diff options
Diffstat (limited to 'theories/ZArith/Zcompare.v')
| -rw-r--r-- | theories/ZArith/Zcompare.v | 713 |
1 files changed, 356 insertions, 357 deletions
diff --git a/theories/ZArith/Zcompare.v b/theories/ZArith/Zcompare.v index 4003c338..6c5b07d2 100644 --- a/theories/ZArith/Zcompare.v +++ b/theories/ZArith/Zcompare.v @@ -8,6 +8,10 @@ (*i $$ i*) +(**********************************************************************) +(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) +(**********************************************************************) + Require Export BinPos. Require Export BinInt. Require Import Lt. @@ -17,485 +21,480 @@ Require Import Mult. Open Local Scope Z_scope. -(**********************************************************************) -(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) -(**********************************************************************) - -(**********************************************************************) -(** Comparison on integers *) +(***************************) +(** * Comparison on integers *) Lemma Zcompare_refl : forall n:Z, (n ?= n) = Eq. Proof. -intro x; destruct x as [| p| p]; simpl in |- *; - [ reflexivity | apply Pcompare_refl | rewrite Pcompare_refl; reflexivity ]. + intro x; destruct x as [| p| p]; simpl in |- *; + [ reflexivity | apply Pcompare_refl | rewrite Pcompare_refl; reflexivity ]. Qed. Lemma Zcompare_Eq_eq : forall n m:Z, (n ?= m) = Eq -> n = m. Proof. -intros x y; destruct x as [| x'| x']; destruct y as [| y'| y']; simpl in |- *; - intro H; reflexivity || (try discriminate H); - [ rewrite (Pcompare_Eq_eq x' y' H); reflexivity - | rewrite (Pcompare_Eq_eq x' y'); - [ reflexivity - | destruct ((x' ?= y')%positive Eq); reflexivity || discriminate ] ]. + intros x y; destruct x as [| x'| x']; destruct y as [| y'| y']; simpl in |- *; + intro H; reflexivity || (try discriminate H); + [ rewrite (Pcompare_Eq_eq x' y' H); reflexivity + | rewrite (Pcompare_Eq_eq x' y'); + [ reflexivity + | destruct ((x' ?= y')%positive Eq); reflexivity || discriminate ] ]. Qed. Lemma Zcompare_Eq_iff_eq : forall n m:Z, (n ?= m) = Eq <-> n = m. Proof. -intros x y; split; intro E; - [ apply Zcompare_Eq_eq; assumption | rewrite E; apply Zcompare_refl ]. + intros x y; split; intro E; + [ apply Zcompare_Eq_eq; assumption | rewrite E; apply Zcompare_refl ]. Qed. Lemma Zcompare_antisym : forall n m:Z, CompOpp (n ?= m) = (m ?= n). Proof. -intros x y; destruct x; destruct y; simpl in |- *; - reflexivity || discriminate H || rewrite Pcompare_antisym; - reflexivity. + intros x y; destruct x; destruct y; simpl in |- *; + reflexivity || discriminate H || rewrite Pcompare_antisym; + reflexivity. Qed. Lemma Zcompare_Gt_Lt_antisym : forall n m:Z, (n ?= m) = Gt <-> (m ?= n) = Lt. Proof. -intros x y; split; intro H; - [ change Lt with (CompOpp Gt) in |- *; rewrite <- Zcompare_antisym; - rewrite H; reflexivity - | change Gt with (CompOpp Lt) in |- *; rewrite <- Zcompare_antisym; - rewrite H; reflexivity ]. + intros x y; split; intro H; + [ change Lt with (CompOpp Gt) in |- *; rewrite <- Zcompare_antisym; + rewrite H; reflexivity + | change Gt with (CompOpp Lt) in |- *; rewrite <- Zcompare_antisym; + rewrite H; reflexivity ]. Qed. -(** Transitivity of comparison *) +(** * Transitivity of comparison *) Lemma Zcompare_Gt_trans : - forall n m p:Z, (n ?