
11 KiB
Coq
315 lines
11 KiB
Coq
(* Projet Coq - WP - MPRI 2.7.1 *)
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(***** Partie 1 : definition de While ****************************************)
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Require Import ZArith.BinInt.
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Import Z.
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Definition Var := nat.
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Definition Mem := Var -> Z.
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Definition Expr:= Mem -> Z.
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Inductive Instr : Type :=
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| skip: Instr
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| abort: Instr
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| assign: Var -> Expr -> Instr
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| seq: Instr -> Instr -> Instr
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| ifelse: Expr -> Instr -> Instr -> Instr
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| while: Expr -> Instr -> Instr.
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Definition ifonly (exp: Expr) (inst: Instr) : Instr :=
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ifelse exp inst skip.
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(***** CPO *******************************************************************)
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Fixpoint nat_eq x y :=
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match x, y with
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| 0, 0 => true
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| 0, S _ => false
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| S _, 0 => false
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| S x0, S y0 => nat_eq x0 y0
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end.
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Definition Sequence (S: Type) := nat -> S.
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Inductive cpo (T: Type): Type :=
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| CpoError: (cpo T)
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| CpoElem: T -> (cpo T).
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Definition cpo_leq: forall (T: Type), cpo T -> cpo T -> Prop :=
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fun T x y => match x, y with
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| CpoError _, _ => True
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| CpoElem _ x0, CpoElem _ y0 => x0 = y0
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| _, _ => False
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end.
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Arguments cpo_leq {T} _ _.
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Infix "cpo<=" := cpo_leq (at level 100).
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Definition is_chain: forall (T: Type), Sequence (cpo T) -> Prop :=
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fun T chain => forall (n: nat), (chain n) cpo<= (chain (S n)).
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Arguments is_chain {T} _.
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Definition is_lub_of: forall (T: Type), Sequence (cpo T) -> cpo T -> Prop :=
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fun T chain elt => forall (n: nat), (chain n) cpo<= elt.
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Arguments is_lub_of {T} _ _.
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Axiom find_lub: forall (T: Type), Sequence (cpo T) -> cpo T.
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Arguments find_lub {T} _.
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Axiom find_lub_correct:
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forall (T: Type), forall (chain: Sequence (cpo T)),
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is_chain chain -> is_lub_of chain (find_lub chain).
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Arguments find_lub_correct {T} {chain} _.
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(***** Interpretation ********************************************************)
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Definition subst: Mem -> Var -> Z -> Mem :=
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fun (m: Mem) (v: Var) (z: Z) (v2: Var) =>
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if nat_eq v v2 then z else m v2.
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Notation "m [ x <- z ]" := (subst m x z) (at level 50, left associativity).
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Definition MemCpo := cpo Mem.
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Definition MemError := CpoError Mem.
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Definition MemElem := CpoElem Mem.
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Fixpoint interp (inst: Instr) (mem: MemCpo) : MemCpo :=
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match mem with
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| CpoError _ => MemError
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| CpoElem _ mem0 =>
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match inst with
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| skip => MemElem mem0
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| abort => MemError
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| assign v e => (MemElem (mem0 [v <- (e mem0)]))
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| seq instr1 instr2 => interp instr2 (interp instr1 (MemElem mem0))
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| ifelse guard instrIf instrElse =>
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if ((guard mem0) =? 0) % Z
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then interp instrElse mem
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else interp instrIf mem
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| while guard body =>
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let fix while_chain (mem: MemCpo) (n: nat): MemCpo :=
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match n with
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| 0 => mem
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| S m =>
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match while_chain (MemElem mem0) m with
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| CpoError _ => MemError
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| CpoElem _ submem =>
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if ((guard submem) =? 0) % Z
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then interp body (MemElem submem)
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else mem
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end
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end
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in find_lub (fun n =>
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match while_chain (MemElem mem0) n with
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| CpoError _ => MemError
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| CpoElem _ submem =>
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if ((guard submem) =? 0) % Z
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then MemError
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else MemElem submem
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end)
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end
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end.
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Fixpoint nth_iterate (instr: Instr) (n: nat) : Instr :=
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match n with
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| 0 => skip
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| S m => seq instr (nth_iterate instr m)
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end.
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Definition satisfies_expr (mem: MemCpo) (expr: Expr) : Prop := match mem with
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| CpoError _ => False
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| CpoElem _ mem0 => (expr mem0 <> 0) % Z
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end.
