diff options
| author | Vincent Laporte | 2019-02-01 15:47:49 +0000 |
|---|---|---|
| committer | Vincent Laporte | 2019-02-01 15:47:49 +0000 |
| commit | f4cf212efd98d01a6470ea7bfd1034d52e928906 (patch) | |
| tree | 360bf6678bf8e1e0095cde7ac5ed17938f87bff1 | |
| parent | 506136d60c0dcc4fc2a2ca83ef3b586fbede55a2 (diff) | |
| parent | 687c7d97971316223e24b637973aa378077b5135 (diff) | |
Merge PR #8062: Add Z.div_mod_to_quot_rem tactic, put it in zify
Ack-by: JasonGross
Reviewed-by: fajb
Reviewed-by: jfehrle
| -rw-r--r-- | CHANGES.md | 8 | ||||
| -rw-r--r-- | doc/sphinx/addendum/micromega.rst | 12 | ||||
| -rw-r--r-- | plugins/omega/PreOmega.v | 132 | ||||
| -rw-r--r-- | test-suite/success/Nia.v | 918 |
4 files changed, 1069 insertions, 1 deletions
diff --git a/CHANGES.md b/CHANGES.md index 4d0063115b..9d912a63b1 100644 --- a/CHANGES.md +++ b/CHANGES.md @@ -83,6 +83,14 @@ Tactics foo : database`). When the database name is omitted, the hint is added to the core database (as previously), but a deprecation warning is emitted. +- There are now tactics in `PreOmega.v` called + `Z.div_mod_to_equations`, `Z.quot_rem_to_equations`, and + `Z.to_euclidean_division_equations` (which combines the `div_mod` + and `quot_rem` variants) which allow `lia`, `nia`, `romega`, etc to + support `Z.div` and `Z.modulo` (`Z.quot` and `Z.rem`, respectively), + by posing the specifying equation for `Z.div` and `Z.modulo` before + replacing them with atoms. + Vernacular commands - `Combined Scheme` can now work when inductive schemes are generated in sort diff --git a/doc/sphinx/addendum/micromega.rst b/doc/sphinx/addendum/micromega.rst index e799677c59..b076aac1ed 100644 --- a/doc/sphinx/addendum/micromega.rst +++ b/doc/sphinx/addendum/micromega.rst @@ -145,7 +145,7 @@ weakness, the :tacn:`lia` tactic is using recursively a combination of: + linear *positivstellensatz* refutations; + cutting plane proofs; + case split. - + Cutting plane proofs ~~~~~~~~~~~~~~~~~~~~~~ @@ -250,6 +250,16 @@ obtain :math:`-1`. By Theorem :ref:`Psatz <psatz_thm>`, the goal is valid. .. [#] Support for :g:`nat` and :g:`N` is obtained by pre-processing the goal with the ``zify`` tactic. +.. [#] Support for :g:`Z.div` and :g:`Z.modulo` may be obtained by + pre-processing the goal with the ``Z.div_mod_to_equations`` tactic (you may + need to manually run ``zify`` first). +.. [#] Support for :g:`Z.quot` and :g:`Z.rem` may be obtained by pre-processing + the goal with the ``Z.quot_rem_to_equations`` tactic (you may need to manually + run ``zify`` first). +.. [#] Note that support for :g:`Z.div`, :g:`Z.modulo`, :g:`Z.quot`, and + :g:`Z.rem` may be simultaneously obtained by pre-processing the goal with the + ``Z.to_euclidean_division_equations`` tactic (you may need to manually run + ``zify`` first). .. [#] Sources and binaries can be found at https://projects.coin-or.org/Csdp .. [#] Variants deal with equalities and strict inequalities. .. [#] In practice, the oracle might fail to produce such a refutation. diff --git a/plugins/omega/PreOmega.v b/plugins/omega/PreOmega.v index 94a3d40441..695f000cb1 100644 --- a/plugins/omega/PreOmega.v +++ b/plugins/omega/PreOmega.v @@ -12,6 +12,120 @@ Require Import Arith Max Min 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 tactic 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_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 = 0. + Proof. intro; subst; destruct x; reflexivity. Qed. + Lemma div_0_r_ext x y : y = 0 -> x / y = 0. + Proof. intro; subst; destruct x; reflexivity. Qed. + + Lemma rem_0_r_ext x y : y = 0 -> Z.rem x y = x. + Proof. intro; subst; destruct x; reflexivity. Qed. + Lemma quot_0_r_ext x y : y = 0 -> Z.quot x y = 0. + Proof. intro; subst; destruct x; reflexivity. Qed. + + Lemma rem_bound_pos_pos x y : 0 < y -> 0 <= x -> 0 <= Z.rem x y < y. + Proof. intros; apply Z.rem_bound_pos; assumption. Qed. + Lemma rem_bound_neg_pos x y : y < 0 -> 0 <= x -> 0 <= Z.rem x y < -y. + Proof. rewrite <- Z.rem_opp_r'; intros; apply Z.rem_bound_pos; rewrite ?Z.opp_pos_neg; assumption. Qed. + Lemma rem_bound_pos_neg x y : 0 < y -> x <= 0 -> -y < Z.rem x y <= 0. + Proof. rewrite <- (Z.opp_involutive x), Z.rem_opp_l', <- Z.opp_lt_mono, and_comm, !Z.opp_nonpos_nonneg; apply rem_bound_pos_pos. Qed. + Lemma rem_bound_neg_neg x y : y < 0 -> x <= 0 -> y < Z.rem x y <= 0. + Proof. rewrite <- (Z.opp_involutive x), <- (Z.opp_involutive y), Z.rem_opp_l', <- Z.opp_lt_mono, and_comm, !Z.opp_nonpos_nonneg, Z.opp_involutive; apply rem_bound_neg_pos. Qed. + + 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 div_mod_to_equations' := repeat div_mod_to_equations_step. + Ltac quot_rem_to_equations' := repeat quot_rem_to_equations_step. + Ltac euclidean_division_equations_cleanup := + repeat match goal with + | [ H : ?x = ?x -> _ |- _ ] => specialize (H eq_refl) + | [ H : ?x <> ?x -> _ |- _ ] => clear H + | [ H : ?x < ?x -> _ |- _ ] => clear H + | [ H : ?T -> _, H' : ?T |- _ ] => specialize (H H') + | [ H : ?T -> _, H' : ~?T |- _ ] => clear H + | [ H : ~?T -> _, H' : ?T |- _ ] => clear H + | [ H : ?A -> ?x = ?x -> _ |- _ ] => specialize (fun a => H a eq_refl) + | [ H : ?A -> ?x <> ?x -> _ |- _ ] => clear H + | [ H : ?A -> ?x < ?x -> _ |- _ ] => clear H + | [ H : ?A -> ?B -> _, H' : ?B |- _ ] => specialize (fun a => H a 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 : 0 <= ?x -> _, H' : ?x <= 0 |- _ ] => specialize (fun pf => H (@Z.eq_le_incl 0 x (eq_sym pf))) + | [ H : ?A -> 0 <= ?x -> _, H' : ?x <= 0 |- _ ] => specialize (fun a pf => H a (@Z.eq_le_incl 0 x (eq_sym pf))) + | [ H : ?x <= 0 -> _, H' : 0 <= ?x |- _ ] => specialize (fun pf => H (@Z.eq_le_incl 0 x pf)) + | [ H : ?A -> ?x <= 0 -> _, H' : 0 <= ?x |- _ ] => specialize (fun a pf => H a (@Z.eq_le_incl x 0 pf)) + | [ 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. + 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 to_euclidean_division_equations := div_mod_to_equations'; quot_rem_to_equations'; euclidean_division_equations_cleanup. +End Z. (** * zify: the Z-ification tactic *) @@ -411,6 +525,24 @@ Ltac zify_N_op := | |- context [ Z.of_N (N.mul ?a ?b) ] => pose proof (N2Z.is_nonneg (N.mul a b)); rewrite (N2Z.inj_mul a b) in * + (* N.div -> Z.div and a positivity hypothesis *) + | H : context [ Z.of_N (N.div ?a ?b) ] |- _ => + pose proof (N2Z.is_nonneg (N.div a b)); rewrite (N2Z.inj_div a b) in * + | |- context [ Z.of_N (N.div ?a ?b) ] => + pose proof (N2Z.is_nonneg (N.div a b)); rewrite (N2Z.inj_div a b) in * + + (* N.modulo -> Z.rem / Z.modulo and a positivity hypothesis (N.modulo agrees with Z.modulo on everything except 0; so we pose both the non-zero proof for this agreement, but also replace things with [Z.rem]) *) + | H : context [ Z.of_N (N.modulo ?a ?b) ] |- _ => + pose proof (N2Z.is_nonneg (N.modulo a b)); + pose proof (@Z.quot_div_nonneg (Z.of_N a) (Z.of_N b) (N2Z.is_nonneg a)); + pose proof (@Z.rem_mod_nonneg (Z.of_N a) (Z.of_N b) (N2Z.is_nonneg a)); + rewrite (N2Z.inj_rem a b) in * + | |- context [ Z.of_N (N.div ?a ?b) ] => + pose proof (N2Z.is_nonneg (N.modulo a b)); + pose proof (@Z.quot_div_nonneg (Z.of_N a) (Z.of_N b) (N2Z.is_nonneg a)); + pose proof (@Z.rem_mod_nonneg (Z.of_N a) (Z.of_N b) (N2Z.is_nonneg a)); + rewrite (N2Z.inj_rem a b) in * + (* atoms of type N : we add a positivity condition (if not already there) *) | _ : 0 <= Z.of_N ?a |- _ => hide_Z_of_N a | _ : context [ Z.of_N ?a ] |- _ => pose proof (N2Z.is_nonneg a); hide_Z_of_N a diff --git a/test-suite/success/Nia.v b/test-suite/success/Nia.v new file mode 100644 index 0000000000..62ecece792 --- /dev/null +++ b/test-suite/success/Nia.v @@ -0,0 +1,918 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.micromega.Lia. +Open Scope Z_scope. + +(** Add [Z.to_euclidean_division_equations] to the end of [zify], just for this + file. *) +Ltac zify ::= repeat (zify_nat; zify_positive; zify_N); zify_op; Z.to_euclidean_division_equations. + +Lemma Z_zerop_or x : x = 0 \/ x <> 0. Proof. nia. Qed. +Lemma Z_eq_dec_or (x y : Z) : x = y \/ x <> y. Proof. nia. Qed. + +Ltac unique_pose_proof pf := + let T := type of pf in + lazymatch goal with + | [ H : T |- _ ] => fail + | _ => pose proof pf + end. + +Ltac saturate_mod_div := + repeat match goal with + | [ |- context[?x mod ?y] ] => unique_pose_proof (Z_zerop_or (x / y)) + | [ H : context[?x mod ?y] |- _ ] => unique_pose_proof (Z_zerop_or (x / y)) + | [ |- context[?x / ?y] ] => unique_pose_proof (Z_zerop_or y) + | [ H : context[?x / ?y] |- _ ] => unique_pose_proof (Z_zerop_or y) + | [ |- context[Z.rem ?x ?y] ] => unique_pose_proof (Z_zerop_or (Z.quot x y)) + | [ H : context[Z.rem ?x ?y] |- _ ] => unique_pose_proof (Z_zerop_or (Z.quot x y)) + | [ |- context[Z.quot ?x ?y] ] => unique_pose_proof (Z_zerop_or y) + | [ H : context[Z.quot ?x ?y] |- _ ] => unique_pose_proof (Z_zerop_or y) + end. + +Ltac t := intros; saturate_mod_div; try nia. + +Ltac destr_step := + match goal with + | [ H : and _ _ |- _ ] => destruct H + | [ H : or _ _ |- _ ] => destruct H + end. + +Example mod_0_l: forall x : Z, 0 mod x = 0. Proof. t. Qed. +Example mod_0_r: forall x : Z, x mod 0 = 0. Proof. intros; nia. Qed. +Example Z_mod_same_full: forall a : Z, a mod a = 0. Proof. t. Qed. +Example Zmod_0_l: forall a : Z, 0 mod a = 0. Proof. t. Qed. +Example Zmod_0_r: forall a : Z, a mod 0 = 0. Proof. intros; nia. Qed. +Example mod_mod_same: forall x y : Z, (x mod y) mod y = x mod y. Proof. t. Qed. +Example Zmod_mod: forall a n : Z, (a mod n) mod n = a mod n. Proof. t. Qed. +Example Zmod_1_r: forall a : Z, a mod 1 = 0. Proof. intros; nia. Qed. +Example Zmod_div: forall a b : Z, a mod b / b = 0. Proof. intros; nia. Qed. +Example Z_mod_1_r: forall a : Z, a mod 1 = 0. Proof. intros; nia. Qed. +Example Z_mod_same: forall a : Z, a > 0 -> a mod a = 0. Proof. t. Qed. +Example Z_mod_mult: forall a b : Z, (a * b) mod b = 0. +Proof. + intros a b. + assert (b = 0 \/ (a * b) / b = a) by nia. + nia. +Qed. +Example Z_mod_same': forall a : Z, a <> 0 -> a mod a = 0. Proof. t. Qed. +Example Z_mod_0_l: forall a : Z, a <> 