= m) = Gt -> (m ?= p) = Gt -> (n ?= p) = Gt. + forall n m p:Z, (n ?= m) = Gt -> (m ?= p) = Gt -> (n ?= p) = Gt. Proof. -intros x y z; case x; case y; case z; simpl in |- *; - try (intros; discriminate H || discriminate H0); auto with arith; - [ intros p q r H H0; apply nat_of_P_gt_Gt_compare_complement_morphism; - unfold gt in |- *; apply lt_trans with (m := nat_of_P q); - apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; - assumption - | intros p q r; do 3 rewrite <- ZC4; intros H H0; - apply nat_of_P_gt_Gt_compare_complement_morphism; - unfold gt in |- *; apply lt_trans with (m := nat_of_P q); - apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; - assumption ]. + intros x y z; case x; case y; case z; simpl in |- *; + try (intros; discriminate H || discriminate H0); auto with arith; + [ intros p q r H H0; apply nat_of_P_gt_Gt_compare_complement_morphism; + unfold gt in |- *; apply lt_trans with (m := nat_of_P q); + apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; + assumption + | intros p q r; do 3 rewrite <- ZC4; intros H H0; + apply nat_of_P_gt_Gt_compare_complement_morphism; + unfold gt in |- *; apply lt_trans with (m := nat_of_P q); + apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; + assumption ]. Qed. -(** Comparison and opposite *) +(** * Comparison and opposite *) Lemma Zcompare_opp : forall n m:Z, (n ?= m) = (- m ?= - n). Proof. -intros x y; case x; case y; simpl in |- *; auto with arith; intros; - rewrite <- ZC4; trivial with arith. + intros x y; case x; case y; simpl in |- *; auto with arith; intros; + rewrite <- ZC4; trivial with arith. Qed. Hint Local Resolve Pcompare_refl. -(** Comparison first-order specification *) +(** * Comparison first-order specification *) Lemma Zcompare_Gt_spec : - forall n m:Z, (n ?= m) = Gt -> exists h : positive, n + - m = Zpos h. + forall n m:Z, (n ?= m) = Gt -> exists h : positive, n + - m = Zpos h. Proof. -intros x y; case x; case y; - [ simpl in |- *; intros H; discriminate H - | simpl in |- *; intros p H; discriminate H - | intros p H; exists p; simpl in |- *; auto with arith - | intros p H; exists p; simpl in |- *; auto with arith - | intros q p H; exists (p - q)%positive; unfold Zplus, Zopp in |- *; - unfold Zcompare in H; rewrite H; trivial with arith - | intros q p H; exists (p + q)%positive; simpl in |- *; trivial with arith - | simpl in |- *; intros p H; discriminate H - | simpl in |- *; intros q p H; discriminate H - | unfold Zcompare in |- *; intros q p; rewrite <- ZC4; intros H; - exists (q - p)%positive; simpl in |- *; rewrite (ZC1 q p H); - trivial with arith ]. + intros x y; case x; case y; + [ simpl in |- *; intros H; discriminate H + | simpl in |- *; intros p H; discriminate H + | intros p H; exists p; simpl in |- *; auto with arith + | intros p H; exists p; simpl in |- *; auto with arith + | intros q p H; exists (p - q)%positive; unfold Zplus, Zopp in |- *; + unfold Zcompare in H; rewrite H; trivial with arith + | intros q p H; exists (p + q)%positive; simpl in |- *; trivial with arith + | simpl in |- *; intros p H; discriminate H + | simpl in |- *; intros q p H; discriminate H + | unfold Zcompare in |- *; intros q p; rewrite <- ZC4; intros H; + exists (q - p)%positive; simpl in |- *; rewrite (ZC1 q p H); + trivial with arith ]. Qed. -(** Comparison and addition *) +(** * Comparison and addition *) Lemma weaken_Zcompare_Zplus_compatible : - (forall (n m:Z) (p:positive), (Zpos p + n ?