diff options
Diffstat (limited to 'theories/Numbers/Natural/Abstract/NOrder.v')
| -rw-r--r-- | theories/Numbers/Natural/Abstract/NOrder.v | 381 |
1 files changed, 47 insertions, 334 deletions
diff --git a/theories/Numbers/Natural/Abstract/NOrder.v b/theories/Numbers/Natural/Abstract/NOrder.v index a5b496ba3..94c68a186 100644 --- a/theories/Numbers/Natural/Abstract/NOrder.v +++ b/theories/Numbers/Natural/Abstract/NOrder.v @@ -10,352 +10,61 @@ (*i $Id$ i*) -Require Export NMul. +Require Export NAdd. -Module NOrderPropFunct (Import NAxiomsMod : NAxiomsSig). -Module Export NMulPropMod := NMulPropFunct NAxiomsMod. -Open Local Scope NatScope. +Module NOrderPropFunct (Import N : NAxiomsSig). +Include NAddPropFunct N. +Local Open Scope NumScope. -(* The tactics le_less, le_equal and le_elim are inherited from NZOrder.v *) - -(* Axioms *) - -Theorem lt_wd : - forall n1 n2 : N, n1 == n2 -> forall m1 m2 : N, m1 == m2 -> (n1 < m1 <-> n2 < m2). -Proof NZlt_wd. - -Theorem le_wd : - forall n1 n2 : N, n1 == n2 -> forall m1 m2 : N, m1 == m2 -> (n1 <= m1 <-> n2 <= m2). -Proof NZle_wd. - -Theorem min_wd : - forall n1 n2 : N, n1 == n2 -> forall m1 m2 : N, m1 == m2 -> min n1 m1 == min n2 m2. -Proof NZmin_wd. - -Theorem max_wd : - forall n1 n2 : N, n1 == n2 -> forall m1 m2 : N, m1 == m2 -> max n1 m1 == max n2 m2. -Proof NZmax_wd. - -Theorem lt_eq_cases : forall n m : N, n <= m <-> n < m \/ n == m. -Proof NZlt_eq_cases. - -Theorem lt_irrefl : forall n : N, ~ n < n. -Proof NZlt_irrefl. - -Theorem lt_succ_r : forall n m : N, n < S m <-> n <= m. -Proof NZlt_succ_r. - -Theorem min_l : forall n m : N, n <= m -> min n m == n. -Proof NZmin_l. - -Theorem min_r : forall n m : N, m <= n -> min n m == m. -Proof NZmin_r. - -Theorem max_l : forall n m : N, m <= n -> max n m == n. -Proof NZmax_l. - -Theorem max_r : forall n m : N, n <= m -> max n m == m. -Proof NZmax_r. - -(* Renaming theorems from NZOrder.v *) - -Theorem lt_le_incl : forall n m : N, n < m -> n <= m. -Proof NZlt_le_incl. - -Theorem eq_le_incl : forall n m : N, n == m -> n <= m. -Proof NZeq_le_incl. - -Theorem lt_neq : forall n m : N, n < m -> n ~= m. -Proof NZlt_neq. - -Theorem le_neq : forall n m : N, n < m <-> n <= m /\ n ~= m. -Proof NZle_neq. - -Theorem le_refl : forall n : N, n <= n. -Proof NZle_refl. - -Theorem lt_succ_diag_r : forall n : N, n < S n. -Proof NZlt_succ_diag_r. - -Theorem le_succ_diag_r : forall n : N, n <= S n. -Proof NZle_succ_diag_r. - -Theorem lt_0_1 : 0 < 1. -Proof NZlt_0_1. - -Theorem le_0_1 : 0 <= 1. -Proof NZle_0_1. - -Theorem lt_lt_succ_r : forall n m : N, n < m -> n < S m. -Proof NZlt_lt_succ_r. - -Theorem le_le_succ_r : forall n m : N, n <= m -> n <= S m. -Proof NZle_le_succ_r. - -Theorem le_succ_r : forall n m : N, n <= S m <-> n <= m \/ n == S m. -Proof NZle_succ_r. - -Theorem neq_succ_diag_l : forall n : N, S n ~= n. -Proof NZneq_succ_diag_l. - -Theorem neq_succ_diag_r : forall n : N, n ~= S n. -Proof NZneq_succ_diag_r. - -Theorem nlt_succ_diag_l : forall n : N, ~ S n < n. -Proof NZnlt_succ_diag_l. - -Theorem nle_succ_diag_l : forall n : N, ~ S n <= n. -Proof NZnle_succ_diag_l. - -Theorem le_succ_l : forall n m : N, S n <= m <-> n < m. -Proof NZle_succ_l. - -Theorem lt_succ_l : forall n m : N, S n < m -> n < m. -Proof NZlt_succ_l. - -Theorem succ_lt_mono : forall