The Coq Proof Assistant

# Library Coq.omega.PreOmega

Require Import Arith BinInt BinNat Znat Nnat.
Local Open Scope Z_scope.

# Z.div_mod_to_equations, Z.quot_rem_to_equations, Z.to_euclidean_division_equations:

the tactics for preprocessing Z.div and Z.modulo, Z.quot and Z.rem
These tactics use the complete specification of Z.div and Z.modulo (Z.quot and Z.rem, respectively) to remove these functions from the goal without losing information. The Z.euclidean_division_equations_find_duplicate_quotients tactic poses facts of the form q₁ = q₂ \/ q₁ <> q₂ for quotients that are likely to be the same, which allows tactics like nia to prove more goals, including those relating Z.div/Z.mod to Z.quot/Z.rem. The Z.euclidean_division_equations_cleanup tactic removes needless hypotheses, which makes tactics like nia run faster. The tactic Z.to_euclidean_division_equations combines the handling of both variants of division/quotient and modulo/remainder.
Module Z.
Lemma mod_0_r_ext x y : y = 0 -> x mod y = x.
Lemma div_0_r_ext x y : y = 0 -> x / y = 0.
Lemma rem_0_r_ext x y : y = 0 -> Z.rem x y = x.
Lemma quot_0_r_ext x y : y = 0 -> Z.quot x y = 0.
Lemma rem_bound_pos_pos x y : 0 < y -> 0 <= x -> 0 <= Z.rem x y < y.
Lemma rem_bound_neg_pos x y : y < 0 -> 0 <= x -> 0 <= Z.rem x y < -y.
Lemma rem_bound_pos_neg x y : 0 < y -> x <= 0 -> -y < Z.rem x y <= 0.
Lemma rem_bound_neg_neg x y : y < 0 -> x <= 0 -> y < Z.rem x y <= 0.
Local Lemma divide_alt x y : Z.divide x y -> exists z, y = x * z.
Ltac div_mod_to_equations_generalize x y :=
pose proof (Z.div_mod x y);
pose proof (Z.mod_pos_bound x y);
pose proof (Z.mod_neg_bound x y);
pose proof (div_0_r_ext x y);
pose proof (mod_0_r_ext x y);
let q := fresh "q" in
let r := fresh "r" in
set (q := x / y) in *;
set (r := x mod y) in *;
clearbody q r.
Ltac quot_rem_to_equations_generalize x y :=
pose proof (Z.quot_rem' x y);
pose proof (rem_bound_pos_pos x y);
pose proof (rem_bound_pos_neg x y);
pose proof (rem_bound_neg_pos x y);
pose proof (rem_bound_neg_neg x y);
pose proof (quot_0_r_ext x y);
pose proof (rem_0_r_ext x y);
let q := fresh "q" in
let r := fresh "r" in
set (q := Z.quot x y) in *;
set (r := Z.rem x y) in *;
clearbody q r.
Ltac div_mod_to_equations_step :=
match goal with
| [ |- context[?x / ?y] ] => div_mod_to_equations_generalize x y
| [ |- context[?x mod ?y] ] => div_mod_to_equations_generalize x y
| [ H : context[?x / ?y] |- _ ] => div_mod_to_equations_generalize x y
| [ H : context[?x mod ?y] |- _ ] => div_mod_to_equations_generalize x y
end.
Ltac quot_rem_to_equations_step :=
match goal with
| [ |- context[Z.quot ?x ?y] ] => quot_rem_to_equations_generalize x y
| [ |- context[Z.rem ?x ?y] ] => quot_rem_to_equations_generalize x y
| [ H : context[Z.quot ?x ?y] |- _ ] => quot_rem_to_equations_generalize x y
| [ H : context[Z.rem ?x ?y] |- _ ] => quot_rem_to_equations_generalize x y
end.
Ltac divide_to_equations_step :=
match goal with | [ H : Z.divide _ _ |- _ ] => apply divide_alt in H; destruct H end.
Ltac div_mod_to_equations' := repeat div_mod_to_equations_step.
Ltac quot_rem_to_equations' := repeat quot_rem_to_equations_step.
Ltac divide_to_equations' := repeat divide_to_equations_step.