= Zpos p + m) = (n ?= m)) -> - forall n m p:Z, (p + n ?= p + m) = (n ?= m). + (forall (n m:Z) (p:positive), (Zpos p + n ?= Zpos p + m) = (n ?= m)) -> + forall n m p:Z, (p + n ?= p + m) = (n ?= m). Proof. -intros H x y z; destruct z; - [ reflexivity - | apply H - | rewrite (Zcompare_opp x y); rewrite Zcompare_opp; - do 2 rewrite Zopp_plus_distr; rewrite Zopp_neg; - apply H ]. + intros H x y z; destruct z; + [ reflexivity + | apply H + | rewrite (Zcompare_opp x y); rewrite Zcompare_opp; + do 2 rewrite Zopp_plus_distr; rewrite Zopp_neg; + apply H ]. Qed. Hint Local Resolve ZC4. Lemma weak_Zcompare_Zplus_compatible : - forall (n m:Z) (p:positive), (Zpos p + n ?= Zpos p + m) = (n ?= m). + forall (n m:Z) (p:positive), (Zpos p + n ?= Zpos p + m) = (n ?= m). Proof. -intros x y z; case x; case y; simpl in |- *; auto with arith; - [ intros p; apply nat_of_P_lt_Lt_compare_complement_morphism; apply ZL17 - | intros p; ElimPcompare z p; intros E; rewrite E; auto with arith; - apply nat_of_P_gt_Gt_compare_complement_morphism; - rewrite nat_of_P_minus_morphism; - [ unfold gt in |- *; apply ZL16 | assumption ] - | intros p; ElimPcompare z p; intros E; auto with arith; - apply nat_of_P_gt_Gt_compare_complement_morphism; - unfold gt in |- *; apply ZL17 - | intros p q; ElimPcompare q p; intros E; rewrite E; - [ rewrite (Pcompare_Eq_eq q p E); apply Pcompare_refl - | apply nat_of_P_lt_Lt_compare_complement_morphism; - do 2 rewrite nat_of_P_plus_morphism; apply plus_lt_compat_l; - apply nat_of_P_lt_Lt_compare_morphism with (1 := E) - | apply nat_of_P_gt_Gt_compare_complement_morphism; unfold gt in |- *; - do 2 rewrite nat_of_P_plus_morphism; apply plus_lt_compat_l; - exact (nat_of_P_gt_Gt_compare_morphism q p E) ] - | intros p q; ElimPcompare z p; intros E; rewrite E; auto with arith; - apply nat_of_P_gt_Gt_compare_complement_morphism; - rewrite nat_of_P_minus_morphism; - [ unfold gt in |- *; apply lt_trans with (m := nat_of_P z); - [ apply ZL16 | apply ZL17 ] - | assumption ] - | intros p; ElimPcompare z p; intros E; rewrite E; auto with arith; - simpl in |- *; apply nat_of_P_lt_Lt_compare_complement_morphism; - rewrite nat_of_P_minus_morphism; [ apply ZL16 | assumption ] - | intros p q; ElimPcompare z q; intros E; rewrite E; auto with arith; - simpl in |- *; apply nat_of_P_lt_Lt_compare_complement_morphism; - rewrite nat_of_P_minus_morphism; - [ apply lt_trans with (m := nat_of_P z); [ apply ZL16 | apply ZL17 ] - | assumption ] - | intros p q; ElimPcompare z q; intros E0; rewrite E0; ElimPcompare z p; - intros E1; rewrite E1; ElimPcompare q p; intros E2; - rewrite E2; auto with arith; - [ absurd ((q ?= p)%positive Eq = Lt); - [ rewrite <- (Pcompare_Eq_eq z q E0); - rewrite <- (Pcompare_Eq_eq z p E1); rewrite (Pcompare_refl z); - discriminate - | assumption ] - | absurd ((q ?