Lvc.Isa.Val
Require Import Arith Util EqDec List.
Require Import Containers.OrderedTypeEx Integers.
Set Implicit Arguments.
Require Import Containers.OrderedTypeEx Integers.
Set Implicit Arguments.
More or less abstract values
A file that abstacts over values and (should) provide all necessary operations. Although we concretely instantiate val to int, we maintain this file as interface between our proofs and Integer library.
Definition val := int.
Definition default_val : val := Int.zero.
Instance inst_val_defaulted : Defaulted val := {
default_el := default_val
}.
Definition val_zero : val := Int.zero.
Instance inst_eq_dec_val : EqDec val eq.
hnf; intros. eapply Int.eq_dec.
Defined.
Inductive binop : Type :=
| BinOpAdd
| BinOpSub
| BinOpMul
| BinOpDiv
| BinOpEq
| BinOpLt.
Instance inst_eq_dec_binop : EqDec binop eq.
Proof.
hnf; intros. change ({x = y} + {x ≠ y}).
decide equality.
Qed.
Inductive unop : Type :=
| UnOpToBool
| UnOpNeg.
Instance inst_eq_dec_unop : EqDec unop eq.
Proof.
hnf; intros. change ({x = y} + {x ≠ y}).
decide equality.
Qed.
Declare ML Module "containers_plugin".
Generate OrderedType unop.
Generate OrderedType binop.
Lemma unop_eq_eq x y
: unop_eq x y → x = y.
Proof.
destruct x,y; inversion 1; eauto.
Qed.
Lemma binop_eq_eq x y
: binop_eq x y → x = y.
Proof.
destruct x,y; inversion 1; eauto.
Qed.
Require Coq.Structures.OrderedTypeEx.
Instance Reflexive_eq_int : Reflexive Int.eq.
Proof.
hnf; intros. hnf. cases; eauto.
rewrite Int.eq_true in Heq. congruence.
Qed.
Instance Symmetric_eq_int : Symmetric Int.eq.
Proof.
hnf; intros. rewrite Int.eq_sym; eauto.
Qed.
Instance Transitive_eq_int : Transitive Int.eq.
Proof.
hnf; intros.
pose proof (Int.eq_spec x y).
pose proof (Int.eq_spec y z).
cases in H1; cases in H2; subst; eauto.
Qed.
Instance Equivalence_eq_int : Equivalence Int.eq.
Proof.
econstructor; eauto with typeclass_instances.
Qed.
Definition int_eq x y := Int.intval x = Int.intval y.
Instance Reflexive_eq_int' : Reflexive int_eq.
Proof.
hnf; intros. hnf. reflexivity.
Qed.
Instance Symmetric_eq_int' : Symmetric int_eq.
Proof.
unfold int_eq. hnf; intros. symmetry; eauto.
Qed.
Instance Transitive_eq_int' : Transitive int_eq.
Proof.
unfold int_eq. hnf; intros. etransitivity; eauto.
Qed.
Instance Equivalence_eq_int' : Equivalence int_eq.
Proof.
econstructor; eauto with typeclass_instances.
Qed.
Instance Transitive_lt_int : Transitive Int.ltu.
Proof.
hnf. unfold Int.ltu. intros ? ? ? A B.
cases in A; invc A; eauto.
cases in B; invc B; eauto.
destruct Z_StrictOrder.
specialize (StrictOrder_Transitive _ _ _ l l0).
rewrite Coqlib.zlt_true; eauto.
Qed.
Instance Irreflexive_lt_int : Irreflexive Int.ltu.
Proof.
unfold Irreflexive, complement, Reflexive, Int.ltu; intros.
cases in H; eauto.
eapply Z.lt_irrefl; eauto.
Qed.
Instance int_eq_eq_lt
: Proper (int_eq ==> int_eq ==> eq) Int.ltu.
Proof.
unfold Proper, respectful, Int.ltu, Int.unsigned, int_eq; intros.
rewrite H, H0. eauto.