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Infix "|=e" := satisfies_expr (at level 32).
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Definition expr_neg (expr: Expr) : Expr :=
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fun mem => match expr mem with
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| 0%Z => 1%Z
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| _ => 0%Z
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end.
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Lemma certain_termination body guard mem :
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interp (while guard body) mem <> MemError ->
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exists n: nat,
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(interp (nth_iterate body n) mem) |=e (expr_neg guard)
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/\ forall p, p < n -> (interp (nth_iterate body p) mem) |=e guard
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/\ interp (while guard body) mem = interp (nth_iterate body n) mem.
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Proof.
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intros notError.
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Admitted.
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(***** Validite, prouvabilite pour Hoare *************************************)
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Definition Assert := Mem -> Prop.
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Delimit Scope assert with assert.
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Definition assertTop : Assert := fun _ => True.
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Definition assertBot : Assert := fun _ => False.
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Definition assertNot : Assert -> Assert :=
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fun orig mem => ~ (orig mem).
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Notation "~ x" := (assertNot x) (at level 75, right associativity) : assert.
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Definition assertAnd : Assert -> Assert -> Assert :=
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fun x1 x2 mem => (x1 mem) /\ (x2 mem).
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Infix "/\" := assertAnd : assert.
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Definition assertOr : Assert -> Assert -> Assert :=
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fun x1 x2 mem => (x1 mem) \/ (x2 mem).
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Infix "\/" := assertOr : assert.
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Definition assertImpl : Assert -> Assert -> Assert :=
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fun x1 x2 => (~x1 \/ x2) % assert.
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Infix "->" := assertImpl : assert.
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Definition assertForall : Var -> Assert -> Assert :=
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fun ident asser mem => forall (z: Z), asser (mem [ident <- z]).
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Notation "\-/ x" := (assertForall x) (at level 90, no associativity) : assert.
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Definition existsForall : Var -> Assert -> Assert :=
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fun ident asser mem => exists (z: Z), asser (mem [ident <- z]).
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Notation "'exists' x" := (existsForall x)
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(at level 87, no associativity): assert.
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Definition assertMemForall : Assert -> Assert :=
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fun asser mem => forall (mem: Mem), asser mem.
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Notation "'\-/m' x" := (assertMemForall x)
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(at level 90, no associativity): assert.
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Definition existsMemForall : Assert -> Assert :=
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fun asser mem => exists (mem: Mem), asser mem.
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Notation "'exists_m' x" := (existsMemForall x)
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(at level 87, no associativity): assert.
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Definition substAssert : Assert -> Var -> Z -> Assert :=
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fun asser ident val mem => asser (mem [ident <- val]).
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Notation "a [[ x <- z ]]" := (substAssert a x z)
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(at level 50, left associativity).
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Definition substAssertExpr : Assert -> Var -> Expr -> Assert :=
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fun asser ident expr mem => asser (mem [ident <- (expr mem)]).
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Notation "a [[ x <- 'expr' z ]]" := (substAssertExpr a x z)
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(at level 50, left associativity).
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Definition assertOfExpr : Expr -> Assert :=
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fun expr mem => (expr mem <> 0)%Z.
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Definition assertImplLogical (a1 a2: Assert): Prop :=
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forall (m: Mem), (a1 m) -> (a2 m).
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(***** Hoare provability *****************************************************)
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Reserved Notation "|- [| x |] y [| z |]" (at level 30).
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Inductive hoare_provability : Assert -> Instr -> Assert -> Prop :=
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| H_skip: forall pre, hoare_provability pre skip pre
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| H_abort: forall pre, forall post, hoare_provability pre abort post
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| H_assign: forall post, forall (x: Var), forall (e: Expr),
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(|- [| post [[ x <- expr e ]] |] (assign x e) [| post |]) % assert
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| H_conseq:
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forall pre, forall post,
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forall pre', forall post',
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forall s,
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(|- [| pre' |] s [| post' |]) % assert ->
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(assertImplLogical pre pre') ->
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(assertImplLogical post' post) ->
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(|- [| pre |] s [| post |]) % assert
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| H_seq:
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forall pre, forall mid, forall post, forall s1, forall s2,
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(|- [|pre|] s1 [|mid|]) % assert ->
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(|- [|mid|] s2 [|post|]) % assert ->
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(|- [|pre|] (seq s1 s2) [|post|]) % assert
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| H_if:
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forall pre, forall post, forall expr, forall sIf, forall sElse,
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(|- [| pre /\ (assertOfExpr expr) |] sIf [| post |]) % assert ->
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(|- [| pre /\ ~(assertOfExpr expr) |] sElse [| post |]) % assert ->
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(|- [| pre |] (ifelse expr sIf sElse) [| post |]) % assert
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| H_while:
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forall inv, forall expr, forall sBody,
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(|- [| inv /\ (assertOfExpr expr) |] sBody [| inv |]) % assert ->
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(|- [| inv |] (while expr sBody)
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[| inv /\ ~ (assertOfExpr expr) |]) % assert
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where "|- [| pre |] instr [| post |]" :=
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(hoare_provability pre instr post) : assert.