= p)%positive Eq = Gt); - [ rewrite <- (Pcompare_Eq_eq z q E0); - rewrite <- (Pcompare_Eq_eq z p E1); rewrite (Pcompare_refl z); - discriminate - | assumption ] - | absurd ((z ?= p)%positive Eq = Lt); - [ rewrite (Pcompare_Eq_eq z q E0); rewrite <- (Pcompare_Eq_eq q p E2); - rewrite (Pcompare_refl q); discriminate - | assumption ] - | absurd ((z ?= p)%positive Eq = Lt); - [ rewrite (Pcompare_Eq_eq z q E0); rewrite E2; discriminate - | assumption ] - | absurd ((z ?= p)%positive Eq = Gt); - [ rewrite (Pcompare_Eq_eq z q E0); rewrite <- (Pcompare_Eq_eq q p E2); - rewrite (Pcompare_refl q); discriminate - | assumption ] - | absurd ((z ?= p)%positive Eq = Gt); - [ rewrite (Pcompare_Eq_eq z q E0); rewrite E2; discriminate - | assumption ] - | absurd ((z ?= q)%positive Eq = Lt); - [ rewrite (Pcompare_Eq_eq z p E1); rewrite (Pcompare_Eq_eq q p E2); - rewrite (Pcompare_refl p); discriminate - | assumption ] - | absurd ((p ?= q)%positive Eq = Gt); - [ rewrite <- (Pcompare_Eq_eq z p E1); rewrite E0; discriminate - | apply ZC2; assumption ] - | simpl in |- *; rewrite (Pcompare_Eq_eq q p E2); - rewrite (Pcompare_refl (p - z)); auto with arith - | simpl in |- *; rewrite <- ZC4; - apply nat_of_P_gt_Gt_compare_complement_morphism; - rewrite nat_of_P_minus_morphism; - [ rewrite nat_of_P_minus_morphism; - [ unfold gt in |- *; apply plus_lt_reg_l with (p := nat_of_P z); - rewrite le_plus_minus_r; - [ rewrite le_plus_minus_r; - [ apply nat_of_P_lt_Lt_compare_morphism; assumption - | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; - assumption ] - | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; - assumption ] - | apply ZC2; assumption ] - | apply ZC2; assumption ] - | simpl in |- *; rewrite <- ZC4; - apply nat_of_P_lt_Lt_compare_complement_morphism; - rewrite nat_of_P_minus_morphism; - [ rewrite nat_of_P_minus_morphism; - [ apply plus_lt_reg_l with (p := nat_of_P z); - rewrite le_plus_minus_r; - [ rewrite le_plus_minus_r; - [ apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; - assumption - | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; - assumption ] - | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; - assumption ] - | apply ZC2; assumption ] - | apply ZC2; assumption ] - | absurd ((z ?= q)%positive Eq = Lt); - [ rewrite (Pcompare_Eq_eq q p E2); rewrite E1; discriminate - | assumption ] - | absurd ((q ?= p)%positive Eq = Lt); - [ cut ((q ?= p)%positive Eq = Gt); - [ intros E; rewrite E; discriminate - | apply nat_of_P_gt_Gt_compare_complement_morphism; - unfold gt in |- *; apply lt_trans with (m := nat_of_P z); - [ apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; assumption - | apply nat_of_P_lt_Lt_compare_morphism; assumption ] ] - | assumption ] - | absurd ((z ?= q)%positive Eq = Gt); - [ rewrite (Pcompare_Eq_eq z p E1); rewrite (Pcompare_Eq_eq q p E2); - rewrite (Pcompare_refl p); discriminate - | assumption ] - | absurd ((z ?= q)%positive Eq = Gt); - [ rewrite (Pcompare_Eq_eq z p E1); rewrite ZC1; - [ discriminate | assumption ] - | assumption ] - | absurd ((z ?= q)%positive Eq = Gt); - [ rewrite (Pcompare_Eq_eq q p E2); rewrite E1; discriminate - | assumption ] - | absurd ((q ?