n m : N, n < m <-> S n < S m. -Proof NZsucc_lt_mono. - -Theorem succ_le_mono : forall n m : N, n <= m <-> S n <= S m. -Proof NZsucc_le_mono. - -Theorem lt_asymm : forall n m : N, n < m -> ~ m < n. -Proof NZlt_asymm. - -Notation lt_ngt := lt_asymm (only parsing). - -Theorem lt_trans : forall n m p : N, n < m -> m < p -> n < p. -Proof NZlt_trans. - -Theorem le_trans : forall n m p : N, n <= m -> m <= p -> n <= p. -Proof NZle_trans. - -Theorem le_lt_trans : forall n m p : N, n <= m -> m < p -> n < p. -Proof NZle_lt_trans. - -Theorem lt_le_trans : forall n m p : N, n < m -> m <= p -> n < p. -Proof NZlt_le_trans. - -Theorem le_antisymm : forall n m : N, n <= m -> m <= n -> n == m. -Proof NZle_antisymm. - -(** Trichotomy, decidability, and double negation elimination *) - -Theorem lt_trichotomy : forall n m : N, n < m \/ n == m \/ m < n. -Proof NZlt_trichotomy. - -Notation lt_eq_gt_cases := lt_trichotomy (only parsing). - -Theorem lt_gt_cases : forall n m : N, n ~= m <-> n < m \/ n > m. -Proof NZlt_gt_cases. - -Theorem le_gt_cases : forall n m : N, n <= m \/ n > m. -Proof NZle_gt_cases. - -Theorem lt_ge_cases : forall n m : N, n < m \/ n >= m. -Proof NZlt_ge_cases. - -Theorem le_ge_cases : forall n m : N, n <= m \/ n >= m. -Proof NZle_ge_cases. - -Theorem le_ngt : forall n m : N, n <= m <-> ~ n > m. -Proof NZle_ngt. - -Theorem nlt_ge : forall n m : N, ~ n < m <-> n >= m. -Proof NZnlt_ge. - -Theorem lt_dec : forall n m : N, decidable (n < m). -Proof NZlt_dec. - -Theorem lt_dne : forall n m : N, ~ ~ n < m <-> n < m. -Proof NZlt_dne. - -Theorem nle_gt : forall n m : N, ~ n <= m <-> n > m. -Proof NZnle_gt. - -Theorem lt_nge : forall n m : N, n < m <-> ~ n >= m. -Proof NZlt_nge. - -Theorem le_dec : forall n m : N, decidable (n <= m). -Proof NZle_dec. - -Theorem le_dne : forall n m : N, ~ ~ n <= m <-> n <= m. -Proof NZle_dne. - -Theorem nlt_succ_r : forall n m : N, ~ m < S n <-> n < m. -Proof NZnlt_succ_r. - -Theorem lt_exists_pred : - forall z n : N, z < n -> exists k : N, n == S k /\ z <= k. -Proof NZlt_exists_pred. - -Theorem lt_succ_iter_r : - forall (n : nat) (m : N), m < NZsucc_iter (Datatypes.S n) m. -Proof NZlt_succ_iter_r. - -Theorem neq_succ_iter_l : - forall (n : nat) (m : N), 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 right_induction : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, A z -> - (forall n : N, z <= n -> A n -> A (S n)) -> - forall n : N, z <= n -> A n. -Proof NZright_induction. - -Theorem left_induction : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, A z -> - (forall n : N, n < z -> A (S n) -> A n) -> - forall n : N, n <= z -> A n. -Proof NZleft_induction. - -Theorem right_induction' : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, - (forall n : N, n <= z -> A n) -> - (forall n : N, z <= n -> A n -> A (S n)) -> - forall n : N, A n. -Proof NZright_induction'. - -Theorem left_induction' : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, - (forall n : N, z <= n -> A n) -> - (forall n : N, n < z -> A (S n) -> A n) -> - forall n : N, A n. -Proof NZleft_induction'. - -Theorem strong_right_induction : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, - (forall n : N, z <= n -> (forall m : N, z <= m -> m < n -> A m) -> A n) -> - forall n : N, z <= n -> A n. -Proof