Ltac euclidean_division_equations_cleanup :=
repeat
(repeat match goal with
| [ H : 0 <= ?x < _ |- _ ] => destruct H
end;
repeat match goal with
| [ H : ?x <> ?x -> _ |- _ ] => clear H
| [ H : ?x < ?x -> _ |- _ ] => clear H
| [ H : ?T -> _, H' : ~?T |- _ ] => clear H
| [ H : ~?T -> _, H' : ?T |- _ ] => clear H
| [ H : ?A -> ?x <> ?x -> _ |- _ ] => clear H
| [ H : ?A -> ?x < ?x -> _ |- _ ] => clear H
| [ H : ?A -> ?B -> _, H' : ~?B |- _ ] => clear H
| [ H : ?A -> ~?B -> _, H' : ?B |- _ ] => clear H
| [ H : 0 < ?x -> _, H' : ?x < 0 |- _ ] => clear H
| [ H : ?x < 0 -> _, H' : 0 < ?x |- _ ] => clear H
| [ H : ?A -> 0 < ?x -> _, H' : ?x < 0 |- _ ] => clear H
| [ H : ?A -> ?x < 0 -> _, H' : 0 < ?x |- _ ] => clear H
| [ H : 0 <= ?x -> _, H' : ?x < 0 |- _ ] => clear H
| [ H : ?x <= 0 -> _, H' : 0 < ?x |- _ ] => clear H
| [ H : ?A -> 0 <= ?x -> _, H' : ?x < 0 |- _ ] => clear H
| [ H : ?A -> ?x <= 0 -> _, H' : 0 < ?x |- _ ] => clear H
| [ H : 0 < ?x -> _, H' : ?x <= 0 |- _ ] => clear H
| [ H : ?x < 0 -> _, H' : 0 <= ?x |- _ ] => clear H
| [ H : ?A -> 0 < ?x -> _, H' : ?x <= 0 |- _ ] => clear H
| [ H : ?A -> ?x < 0 -> _, H' : 0 <= ?x |- _ ] => clear H
| [ H : ?x < ?y -> _, H' : ?x = ?y |- _ ] => clear H
| [ H : ?x < ?y -> _, H' : ?y = ?x |- _ ] => clear H
| [ H : ?A -> ?x < ?y -> _, H' : ?x = ?y |- _ ] => clear H
| [ H : ?A -> ?x < ?y -> _, H' : ?y = ?x |- _ ] => clear H
| [ H : ?x = ?y -> _, H' : ?x < ?y |- _ ] => clear H
| [ H : ?x = ?y -> _, H' : ?y < ?x |- _ ] => clear H
| [ H : ?A -> ?x = ?y -> _, H' : ?x < ?y |- _ ] => clear H
| [ H : ?A -> ?x = ?y -> _, H' : ?y < ?x |- _ ] => clear H
end;
repeat match goal with
| [ H : ?x = ?x -> ?Q |- _ ] => specialize (H eq_refl)
| [ H : ?T -> ?Q, H' : ?T |- _ ] => specialize (H H')
| [ H : ?A -> ?x = ?x -> ?Q |- _ ] => specialize (fun a => H a eq_refl)
| [ H : ?A -> ?B -> ?Q, H' : ?B |- _ ] => specialize (fun a => H a H')
| [ H : 0 <= ?x -> ?Q, H' : ?x <= 0 |- _ ] => specialize (fun pf => H (@Z.eq_le_incl 0 x (eq_sym pf)))
| [ H : ?A -> 0 <= ?x -> ?Q, H' : ?x <= 0 |- _ ] => specialize (fun a pf => H a (@Z.eq_le_incl 0 x (eq_sym pf)))
| [ H : ?x <= 0 -> ?Q, H' : 0 <= ?x |- _ ] => specialize (fun pf => H (@Z.eq_le_incl 0 x pf))
| [ H : ?A -> ?x <= 0 -> ?Q, H' : 0 <= ?x |- _ ] => specialize (fun a pf => H a (@Z.eq_le_incl x 0 pf))
end).
poses x = y \/ x <> y unless that is redundant or contradictory
Ltac euclidean_division_equations_pose_eq_fact x y :=
assert_fails constr_eq x y;
lazymatch goal with
| [ H : x = y |- _ ] => fail
| [ H : y = x |- _ ] => fail
| [ H : x = y \/ x <> y |- _ ] => fail
| [ H : y = x \/ y <> x |- _ ] => fail
| [ H : x < y |- _ ] => fail
| [ H : y < x |- _ ] => fail
| [ H : x <> y |- _ ] => fail
| [ H : y <> x |- _ ] => fail
| _ => pose proof (Z.eq_decidable x y : x = y \/ x <> y)
end.