Qed.
Instance StrictOrder_lt_int : StrictOrder Int.ltu eq.
Proof.
econstructor; eauto with typeclass_instances.
- intros ? ? LE EQ.
rewrite <- EQ in LE.
eapply (Irreflexive_lt_int x); eauto.
Defined.
Definition int_compare (x y : int) : Datatypes.comparison :=
Z.compare (Int.intval x) (Int.intval y).
Unset Printing Records.
Require Import ProofIrrelevance.
Lemma compare_spec_int x y
: compare_spec eq Int.ltu x y (int_compare x y).
Proof.
pose proof (Z.compare_spec (Int.intval x) (Int.intval y)).
case_eq (int_compare x y); intros EQ; econstructor; unfold int_compare in EQ;
rewrite EQ in H; inv H; eauto.
- destruct x,y; simpl in *; subst.
f_equal. eapply ProofIrrelevance.proof_irrelevance.
- hnf. unfold Int.ltu.
rewrite Coqlib.zlt_true; eauto.
- hnf. unfold Int.ltu.
rewrite Coqlib.zlt_true; eauto.
Qed.
Lemma compare_spec_int_eq x y
: int_compare x y = Eq → x = y.
Proof.
intros.
pose proof (compare_spec_int x y). rewrite H in H0.
inv H0. eauto.
Qed.
Instance int_eq_dec : EqDec val int_eq.
hnf; intros. destruct x,y.
eapply Z.eq_dec.
Defined.
Instance int_eq_dec' x y : Computable (int_eq x y).
hnf; intros. destruct x,y.
eapply Z.eq_dec.
Defined.
Lemma int_eq_eq x y
: int_eq x y → x = y.
Proof.
unfold int_eq; destruct x,y; simpl; intros; subst.
eapply Integers.Int.mkint_eq. reflexivity.
Qed.
Hint Resolve int_eq_eq.
Smpl Add 100
match goal with
| [ H : int_eq _ _ |- _ ] ⇒ eapply int_eq_eq in H
end : inv_trivial.
Instance OrderedType_int : OrderedType int.
Proof.
eapply (Build_OrderedType _ eq Int.ltu); eauto with typeclass_instances.
eapply compare_spec_int.
Defined.
Definition option_lift1 A B (f:A → B) := fun x ⇒ Some (f x).
Definition option_lift2 A B C (f:A → B → C) := fun x y ⇒ Some (f x y).
Definition val_true := Int.one.
Definition val_false := Int.zero.
Lemma val_true_false_neq
: val_true ≠ val_false.
Proof.
eapply Int.one_not_zero.
Qed.
Lemma val_true_false_nequiv
: val_true =/= val_false.
Proof.
inversion 1.
Qed.
Definition bool2val (b:bool) :=
match b with
| true ⇒ val_true
| false ⇒ val_false
end.
Definition val2bool (v: val) : bool := if [ v = val_zero ] then false else true.
Lemma val2bool_true
: val2bool val_true = true.
Proof.
unfold val2bool.
cases; eauto.
exfalso; eauto using val_true_false_neq.
Qed.
Lemma val2bool_false
: val2bool val_false = false.
Proof.
unfold val2bool.
cases; eauto.
Qed.
Definition binop_eval (o:binop) :=
match o with
| BinOpAdd ⇒ option_lift2 Int.add
| BinOpSub ⇒ option_lift2 Int.sub
| BinOpMul ⇒ option_lift2 Int.mul
| BinOpDiv ⇒ fun x y ⇒ if [ y = val_zero] then None else (Some (Int.divs x y))
| BinOpEq ⇒ option_lift2 (fun x y ⇒ bool2val(Int.eq x y))
| BinOpLt ⇒ option_lift2 (fun x y ⇒ bool2val(Int.lt x y))
end.
Lemma binop_eval_div_zero x y
: y = val_zero
↔ binop_eval BinOpDiv x y = None.