0 -> 0 mod a = 0. Proof. t. Qed. +Example Zmod_opp_opp: forall a b : Z, - a mod - b = - (a mod b). +Proof. + intros a b. + pose proof (Z_eq_dec_or ((-a)/(-b)) (a/b)). + nia. +Qed. +Example Z_mod_le: forall a b : Z, 0 <= a -> 0 < b -> a mod b <= a. Proof. t. Qed. +Example Zmod_le: forall a b : Z, 0 < b -> 0 <= a -> a mod b <= a. Proof. t. Qed. +Example Zplus_mod_idemp_r: forall a b n : Z, (b + a mod n) mod n = (b + a) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert ((b + a mod n) / n = (b / n) + (b mod n + a mod n) / n) + by nia. + assert ((b + a) / n = (b / n) + (a / n) + (b mod n + a mod n) / n) + by nia. + nia. +Qed. +Example Zplus_mod_idemp_l: forall a b n : Z, (a mod n + b) mod n = (a + b) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert ((a mod n + b) / n = (b / n) + (b mod n + a mod n) / n) by nia. + assert ((a + b) / n = (b / n) + (a / n) + (b mod n + a mod n) / n) by nia. + nia. +Qed. +Example Zmult_mod_distr_r: forall a b c : Z, (a * c) mod (b * c) = a mod b * c. +Proof. + intros a b c. + destruct (Z_zerop c); try nia. + pose proof (Z_eq_dec_or ((a * c) / (b * c)) (a / b)). + nia. +Qed. +Example Z_mod_zero_opp_full: forall a b : Z, a mod b = 0 -> - a mod b = 0. +Proof. + intros a b. + pose proof (Z_eq_dec_or (a/b) (-(-a/b))). + nia. +Qed. +Example Zmult_mod_idemp_r: forall a b n : Z, (b * (a mod n)) mod n = (b * a) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert ((b * (a mod n)) / n = (b / n) * (a mod n) + ((b mod n) * (a mod n)) / n) + by nia. + assert ((b * a) / n = (b / n) * (a / n) * n + (b / n) * (a mod n) + (b mod n) * (a / n) + ((b mod n) * (a mod n)) / n) + by nia. + nia. +Qed. +Example Zmult_mod_idemp_l: forall a b n : Z, (a mod n * b) mod n = (a * b) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert (((a mod n) * b) / n = (b / n) * (a mod n) + ((b mod n) * (a mod n)) / n) + by nia. + assert ((a * b) / n = (b / n) * (a / n) * n + (b / n) * (a mod n) + (b mod n) * (a / n) + ((b mod n) * (a mod n)) / n) + by nia. + nia. +Qed. +Example Zminus_mod_idemp_r: forall a b n : Z, (a - b mod n) mod n = (a - b) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert ((a - b mod n) / n = a / n + ((a mod n) - (b mod n)) / n) by nia. + assert ((a - b) / n = a / n - b / n + ((a mod n) - (b mod n)) / n) by nia. + nia. +Qed. +Example Zminus_mod_idemp_l: forall a b n : Z, (a mod n - b) mod n = (a - b) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert ((a mod n - b) / n = - (b / n) + ((a mod n) - (b mod n)) / n) by nia. + assert ((a - b) / n = a / n - b / n + ((a mod n) - (b mod n)) / n) by nia. + nia. +Qed. +Example Z_mod_plus_full: forall a b c : Z, (a + b * c) mod c = a mod c. +Proof. + intros a b c. + pose proof (Z_eq_dec_or ((a+b*c)/c) (a/c + b)). + nia. +Qed. +Example Zmult_mod_distr_l: forall a b c : Z, (c * a) mod (c * b) = c * (a mod b). +Proof. + intros a b c. + destruct (Z_zerop c); try nia. + pose proof (Z_eq_dec_or ((c * a) / (c * b)) (a / b)). + nia. +Qed. +Example Z_mod_zero_opp_r: forall a b : Z, a mod b = 0 -> a mod - b = 0. +Proof. + intros a b. + pose proof (Z_eq_dec_or (a/b) (-(a/-b))). + nia. +Qed. +Example Zmod_1_l: forall a : Z, 1 < a -> 1 mod a = 1. Proof. t. Qed. +Example Z_mod_1_l: forall a : Z, 1 < a -> 1 mod a = 1. Proof. t. Qed. +Example Z_mod_mul: forall a b : Z, b <> 0 -> (a * b) mod b = 0. +Proof. + intros a b. + pose proof (Z_eq_dec_or ((a*b)/b) a). + nia. +Qed. +Example Zminus_mod: forall a b n : Z, (a - b) mod n = (a mod n - b mod n) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert ((a - b) / n = (a / n) - (b / n) + ((a mod n) - (b mod n)) / n) by nia. + nia. +Qed. +Example Zplus_mod: forall a b n : Z, (a + b) mod n = (a mod n + b mod n) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert ((a + b) / n = (a / n) + (b / n) + ((a mod n) + (b mod n)) / n) by nia. + nia. +Qed. +Example Zmult_mod: forall a b n : Z, (a * b) mod n = (a mod n * (b mod n)) mod n. +Proof. + intros a b n. + destruct (Z_zerop n); [ subst; nia | ]. + assert ((a * b) / n = n * (a / n) * (b / n) + (a mod n) * (b / n) + (a / n) * (b mod n) + ((a mod n) * (b mod n)) / n) + by nia. + nia. +Qed. +Example Z_mod_mod: forall a n : Z, n <> 0 -> (a mod n) mod n = a mod n. Proof. t. Qed. +Example Z_mod_div: forall a b : Z, b <> 0 -> a mod b / b = 0. Proof. intros; nia. Qed. +Example Z_div_exact_full_1: forall a b : Z, a = b * (a / b) -> a mod b = 0. Proof. intros; nia. Qed. +Example Z_mod_pos_bound: forall a b : Z, 0 < b -> 0 <= a mod b < b. Proof. intros; nia. Qed. +Example Z_mod_sign_mul: forall a b : Z, b <> 0 -> 0 <= a mod b * b. Proof. intros; nia. Qed. +Example Z_mod_neg_bound: forall a b : Z, b < 0 -> b < a mod b <= 0. Proof. intros; nia. Qed. +Example Z_mod_neg: forall a b : Z, b < 0 -> b < a mod b <= 0. Proof. intros; nia. Qed. +Example div_mod_small: forall x y : Z, 0 <= x < y -> x mod y = x. Proof. t. Qed. +Example Zmod_small: forall a n : Z, 0 <= a < n -> a mod n = a. Proof. t. Qed. +Example Z_mod_small: forall a b : Z, 0 <= a < b -> a mod b = a. Proof. t. Qed. +Example Z_div_zero_opp_full: forall a b : Z, a mod b = 0 -> - a / b = - (a / b). Proof. intros; nia. Qed. +Example Z_mod_zero_opp: forall a b : Z, b > 0 -> a mod b = 0 -> - a mod b = 0. +Proof. + intros a b. + pose proof (Z_eq_dec_or (a/b) (-(-a/b))). + nia. +Qed. +Example Z_div_zero_opp_r: forall a b : Z, a mod b = 0 -> a / - b = - (a / b). Proof. intros; nia. Qed. +Example Z_mod_lt: forall a b : Z, b > 0 -> 0 <= a mod b < b. Proof. intros; nia. Qed. +Example Z_mod_opp_opp: forall a b : Z, b <> 0 -> - a mod - b = - (a mod b). +Proof. + intros a b. + pose proof (Z_eq_dec_or ((-a)/(-b)) ((a/b))). + nia. +Qed. +Example Z_mod_bound_pos: forall a b : Z, 0 <= a -> 0 < b -> 0 <= a mod b < b. Proof. intros; nia. Qed. +Example Z_mod_opp_l_z: forall a b : Z, b <> 0 -> a mod b = 0 -> - a mod b = 0. +Proof. + intros a b. + pose proof (Z_eq_dec_or (a/b) (-(-a/b))). + nia. +Qed. +Example Z_mod_plus: forall a b c : Z, c > 0 -> (a + b * c) mod c = a mod c. +Proof. + intros a b c. + pose proof (Z_eq_dec_or ((a+b*c)/c) (a/c+b)). + nia. +Qed. +Example Z_mod_opp_r_z: forall a b : Z, b <> 0 -> a mod b = 0 -> a mod - b = 0. +Proof. + intros a b. + pose proof (Z_eq_dec_or (a/b) (-(a/-b))). + nia. +Qed. +Example Zmod_eq: forall a b : Z, b > 0 -> a mod b = a - a / b * b. Proof. intros; nia. Qed. +Example Z_div_exact_2: forall a b : Z, b > 0 -> a mod b = 0 -> a = b * (a / b). Proof. intros; nia. Qed. +Example Z_div_mod_eq: forall a b : Z, b > 0 -> a = b * (a / b) + a mod b. Proof. intros; nia. Qed. +Example Z_div_exact_1: forall a b : Z, b > 0 -> a = b * (a / b) -> a mod b = 0. Proof. intros; nia. Qed. +Example Z_mod_add: forall a b c : Z, c <> 0 -> (a + b * c) mod c = a mod c. +Proof. + intros a b c. + pose proof (Z_eq_dec_or ((a+b*c)/c) (a/c+b)). + nia. +Qed. +Example Z_mod_nz_opp_r: forall a b : Z, a mod b <> 0 -> a mod - b = a mod b - b. +Proof. + intros a b. + assert (a mod b <> 0 -> a / -b = -(a/b)-1) by t. + nia. +Qed. +Example Z_mul_mod_idemp_l: forall a b n : Z, n <> 0 -> (a mod n * b) mod n = (a * b) mod n. +Proof. + intros a b n ?. + assert (((a mod n) * b) / n = (b / n) * (a mod n) + ((b mod n) * (a mod n)) / n) + by nia. + assert ((a * b) / n = (b / n) * (a / n) * n + (b / n) * (a mod n) + (b mod n) * (a / n) + ((b mod n) * (a mod n)) / n) + by nia. + nia. +Qed. +Example Z_mod_nz_opp_full: forall a b : Z, a mod b <> 0 -> - a mod b = b - a mod b. +Proof. + intros a b. + assert (a mod b <> 0 -> -a/b = -1-a/b) by nia. + nia. +Qed. +Example Z_add_mod_idemp_r: forall a b n : Z, n <> 0 -> (a + b mod n) mod n = (a + b) mod n. +Proof. + intros a b n ?. + assert ((a + b mod n) / n = (a / n) + (a mod n + b mod n) / n) by nia. + assert ((a + b) / n = (a / n) + (b / n) + (a mod n + b mod n) / n) by nia. + nia. +Qed. +Example Z_add_mod_idemp_l: forall a b n : Z, n <> 0 -> (a mod n + b) mod n = (a + b) mod n. +Proof. + intros a b n ?. + assert ((a mod n + b) / n = (b / n) + (a mod n + b mod n) / n) by nia. + assert ((a + b) / n = (a / n) + (b / n) + (a mod n + b mod n) / n) by nia. + nia. +Qed. +Example Z_mul_mod_idemp_r: forall a b n : Z, n <> 0 -> (a * (b mod n)) mod n = (a * b) mod n. +Proof. + intros a b n ?. + assert ((a * (b mod n)) / n = (a / n) * (b mod n) + ((a mod n) * (b mod n)) / n) + by nia. + assert ((a * b) / n = (b / n) * (a / n) * n + (b / n) * (a mod n) + (b mod n) * (a / n) + ((a mod n) * (b mod n)) / n) + by nia. + nia. +Qed. +Example Zmod_eq_full: forall a b : Z, b <> 0 -> a mod b = a - a / b * b. Proof. intros; nia. Qed. +Example div_eq: forall x y : Z, y <> 0 -> x mod y = 0 -> x / y * y = x. Proof. intros; nia. Qed. +Example Z_mod_eq: forall a b : Z, b <> 0 -> a mod b = a - b * (a / b). Proof. intros; nia. Qed. +Example Z_mod_sign_nz: forall a b : Z, b <> 0 -> a mod b <> 0 -> Z.sgn (a mod b) = Z.sgn b. Proof. intros; nia. Qed. +Example Z_div_exact_full_2: forall a b : Z, b <> 0 -> a mod b = 0 -> a = b * (a / b). Proof. intros; nia. Qed. +Example Z_div_mod: forall a b : Z, b <> 0 -> a = b * (a / b) + a mod b. Proof. intros; nia. Qed. +Example Z_add_mod: forall a b n : Z, n <> 0 -> (a + b) mod n = (a mod n + b mod n) mod n. +Proof. + intros a b n ?. + assert ((a + b) / n = (a / n) + (b / n) + (a mod n + b mod n) / n) by nia. + nia. +Qed. +Example Z_mul_mod: forall a b n : Z, n <> 0 -> (a * b) mod n = (a mod n * (b mod n)) mod n. +Proof. + intros a b n ?. + assert ((a * b) / n = (b / n) * (a / n) * n + (b / n) * (a mod n) + (b mod n) * (a / n) + ((a mod n) * (b mod n)) / n) + by nia. + nia. +Qed. +Example Z_div_exact: forall a b : Z, b <> 0 -> a = b * (a / b) <-> a mod b = 0. Proof. intros; nia. Qed. +Example Z_div_opp_l_z: forall a b : Z, b <> 0 -> a mod b = 0 -> - a / b = - (a / b). Proof. intros; nia. Qed. +Example Z_div_opp_r_z: forall a b : Z, b <> 0 -> a mod b = 0 -> a / - b = - (a / b). Proof. intros; nia. Qed. +Example Z_mod_opp_r_nz: forall a b : Z, b <> 0 -> a mod b <> 0 -> a mod - b = a mod b - b. +Proof. + intros a b. + assert (a mod b <> 0 -> a/(-b) = -1-a/b) by nia. + nia. +Qed. +Example Z_mul_mod_distr_r: forall a b