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+(************************************************************************)
+(* v * The Coq Proof Assistant / The Coq Development Team *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(* \VV/ **************************************************************)
+(* // * This file is distributed under the terms of the *)
+(* * GNU Lesser General Public License Version 2.1 *)
+(************************************************************************)
+(* Evgeny Makarov, INRIA, 2007 *)
+(************************************************************************)
+
+(*i $Id: ZLt.v 11040 2008-06-03 00:04:16Z letouzey $ i*)
+
+Require Export ZMul.
+
+Module ZOrderPropFunct (Import ZAxiomsMod : ZAxiomsSig).
+Module Export ZMulPropMod := ZMulPropFunct ZAxiomsMod.
+Open Local Scope IntScope.
+
+(* Axioms *)
+
+Theorem Zlt_wd :
+ forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> (n1 < m1 <-> n2 < m2).
+Proof NZlt_wd.
+
+Theorem Zle_wd :
+ forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> (n1 <= m1 <-> n2 <= m2).
+Proof NZle_wd.
+
+Theorem Zmin_wd :
+ forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> Zmin n1 m1 == Zmin n2 m2.
+Proof NZmin_wd.
+
+Theorem Zmax_wd :
+ forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> Zmax n1 m1 == Zmax n2 m2.
+Proof NZmax_wd.
+
+Theorem Zlt_eq_cases : forall n m : Z, n <= m <-> n < m \/ n == m.
+Proof NZlt_eq_cases.
+
+Theorem Zlt_irrefl : forall n : Z, ~ n < n.
+Proof NZlt_irrefl.
+
+Theorem Zlt_succ_r : forall n m : Z, n < S m <-> n <= m.
+Proof NZlt_succ_r.
+
+Theorem Zmin_l : forall n m : Z, n <= m -> Zmin n m == n.
+Proof NZmin_l.
+
+Theorem Zmin_r : forall n m : Z, m <= n -> Zmin n m == m.
+Proof NZmin_r.
+
+Theorem Zmax_l : forall n m : Z, m <= n -> Zmax n m == n.
+Proof NZmax_l.
+
+Theorem Zmax_r : forall n m : Z, n <= m -> Zmax n m == m.
+Proof NZmax_r.
+
+(* Renaming theorems from NZOrder.v *)
+
+Theorem Zlt_le_incl : forall n m : Z, n < m -> n <= m.
+Proof NZlt_le_incl.
+
+Theorem Zlt_neq : forall n m : Z, n < m -> n ~= m.
+Proof NZlt_neq.
+
+Theorem Zle_neq : forall n m : Z, n < m <-> n <= m /\ n ~= m.
+Proof NZle_neq.
+
+Theorem Zle_refl : forall n : Z, n <= n.
+Proof NZle_refl.
+
+Theorem Zlt_succ_diag_r : forall n : Z, n < S n.
+Proof NZlt_succ_diag_r.
+
+Theorem Zle_succ_diag_r : forall n : Z, n <= S n.
+Proof NZle_succ_diag_r.
+
+Theorem Zlt_0_1 : 0 < 1.
+Proof NZlt_0_1.
+
+Theorem Zle_0_1 : 0 <= 1.
+Proof NZle_0_1.
+
+Theorem Zlt_lt_succ_r : forall n m : Z, n < m -> n < S m.
+Proof NZlt_lt_succ_r.
+
+Theorem Zle_le_succ_r : forall n m : Z, n <= m -> n <= S m.
+Proof NZle_le_succ_r.
+
+Theorem Zle_succ_r : forall n m : Z, n <= S m <-> n <= m \/ n == S m.
+Proof NZle_succ_r.
+
+Theorem Zneq_succ_diag_l : forall n : Z, S n ~= n.
+Proof NZneq_succ_diag_l.
+
+Theorem Zneq_succ_diag_r : forall n : Z, n ~= S n.
+Proof NZneq_succ_diag_r.
+
+Theorem Znlt_succ_diag_l : forall n : Z, ~ S n < n.
+Proof NZnlt_succ_diag_l.
+
+Theorem Znle_succ_diag_l : forall n : Z, ~ S n <= n.
+Proof NZnle_succ_diag_l.
+
+Theorem Zle_succ_l : forall n m : Z, S n <= m <-> n < m.
+Proof NZle_succ_l.
+
+Theorem Zlt_succ_l : forall n m : Z, S n < m -> n < m.