Proof.
intros; simpl; cases; split; intros; eauto; exfalso; congruence.
Qed.
Lemma binop_eval_div_nonzero x y
: y ≠ val_zero
→ ∃ v, binop_eval BinOpDiv x y = Some v.
Proof.
intros; simpl; cases; eauto.
Qed.
Lemma binop_eval_not_div op x y
: op ≠ BinOpDiv
→ ∃ v, binop_eval op x y = Some v.
Proof.
intros; destruct op; simpl; unfold option_lift2; eauto.
congruence.
Qed.
Definition unop_eval (o:unop) :=
match o with
| UnOpToBool ⇒ option_lift1 (fun a ⇒ bool2val(val2bool a))
| UnOpNeg ⇒ option_lift1 Int.notbool
end
.
Lemma unop_eval_total op x
: ∃ v, unop_eval op x = Some v.
Proof.
intros; destruct op; simpl; unfold option_lift1; eauto.
Qed.
Opaque val.
Opaque default_val.
Opaque val_true.
Opaque val_false.
Opaque val_zero.
Opaque val2bool.
Lemma int_eq_true_false_absurd
: int_eq val_true val_false → False.
Proof.
intros. eapply int_eq_eq in H. inv H.
Qed.
Smpl Add 100
match goal with
| [ H : @Equivalence.equiv val int_eq Equivalence_eq_int' _ _ |- _ ] ⇒
eapply int_eq_eq in H
end : inv_trivial.
Smpl Add match goal with
| [ H : int_eq val_true val_false |- _ ] ⇒
exfalso; eapply int_eq_true_false_absurd in H; eauto
| [ H : val_true = val_false |- _ ] ⇒ inv H
| [ H : val_false = val_true |- _ ] ⇒ inv H
| [ H : val_true === val_false |- _ ] ⇒ inv H
| [ H : val_false === val_true |- _ ] ⇒ inv H
end : inv_trivial.
Instance int_eq_trans : Transitive (int_eq).
Proof.
hnf; intros. rewrite H ;eauto.
Qed.
Instance int_eq_sym : Symmetric (int_eq).
Proof.
hnf; intros. rewrite H ;eauto. reflexivity.
Qed.
Definition default_val : val := Int.zero.
Instance inst_val_defaulted : Defaulted val := {
default_el := default_val
}.
Definition val_zero : val := Int.zero.
Instance inst_eq_dec_val : EqDec val eq.
hnf; intros. eapply Int.eq_dec.
Defined.
Inductive binop : Type :=
| BinOpAdd
| BinOpSub
| BinOpMul
| BinOpDiv
| BinOpEq
| BinOpLt.
Instance inst_eq_dec_binop : EqDec binop eq.
Proof.
hnf; intros. change ({x = y} + {x ≠ y}).
decide equality.
Qed.
Inductive unop : Type :=
| UnOpToBool
| UnOpNeg.
Instance inst_eq_dec_unop : EqDec unop eq.
Proof.
hnf; intros. change ({x = y} + {x ≠ y}).
decide equality.
Qed.
Declare ML Module "containers_plugin".
Generate OrderedType unop.
Generate OrderedType binop.
Lemma unop_eq_eq x y
: unop_eq x y → x = y.
Proof.
destruct x,y; inversion 1; eauto.
Qed.
Lemma binop_eq_eq x y
: binop_eq x y → x = y.
Proof.
destruct x,y; inversion 1; eauto.
Qed.
Require Coq.Structures.OrderedTypeEx.
Instance Reflexive_eq_int : Reflexive Int.eq.
Proof.
hnf; intros. hnf. cases; eauto.
rewrite Int.eq_true in Heq. congruence.
Qed.
Instance Symmetric_eq_int : Symmetric Int.eq.
Proof.
hnf; intros. rewrite Int.eq_sym; eauto.
Qed.
Instance Transitive_eq_int : Transitive Int.eq.