c : Z, b <> 0 -> c <> 0 -> (a * c) mod (b * c) = a mod b * c. +Proof. + intros a b c. + pose proof (Z_eq_dec_or ((a*c)/(b*c)) (a/b)). + nia. +Qed. +Example Z_mul_mod_distr_l: forall a b c : Z, b <> 0 -> c <> 0 -> (c * a) mod (c * b) = c * (a mod b). +Proof. + intros a b c. + pose proof (Z_eq_dec_or ((c*a)/(c*b)) (a/b)). + nia. +Qed. +Example Z_mod_opp_l_nz: forall a b : Z, b <> 0 -> a mod b <> 0 -> - a mod b = b - a mod b. +Proof. + intros a b. + assert (a mod b <> 0 -> -a/b = -1-a/b) by nia. + nia. +Qed. +Example mod_eq: forall x x' y : Z, x / y = x' / y -> x mod y = x' mod y -> y <> 0 -> x = x'. Proof. intros; nia. Qed. +Example Z_div_nz_opp_r: forall a b : Z, a mod b <> 0 -> a / - b = - (a / b) - 1. Proof. intros; nia. Qed. +Example Z_div_nz_opp_full: forall a b : Z, a mod b <> 0 -> - a / b = - (a / b) - 1. Proof. intros; nia. Qed. +Example Zmod_unique: forall a b q r : Z, 0 <= r < b -> a = b * q + r -> r = a mod b. +Proof. + intros a b q r ??. + assert (q = a / b) by nia. + nia. +Qed. +Example Z_mod_unique_neg: forall a b q r : Z, b < r <= 0 -> a = b * q + r -> r = a mod b. +Proof. + intros a b q r ??. + assert (q = a / b) by nia. + nia. +Qed. +Example Z_mod_unique_pos: forall a b q r : Z, 0 <= r < b -> a = b * q + r -> r = a mod b. +Proof. + intros a b q r ??. + assert (q = a / b) by nia. + nia. +Qed. +Example Z_rem_mul_r: forall a b c : Z, b <> 0 -> 0 < c -> a mod (b * c) = a mod b + b * ((a / b) mod c). +Proof. + intros a b c ??. + assert (a / (b * c) = ((a / b) / c)) by nia. + nia. +Qed. +Example Z_mod_bound_or: forall a b : Z, b <> 0 -> 0 <= a mod b < b \/ b < a mod b <= 0. Proof. intros; nia. Qed. +Example Z_div_opp_l_nz: forall a b : Z, b <> 0 -> a mod b <> 0 -> - a / b = - (a / b) - 1. Proof. intros; nia. Qed. +Example Z_div_opp_r_nz: forall a b : Z, b <> 0 -> a mod b <> 0 -> a / - b = - (a / b) - 1. Proof. intros; nia. Qed. +Example Z_mod_small_iff: forall a b : Z, b <> 0 -> a mod b = a <-> 0 <= a < b \/ b < a <= 0. Proof. t. Qed. +Example Z_mod_unique: forall a b q r : Z, 0 <= r < b \/ b < r <= 0 -> a = b * q + r -> r = a mod b. +Proof. + intros a b q r ??. + assert (q = a/b) by nia. + nia. +Qed. +Example Z_opp_mod_bound_or: forall a b : Z, b <> 0 -> 0 <= - (a mod b) < - b \/ - b < - (a mod b) <= 0. Proof. intros; nia. Qed. + +Example Zdiv_0_r: forall a : Z, a / 0 = 0. Proof. intros; nia. Qed. +Example Zdiv_0_l: forall a : Z, 0 / a = 0. Proof. intros; nia. Qed. +Example Z_div_1_r: forall a : Z, a / 1 = a. Proof. intros; nia. Qed. +Example Zdiv_1_r: forall a : Z, a / 1 = a. Proof. intros; nia. Qed. +Example Zdiv_opp_opp: forall a b : Z, - a / - b = a / b. Proof. intros; nia. Qed. +Example Z_div_0_l: forall a : Z, a <> 0 -> 0 / a = 0. Proof. intros; nia. Qed. +Example Z_div_pos: forall a b : Z, b > 0 -> 0 <= a -> 0 <= a / b. Proof. intros; nia. Qed. +Example Z_div_ge0: forall a b : Z, b > 0 -> a >= 0 -> a / b >= 0. Proof. intros; nia. Qed. +Example Z_div_pos': forall a b : Z, 0 <= a -> 0 < b -> 0 <= a / b. Proof. intros; nia. Qed. +Example Z_mult_div_ge: forall a b : Z, b > 0 -> b * (a / b) <= a. Proof. intros; nia. Qed. +Example Z_mult_div_ge_neg: forall a b : Z, b < 0 -> b * (a / b) >= a. Proof. intros; nia. Qed. +Example Z_mul_div_le: forall a b : Z, 0 < b -> b * (a / b) <= a. Proof. intros; nia. Qed. +Example Z_mul_div_ge: forall a b : Z, b < 0 -> a <= b * (a / b). Proof. intros; nia. Qed. +Example Z_div_same: forall a : Z, a > 0 -> a / a = 1. Proof. intros; nia. Qed. +Example Z_div_mult: forall a b : Z, b > 0 -> a * b / b = a. Proof. intros; nia. Qed. +Example Z_mul_succ_div_gt: forall a b : Z, 0 < b -> a < b * Z.succ (a / b). Proof. intros; nia. Qed. +Example Z_mul_succ_div_lt: forall a b : Z, b < 0 -> b * Z.succ (a / b) < a. Proof. intros; nia. Qed. +Example Zdiv_1_l: forall a : Z, 1 < a -> 1 / a = 0. Proof. intros; nia. Qed. +Example Z_div_1_l: forall a : Z, 1 < a -> 1 / a = 0. Proof. intros; nia. Qed. +Example Z_div_str_pos: forall a b : Z, 0 < b <= a -> 0 < a / b. Proof. intros; nia. Qed. +Example Z_div_ge: forall a b c : Z, c > 0 -> a >= b -> a / c >= b / c. Proof. intros; nia. Qed. +Example Z_div_mult_full: forall a b : Z, b <> 0 -> a * b / b = a. Proof. intros; nia. Qed. +Example Z_div_same': forall a : Z, a <> 0 -> a / a = 1. Proof. intros; nia. Qed. +Example Zdiv_lt_upper_bound: forall a b q : Z, 0 < b -> a < q * b -> a / b < q. Proof. intros; nia. Qed. +Example Z_div_mul: forall a b : Z, b <> 0 -> a * b / b = a. Proof. intros; nia. Qed. +Example Z_div_lt: forall a b : Z, 0 < a -> 1 < b -> a / b < a. Proof. intros; nia. Qed. +Example Z_div_le_mono: forall a b c : Z, 0 < c -> a <= b -> a / c <= b / c. Proof. intros; nia. Qed. +Example Zdiv_sgn: forall a b : Z, 0 <= Z.sgn (a / b) * Z.sgn a * Z.sgn b. Proof. intros; nia. Qed. +Example Z_div_same_full: forall a : Z, a <> 0 -> a / a = 1. Proof. intros; nia. Qed. +Example Z_div_lt_upper_bound: forall a b q : Z, 0 < b -> a < b * q -> a / b < q. Proof. intros; nia. Qed. +Example Z_div_le: forall a b c : Z, c > 0 -> a <= b -> a / c <= b / c. Proof. intros; nia. Qed. +Example Z_div_le_lower_bound: forall a b q : Z, 0 < b -> b * q <= a -> q <= a / b. Proof. intros; nia. Qed. +Example Zdiv_le_lower_bound: forall a b q : Z, 0 < b -> q * b <= a -> q <= a / b. Proof. intros; nia. Qed. +Example Zdiv_le_upper_bound: forall a b q : Z, 0 < b -> a <= q * b -> a / b <= q. Proof. intros; nia. Qed. +Example Z_div_le_upper_bound: forall a b q : Z, 0 < b -> a <= b * q -> a / b <= q. Proof. intros; nia. Qed. +Example Z_div_small: forall a b : Z, 0 <= a < b -> a / b = 0. Proof. intros; nia. Qed. +Example Zdiv_small: forall a b : Z, 0 <= a < b -> a / b = 0. Proof. intros; nia. Qed. +Example Z_div_opp_opp: forall a b : Z, b <> 0 -> - a / - b = a / b. Proof. intros; nia. Qed. +Example Zdiv_mult_cancel_r: forall a b c : Z, c <> 0 -> a * c / (b * c) = a / b. Proof. intros; nia. Qed. +Example Z_div_unique_exact: forall a b q : Z, b <> 0 -> a = b * q -> q = a / b. Proof. intros; nia. Qed. +Example Zdiv_mult_cancel_l: forall a b c : Z, c <> 0 -> c * a / (c * b) = a / b. Proof. intros; nia. Qed. +Example Zdiv_le_compat_l: forall p q r : Z, 0 <= p -> 0 < q < r -> p / r <= p / q. +Proof. + intros p q r ??. + assert (p mod r <= p mod q \/ p mod q <= p mod r) by nia. + assert (0 <= p / r) by nia. + assert (0 <= p / q) by nia. + nia. +Qed. +Example Z_div_le_compat_l: forall p q r : Z, 0 <= p -> 0 < q <= r -> p / r <= p / q. +Proof. + intros p q r ??. + assert (p mod r <= p mod q \/ p mod q <= p mod r) by nia. + assert (0 <= p / r) by nia. + assert (0 <= p / q) by nia. + nia. +Qed. +Example Zdiv_Zdiv: forall a b c : Z, 0 <= b -> 0 <= c -> a / b / c = a / (b * c). Proof. intros; nia. Qed. +Example Z_div_plus: forall a b c : Z, c > 0 -> (a + b * c) / c = a / c + b. Proof. intros; nia. Qed. +Example Z_div_lt': forall a b : Z, b >= 2 -> a > 0 -> a / b < a. Proof. intros; nia. Qed. +Example Zdiv_mult_le: forall a b c : Z, 0 <= a -> 0 <= b -> 0 <= c -> c * (a / b) <= c * a / b. Proof. intros; nia. Qed. +Example Z_div_add_l: forall a b c : Z, b <> 0 -> (a * b + c) / b = a + c / b. Proof. intros; nia. Qed. +Example Z_div_plus_full_l: forall a b c : Z, b <> 0 -> (a * b + c) / b = a + c / b. Proof. intros; nia. Qed. +Example Z_div_add: forall a b c : Z, c <> 0 -> (a + b * c) / c = a / c + b. Proof. intros; nia. Qed. +Example Z_div_plus_full: forall a b c : Z, c <> 0 -> (a + b * c) / c = a / c + b. Proof. intros; nia. Qed. +Example Z_div_mul_le: forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a / b) <= c * a / b. Proof. intros; nia. Qed. +Example Z_div_mul_cancel_r: forall a b c : Z, b <> 0 -> c <> 0 -> a * c / (b * c) = a / b. Proof. intros; nia. Qed. +Example Z_div_div: forall a b c : Z, b <> 0 -> 0 < c -> a / b / c = a / (b * c). Proof. intros; nia. Qed. +Example Z_div_mul_cancel_l: forall a b c : Z, b <> 0 -> c <> 0 -> c * a / (c * b) = a / b. Proof. intros; nia. Qed. +Example Z_div_unique_neg: forall a b q r : Z, b < r <= 0 -> a = b * q + r -> q = a / b. Proof. intros; nia. Qed. +Example Zdiv_unique: forall a b q r : Z, 0 <= r < b -> a = b * q + r -> q = a / b. Proof. intros; nia. Qed. +Example Z_div_unique_pos: forall a b q r : Z, 0 <= r < b -> a = b * q + r -> q = a / b. Proof. intros; nia. Qed. +Example Z_div_small_iff: forall a b : Z, b <> 0 -> a / b = 0 <-> 0 <= a < b \/ b < a <= 0. Proof. intros; nia. Qed. +Example Z_div_unique: forall a b q r : Z, 0 <= r < b \/ b < r <= 0 -> a = b * q + r -> q = a / b. Proof. intros; nia. Qed. + +(** Now we do the same, but with [Z.quot] and [Z.rem] instead. *) +Lemma N2Z_inj_quot : forall n m : N, Z.of_N (n / m) = Z.of_N n ÷ Z.of_N m. Proof. intros; nia. Qed. +Lemma N2Z_inj_rem : forall n m : N, Z.of_N (n mod m) = Z.rem (Z.of_N n) (Z.of_N m). Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_mul_quot_ge : forall a b : Z, a <= 0 -> b <> 0 -> a <= b * (a ÷ b) <= 0. +Proof. intros; destruct (Z_zerop (a ÷ b)); nia. Qed. +Lemma OrdersEx_Z_as_DT_mul_quot_le : forall a b : Z, 0 <= a -> b <> 0 -> 0 <= b * (a ÷ b) <= a. Proof. intros; destruct (Z_zerop (a ÷ b)); nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_0_l : forall a : Z, 0 < a -> 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_1_l : forall a : Z, 1 < a -> 1 ÷ a = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_1_r : forall a : Z, 0 <= a -> a ÷ 1 = a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_add : forall a b c : Z, 0 <= a -> 0 <= a + b * c -> 0 < c -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_add_l : forall a b c : Z, 0 <= c -> 0 <= a * b + c -> 0 < b -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_div : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> a ÷ b ÷ c = a ÷ (b * c). +Proof. intros; assert (0 < b * c) by nia; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_le_compat_l : forall p q r : Z, 0 <= p -> 0 < q <= r -> p ÷ r <= p ÷ q. +Proof. + intros. + destruct (Z_zerop p), (Z_zerop (p ÷ r)), (Z_zerop (p ÷ q)); subst; [ nia.. | ]. + assert (0 < q) by nia; assert (0 < r) by nia; assert (0 < p) by nia. + nia. +Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_le_lower_bound : forall a b q : Z, 0 <= a -> 0 < b -> b * q <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_le_mono : forall a b c : Z, 0 < c -> 0 <= a <= b -> a ÷ c <= b ÷ c. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_le_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a <= b * q -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_lt : forall a b : Z, 0 < a -> 1 < b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < b * q -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_mul_cancel_l : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> c * a ÷ (c * b) = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_mul_cancel_r : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> a * c ÷ (b * c) = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_mul : forall a b : Z, 0 <= a -> 0 < b -> a * b ÷ b = a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_mul_le : forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_same : forall a : Z, 0 < a -> a ÷ a = 1. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_small : forall a b : Z, 0 <= a < b -> a ÷ b = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_small_iff : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = 0 <-> a < b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_str_pos : forall a b : Z, 0 < b <= a -> 0 < a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_str_pos_iff : forall a b : Z, 0 <= a -> 0 < b -> 0 < a ÷ b <-> b <= a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_unique_exact : forall a b q : Z, 0 <= a -> 0 < b -> a = b * q -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_div_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_Private_Div_NZQuot_mul_div_le : forall a b : Z, 0 <= a -> 0 < b -> b * (a ÷ b) <= a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_0_l : forall a : Z, a <> 0 -> 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_1_l : forall a : Z, 1 < a -> 1 ÷ a = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_1_r : forall a : Z, a ÷ 1 = a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_add : forall a b c : Z, c <> 0 -> 0 <= (a + b * c) * a -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_add_l : forall a b c : Z, b <> 0 -> 0 <= (a * b + c) * c -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_div_nonneg : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = a / b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_le_compat_l : forall p q r : Z, 0 <= p -> 0 < q <= r -> p ÷ r <= p ÷ q. +Proof. + intros. + destruct (Z_zerop p), (Z_zerop (p ÷ r)), (Z_zerop (p ÷ q)); [ subst; nia.. | ]. + assert (0 < p) by nia; assert (0 < r) by nia. + nia. +Qed. +Lemma OrdersEx_Z_as_DT_quot_le_lower_bound : forall a b q : Z, 0 < b -> b * q <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_le_mono : forall a b c : Z, 0 < c -> a <= b -> a ÷ c <= b ÷ c. +Proof. + intros. + destruct (Z_zerop a), (Z_zerop b), (Z_zerop (a ÷ c)), (Z_zerop (b ÷ c)); [ subst; nia.. | ]. + nia. +Qed. +Lemma OrdersEx_Z_as_DT_quot_le_upper_bound : forall a b q : Z, 0 < b -> a <= b * q -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_lt : forall a b : Z, 0 < a -> 1 < b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < b * q -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_mul_cancel_l : forall a b c : Z, b <> 0 -> c <> 0 -> c * a ÷ (c * b) = a ÷ b. +Proof. + intros. + assert (c * b <> 0) by nia. + destruct (Z_zerop a), (Z_zerop (c * a)); subst; [ nia | exfalso; nia.. | ]. +Abort. +Lemma OrdersEx_Z_as_DT_quot_mul_cancel_r : forall a b c : Z, b <> 0 -> c <> 0 -> a * c ÷ (b * c) = a ÷ b. Proof. Abort. +Lemma OrdersEx_Z_as_DT_quot_mul : forall a b : Z, b <> 0 -> a * b ÷ b = a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_mul_le : forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_opp_l : forall a b : Z, b <> 0 -> - a ÷ b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_opp_opp : forall a b : Z, b <> 0 -> - a ÷ - b = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_opp_r : forall a b : Z, b <> 0 -> a ÷ - b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_quot : forall a b c : Z, b <> 0 -> c <> 0 -> a ÷ b ÷ c = a ÷ (b * c). Proof. intros; assert (b * c <> 0) by nia. Abort. +Lemma OrdersEx_Z_as_DT_quot_same : forall a : Z, a <> 0 -> a ÷ a = 1. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_small : forall a b : Z, 0 <= a < b -> a ÷ b = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_str_pos : forall a b : Z, 0 < b <= a -> 0 < a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_unique_exact : forall a b q : Z, b <> 0 -> a = b * q -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_DT_quot_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_mul_quot_ge : forall a b : Z, a <= 0 -> b <> 0 -> a <= b * (a ÷ b) <= 0. Proof. intros. Fail nia. Abort. +Lemma OrdersEx_Z_as_OT_mul_quot_le : forall a b : Z, 0 <= a -> b <> 0 -> 0 <= b * (a ÷ b) <= a. Proof. intros. Fail nia. Abort. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_0_l : forall a : Z, 0 < a -> 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_1_l : forall a : Z, 1 < a -> 1 ÷ a = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_1_r : forall a : Z, 0 <= a -> a ÷ 1 = a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_add : forall a b c : Z, 0 <= a -> 0 <= a + b * c -> 0 < c -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_add_l : forall a b c : Z, 0 <= c -> 0 <= a * b + c -> 0 < b -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_div : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> a ÷ b ÷ c = a ÷ (b * c). Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_le_compat_l : forall p q r : Z, 0 <= p -> 0 < q <= r -> p ÷ r <= p ÷ q. Proof. intros. Abort. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_le_lower_bound : forall a b q : Z, 0 <= a -> 0 < b -> b * q <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_le_mono : forall a b c : Z, 0 < c -> 0 <= a <= b -> a ÷ c <= b ÷ c. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_le_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a <= b * q -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_lt : forall a b : Z, 0 < a -> 1 < b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < b * q -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_mul_cancel_l : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> c * a ÷ (c * b) = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_mul_cancel_r : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> a * c ÷ (b * c) = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_mul : forall a b : Z, 0 <= a -> 0 < b -> a * b ÷ b = a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_mul_le : forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_same : forall a : Z, 0 < a -> a ÷ a = 1. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_small : forall a b : Z, 0 <= a < b -> a ÷ b = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_small_iff : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = 0 <-> a < b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_str_pos : forall a b : Z, 0 < b <= a -> 0 < a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_str_pos_iff : forall a b : Z, 0 <= a -> 0 < b -> 0 < a ÷ b <-> b <= a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_unique_exact : forall a b q : Z, 0 <= a -> 0 < b -> a = b * q -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_div_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_Private_Div_NZQuot_mul_div_le : forall a b : Z, 0 <= a -> 0 < b -> b * (a ÷ b) <= a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_0_l : forall a : Z, a <> 0 -> 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_1_l : forall a : Z, 1 < a -> 1 ÷ a = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_1_r : forall a : Z, a ÷ 1 = a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_add : forall a b c : Z, c <> 0 -> 0 <= (a + b * c) * a -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_add_l : forall a b c : Z, b <> 0 -> 0 <= (a * b + c) * c -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_div_nonneg : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = a / b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_le_compat_l : forall p q r : Z, 0 <= p -> 0 < q <= r -> p ÷ r <= p ÷ q. Proof. intros. Fail nia. Abort. +Lemma OrdersEx_Z_as_OT_quot_le_lower_bound : forall a b q : Z, 0 < b -> b * q <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_le_mono : forall a b c : Z, 0 < c -> a <= b -> a ÷ c <= b ÷ c. Proof. intros. Fail nia. Abort. +Lemma OrdersEx_Z_as_OT_quot_le_upper_bound : forall a b q : Z, 0 < b -> a <= b * q -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_lt : forall a b : Z, 0 < a -> 1 < b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < b * q -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_mul_cancel_l : forall a b c : Z, b <> 0 -> c <> 0 -> c * a ÷ (c * b) = a ÷ b. Proof. intros. Abort. +Lemma OrdersEx_Z_as_OT_quot_mul_cancel_r : forall a b c : Z, b <> 0 -> c <> 0 -> a * c ÷ (b * c) = a ÷ b. Proof. intros. Abort. +Lemma OrdersEx_Z_as_OT_quot_mul : forall a b : Z, b <> 0 -> a * b ÷ b = a. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_mul_le : forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_opp_l : forall a b : Z, b <> 0 -> - a ÷ b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_opp_opp : forall a b : Z, b <> 0 -> - a ÷ - b = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_opp_r : forall a b : Z, b <> 0 -> a ÷ - b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_quot : forall a b c : Z, b <> 0 -> c <> 0 -> a ÷ b ÷ c = a ÷ (b * c). Proof. intros. Abort. +Lemma OrdersEx_Z_as_OT_quot_same : forall a : Z, a <> 0 -> a ÷ a = 1. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_small : forall a b : Z, 0 <= a < b -> a ÷ b = 0. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_str_pos : forall a b : Z, 0 < b <= a -> 0 < a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_unique_exact : forall a b q : Z, b <> 0 -> a = b * q -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma OrdersEx_Z_as_OT_quot_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma Z2N_inj_quot : forall n m : Z, 0 <= n -> 0 <= m -> Z.to_N (n ÷ m) = (Z.to_N n / Z.to_N m)%N. +Proof. intros; destruct (Z_zerop n), (Z_zerop m), (Z_zerop (n ÷ m)); [ subst; try nia.. | ]. Abort. +Lemma Z2N_inj_rem : forall n m : Z, 0 <= n -> 0 <= m -> Z.to_N (Z.rem n m) = (Z.to_N n mod Z.to_N m)%N. Proof. intros. Abort. +Lemma Zabs2N_inj_quot : forall n m : Z, Z.abs_N (n ÷ m) = (Z.abs_N n / Z.abs_N m)%N. Proof. intros. Abort. +Lemma Zabs2N_inj_rem : forall n m : Z, Z.abs_N (Z.rem n m) = (Z.abs_N n mod Z.abs_N m)%N. Proof. intros. Abort. +(* Some of these don't work, and I haven't gone through and figured out which ones yet, so they're all commented out for now *) +(* +Lemma Z_add_rem : forall a b n : Z, n <> 0 -> 0 <= a * b -> Z.rem (a + b) n = Z.rem (Z.rem a n + Z.rem b n) n. Proof. intros; nia. Qed. +Lemma Z_add_rem_idemp_l : forall a b n : Z, n <> 0 -> 0 <= a * b -> Z.rem (Z.rem a n + b) n = Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma Z_add_rem_idemp_r : forall a b n : Z, n <> 0 -> 0 <= a * b -> Z.rem (a + Z.rem b n) n = Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_add_rem : forall a b n : Z, n <> 0 -> 0 <= a * b -> ZBinary.Z.rem (a + b) n = ZBinary.Z.rem (ZBinary.Z.rem a n + ZBinary.Z.rem b n) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_add_rem_idemp_l : forall a b n : Z, n <> 0 -> 0 <= a * b -> ZBinary.Z.rem (ZBinary.Z.rem a n + b) n = ZBinary.Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_add_rem_idemp_r : forall a b n : Z, n <> 0 -> 0 <= a * b -> ZBinary.Z.rem (a + ZBinary.Z.rem b n) n = ZBinary.Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_gcd_quot_gcd : forall a b g : Z, g <> 0 -> g = ZBinary.Z.gcd a b -> ZBinary.Z.gcd (a ÷ g) (b ÷ g) = 1. Proof. intros; nia. Qed. +Lemma ZBinary_Z_gcd_rem : forall a b : Z, b <> 0 -> ZBinary.Z.gcd (ZBinary.Z.rem a b) b = ZBinary.Z.gcd b a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mod_mul_r : forall a b c : Z, b <> 0 -> c <> 0 -> ZBinary.Z.rem a (b * c) = ZBinary.Z.rem a b + b * ZBinary.Z.rem (a ÷ b) c. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_pred_quot_gt : forall a b : Z, 0 <= a -> b < 0 -> a < b * ZBinary.Z.pred (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_pred_quot_lt : forall a b : Z, a <= 0 -> 0 < b -> b * ZBinary.Z.pred (a ÷ b) < a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_quot_ge : forall a b : Z, a <= 0 -> b <> 0 -> a <= b * (a ÷ b) <= 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_quot_le : forall a b : Z, 0 <= a -> b <> 0 -> 0 <= b * (a ÷ b) <= a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_rem_distr_l : forall a b c : Z, b <> 0 -> c <> 0 -> ZBinary.Z.rem (c * a) (c * b) = c * ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_rem_distr_r : forall a b c : Z, b <> 0 -> c <> 0 -> ZBinary.Z.rem (a * c) (b * c) = ZBinary.Z.rem a b * c. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_rem : forall a b n : Z, n <> 0 -> ZBinary.Z.rem (a * b) n = ZBinary.Z.rem (ZBinary.Z.rem a n * ZBinary.Z.rem b n) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_rem_idemp_l : forall a b n : Z, n <> 0 -> ZBinary.Z.rem (ZBinary.Z.rem a n * b) n = ZBinary.Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_rem_idemp_r : forall a b n : Z, n <> 0 -> ZBinary.Z.rem (a * ZBinary.Z.rem b n) n = ZBinary.Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_succ_quot_gt : forall a b : Z, 0 <= a -> 0 < b -> a < b * ZBinary.Z.succ (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_mul_succ_quot_lt : forall a b : Z, a <= 0 -> b < 0 -> b * ZBinary.Z.succ (a ÷ b) < a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_add_mod : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> ZBinary.Z.rem (a + b) n = ZBinary.Z.rem (ZBinary.Z.rem a n + ZBinary.Z.rem b n) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_add_mod_idemp_l : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> ZBinary.Z.rem (ZBinary.Z.rem a n + b) n = ZBinary.Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_add_mod_idemp_r : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> ZBinary.Z.rem (a + ZBinary.Z.rem b n) n = ZBinary.Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_0_l : forall a : Z, 0 < a -> 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_1_l : forall a : Z, 1 < a -> 1 ÷ a = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_1_r : forall a : Z, 0 <= a -> a ÷ 1 = a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_add : forall a b c : Z, 0 <= a -> 0 <= a + b * c -> 0 < c -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_add_l : forall a b c : Z, 0 <= c -> 0 <= a * b + c -> 0 < b -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_div : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> a ÷ b ÷ c = a ÷ (b * c). Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_exact : forall a b : Z, 0 <= a -> 0 < b -> a = b * (a ÷ b) <-> ZBinary.Z.rem a b = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_le_compat_l : forall p q r : Z, 0 <= p -> 0 < q <= r -> p ÷ r <= p ÷ q. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_le_lower_bound : forall a b q : Z, 0 <= a -> 0 < b -> b * q <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_le_mono : forall a b c : Z, 0 < c -> 0 <= a <= b -> a ÷ c <= b ÷ c. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_le_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a <= b * q -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_lt : forall a b : Z, 0 < a -> 1 < b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < b * q -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_mul_cancel_l : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> c * a ÷ (c * b) = a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_mul_cancel_r : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> a * c ÷ (b * c) = a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_mul : forall a b : Z, 0 <= a -> 0 < b -> a * b ÷ b = a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_mul_le : forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_same : forall a : Z, 0 < a -> a ÷ a = 1. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_small : forall a b : Z, 0 <= a < b -> a ÷ b = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_small_iff : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = 0 <-> a < b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_str_pos : forall a b : Z, 0 < b <= a -> 0 < a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_str_pos_iff : forall a b : Z, 0 <= a -> 0 < b -> 0 < a ÷ b <-> b <= a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_unique_exact : forall a b q : Z, 0 <= a -> 0 < b -> a = b * q -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_div_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_0_l : forall a : Z, 0 < a -> ZBinary.Z.rem 0 a = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_1_l : forall a : Z, 1 < a -> ZBinary.Z.rem 1 a = 1. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_1_r : forall a : Z, 0 <= a -> ZBinary.Z.rem a 1 = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_add : forall a b c : Z, 0 <= a -> 0 <= a + b * c -> 0 < c -> ZBinary.Z.rem (a + b * c) c = ZBinary.Z.rem a c. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_divides : forall a b : Z, 0 <= a -> 0 < b -> ZBinary.Z.rem a b = 0 <-> (exists c : Z, a = b * c). Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_le : forall a b : Z, 0 <= a -> 0 < b -> ZBinary.Z.rem a b <= a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_mod : forall a n : Z, 0 <= a -> 0 < n -> ZBinary.Z.rem (ZBinary.Z.rem a n) n = ZBinary.Z.rem a n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_mul : forall a b : Z, 0 <= a -> 0 < b -> ZBinary.Z.rem (a * b) b = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_mul_r : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> ZBinary.Z.rem a (b * c) = ZBinary.Z.rem a b + b * ZBinary.Z.rem (a ÷ b) c. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_same : forall a : Z, 0 < a -> ZBinary.Z.rem a a = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_small : forall a b : Z, 0 <= a < b -> ZBinary.Z.rem a b = a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_small_iff : forall a b : Z, 0 <= a -> 0 < b -> ZBinary.Z.rem a b = a <-> a < b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mod_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> r = ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mul_div_le : forall a b : Z, 0 <= a -> 0 < b -> b * (a ÷ b) <= a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mul_mod_distr_l : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> ZBinary.Z.rem (c * a) (c * b) = c * ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mul_mod_distr_r : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> ZBinary.Z.rem (a * c) (b * c) = ZBinary.Z.rem a b * c. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mul_mod : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> ZBinary.Z.rem (a * b) n = ZBinary.Z.rem (ZBinary.Z.rem a n * ZBinary.Z.rem b n) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mul_mod_idemp_l : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> ZBinary.Z.rem (ZBinary.Z.rem a n * b) n = ZBinary.Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mul_mod_idemp_r : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> ZBinary.Z.rem (a * ZBinary.Z.rem b n) n = ZBinary.Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_NZQuot_mul_succ_div_gt : forall a b : Z, 0 <= a -> 0 < b -> a < b * ZBinary.Z.succ (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_Private_Div_Quot2Div_div_mod : forall a b : Z, b <> 0 -> a = b * (a ÷ b) + ZBinary.Z.rem a b. Proof. intros; nia. Qed. +ZBinary_Z_Private_Div_Quot2Div_div_wd Morphisms.Proper (Morphisms.respectful eq (Morphisms.respectful eq eq)) ZBinary.Z.quot +Lemma ZBinary_Z_Private_Div_Quot2Div_mod_bound_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= ZBinary.Z.rem a b < b. Proof. intros; nia. Qed. +ZBinary_Z_Private_Div_Quot2Div_mod_wd Morphisms.Proper (Morphisms.respectful eq (Morphisms.respectful eq eq)) ZBinary.Z.rem +Lemma ZBinary_Z_quot_0_l : forall a : Z, a <> 0 -> 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_1_l : forall a : Z, 1 < a -> 1 ÷ a = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_1_r : forall a : Z, a ÷ 1 = a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_abs : forall a b : Z, b <> 0 -> ZBinary.Z.abs a ÷ ZBinary.Z.abs b = ZBinary.Z.abs (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_abs_l : forall a b : Z, b <> 0 -> ZBinary.Z.abs a ÷ b = ZBinary.Z.sgn a * (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_abs_r : forall a b : Z, b <> 0 -> a ÷ ZBinary.Z.abs b = ZBinary.Z.sgn b * (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_add : forall a b c : Z, c <> 0 -> 0 <= (a + b * c) * a -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_add_l : forall a b c : Z, b <> 0 -> 0 <= (a * b + c) * c -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_div : forall a b : Z, b <> 0 -> a ÷ b = ZBinary.Z.sgn a * ZBinary.Z.sgn b * (ZBinary.Z.abs a / ZBinary.Z.abs b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_div_nonneg : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = a / b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_exact : forall a b : Z, b <> 0 -> a = b * (a ÷ b) <-> ZBinary.Z.rem a b = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_le_compat_l : forall p q r : Z, 0 <= p -> 0 < q <= r -> p ÷ r <= p ÷ q. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_le_lower_bound : forall a b q : Z, 0 < b -> b * q <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_le_mono : forall a b c : Z, 0 < c -> a <= b -> a ÷ c <= b ÷ c. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_le_upper_bound : forall a b q : Z, 0 < b -> a <= b * q -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_lt : forall a b : Z, 0 < a -> 1 < b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < b * q -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_mul_cancel_l : forall a b c : Z, b <> 0 -> c <> 0 -> c * a ÷ (c * b) = a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_mul_cancel_r : forall a b c : Z, b <> 0 -> c <> 0 -> a * c ÷ (b * c) = a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_mul : forall a b : Z, b <> 0 -> a * b ÷ b = a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_mul_le : forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_opp_l : forall a b : Z, b <> 0 -> - a ÷ b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_opp_opp : forall a b : Z, b <> 0 -> - a ÷ - b = a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_opp_r : forall a b : Z, b <> 0 -> a ÷ - b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_quot : forall a b c : Z, b <> 0 -> c <> 0 -> a ÷ b ÷ c = a ÷ (b * c). Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_rem' : forall a b : Z, a = b * (a ÷ b) + ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_rem : forall a b : Z, b <> 0 -> a = b * (a ÷ b) + ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_same : forall a : Z, a <> 0 -> a ÷ a = 1. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_small : forall a b : Z, 0 <= a < b -> a ÷ b = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_small_iff : forall a b : Z, b <> 0 -> a ÷ b = 0 <-> ZBinary.Z.abs a < ZBinary.Z.abs b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_str_pos : forall a b : Z, 0 < b <= a -> 0 < a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_unique_exact : forall a b q : Z, b <> 0 -> a = b * q -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_quot_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +ZBinary_Z_quot_wd Morphisms.Proper (Morphisms.respectful ZBinary.Z.eq (Morphisms.respectful ZBinary.Z.eq ZBinary.Z.eq)) ZBinary.Z.quot +Lemma ZBinary_Z_rem_0_l : forall a : Z, a <> 0 -> ZBinary.Z.rem 0 a = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_1_l : forall a : Z, 1 < a -> ZBinary.Z.rem 1 a = 1. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_1_r : forall a : Z, ZBinary.Z.rem a 1 = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_abs : forall a b : Z, b <> 0 -> ZBinary.Z.rem (ZBinary.Z.abs a) (ZBinary.Z.abs b) = ZBinary.Z.abs (ZBinary.Z.rem a b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_abs_l : forall a b : Z, b <> 0 -> ZBinary.Z.rem (ZBinary.Z.abs a) b = ZBinary.Z.abs (ZBinary.Z.rem a b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_abs_r : forall a b : Z, b <> 0 -> ZBinary.Z.rem a (ZBinary.Z.abs b) = ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_add : forall a b c : Z, c <> 0 -> 0 <= (a + b * c) * a -> ZBinary.Z.rem (a + b * c) c = ZBinary.Z.rem a c. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_bound_abs : forall a b : Z, b <> 0 -> ZBinary.Z.abs (ZBinary.Z.rem a b) < ZBinary.Z.abs b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_bound_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= ZBinary.Z.rem a b < b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_eq : forall a b : Z, b <> 0 -> ZBinary.Z.rem a b = a - b * (a ÷ b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_le : forall a b : Z, 0 <= a -> 0 < b -> ZBinary.Z.rem a b <= a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_mod_eq_0 : forall a b : Z, b <> 0 -> ZBinary.Z.rem a b = 0 <-> a mod b = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_mod : forall a b : Z, b <> 0 -> ZBinary.Z.rem a b = ZBinary.Z.sgn a * (ZBinary.Z.abs a mod ZBinary.Z.abs b). Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_mod_nonneg : forall a b : Z, 0 <= a -> 0 < b -> ZBinary.Z.rem a b = a mod b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_mul : forall a b : Z, b <> 0 -> ZBinary.Z.rem (a * b) b = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_nonneg : forall a b : Z, b <> 0 -> 0 <= a -> 0 <= ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_nonpos : forall a b : Z, b <> 0 -> a <= 0 -> ZBinary.Z.rem a b <= 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_opp_l : forall a b : Z, b <> 0 -> ZBinary.Z.rem (- a) b = - ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_opp_l' : forall a b : Z, ZBinary.Z.rem (- a) b = - ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_opp_opp : forall a b : Z, b <> 0 -> ZBinary.Z.rem (- a) (- b) = - ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_opp_r : forall a b : Z, b <> 0 -> ZBinary.Z.rem a (- b) = ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_opp_r' : forall a b : Z, ZBinary.Z.rem a (- b) = ZBinary.Z.rem a b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_quot : forall a b : Z, b <> 0 -> ZBinary.Z.rem a b ÷ b = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_rem : forall a n : Z, n <> 0 -> ZBinary.Z.rem (ZBinary.Z.rem a n) n = ZBinary.Z.rem a n. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_same : forall a : Z, a <> 0 -> ZBinary.Z.rem a a = 0. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_sign : forall a b : Z, a <> 0 -> b <> 0 -> ZBinary.Z.sgn (ZBinary.Z.rem a b) <> - ZBinary.Z.sgn a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_sign_mul : forall a b : Z, b <> 0 -> 0 <= ZBinary.Z.rem a b * a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_sign_nz : forall a b : Z, b <> 0 -> ZBinary.Z.rem a b <> 0 -> ZBinary.Z.sgn (ZBinary.Z.rem a b) = ZBinary.Z.sgn a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_small : forall a b : Z, 0 <= a < b -> ZBinary.Z.rem a b = a. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_small_iff : forall a b : Z, b <> 0 -> ZBinary.Z.rem a b = a <-> ZBinary.Z.abs a < ZBinary.Z.abs b. Proof. intros; nia. Qed. +Lemma ZBinary_Z_rem_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> r = ZBinary.Z.rem a b. Proof. intros; nia. Qed. +ZBinary_Z_rem_wd Morphisms.Proper (Morphisms.respectful ZBinary.Z.eq (Morphisms.respectful ZBinary.Z.eq ZBinary.Z.eq)) ZBinary.Z.rem +Lemma Z_gcd_quot_gcd : forall a b g : Z, g <> 0 -> g = Z.gcd a b -> Z.gcd (a ÷ g) (b ÷ g) = 1. Proof. intros; nia. Qed. +Lemma Z_gcd_rem : forall a b : Z, b <> 0 -> Z.gcd (Z.rem a b) b = Z.gcd b a. Proof. intros; nia. Qed. +Lemma Z_mod_mul_r : forall a b c : Z, b <> 0 -> c <> 0 -> Z.rem a (b * c) = Z.rem a b + b * Z.rem (a ÷ b) c. Proof. intros; nia. Qed. +Lemma Z_mul_pred_quot_gt : forall a b : Z, 0 <= a -> b < 0 -> a < b * Z.pred (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_mul_pred_quot_lt : forall a b : Z, a <= 0 -> 0 < b -> b * Z.pred (a ÷ b) < a. Proof. intros; nia. Qed. +Lemma Z_mul_quot_ge : forall a b : Z, a <= 0 -> b <> 0 -> a <= b * (a ÷ b) <= 0. Proof. intros; nia. Qed. +Lemma Z_mul_quot_le : forall a b : Z, 0 <= a -> b <> 0 -> 0 <= b * (a ÷ b) <= a. Proof. intros; nia. Qed. +Lemma Z_mul_rem_distr_l : forall a b c : Z, b <> 0 -> c <> 0 -> Z.rem (c * a) (c * b) = c * Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_mul_rem_distr_r : forall a b c : Z, b <> 0 -> c <> 0 -> Z.rem (a * c) (b * c) = Z.rem a b * c. Proof. intros; nia. Qed. +Lemma Z_mul_rem : forall a b n : Z, n <> 0 -> Z.rem (a * b) n = Z.rem (Z.rem a n * Z.rem b n) n. Proof. intros; nia. Qed. +Lemma Z_mul_rem_idemp_l : forall a b n : Z, n <> 0 -> Z.rem (Z.rem a n * b) n = Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma Z_mul_rem_idemp_r : forall a b n : Z, n <> 0 -> Z.rem (a * Z.rem b n) n = Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma Z_mul_succ_quot_gt : forall a b : Z, 0 <= a -> 0 < b -> a < b * Z.succ (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_mul_succ_quot_lt : forall a b : Z, a <= 0 -> b < 0 -> b * Z.succ (a ÷ b) < a. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_add_mod : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> Z.rem (a + b) n = Z.rem (Z.rem a n + Z.rem b n) n. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_add_mod_idemp_l : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> Z.rem (Z.rem a n + b) n = Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_add_mod_idemp_r : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> Z.rem (a + Z.rem b n) n = Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_0_l : forall a : Z, 0 < a -> 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_1_l : forall a : Z, 1 < a -> 1 ÷ a = 0. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_1_r : forall a : Z, 0 <= a -> a ÷ 1 = a. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_add : forall a b c : Z, 0 <= a -> 0 <= a + b * c -> 0 < c -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_add_l : forall a b c : Z, 0 <= c -> 0 <= a * b + c -> 0 < b -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_div : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> a ÷ b ÷ c = a ÷ (b * c). Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_exact : forall a b : Z, 0 <= a -> 0 < b -> a = b * (a ÷ b) <-> Z.rem a b = 0. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_le_compat_l : forall p q r : Z, 0 <= p -> 0 < q <= r -> p ÷ r <= p ÷ q. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_le_lower_bound : forall a b q : Z, 0 <= a -> 0 < b -> b * q <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_le_mono : forall a b c : Z, 0 < c -> 0 <= a <= b -> a ÷ c <= b ÷ c. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_le_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a <= b * q -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_lt : forall a b : Z, 0 < a -> 1 < b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < b * q -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_mul_cancel_l : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> c * a ÷ (c * b) = a ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_mul_cancel_r : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> a * c ÷ (b * c) = a ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_mul : forall a b : Z, 0 <= a -> 0 < b -> a * b ÷ b = a. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_mul_le : forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_same : forall a : Z, 0 < a -> a ÷ a = 1. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_small : forall a b : Z, 0 <= a < b -> a ÷ b = 0. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_small_iff : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = 0 <-> a < b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_str_pos : forall a b : Z, 0 < b <= a -> 0 < a ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_str_pos_iff : forall a b : Z, 0 <= a -> 0 < b -> 0 < a ÷ b <-> b <= a. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_unique_exact : forall a b q : Z, 0 <= a -> 0 < b -> a = b * q -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_div_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_0_l : forall a : Z, 0 < a -> Z.rem 0 a = 0. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_1_l : forall a : Z, 1 < a -> Z.rem 1 a = 1. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_1_r : forall a : Z, 0 <= a -> Z.rem a 1 = 0. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_add : forall a b c : Z, 0 <= a -> 0 <= a + b * c -> 0 < c -> Z.rem (a + b * c) c = Z.rem a c. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_divides : forall a b : Z, 0 <= a -> 0 < b -> Z.rem a b = 0 <-> (exists c : Z, a = b * c). Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_le : forall a b : Z, 0 <= a -> 0 < b -> Z.rem a b <= a. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_mod : forall a n : Z, 0 <= a -> 0 < n -> Z.rem (Z.rem a n) n = Z.rem a n. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_mul : forall a b : Z, 0 <= a -> 0 < b -> Z.rem (a * b) b = 0. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_mul_r : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> Z.rem a (b * c) = Z.rem a b + b * Z.rem (a ÷ b) c. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_same : forall a : Z, 0 < a -> Z.rem a a = 0. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_small : forall a b : Z, 0 <= a < b -> Z.rem a b = a. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_small_iff : forall a b : Z, 0 <= a -> 0 < b -> Z.rem a b = a <-> a < b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mod_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> r = Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mul_div_le : forall a b : Z, 0 <= a -> 0 < b -> b * (a ÷ b) <= a. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mul_mod_distr_l : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> Z.rem (c * a) (c * b) = c * Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mul_mod_distr_r : forall a b c : Z, 0 <= a -> 0 < b -> 0 < c -> Z.rem (a * c) (b * c) = Z.rem a b * c. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mul_mod : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> Z.rem (a * b) n = Z.rem (Z.rem a n * Z.rem b n) n. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mul_mod_idemp_l : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> Z.rem (Z.rem a n * b) n = Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mul_mod_idemp_r : forall a b n : Z, 0 <= a -> 0 <= b -> 0 < n -> Z.rem (a * Z.rem b n) n = Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma Z_Private_Div_NZQuot_mul_succ_div_gt : forall a b : Z, 0 <= a -> 0 < b -> a < b * Z.succ (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_Private_Div_Quot2Div_div_mod : forall a b : Z, b <> 0 -> a = b * (a ÷ b) + Z.rem a b. Proof. intros; nia. Qed. +Z_Private_Div_Quot2Div_div_wd Morphisms.Proper (Morphisms.respectful eq (Morphisms.respectful eq eq)) Z.quot +Lemma Z_Private_Div_Quot2Div_mod_bound_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= Z.rem a b < b. Proof. intros; nia. Qed. +Z_Private_Div_Quot2Div_mod_wd Morphisms.Proper (Morphisms.respectful eq (Morphisms.respectful eq eq)) Z.rem +Lemma Z_quot_0_l : forall a : Z, a <> 0 -> 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma Z_quot_0_r_ext : forall x y : Z, y = 0 -> x ÷ y = 0. Proof. intros; nia. Qed. +Lemma Z_quot_1_l : forall a : Z, 1 < a -> 1 ÷ a = 0. Proof. intros; nia. Qed. +Lemma Z_quot_1_r : forall a : Z, a ÷ 1 = a. Proof. intros; nia. Qed. +Lemma Zquot2_quot : forall n : Z, Z.quot2 n = n ÷ 2. Proof. intros; nia. Qed. +Lemma Z_quot_abs : forall a b : Z, b <> 0 -> Z.abs a ÷ Z.abs b = Z.abs (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_quot_abs_l : forall a b : Z, b <> 0 -> Z.abs a ÷ b = Z.sgn a * (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_quot_abs_r : forall a b : Z, b <> 0 -> a ÷ Z.abs b = Z.sgn b * (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_quot_add : forall a b c : Z, c <> 0 -> 0 <= (a + b * c) * a -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma Z_quot_add_l : forall a b c : Z, b <> 0 -> 0 <= (a * b + c) * c -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_div : forall a b : Z, b <> 0 -> a ÷ b = Z.sgn a * Z.sgn b * (Z.abs a / Z.abs b). Proof. intros; nia. Qed. +Lemma Z_quot_div_nonneg : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = a / b. Proof. intros; nia. Qed. +Lemma Z_quot_exact : forall a b : Z, b <> 0 -> a = b * (a ÷ b) <-> Z.rem a b = 0. Proof. intros; nia. Qed. +Lemma Z_quot_le_compat_l : forall p q r : Z, 0 <= p -> 0 < q <= r -> p ÷ r <= p ÷ q. Proof. intros; nia. Qed. +Lemma Z_quot_le_lower_bound : forall a b q : Z, 0 < b -> b * q <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_le_mono : forall a b c : Z, 0 < c -> a <= b -> a ÷ c <= b ÷ c. Proof. intros; nia. Qed. +Lemma Z_quot_le_upper_bound : forall a b q : Z, 0 < b -> a <= b * q -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma Z_quot_lt : forall a b : Z, 0 < a -> 1 < b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma Z_quot_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < b * q -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma Z_quot_mul_cancel_l : forall a b c : Z, b <> 0 -> c <> 0 -> c * a ÷ (c * b) = a ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_mul_cancel_r : forall a b c : Z, b <> 0 -> c <> 0 -> a * c ÷ (b * c) = a ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_mul : forall a b : Z, b <> 0 -> a * b ÷ b = a. Proof. intros; nia. Qed. +Lemma Z_quot_mul_le : forall a b c : Z, 0 <= a -> 0 < b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_opp_l : forall a b : Z, b <> 0 -> - a ÷ b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_quot_opp_opp : forall a b : Z, b <> 0 -> - a ÷ - b = a ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_opp_r : forall a b : Z, b <> 0 -> a ÷ - b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_quot_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_quot : forall a b c : Z, b <> 0 -> c <> 0 -> a ÷ b ÷ c = a ÷ (b * c). Proof. intros; nia. Qed. +Lemma Z_quot_rem' : forall a b : Z, a = b * (a ÷ b) + Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_quot_rem : forall a b : Z, b <> 0 -> a = b * (a ÷ b) + Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_quot_same : forall a : Z, a <> 0 -> a ÷ a = 1. Proof. intros; nia. Qed. +Lemma Z_quot_small : forall a b : Z, 0 <= a < b -> a ÷ b = 0. Proof. intros; nia. Qed. +Lemma Z_quot_small_iff : forall a b : Z, b <> 0 -> a ÷ b = 0 <-> Z.abs a < Z.abs b. Proof. intros; nia. Qed. +Lemma Z_quot_str_pos : forall a b : Z, 0 < b <= a -> 0 < a ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_unique_exact : forall a b q : Z, b <> 0 -> a = b * q -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma Z_quot_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +Z_quot_wd Morphisms.Proper (Morphisms.respectful Z.eq (Morphisms.respectful Z.eq Z.eq)) Z.quot +Lemma Zquot_Zeven_rem : forall a : Z, Z.even a = (Z.rem a 2 =? 0). Proof. intros; nia. Qed. +Lemma Zquot_Z_mult_quot_ge : forall a b : Z, a <= 0 -> a <= b * (a ÷ b) <= 0. Proof. intros; nia. Qed. +Lemma Zquot_Z_mult_quot_le : forall a b : Z, 0 <= a -> 0 <= b * (a ÷ b) <= a. Proof. intros; nia. Qed. +Lemma Zquot_Zmult_rem_distr_l : forall a b c : Z, Z.rem (c * a) (c * b) = c * Z.rem a b. Proof. intros; nia. Qed. +Lemma Zquot_Zmult_rem_distr_r : forall a b c : Z, Z.rem (a * c) (b * c) = Z.rem a b * c. Proof. intros; nia. Qed. +Lemma Zquot_Zmult_rem : forall a b n : Z, Z.rem (a * b) n = Z.rem (Z.rem a n * Z.rem b n) n. Proof. intros; nia. Qed. +Lemma Zquot_Zmult_rem_idemp_l : forall a b n : Z, Z.rem (Z.rem a n * b) n = Z.rem (a * b) n. Proof. intros; nia. Qed. +Lemma Zquot_Zmult_rem_idemp_r : forall a b n : Z, Z.rem (b * Z.rem a n) n = Z.rem (b * a) n. Proof. intros; nia. Qed. +Lemma Zquot_Zodd_rem : forall a : Z, Z.odd a = negb (Z.rem a 2 =? 