+Proof NZlt_succ_l.
+
+Theorem Zsucc_lt_mono : forall n m : Z, n < m <-> S n < S m.
+Proof NZsucc_lt_mono.
+
+Theorem Zsucc_le_mono : forall n m : Z, n <= m <-> S n <= S m.
+Proof NZsucc_le_mono.
+
+Theorem Zlt_asymm : forall n m, n < m -> ~ m < n.
+Proof NZlt_asymm.
+
+Notation Zlt_ngt := Zlt_asymm (only parsing).
+
+Theorem Zlt_trans : forall n m p : Z, n < m -> m < p -> n < p.
+Proof NZlt_trans.
+
+Theorem Zle_trans : forall n m p : Z, n <= m -> m <= p -> n <= p.
+Proof NZle_trans.
+
+Theorem Zle_lt_trans : forall n m p : Z, n <= m -> m < p -> n < p.
+Proof NZle_lt_trans.
+
+Theorem Zlt_le_trans : forall n m p : Z, n < m -> m <= p -> n < p.
+Proof NZlt_le_trans.
+
+Theorem Zle_antisymm : forall n m : Z, n <= m -> m <= n -> n == m.
+Proof NZle_antisymm.
+
+Theorem Zlt_1_l : forall n m : Z, 0 < n -> n < m -> 1 < m.
+Proof NZlt_1_l.
+
+(** Trichotomy, decidability, and double negation elimination *)
+
+Theorem Zlt_trichotomy : forall n m : Z, n < m \/ n == m \/ m < n.
+Proof NZlt_trichotomy.
+
+Notation Zlt_eq_gt_cases := Zlt_trichotomy (only parsing).
+
+Theorem Zlt_gt_cases : forall n m : Z, n ~= m <-> n < m \/ n > m.
+Proof NZlt_gt_cases.
+
+Theorem Zle_gt_cases : forall n m : Z, n <= m \/ n > m.
+Proof NZle_gt_cases.
+
+Theorem Zlt_ge_cases : forall n m : Z, n < m \/ n >= m.
+Proof NZlt_ge_cases.
+
+Theorem Zle_ge_cases : forall n m : Z, n <= m \/ n >= m.
+Proof NZle_ge_cases.
+
+(** Instances of the previous theorems for m == 0 *)
+
+Theorem Zneg_pos_cases : forall n : Z, n ~= 0 <-> n < 0 \/ n > 0.
+Proof.
+intro; apply Zlt_gt_cases.
+Qed.
+
+Theorem Znonpos_pos_cases : forall n : Z, n <= 0 \/ n > 0.
+Proof.
+intro; apply Zle_gt_cases.
+Qed.
+
+Theorem Zneg_nonneg_cases : forall n : Z, n < 0 \/ n >= 0.
+Proof.
+intro; apply Zlt_ge_cases.
+Qed.
+
+Theorem Znonpos_nonneg_cases : forall n : Z, n <= 0 \/ n >= 0.
+Proof.
+intro; apply Zle_ge_cases.
+Qed.
+
+Theorem Zle_ngt : forall n m : Z, n <= m <-> ~ n > m.
+Proof NZle_ngt.
+
+Theorem Znlt_ge : forall n m : Z, ~ n < m <-> n >= m.
+Proof NZnlt_ge.
+
+Theorem Zlt_dec : forall n m : Z, decidable (n < m).
+Proof NZlt_dec.
+
+Theorem Zlt_dne : forall n m, ~ ~ n < m <-> n < m.
+Proof NZlt_dne.
+
+Theorem Znle_gt : forall n m : Z, ~ n <= m <-> n > m.
+Proof NZnle_gt.
+
+Theorem Zlt_nge : forall n m : Z, n < m <-> ~ n >= m.
+Proof NZlt_nge.
+
+Theorem Zle_dec : forall n m : Z, decidable (n <= m).
+Proof NZle_dec.
+
+Theorem Zle_dne : forall n m : Z, ~ ~ n <= m <-> n <= m.
+Proof NZle_dne.
+
+Theorem Znlt_succ_r : forall n m : Z, ~ m < S n <-> n < m.
+Proof NZnlt_succ_r.
+
+Theorem Zlt_exists_pred :
+ forall z n : Z, z < n -> exists k : Z, n == S k /\ z <= k.
+Proof NZlt_exists_pred.