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(***** Hoare: provability implies consequence ********************************)
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Definition conseq_or_bottom (y: Assert) (m: MemCpo) :=
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match m with
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| CpoError _ => True
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| CpoElem _ m0 => y m0
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end.
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Definition hoare_consequence (pre: Assert) (instr: Instr) (post: Assert) :=
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forall mem: Mem,
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(pre mem) -> (conseq_or_bottom post (interp instr (MemElem mem))).
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Notation "|= [| pre |] instr [| post |]" :=
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(hoare_consequence pre instr post) (at level 30): assert.
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Lemma weaken_in_conseq:
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forall a1, forall a2, forall m,
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assertImplLogical a1 a2 -> conseq_or_bottom a1 m ->
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conseq_or_bottom a2 m.
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Proof.
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intros a1 a2 m impl conseq. destruct m.
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- apply conseq.
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- simpl. unfold assertImplLogical in impl. apply (impl m). apply conseq.
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Qed.
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Lemma interp_of_error (s: Instr):
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interp s (MemError) = MemError.
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Proof.
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unfold MemError. destruct s; cbv; trivial.
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Qed.
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Theorem hoare_provability_implies_consequence :
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forall (pre: Assert), forall (s: Instr), forall (post: Assert),
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( |- [| pre |] s [| post |] ) % assert
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-> ( |= [| pre |] s [| post |] ) % assert.
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Proof.
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intros pre instr post. intros deduction.
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induction deduction; intros mem preInMem.
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- exact preInMem.
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- simpl; trivial.
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- exact preInMem.
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- apply (weaken_in_conseq post' post (interp s (MemElem mem)) H0).
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apply IHdeduction. apply H. exact preInMem.
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- simpl; destruct (interp s1 (MemElem mem)) eqn:mRel.
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* fold MemError. rewrite (interp_of_error s2); simpl; trivial.
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* apply (IHdeduction2 m). unfold hoare_consequence in IHdeduction1.
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specialize IHdeduction1 with mem as IH1_mem.
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rewrite mRel in IH1_mem. apply IH1_mem. assumption.
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- simpl; destruct (expr mem =? 0)%Z eqn:branchEqn.
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* apply (IHdeduction2 mem). unfold assertOfExpr.
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unfold assertAnd. split.
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+ assumption.
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+ unfold assertNot. rewrite <- Z.eqb_eq. congruence.
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* apply (IHdeduction1 mem). unfold assertOfExpr.
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unfold assertAnd. split.
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+ assumption.
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+ rewrite <- Z.eqb_eq. congruence.
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- unfold conseq_or_bottom.
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destruct (interp (while expr sBody) (MemElem mem)) eqn:interpRel.
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* trivial.
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* unfold assertAnd. split.
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+ specialize (IHdeduction m) as IHdeduction_m.
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unfold conseq_or_bottom in IHdeduction_m.
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Admitted.
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(***** Weakest precondition **************************************************)
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Fixpoint wp (instr: Instr) (cond: Assert) : Assert := match instr with
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| skip =>
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cond
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| abort =>
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assertTop
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| assign x expr =>
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cond [[ x <- expr expr ]]
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| seq s1 s2 =>
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wp s1 (wp s2 cond)
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| ifelse guard sIf sElse =>
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(assertOfExpr guard -> wp sIf cond
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/\ (~ (assertOfExpr guard) -> wp sElse cond)) % assert
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| while guard body => assertTop
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end.
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Theorem wp_correctness (instr: Instr) (post: Assert) :
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( |= [| wp instr post |] instr [| post |] ) % assert.
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Proof.
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(* TODO *)
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Admitted.
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