= p)%positive Eq = Gt); - [ rewrite ZC1; - [ discriminate - | apply nat_of_P_gt_Gt_compare_complement_morphism; - unfold gt in |- *; apply lt_trans with (m := nat_of_P z); - [ apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; assumption - | apply nat_of_P_lt_Lt_compare_morphism; assumption ] ] - | assumption ] - | simpl in |- *; rewrite (Pcompare_Eq_eq q p E2); apply Pcompare_refl - | simpl in |- *; apply nat_of_P_gt_Gt_compare_complement_morphism; - unfold gt in |- *; rewrite nat_of_P_minus_morphism; - [ rewrite nat_of_P_minus_morphism; - [ apply plus_lt_reg_l with (p := nat_of_P p); - rewrite le_plus_minus_r; - [ rewrite plus_comm; apply plus_lt_reg_l with (p := nat_of_P q); - rewrite plus_assoc; rewrite le_plus_minus_r; - [ rewrite (plus_comm (nat_of_P q)); apply plus_lt_compat_l; - apply nat_of_P_lt_Lt_compare_morphism; - assumption - | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; - apply ZC1; assumption ] - | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; - apply ZC1; assumption ] - | assumption ] - | assumption ] - | simpl in |- *; apply nat_of_P_lt_Lt_compare_complement_morphism; - rewrite nat_of_P_minus_morphism; - [ rewrite nat_of_P_minus_morphism; - [ apply plus_lt_reg_l with (p := nat_of_P q); - rewrite le_plus_minus_r; - [ rewrite plus_comm; apply plus_lt_reg_l with (p := nat_of_P p); - rewrite plus_assoc; rewrite le_plus_minus_r; - [ rewrite (plus_comm (nat_of_P p)); apply plus_lt_compat_l; - apply nat_of_P_lt_Lt_compare_morphism; - apply ZC1; assumption - | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; - apply ZC1; assumption ] - | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; - apply ZC1; assumption ] - | assumption ] - | assumption ] ] ]. + intros x y z; case x; case y; simpl in |- *; auto with arith; + [ intros p; apply nat_of_P_lt_Lt_compare_complement_morphism; apply ZL17 + | intros p; ElimPcompare z p; intros E; rewrite E; auto with arith; + apply nat_of_P_gt_Gt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; + [ unfold gt in |- *; apply ZL16 | assumption ] + | intros p; ElimPcompare z p; intros E; auto with arith; + apply nat_of_P_gt_Gt_compare_complement_morphism; + unfold gt in |- *; apply ZL17 + | intros p q; ElimPcompare q p; intros E; rewrite E; + [ rewrite (Pcompare_Eq_eq q p E); apply Pcompare_refl + | apply nat_of_P_lt_Lt_compare_complement_morphism; + do 2 rewrite nat_of_P_plus_morphism; apply plus_lt_compat_l; + apply nat_of_P_lt_Lt_compare_morphism with (1 := E) + | apply nat_of_P_gt_Gt_compare_complement_morphism; unfold gt in |- *; + do 2 rewrite nat_of_P_plus_morphism; apply plus_lt_compat_l; + exact (nat_of_P_gt_Gt_compare_morphism q p E) ] + | intros p q; ElimPcompare z p; intros E; rewrite E; auto with arith; + apply nat_of_P_gt_Gt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; + [ unfold gt in |- *; apply lt_trans with (m := nat_of_P z); + [ apply ZL16 | apply ZL17 ] + | assumption ] + | intros p; ElimPcompare z p; intros E; rewrite E; auto with arith; + simpl in |- *; apply nat_of_P_lt_Lt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; [ apply ZL16 | assumption ] + | intros p q; ElimPcompare z q; intros E; rewrite E; auto with arith; + simpl in |- *; apply nat_of_P_lt_Lt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; + [ apply lt_trans with (m := nat_of_P z); [ apply ZL16 | apply ZL17 ] + | assumption ] + | intros p q; ElimPcompare z q; intros E0; rewrite E0; ElimPcompare z p; + intros E1; rewrite E1; ElimPcompare q p; intros E2; + rewrite E2; auto with arith; + [ absurd ((q ?