+
+Theorem Zlt_succ_iter_r :
+ forall (n : nat) (m : Z), m < NZsucc_iter (Datatypes.S n) m.
+Proof NZlt_succ_iter_r.
+
+Theorem Zneq_succ_iter_l :
+ forall (n : nat) (m : Z), NZsucc_iter (Datatypes.S n) m ~= m.
+Proof NZneq_succ_iter_l.
+
+(** Stronger variant of induction with assumptions n >= 0 (n < 0)
+in the induction step *)
+
+Theorem Zright_induction :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z, A z ->
+ (forall n : Z, z <= n -> A n -> A (S n)) ->
+ forall n : Z, z <= n -> A n.
+Proof NZright_induction.
+
+Theorem Zleft_induction :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z, A z ->
+ (forall n : Z, n < z -> A (S n) -> A n) ->
+ forall n : Z, n <= z -> A n.
+Proof NZleft_induction.
+
+Theorem Zright_induction' :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z,
+ (forall n : Z, n <= z -> A n) ->
+ (forall n : Z, z <= n -> A n -> A (S n)) ->
+ forall n : Z, A n.
+Proof NZright_induction'.
+
+Theorem Zleft_induction' :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z,
+ (forall n : Z, z <= n -> A n) ->
+ (forall n : Z, n < z -> A (S n) -> A n) ->
+ forall n : Z, A n.
+Proof NZleft_induction'.
+
+Theorem Zstrong_right_induction :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z,
+ (forall n : Z, z <= n -> (forall m : Z, z <= m -> m < n -> A m) -> A n) ->
+ forall n : Z, z <= n -> A n.
+Proof NZstrong_right_induction.
+
+Theorem Zstrong_left_induction :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z,
+ (forall n : Z, n <= z -> (forall m : Z, m <= z -> S n <= m -> A m) -> A n) ->
+ forall n : Z, n <= z -> A n.
+Proof NZstrong_left_induction.
+
+Theorem Zstrong_right_induction' :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z,
+ (forall n : Z, n <= z -> A n) ->
+ (forall n : Z, z <= n -> (forall m : Z, z <= m -> m < n -> A m) -> A n) ->
+ forall n : Z, A n.
+Proof NZstrong_right_induction'.
+
+Theorem Zstrong_left_induction' :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z,
+ (forall n : Z, z <= n -> A n) ->
+ (forall n : Z, n <= z -> (forall m : Z, m <= z -> S n <= m -> A m) -> A n) ->
+ forall n : Z, A n.
+Proof NZstrong_left_induction'.
+
+Theorem Zorder_induction :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z, A z ->
+ (forall n : Z, z <= n -> A n -> A (S n)) ->
+ (forall n : Z, n < z -> A (S n) -> A n) ->
+ forall n : Z, A n.
+Proof NZorder_induction.
+
+Theorem Zorder_induction' :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall z : Z, A z ->
+ (forall n : Z, z <= n -> A n -> A (S n)) ->
+ (forall n : Z, n <= z -> A n -> A (P n)) ->
+ forall n : Z, A n.
+Proof NZorder_induction'.
+
+Theorem Zorder_induction_0 :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ A 0 ->
+ (forall n : Z, 0 <= n -> A n -> A (S n)) ->
+ (forall n : Z, n < 0 -> A (S n) -> A n) ->
+ forall n : Z, A n.
+Proof NZorder_induction_0.
+
+Theorem Zorder_induction'_0 :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ A 0 ->
+ (forall n : Z, 0 <= n -> A n -> A (S n)) ->
+ (forall n : Z, n <= 0 -> A n -> A (P n)) ->
+ forall n : Z, A n.
+Proof NZorder_induction'_0.
+
+Ltac Zinduct n := induction_maker n ltac:(apply Zorder_induction_0).
+
+(** Elimintation principle for < *)
+
+Theorem Zlt_ind :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall n : Z, A (S n) ->
+ (forall m : Z, n < m -> A m -> A (S m)) -> forall m : Z, n < m -> A m.
+Proof NZlt_ind.
+
+(** Elimintation principle for <= *)
+
+Theorem Zle_ind :
+ forall A : Z -> Prop, predicate_wd Zeq A ->
+ forall n : Z, A n ->
+ (forall m : Z, n <= m -> A m -> A (S m)) -> forall m : Z, n <= m -> A m.
+Proof NZle_ind.