Ltac euclidean_division_equations_find_duplicate_quotients_step :=
let pose_eq_fact x y := euclidean_division_equations_pose_eq_fact x y in
match goal with
| [ H : context[?x = ?y * ?q1], H' : context[?x = ?y * ?q2] |- _ ] => pose_eq_fact q1 q2
| [ H : context[?x = ?y * ?q1 + _], H' : context[?x = ?y * ?q2] |- _ ] => pose_eq_fact q1 q2
| [ H : context[?x = ?y * ?q1 + _], H' : context[?x = ?y * ?q2 + _] |- _ ] => pose_eq_fact q1 q2
| [ H : context[?y * ?q2 + _ = ?y * ?q1 + _] |- _ ] => pose_eq_fact q1 q2
| [ H : context[?x * ?y = ?y * ?q1 + _] |- _ ] => pose_eq_fact x q1
| [ H : context[?y * ?x = ?y * ?q1 + _] |- _ ] => pose_eq_fact x q1
end.
Ltac euclidean_division_equations_find_duplicate_quotients :=
repeat euclidean_division_equations_find_duplicate_quotients_step.
Ltac div_mod_to_equations := div_mod_to_equations'; euclidean_division_equations_cleanup.
Ltac quot_rem_to_equations := quot_rem_to_equations'; euclidean_division_equations_cleanup.
Ltac divide_to_equations := divide_to_equations'; euclidean_division_equations_cleanup.
Module euclidean_division_equations_flags.
#[local] Set Primitive Projections.
Record t :=
{ find_duplicate_quotients : bool }.
Ltac default_find_duplicate_quotients := constr:(true).
Ltac default :=
let find_duplicate_quotients_value := default_find_duplicate_quotients in
constr:({| find_duplicate_quotients := find_duplicate_quotients_value
|}).
Module Import DefaultHelpers.
Ltac try_unify_args x y :=
tryif first [ has_evar x | has_evar y ]
then (tryif unify x y
then idtac
else (lazymatch x with
| ?f ?x
=> lazymatch y with
| ?g ?y
=> try_unify_args f g; try_unify_args x y
| ?y => fail 0 "Z.euclidean_division_equations_flags: try_unify_args: cannot unify application" x "with non-application" y
end
| ?x
=> (tryif has_evar x
then fail 0 "Z.euclidean_division_equations_flags: try_unify_args: cannot unify evar-containing non-application" x "with" y
else (tryif has_evar y
then fail 0 "Z.euclidean_division_equations_flags: try_unify_args: cannot unify non-application" x "with evar-containing" y
else fail 100 "Z.euclidean_division_equations_flags: try_unify_args: Impossible inconsistent state of has_evar in try_unify_args" x y))
end))
else idtac.
End DefaultHelpers.
Ltac flags_with orig_flags proj value :=
let flags := open_constr:(match True return t with _ => ltac:(econstructor) end) in
let __unif := constr:(eq_refl : proj flags = value) in
let __force := lazymatch goal with _ => try_unify_args flags orig_flags end in
flags.
Ltac default_with proj value := flags_with default proj value.
Ltac guard_with proj flags tac :=
lazymatch (eval cbv in (proj flags)) with
| true => tac
| false => idtac
| ?v => let ctrue := constr:(true) in
let cfalse := constr:(false) in
fail 0 "Invalid flag value for" proj "in" flags "(got" v "expected" ctrue "or" cfalse ")"
end.
End euclidean_division_equations_flags.
Import euclidean_division_equations_flags (find_duplicate_quotients).
Ltac to_euclidean_division_equations_with flags :=
divide_to_equations'; div_mod_to_equations'; quot_rem_to_equations';
euclidean_division_equations_cleanup;
euclidean_division_equations_flags.guard_with find_duplicate_quotients flags euclidean_division_equations_find_duplicate_quotients.
Ltac to_euclidean_division_equations :=
to_euclidean_division_equations_with euclidean_division_equations_flags.default.
End Z.
Require Import ZifyClasses ZifyInst.
Require Zify.
Ltac Zify.zify_internal_to_euclidean_division_equations ::= Z.to_euclidean_division_equations.
Ltac zify := Zify.zify.