= p)%positive Eq = Lt); + [ rewrite <- (Pcompare_Eq_eq z q E0); + rewrite <- (Pcompare_Eq_eq z p E1); rewrite (Pcompare_refl z); + discriminate + | assumption ] + | absurd ((q ?= p)%positive Eq = Gt); + [ rewrite <- (Pcompare_Eq_eq z q E0); + rewrite <- (Pcompare_Eq_eq z p E1); rewrite (Pcompare_refl z); + discriminate + | assumption ] + | absurd ((z ?= p)%positive Eq = Lt); + [ rewrite (Pcompare_Eq_eq z q E0); rewrite <- (Pcompare_Eq_eq q p E2); + rewrite (Pcompare_refl q); discriminate + | assumption ] + | absurd ((z ?= p)%positive Eq = Lt); + [ rewrite (Pcompare_Eq_eq z q E0); rewrite E2; discriminate + | assumption ] + | absurd ((z ?= p)%positive Eq = Gt); + [ rewrite (Pcompare_Eq_eq z q E0); rewrite <- (Pcompare_Eq_eq q p E2); + rewrite (Pcompare_refl q); discriminate + | assumption ] + | absurd ((z ?= p)%positive Eq = Gt); + [ rewrite (Pcompare_Eq_eq z q E0); rewrite E2; discriminate + | assumption ] + | absurd ((z ?= q)%positive Eq = Lt); + [ rewrite (Pcompare_Eq_eq z p E1); rewrite (Pcompare_Eq_eq q p E2); + rewrite (Pcompare_refl p); discriminate + | assumption ] + | absurd ((p ?= q)%positive Eq = Gt); + [ rewrite <- (Pcompare_Eq_eq z p E1); rewrite E0; discriminate + | apply ZC2; assumption ] + | simpl in |- *; rewrite (Pcompare_Eq_eq q p E2); + rewrite (Pcompare_refl (p - z)); auto with arith + | simpl in |- *; rewrite <- ZC4; + apply nat_of_P_gt_Gt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; + [ rewrite nat_of_P_minus_morphism; + [ unfold gt in |- *; apply plus_lt_reg_l with (p := nat_of_P z); + rewrite le_plus_minus_r; + [ rewrite le_plus_minus_r; + [ apply nat_of_P_lt_Lt_compare_morphism; assumption + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + assumption ] + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + assumption ] + | apply ZC2; assumption ] + | apply ZC2; assumption ] + | simpl in |- *; rewrite <- ZC4; + apply nat_of_P_lt_Lt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; + [ rewrite nat_of_P_minus_morphism; + [ apply plus_lt_reg_l with (p := nat_of_P z); + rewrite le_plus_minus_r; + [ rewrite le_plus_minus_r; + [ apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; + assumption + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + assumption ] + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + assumption ] + | apply ZC2; assumption ] + | apply ZC2; assumption ] + | absurd ((z ?= q)%positive Eq = Lt); + [ rewrite (Pcompare_Eq_eq q p E2); rewrite E1; discriminate + | assumption ] + | absurd ((q ?= p)%positive Eq = Lt); + [ cut ((q ?= p)%positive Eq = Gt); + [ intros E; rewrite E; discriminate + | apply nat_of_P_gt_Gt_compare_complement_morphism; + unfold gt in |- *; apply lt_trans with (m := nat_of_P z); + [ apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; assumption + | apply nat_of_P_lt_Lt_compare_morphism; assumption ] ] + | assumption ] + | absurd ((z ?