+
+(** Well-founded relations *)
+
+Theorem Zlt_wf : forall z : Z, well_founded (fun n m : Z => z <= n /\ n < m).
+Proof NZlt_wf.
+
+Theorem Zgt_wf : forall z : Z, well_founded (fun n m : Z => m < n /\ n <= z).
+Proof NZgt_wf.
+
+(* Theorems that are either not valid on N or have different proofs on N and Z *)
+
+Theorem Zlt_pred_l : forall n : Z, P n < n.
+Proof.
+intro n; rewrite <- (Zsucc_pred n) at 2; apply Zlt_succ_diag_r.
+Qed.
+
+Theorem Zle_pred_l : forall n : Z, P n <= n.
+Proof.
+intro; apply Zlt_le_incl; apply Zlt_pred_l.
+Qed.
+
+Theorem Zlt_le_pred : forall n m : Z, n < m <-> n <= P m.
+Proof.
+intros n m; rewrite <- (Zsucc_pred m); rewrite Zpred_succ. apply Zlt_succ_r.
+Qed.
+
+Theorem Znle_pred_r : forall n : Z, ~ n <= P n.
+Proof.
+intro; rewrite <- Zlt_le_pred; apply Zlt_irrefl.
+Qed.
+
+Theorem Zlt_pred_le : forall n m : Z, P n < m <-> n <= m.
+Proof.
+intros n m; rewrite <- (Zsucc_pred n) at 2.
+symmetry; apply Zle_succ_l.
+Qed.
+
+Theorem Zlt_lt_pred : forall n m : Z, n < m -> P n < m.
+Proof.
+intros; apply <- Zlt_pred_le; now apply Zlt_le_incl.
+Qed.
+
+Theorem Zle_le_pred : forall n m : Z, n <= m -> P n <= m.
+Proof.
+intros; apply Zlt_le_incl; now apply <- Zlt_pred_le.
+Qed.
+
+Theorem Zlt_pred_lt : forall n m : Z, n < P m -> n < m.
+Proof.
+intros n m H; apply Zlt_trans with (P m); [assumption | apply Zlt_pred_l].
+Qed.
+
+Theorem Zle_pred_lt : forall n m : Z, n <= P m -> n <= m.
+Proof.
+intros; apply Zlt_le_incl; now apply <- Zlt_le_pred.
+Qed.
+
+Theorem Zpred_lt_mono : forall n m : Z, n < m <-> P n < P m.
+Proof.
+intros; rewrite Zlt_le_pred; symmetry; apply Zlt_pred_le.
+Qed.
+
+Theorem Zpred_le_mono : forall n m : Z, n <= m <-> P n <= P m.
+Proof.
+intros; rewrite <- Zlt_pred_le; now rewrite Zlt_le_pred.
+Qed.
+
+Theorem Zlt_succ_lt_pred : forall n m : Z, S n < m <-> n < P m.
+Proof.
+intros n m; now rewrite (Zpred_lt_mono (S n) m), Zpred_succ.
+Qed.
+
+Theorem Zle_succ_le_pred : forall n m : Z, S n <= m <-> n <= P m.
+Proof.
+intros n m; now rewrite (Zpred_le_mono (S n) m), Zpred_succ.
+Qed.
+
+Theorem Zlt_pred_lt_succ : forall n m : Z, P n < m <-> n < S m.
+Proof.
+intros; rewrite Zlt_pred_le; symmetry; apply Zlt_succ_r.
+Qed.
+
+Theorem Zle_pred_lt_succ : forall n m : Z, P n <= m <-> n <= S m.
+Proof.
+intros n m; now rewrite (Zpred_le_mono n (S m)), Zpred_succ.
+Qed.
+
+Theorem Zneq_pred_l : forall n : Z, P n ~= n.
+Proof.
+intro; apply Zlt_neq; apply Zlt_pred_l.
+Qed.
+
+Theorem Zlt_n1_r : forall n m : Z, n < m -> m < 0 -> n < -1.
+Proof.
+intros n m H1 H2. apply -> Zlt_le_pred in H2.
+setoid_replace (P 0) with (-1) in H2. now apply NZlt_le_trans with m.
+apply <- Zeq_opp_r. now rewrite Zopp_pred, Zopp_0.
+Qed.
+
+End ZOrderPropFunct.
+
generated by cgit v1.2.3 (git 2.39.1) at 2025-01-29 04:41:56 +0000