NZstrong_right_induction. - -Theorem strong_left_induction : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, - (forall n : N, n <= z -> (forall m : N, m <= z -> S n <= m -> A m) -> A n) -> - forall n : N, n <= z -> A n. -Proof NZstrong_left_induction. - -Theorem strong_right_induction' : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, - (forall n : N, n <= z -> A n) -> - (forall n : N, z <= n -> (forall m : N, z <= m -> m < n -> A m) -> A n) -> - forall n : N, A n. -Proof NZstrong_right_induction'. - -Theorem strong_left_induction' : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, - (forall n : N, z <= n -> A n) -> - (forall n : N, n <= z -> (forall m : N, m <= z -> S n <= m -> A m) -> A n) -> - forall n : N, A n. -Proof NZstrong_left_induction'. - -Theorem order_induction : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, A z -> - (forall n : N, z <= n -> A n -> A (S n)) -> - (forall n : N, n < z -> A (S n) -> A n) -> - forall n : N, A n. -Proof NZorder_induction. - -Theorem order_induction' : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall z : N, A z -> - (forall n : N, z <= n -> A n -> A (S n)) -> - (forall n : N, n <= z -> A n -> A (P n)) -> - forall n : N, A n. -Proof NZorder_induction'. - -(* We don't need order_induction_0 and order_induction'_0 (see NZOrder and -ZOrder) since they boil down to regular induction *) - -(** Elimintation principle for < *) - -Theorem lt_ind : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall n : N, - A (S n) -> - (forall m : N, n < m -> A m -> A (S m)) -> - forall m : N, n < m -> A m. -Proof NZlt_ind. - -(** Elimintation principle for <= *) - -Theorem le_ind : - forall A : N -> Prop, Proper (Neq==>iff) A -> - forall n : N, - A n -> - (forall m : N, n <= m -> A m -> A (S m)) -> - forall m : N, n <= m -> A m. -Proof NZle_ind. - -(** Well-founded relations *) - -Theorem lt_wf : forall z : N, well_founded (fun n m : N => z <= n /\ n < m). -Proof NZlt_wf. - -Theorem gt_wf : forall z : N, well_founded (fun n m : N => m < n /\ n <= z). -Proof NZgt_wf. +(* Theorems that are true for natural numbers but not for integers *) Theorem lt_wf_0 : well_founded lt. Proof. -setoid_replace lt with (fun n m : N => 0 <= n /\ n < m). +setoid_replace lt with (fun n m => 0 <= n /\ n < m). apply lt_wf. intros x y; split. intro H; split; [apply le_0_l | assumption]. now intros [_ H]. Defined. -(* Theorems that are true for natural numbers but not for integers *) - (* "le_0_l : forall n : N, 0 <= n" was proved in NBase.v *) -Theorem nlt_0_r : forall n : N, ~ n < 0. +Theorem nlt_0_r : forall n, ~ n < 0. Proof. intro n; apply -> le_ngt. apply le_0_l. Qed. -Theorem nle_succ_0 : forall n : N, ~ (S n <= 0). +Theorem nle_succ_0 : forall n, ~ (S n <= 0). Proof. intros n H; apply -> le_succ_l in H; false_hyp H nlt_0_r. Qed. -Theorem le_0_r : forall n : N, n <= 0 <-> n == 0. +Theorem le_0_r : forall n, n <= 0 <-> n == 0. Proof. intros n; split; intro H. le_elim H; [false_hyp H nlt_0_r | assumption]. now apply eq_le_incl. Qed. -Theorem lt_0_succ : forall n : N, 0 < S n. +Theorem lt_0_succ : forall n, 0 < S n. Proof. induct n; [apply lt_succ_diag_r | intros n H; now apply lt_lt_succ_r]. Qed. -Theorem neq_0_lt_0 : forall n : N, n ~= 0 <-> 0 < n. +Theorem neq_0_lt_0 : forall n, n ~= 0 <-> 0 < n. Proof. cases