The Coq Proof Assistant

# Library Coq.micromega.ZifySint63

Require Import ZArith.
Require Import Sint63.
Require Import ZifyBool.
Import ZifyClasses.
Lemma to_Z_bounded (x : int) :
(-4611686018427387904 <= to_Z x <= 4611686018427387903)%Z.
#[global]
Instance Inj_int_Z : InjTyp int Z :=
mkinj _ _ to_Z (fun x => -4611686018427387904 <= x <= 4611686018427387903)%Z
to_Z_bounded.
Add Zify InjTyp Inj_int_Z.
#[global]
Instance Op_max_int : CstOp max_int :=
{ TCst := 4611686018427387903 ; TCstInj := eq_refl }.
Add Zify CstOp Op_max_int.
#[global]
Instance Op_min_int : CstOp min_int :=
{ TCst := -4611686018427387904 ; TCstInj := eq_refl }.
Add Zify CstOp Op_min_int.
#[global]
Instance Op_digits : CstOp digits :=
{ TCst := 63 ; TCstInj := eq_refl }.
Add Zify CstOp Op_digits.
#[global]
Instance Op_size : CstOp size :=
{ TCst := 63 ; TCstInj := eq_refl }.
Add Zify CstOp Op_size.
#[global]
Instance Op_wB : CstOp wB :=
{ TCst := 2^63 ; TCstInj := eq_refl }.
Add Zify CstOp Op_wB.
Lemma ltb_lt : forall n m,
(n <? m)%sint63 = (to_Z n <? to_Z m)%Z.
#[global]
Instance Op_ltb : BinOp ltb :=
{| TBOp := Z.ltb; TBOpInj := ltb_lt |}.
Add Zify BinOp Op_ltb.
Lemma leb_le : forall n m,
(n <=? m)%sint63 = (to_Z n <=? to_Z m)%Z.
#[global]
Instance Op_leb : BinOp leb :=
{| TBOp := Z.leb; TBOpInj := leb_le |}.
Add Zify BinOp Op_leb.
Lemma eqb_eq : forall n m,
(n =? m)%sint63 = (to_Z n =? to_Z m)%Z.
#[global]
Instance Op_eqb : BinOp eqb :=
{| TBOp := Z.eqb; TBOpInj := eqb_eq |}.
Add Zify BinOp Op_eqb.
Lemma eq_int_inj : forall n m : int, n = m <-> (to_Z n = to_Z m)%sint63.
#[global]
Instance Op_eq : BinRel (@eq int) :=
{| TR := @eq Z; TRInj := eq_int_inj |}.
Add Zify BinRel Op_eq.
Notation cmodwB x :=
((x + 4611686018427387904) mod 9223372036854775808 - 4611686018427387904)%Z.
#[global]
Instance Op_add : BinOp add :=
{| TBOp := fun x y => cmodwB (x + y); TBOpInj := add_spec |}%Z.
Add Zify BinOp Op_add.
#[global]
Instance Op_sub : BinOp sub :=
{| TBOp := fun x y => cmodwB (x - y); TBOpInj := sub_spec |}%Z.
Add Zify BinOp Op_sub.
#[global]
Instance Op_opp : UnOp Uint63.opp :=
{| TUOp := fun x => cmodwB (- x); TUOpInj := (sub_spec 0) |}%Z.
Add Zify UnOp Op_opp.
#[global]
Instance Op_succ : UnOp succ :=
{| TUOp := fun x => cmodwB (x + 1); TUOpInj := succ_spec |}%Z.
Add Zify UnOp Op_succ.
#[global]
Instance Op_pred : UnOp Uint63.pred :=
{| TUOp := fun x => cmodwB (x - 1); TUOpInj := pred_spec |}%Z.
Add Zify UnOp Op_pred.
#[global]
Instance Op_mul : BinOp mul :=
{| TBOp := fun x y => cmodwB (x * y); TBOpInj := mul_spec |}%Z.
Add Zify BinOp Op_mul.
#[global]
Instance Op_mod : BinOp PrimInt63.mods :=
{| TBOp := Z.rem ; TBOpInj := mod_spec |}.
Add Zify BinOp Op_mod.
#[global]
Instance Op_asr : BinOp asr :=
{| TBOp := fun x y => x / 2^ y ; TBOpInj := asr_spec |}%Z.
Add Zify BinOp Op_asr.
Definition quots (x d : Z) : Z :=
if ((x =? -4611686018427387904)%Z && (d =? -1)%Z)%bool then
-4611686018427387904
else
Z.quot x d.
Lemma div_quots (x y : int) : to_Z (x / y) = quots (to_Z x) (to_Z y).
#[global]
Instance Op_div : BinOp div :=
{| TBOp := quots ; TBOpInj := div_quots |}.
Add Zify BinOp Op_div.
Lemma quots_spec (x y : Z) :
((x = -4611686018427387904 /\ y = -1 /\ quots x y = -4611686018427387904)
\/ ((x <> -4611686018427387904 \/ y <> -1) /\ quots x y = Z.quot x y))%Z.
#[global]
Instance quotsSpec : BinOpSpec quots :=
{| BPred := fun x d r : Z =>
((x = -4611686018427387904 /\ d = -1 /\ r = -4611686018427387904)
\/ ((x <> -4611686018427387904 \/ d <> -1) /\ r = Z.quot x d))%Z;
BSpec := quots_spec |}.
Add Zify BinOpSpec quotsSpec.
#[global]
Instance Op_of_Z : UnOp of_Z :=
{ TUOp := fun x => cmodwB x; TUOpInj := of_Z_spec }.
Add Zify UnOp Op_of_Z.
#[global]
Instance Op_to_Z : UnOp to_Z :=
{ TUOp := fun x => x ; TUOpInj := fun x : int => eq_refl }.
Add Zify UnOp Op_to_Z.
Lemma is_zeroE : forall n : int, is_zero n = (to_Z n =? 0)%Z.
#[global]
Instance Op_is_zero : UnOp is_zero :=
{ TUOp := (Z.eqb 0) ; TUOpInj := is_zeroE }.
Add Zify UnOp Op_is_zero.
#[global]
Instance Op_abs : UnOp abs :=
{ TUOp := fun x => cmodwB (Z.abs x) ; TUOpInj := abs_spec }.
Add Zify UnOp Op_abs.
Ltac Zify.zify_convert_to_euclidean_division_equations_flag ::= constr:(true).