0). Proof. intros; nia. Qed. +Lemma Zquot_Zplus_rem : forall a b n : Z, 0 <= a * b -> Z.rem (a + b) n = Z.rem (Z.rem a n + Z.rem b n) n. Proof. intros; nia. Qed. +Lemma Zquot_Zplus_rem_idemp_l : forall a b n : Z, 0 <= a * b -> Z.rem (Z.rem a n + b) n = Z.rem (a + b) n. Proof. intros; nia. Qed. +Lemma Zquot_Zplus_rem_idemp_r : forall a b n : Z, 0 <= a * b -> Z.rem (b + Z.rem a n) n = Z.rem (b + a) n. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_0_l : forall a : Z, 0 ÷ a = 0. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_0_r : forall a : Z, a ÷ 0 = 0. Proof. intros; nia. Qed. +Lemma Zquot_Z_quot_exact_full : forall a b : Z, a = b * (a ÷ b) <-> Z.rem a b = 0. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_le_lower_bound : forall a b q : Z, 0 < b -> q * b <= a -> q <= a ÷ b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_le_upper_bound : forall a b q : Z, 0 < b -> a <= q * b -> a ÷ b <= q. Proof. intros; nia. Qed. +Lemma Zquot_Z_quot_lt : forall a b : Z, 0 < a -> 2 <= b -> a ÷ b < a. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_lt_upper_bound : forall a b q : Z, 0 <= a -> 0 < b -> a < q * b -> a ÷ b < q. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_mod_unique_full : forall a b q r : Z, Zquot.Remainder a b r -> a = b * q + r -> q = a ÷ b /\ r = Z.rem a b. Proof. intros; nia. Qed. +Lemma Zquot_Z_quot_monotone : forall a b c : Z, 0 <= c -> a <= b -> a ÷ c <= b ÷ c. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_mult_cancel_l : forall a b c : Z, c <> 0 -> c * a ÷ (c * b) = a ÷ b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_mult_cancel_r : forall a b c : Z, c <> 0 -> a * c ÷ (b * c) = a ÷ b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_mult_le : forall a b c : Z, 0 <= a -> 0 <= b -> 0 <= c -> c * (a ÷ b) <= c * a ÷ b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_opp_l : forall a b : Z, - a ÷ b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma Zquot_Zquot_opp_opp : forall a b : Z, - a ÷ - b = a ÷ b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_opp_r : forall a b : Z, a ÷ - b = - (a ÷ b). Proof. intros; nia. Qed. +Lemma Zquot_Z_quot_plus : forall a b c : Z, 0 <= (a + b * c) * a -> c <> 0 -> (a + b * c) ÷ c = a ÷ c + b. Proof. intros; nia. Qed. +Lemma Zquot_Z_quot_plus_l : forall a b c : Z, 0 <= (a * b + c) * c -> b <> 0 -> b <> 0 -> (a * b + c) ÷ b = a + c ÷ b. Proof. intros; nia. Qed. +Lemma Zquot_Z_quot_pos : forall a b : Z, 0 <= a -> 0 <= b -> 0 <= a ÷ b. Proof. intros; nia. Qed. +Lemma Zquot_Zquotrem_Zdiv_eucl_pos : forall a b : Z, 0 <= a -> 0 < b -> a ÷ b = a / b /\ Z.rem a b = a mod b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_sgn : forall a b : Z, 0 <= Z.sgn (a ÷ b) * Z.sgn a * Z.sgn b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_unique_full : forall a b q r : Z, Zquot.Remainder a b r -> a = b * q + r -> q = a ÷ b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_Zdiv_pos : forall a b : Z, 0 <= a -> 0 <= b -> a ÷ b = a / b. Proof. intros; nia. Qed. +Lemma Zquot_Zquot_Zquot : forall a b c : Z, a ÷ b ÷ c = a ÷ (b * c). Proof. intros; nia. Qed. +Lemma Zquot_Zrem_0_l : forall a : Z, Z.rem 0 a = 0. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_0_r : forall a : Z, Z.rem a 0 = a. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_divides : forall a b : Z, Z.rem a b = 0 <-> (exists c : Z, a = b * c). Proof. intros; nia. Qed. +Lemma Zquot_Zrem_even : forall a : Z, Z.rem a 2 = (if Z.even a then 0 else Z.sgn a). Proof. intros; nia. Qed. +Lemma Zquot_Zrem_le : forall a b : Z, 0 <= a -> 0 <= b -> Z.rem a b <= a. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_lt_neg : forall a b : Z, a <= 0 -> b <> 0 -> - Z.abs b < Z.rem a b <= 0. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_lt_neg_neg : forall a b : Z, a <= 0 -> b < 0 -> b < Z.rem a b <= 0. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_lt_neg_pos : forall a b : Z, a <= 0 -> 0 < b -> - b < Z.rem a b <= 0. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_lt_pos : forall a b : Z, 0 <= a -> b <> 0 -> 0 <= Z.rem a b < Z.abs b. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_lt_pos_neg : forall a b : Z, 0 <= a -> b < 0 -> 0 <= Z.rem a b < - b. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_lt_pos_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= Z.rem a b < b. Proof. intros; nia. Qed. +Lemma Zquot_Z_rem_mult : forall a b : Z, Z.rem (a * b) b = 0. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_odd : forall a : Z, Z.rem a 2 = (if Z.odd a then Z.sgn a else 0). Proof. intros; nia. Qed. +Lemma Zquot_Zrem_opp_l : forall a b : Z, Z.rem (- a) b = - Z.rem a b. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_opp_opp : forall a b : Z, Z.rem (- a) (- b) = - Z.rem a b. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_opp_r : forall a b : Z, Z.rem a (- b) = Z.rem a b. Proof. intros; nia. Qed. +Lemma Zquot_Z_rem_plus : forall a b c : Z, 0 <= (a + b * c) * a -> Z.rem (a + b * c) c = Z.rem a c. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_rem : forall a n : Z, Z.rem (Z.rem a n) n = Z.rem a n. Proof. intros; nia. Qed. +Lemma Zquot_Z_rem_same : forall a : Z, Z.rem a a = 0. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_sgn2 : forall a b : Z, 0 <= Z.rem a b * a. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_sgn : forall a b : Z, 0 <= Z.sgn (Z.rem a b) * Z.sgn a. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_unique_full : forall a b q r : Z, Zquot.Remainder a b r -> a = b * q + r -> r = Z.rem a b. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_Zmod_pos : forall a b : Z, 0 <= a -> 0 < b -> Z.rem a b = a mod b. Proof. intros; nia. Qed. +Lemma Zquot_Zrem_Zmod_zero : forall a b : Z, b <> 0 -> Z.rem a b = 0 <-> a mod b = 0. Proof. intros; nia. Qed. +Lemma Z_rem_0_l : forall a : Z, a <> 0 -> Z.rem 0 a = 0. Proof. intros; nia. Qed. +Lemma Z_rem_0_r_ext : forall x y : Z, y = 0 -> Z.rem x y = x. Proof. intros; nia. Qed. +Lemma Z_rem_1_l : forall a : Z, 1 < a -> Z.rem 1 a = 1. Proof. intros; nia. Qed. +Lemma Z_rem_1_r : forall a : Z, Z.rem a 1 = 0. Proof. intros; nia. Qed. +Lemma Z_rem_abs : forall a b : Z, b <> 0 -> Z.rem (Z.abs a) (Z.abs b) = Z.abs (Z.rem a b). Proof. intros; nia. Qed. +Lemma Z_rem_abs_l : forall a b : Z, b <> 0 -> Z.rem (Z.abs a) b = Z.abs (Z.rem a b). Proof. intros; nia. Qed. +Lemma Z_rem_abs_r : forall a b : Z, b <> 0 -> Z.rem a (Z.abs b) = Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_rem_add : forall a b c : Z, c <> 0 -> 0 <= (a + b * c) * a -> Z.rem (a + b * c) c = Z.rem a c. Proof. intros; nia. Qed. +Lemma Z_rem_bound_abs : forall a b : Z, b <> 0 -> Z.abs (Z.rem a b) < Z.abs b. Proof. intros; nia. Qed. +Lemma Z_rem_bound_neg_neg : forall x y : Z, y < 0 -> x <= 0 -> y < Z.rem x y <= 0. Proof. intros; nia. Qed. +Lemma Z_rem_bound_neg_pos : forall x y : Z, y < 0 -> 0 <= x -> 0 <= Z.rem x y < - y. Proof. intros; nia. Qed. +Lemma Z_rem_bound_pos : forall a b : Z, 0 <= a -> 0 < b -> 0 <= Z.rem a b < b. Proof. intros; nia. Qed. +Lemma Z_rem_bound_pos_neg : forall x y : Z, 0 < y -> x <= 0 -> - y < Z.rem x y <= 0. Proof. intros; nia. Qed. +Lemma Z_rem_bound_pos_pos : forall x y : Z, 0 < y -> 0 <= x -> 0 <= Z.rem x y < y. Proof. intros; nia. Qed. +Lemma Z_rem_eq : forall a b : Z, b <> 0 -> Z.rem a b = a - b * (a ÷ b). Proof. intros; nia. Qed. +Lemma Z_rem_le : forall a b : Z, 0 <= a -> 0 < b -> Z.rem a b <= a. Proof. intros; nia. Qed. +Lemma Z_rem_mod_eq_0 : forall a b : Z, b <> 0 -> Z.rem a b = 0 <-> a mod b = 0. Proof. intros; nia. Qed. +Lemma Z_rem_mod : forall a b : Z, b <> 0 -> Z.rem a b = Z.sgn a * (Z.abs a mod Z.abs b). Proof. intros; nia. Qed. +Lemma Z_rem_mod_nonneg : forall a b : Z, 0 <= a -> 0 < b -> Z.rem a b = a mod b. Proof. intros; nia. Qed. +Lemma Z_rem_mul : forall a b : Z, b <> 0 -> Z.rem (a * b) b = 0. Proof. intros; nia. Qed. +Lemma Z_rem_nonneg : forall a b : Z, b <> 0 -> 0 <= a -> 0 <= Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_rem_nonpos : forall a b : Z, b <> 0 -> a <= 0 -> Z.rem a b <= 0. Proof. intros; nia. Qed. +Lemma Z_rem_opp_l : forall a b : Z, b <> 0 -> Z.rem (- a) b = - Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_rem_opp_l' : forall a b : Z, Z.rem (- a) b = - Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_rem_opp_opp : forall a b : Z, b <> 0 -> Z.rem (- a) (- b) = - Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_rem_opp_r : forall a b : Z, b <> 0 -> Z.rem a (- b) = Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_rem_opp_r' : forall a b : Z, Z.rem a (- b) = Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_rem_quot : forall a b : Z, b <> 0 -> Z.rem a b ÷ b = 0. Proof. intros; nia. Qed. +Lemma Z_rem_rem : forall a n : Z, n <> 0 -> Z.rem (Z.rem a n) n = Z.rem a n. Proof. intros; nia. Qed. +Lemma Z_rem_same : forall a : Z, a <> 0 -> Z.rem a a = 0. Proof. intros; nia. Qed. +Lemma Z_rem_sign : forall a b : Z, a <> 0 -> b <> 0 -> Z.sgn (Z.rem a b) <> - Z.sgn a. Proof. intros; nia. Qed. +Lemma Z_rem_sign_mul : forall a b : Z, b <> 0 -> 0 <= Z.rem a b * a. Proof. intros; nia. Qed. +Lemma Z_rem_sign_nz : forall a b : Z, b <> 0 -> Z.rem a b <> 0 -> Z.sgn (Z.rem a b) = Z.sgn a. Proof. intros; nia. Qed. +Lemma Z_rem_small : forall a b : Z, 0 <= a < b -> Z.rem a b = a. Proof. intros; nia. Qed. +Lemma Z_rem_small_iff : forall a b : Z, b <> 0 -> Z.rem a b = a <-> Z.abs a < Z.abs b. Proof. intros; nia. Qed. +Lemma Z_rem_unique : forall a b q r : Z, 0 <= a -> 0 <= r < b -> a = b * q + r -> r = Z.rem a b. Proof. intros; nia. Qed. +Lemma Z_rem_wd : Morphisms.Proper (Morphisms.respectful Z.eq (Morphisms.respectful Z.eq Z.eq)) Z.rem. Proof. intros; nia. Qed. +*) |