Proof.
hnf; intros.
pose proof (Int.eq_spec x y).
pose proof (Int.eq_spec y z).
cases in H1; cases in H2; subst; eauto.
Qed.
Instance Equivalence_eq_int : Equivalence Int.eq.
Proof.
econstructor; eauto with typeclass_instances.
Qed.
Definition int_eq x y := Int.intval x = Int.intval y.
Instance Reflexive_eq_int' : Reflexive int_eq.
Proof.
hnf; intros. hnf. reflexivity.
Qed.
Instance Symmetric_eq_int' : Symmetric int_eq.
Proof.
unfold int_eq. hnf; intros. symmetry; eauto.
Qed.
Instance Transitive_eq_int' : Transitive int_eq.
Proof.
unfold int_eq. hnf; intros. etransitivity; eauto.
Qed.
Instance Equivalence_eq_int' : Equivalence int_eq.
Proof.
econstructor; eauto with typeclass_instances.
Qed.
Instance Transitive_lt_int : Transitive Int.ltu.
Proof.
hnf. unfold Int.ltu. intros ? ? ? A B.
cases in A; invc A; eauto.
cases in B; invc B; eauto.
destruct Z_StrictOrder.
specialize (StrictOrder_Transitive _ _ _ l l0).
rewrite Coqlib.zlt_true; eauto.
Qed.
Instance Irreflexive_lt_int : Irreflexive Int.ltu.
Proof.
unfold Irreflexive, complement, Reflexive, Int.ltu; intros.
cases in H; eauto.
eapply Z.lt_irrefl; eauto.
Qed.
Instance int_eq_eq_lt
: Proper (int_eq ==> int_eq ==> eq) Int.ltu.
Proof.
unfold Proper, respectful, Int.ltu, Int.unsigned, int_eq; intros.
rewrite H, H0. eauto.
Qed.
Instance StrictOrder_lt_int : StrictOrder Int.ltu eq.
Proof.
econstructor; eauto with typeclass_instances.
- intros ? ? LE EQ.
rewrite <- EQ in LE.
eapply (Irreflexive_lt_int x); eauto.
Defined.
Definition int_compare (x y : int) : Datatypes.comparison :=
Z.compare (Int.intval x) (Int.intval y).
Unset Printing Records.
Require Import ProofIrrelevance.
Lemma compare_spec_int x y
: compare_spec eq Int.ltu x y (int_compare x y).
Proof.
pose proof (Z.compare_spec (Int.intval x) (Int.intval y)).
case_eq (int_compare x y); intros EQ; econstructor; unfold int_compare in EQ;
rewrite EQ in H; inv H; eauto.
- destruct x,y; simpl in *; subst.
f_equal. eapply ProofIrrelevance.proof_irrelevance.
- hnf. unfold Int.ltu.
rewrite Coqlib.zlt_true; eauto.
- hnf. unfold Int.ltu.
rewrite Coqlib.zlt_true; eauto.
Qed.
Lemma compare_spec_int_eq x y
: int_compare x y = Eq → x = y.
Proof.
intros.
pose proof (compare_spec_int x y). rewrite H in H0.
inv H0. eauto.
Qed.
Instance int_eq_dec : EqDec val int_eq.
hnf; intros. destruct x,y.
eapply Z.eq_dec.
Defined.
Instance int_eq_dec' x y : Computable (int_eq x y).
hnf; intros. destruct x,y.
eapply Z.eq_dec.
Defined.
Lemma int_eq_eq x y
: int_eq x y → x = y.
Proof.
unfold int_eq; destruct x,y; simpl; intros; subst.
eapply Integers.Int.mkint_eq. reflexivity.
Qed.
Hint Resolve int_eq_eq.
Smpl Add 100
match goal with
| [ H : int_eq _ _ |- _ ] ⇒ eapply int_eq_eq in H
end : inv_trivial.
Instance OrderedType_int : OrderedType int.
Proof.
eapply (Build_OrderedType _ eq Int.ltu); eauto with typeclass_instances.
eapply compare_spec_int.