n. split; intro H; [now elim H | intro; now apply lt_irrefl with 0]. intro n; split; intro H; [apply lt_0_succ | apply neq_succ_0]. Qed. -Theorem eq_0_gt_0_cases : forall n : N, n == 0 \/ 0 < n. +Theorem eq_0_gt_0_cases : forall n, n == 0 \/ 0 < n. Proof. cases n. now left. intro; right; apply lt_0_succ. Qed. -Theorem zero_one : forall n : N, n == 0 \/ n == 1 \/ 1 < n. +Theorem zero_one : forall n, n == 0 \/ n == 1 \/ 1 < n. Proof. induct n. now left. cases n. intros; right; now left. @@ -365,7 +74,7 @@ right; right. rewrite H. apply lt_succ_diag_r. right; right. now apply lt_lt_succ_r. Qed. -Theorem lt_1_r : forall n : N, n < 1 <-> n == 0. +Theorem lt_1_r : forall n, n < 1 <-> n == 0. Proof. cases n. split; intro; [reflexivity | apply lt_succ_diag_r]. @@ -373,7 +82,7 @@ intros n. rewrite <- succ_lt_mono. split; intro H; [false_hyp H nlt_0_r | false_hyp H neq_succ_0]. Qed. -Theorem le_1_r : forall n : N, n <= 1 <-> n == 0 \/ n == 1. +Theorem le_1_r : forall n, n <= 1 <-> n == 0 \/ n == 1. Proof. cases n. split; intro; [now left | apply le_succ_diag_r]. @@ -381,31 +90,30 @@ intro n. rewrite <- succ_le_mono, le_0_r, succ_inj_wd. split; [intro; now right | intros [H | H]; [false_hyp H neq_succ_0 | assumption]]. Qed. -Theorem lt_lt_0 : forall n m : N, n < m -> 0 < m. +Theorem lt_lt_0 : forall n m, n < m -> 0 < m. Proof. intros n m; induct n. trivial. intros n IH H. apply IH; now apply lt_succ_l. Qed. -Theorem lt_1_l : forall n m p : N, n < m -> m < p -> 1 < p. +Theorem lt_1_l' : forall n m p, n < m -> m < p -> 1 < p. Proof. -intros n m p H1 H2. -apply le_lt_trans with m. apply <- le_succ_l. apply le_lt_trans with n. -apply le_0_l. assumption. assumption. +intros. apply lt_1_l with m; auto. +apply le_lt_trans with n; auto. now apply le_0_l. Qed. (** Elimination principlies for < and <= for relations *) Section RelElim. -Variable R : relation N. -Hypothesis R_wd : Proper (Neq==>Neq==>iff) R. +Variable R : relation N.t. +Hypothesis R_wd : Proper (N.eq==>N.eq==>iff) R. Theorem le_ind_rel : - (forall m : N, R 0 m) -> - (forall n m : N, n <= m -> R n m -> R (S n) (S m)) -> - forall n m : N, n <= m -> R n m. + (forall m, R 0 m) -> + (forall n m, n <= m -> R n m -> R (S n) (S m)) -> + forall n m, n <= m -> R n m. Proof. intros Base Step; induct n. intros; apply Base. @@ -416,9 +124,9 @@ intros k H1 H2. apply -> le_succ_l in H1. apply lt_le_incl in H1. auto. Qed. Theorem lt_ind_rel : - (forall m : N, R 0 (S m)) -> - (forall n m : N, n < m -> R n m -> R (S n) (S m)) -> - forall n m : N, n < m -> R n m. + (forall m, R 0 (S m)) -> + (forall n m, n < m -> R n m -> R (S n) (S m)) -> + forall n m, n < m -> R n m. Proof. intros Base Step; induct n. intros m H. apply lt_exists_pred in H; destruct H as [m' [H _]]. @@ -433,61 +141,64 @@ End RelElim. (** Predecessor and order *) -Theorem succ_pred_pos : forall n : N, 0 < n -> S (P n) == n. +Theorem succ_pred_pos : forall n, 0 < n -> S (P n) == n. Proof. intros n H; apply succ_pred; intro H1; rewrite H1 in H. false_hyp H lt_irrefl. Qed. -Theorem le_pred_l : forall n : N, P n <= n. +Theorem le_pred_l : forall n, P n <= n. Proof. cases n. rewrite pred_0; now apply eq_le_incl. intros; rewrite pred_succ; apply le_succ_diag_r. Qed. -Theorem lt_pred_l : forall n : N, n ~= 0 -> P n < n. +Theorem lt_pred_l : forall n, n ~= 0 -> P n < n. Proof. cases n. intro H; exfalso; now apply H. intros; rewrite pred_succ; apply lt_succ_diag_r. Qed. -Theorem le_le_pred : forall n m : N, n <= m -> P n <= m. +Theorem le_le_pred : forall n m, n <= m -> P n <= m. Proof. intros n m H; apply le_trans with n. apply le_pred_l. assumption. Qed. -Theorem lt_lt_pred : forall n m : N, n < m -> P n < m. +Theorem lt_lt_pred : forall n m, n < m -> P n < m. Proof. intros n m H; apply le_lt_trans with n. apply le_pred_l. assumption. Qed. -Theorem lt_le_pred : forall n m : N, n < m -> n <= P m. (* Converse is false for n == m == 0 *) +Theorem lt_le_pred : forall n m, n < m -> n <= P m. + (* Converse is false for n == m == 0 *) Proof. intro n; cases m. intro H; false_hyp H nlt_0_r. intros m IH. rewrite pred_succ; now apply -> lt_succ_r. Qed. -Theorem lt_pred_le : forall n m : N, P n < m -> n <= m. (* Converse is false for n == m == 0 *) +Theorem lt_pred_le : forall n m, P n < m -> n <= m. + (* Converse is false for n == m == 0 *) Proof. intros n m; cases n. rewrite pred_0; intro H; now apply lt_le_incl. intros n IH. rewrite pred_succ in IH. now apply <- le_succ_l. Qed. -Theorem lt_pred_lt : forall n m : N, n < P m -> n < m. +Theorem lt_pred_lt : forall n m, n < P m -> n < m. Proof. intros n m H; apply lt_le_trans with (P m); [assumption | apply le_pred_l]. Qed. -Theorem le_pred_le : forall n m : N, n <= P m -> n <= m. +Theorem le_pred_le : forall n m, n <= P m -> n <= m. Proof. intros n m H; apply le_trans with (P m); [assumption | apply le_pred_l]. Qed. -Theorem pred_le_mono : forall n m : N, n <= m -> P n <= P m. (* Converse is false for n == 1, m == 0 *) +Theorem pred_le_mono : forall n m, n <= m -> P n <= P m. + (* Converse is false for n == 1, m == 0 *) Proof. intros n m H; elim H using le_ind_rel. solve_relation_wd. @@ -495,7 +206,7 @@ intro; rewrite pred_0; apply le_0_l. intros p q H1 _; now do 2 rewrite pred_succ. Qed. -Theorem pred_lt_mono : forall n m : N, n ~= 0 -> (n < m <-> P n < P m). +Theorem pred_lt_mono : forall n m, n ~= 0 -> (n < m <-> P n < P m). Proof. intros n m H1; split; intro H2. assert (m ~= 0). apply <- neq_0_lt_0. now apply lt_lt_0 with n. @@ -506,22 +217,24 @@ apply lt_le_trans with (P m). assumption. apply le_pred_l. apply -> succ_lt_mono in H2. now do 2 rewrite succ_pred in H2. Qed. -Theorem lt_succ_lt_pred : forall n m : N, S n < m <-> n < P m. +Theorem lt_succ_lt_pred : forall n m, S n < m <-> n < P m. Proof. intros n m. rewrite pred_lt_mono by apply neq_succ_0. now rewrite pred_succ. Qed. -Theorem le_succ_le_pred : forall n m : N, S n <= m -> n <= P m. (* Converse is false for n == m == 0 *) +Theorem le_succ_le_pred : forall n m, S n <= m -> n <= P m. + (* Converse is false for n == m == 0 *) Proof. intros n m H. apply lt_le_pred. now apply -> le_succ_l. Qed. -Theorem lt_pred_lt_succ : forall n m : N, P n < m -> n < S m. (* Converse is false for n == m == 0 *) +Theorem lt_pred_lt_succ : forall n m, P n < m -> n < S m. + (* Converse is false for n == m == 0 *) Proof. intros n m H. apply <- lt_succ_r. now apply lt_pred_le. Qed. -Theorem le_pred_le_succ : forall n m : N, P n <= m <-> n <= S m. +Theorem le_pred_le_succ : forall n m, P n <= m <-> n <= S m. Proof. intros n m; cases n. rewrite pred_0. split; intro H; apply le_0_l. |