Defined.
Definition option_lift1 A B (f:A → B) := fun x ⇒ Some (f x).
Definition option_lift2 A B C (f:A → B → C) := fun x y ⇒ Some (f x y).
Definition val_true := Int.one.
Definition val_false := Int.zero.
Lemma val_true_false_neq
: val_true ≠ val_false.
Proof.
eapply Int.one_not_zero.
Qed.
Lemma val_true_false_nequiv
: val_true =/= val_false.
Proof.
inversion 1.
Qed.
Definition bool2val (b:bool) :=
match b with
| true ⇒ val_true
| false ⇒ val_false
end.
Definition val2bool (v: val) : bool := if [ v = val_zero ] then false else true.
Lemma val2bool_true
: val2bool val_true = true.
Proof.
unfold val2bool.
cases; eauto.
exfalso; eauto using val_true_false_neq.
Qed.
Lemma val2bool_false
: val2bool val_false = false.
Proof.
unfold val2bool.
cases; eauto.
Qed.
Definition binop_eval (o:binop) :=
match o with
| BinOpAdd ⇒ option_lift2 Int.add
| BinOpSub ⇒ option_lift2 Int.sub
| BinOpMul ⇒ option_lift2 Int.mul
| BinOpDiv ⇒ fun x y ⇒ if [ y = val_zero] then None else (Some (Int.divs x y))
| BinOpEq ⇒ option_lift2 (fun x y ⇒ bool2val(Int.eq x y))
| BinOpLt ⇒ option_lift2 (fun x y ⇒ bool2val(Int.lt x y))
end.
Lemma binop_eval_div_zero x y
: y = val_zero
↔ binop_eval BinOpDiv x y = None.
Proof.
intros; simpl; cases; split; intros; eauto; exfalso; congruence.
Qed.
Lemma binop_eval_div_nonzero x y
: y ≠ val_zero
→ ∃ v, binop_eval BinOpDiv x y = Some v.
Proof.
intros; simpl; cases; eauto.
Qed.
Lemma binop_eval_not_div op x y
: op ≠ BinOpDiv
→ ∃ v, binop_eval op x y = Some v.
Proof.
intros; destruct op; simpl; unfold option_lift2; eauto.
congruence.
Qed.
Definition unop_eval (o:unop) :=
match o with
| UnOpToBool ⇒ option_lift1 (fun a ⇒ bool2val(val2bool a))
| UnOpNeg ⇒ option_lift1 Int.notbool
end
.
Lemma unop_eval_total op x
: ∃ v, unop_eval op x = Some v.
Proof.
intros; destruct op; simpl; unfold option_lift1; eauto.
Qed.
Opaque val.
Opaque default_val.
Opaque val_true.
Opaque val_false.
Opaque val_zero.
Opaque val2bool.
Lemma int_eq_true_false_absurd
: int_eq val_true val_false → False.
Proof.
intros. eapply int_eq_eq in H. inv H.
Qed.
Smpl Add 100
match goal with
| [ H : @Equivalence.equiv val int_eq Equivalence_eq_int' _ _ |- _ ] ⇒
eapply int_eq_eq in H
end : inv_trivial.
Smpl Add match goal with
| [ H : int_eq val_true val_false |- _ ] ⇒
exfalso; eapply int_eq_true_false_absurd in H; eauto
| [ H : val_true = val_false |- _ ] ⇒ inv H
| [ H : val_false = val_true |- _ ] ⇒ inv H
| [ H : val_true === val_false |- _ ] ⇒ inv H
| [ H : val_false === val_true |- _ ] ⇒ inv H
end : inv_trivial.
Instance int_eq_trans : Transitive (int_eq).
Proof.
hnf; intros. rewrite H ;eauto.
Qed.
Instance int_eq_sym : Symmetric (int_eq).
Proof.
hnf; intros. rewrite H ;eauto. reflexivity.
Qed.