= q)%positive Eq = Gt); + [ rewrite (Pcompare_Eq_eq z p E1); rewrite (Pcompare_Eq_eq q p E2); + rewrite (Pcompare_refl p); discriminate + | assumption ] + | absurd ((z ?= q)%positive Eq = Gt); + [ rewrite (Pcompare_Eq_eq z p E1); rewrite ZC1; + [ discriminate | assumption ] + | assumption ] + | absurd ((z ?= q)%positive Eq = Gt); + [ rewrite (Pcompare_Eq_eq q p E2); rewrite E1; discriminate + | assumption ] + | absurd ((q ?= p)%positive Eq = Gt); + [ rewrite ZC1; + [ discriminate + | apply nat_of_P_gt_Gt_compare_complement_morphism; + unfold gt in |- *; apply lt_trans with (m := nat_of_P z); + [ apply nat_of_P_lt_Lt_compare_morphism; apply ZC1; assumption + | apply nat_of_P_lt_Lt_compare_morphism; assumption ] ] + | assumption ] + | simpl in |- *; rewrite (Pcompare_Eq_eq q p E2); apply Pcompare_refl + | simpl in |- *; apply nat_of_P_gt_Gt_compare_complement_morphism; + unfold gt in |- *; rewrite nat_of_P_minus_morphism; + [ rewrite nat_of_P_minus_morphism; + [ apply plus_lt_reg_l with (p := nat_of_P p); + rewrite le_plus_minus_r; + [ rewrite plus_comm; apply plus_lt_reg_l with (p := nat_of_P q); + rewrite plus_assoc; rewrite le_plus_minus_r; + [ rewrite (plus_comm (nat_of_P q)); apply plus_lt_compat_l; + apply nat_of_P_lt_Lt_compare_morphism; + assumption + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + apply ZC1; assumption ] + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + apply ZC1; assumption ] + | assumption ] + | assumption ] + | simpl in |- *; apply nat_of_P_lt_Lt_compare_complement_morphism; + rewrite nat_of_P_minus_morphism; + [ rewrite nat_of_P_minus_morphism; + [ apply plus_lt_reg_l with (p := nat_of_P q); + rewrite le_plus_minus_r; + [ rewrite plus_comm; apply plus_lt_reg_l with (p := nat_of_P p); + rewrite plus_assoc; rewrite le_plus_minus_r; + [ rewrite (plus_comm (nat_of_P p)); apply plus_lt_compat_l; + apply nat_of_P_lt_Lt_compare_morphism; + apply ZC1; assumption + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + apply ZC1; assumption ] + | apply lt_le_weak; apply nat_of_P_lt_Lt_compare_morphism; + apply ZC1; assumption ] + | assumption ] + | assumption ] ] ]. Qed. Lemma Zcompare_plus_compat : forall n m p:Z, (p + n ?= p + m) = (n ?= m). Proof. -exact (weaken_Zcompare_Zplus_compatible weak_Zcompare_Zplus_compatible). + exact (weaken_Zcompare_Zplus_compatible weak_Zcompare_Zplus_compatible). Qed. Lemma Zplus_compare_compat : - forall (r:comparison) (n m p q:Z), - (n ?= m) = r -> (p ?= q) = r -> (n + p ?= m + q) = r. + forall (r:comparison) (n m p q:Z), + (n ?= m) = r -> (p ?= q) = r -> (n + p ?= m + q) = r. Proof. -intros r x y z t; case r; - [ intros H1 H2; elim (Zcompare_Eq_iff_eq x y); elim (Zcompare_Eq_iff_eq z t); - intros H3 H4 H5 H6; rewrite H3; - [ rewrite H5; - [ elim (Zcompare_Eq_iff_eq (y + t) (y + t)); auto with arith - | auto with arith ] - | auto with arith ] - | intros H1 H2; elim (Zcompare_Gt_Lt_antisym (y + t) (x + z)); intros H3 H4; - apply H3; apply Zcompare_Gt_trans with (m := y + z); - [ rewrite Zcompare_plus_compat; elim (Zcompare_Gt_Lt_antisym t z); - auto with arith - | do 2 rewrite <- (Zplus_comm z); rewrite Zcompare_plus_compat; - elim (Zcompare_Gt_Lt_antisym y x); auto with arith ] - | intros H1 H2; apply Zcompare_Gt_trans with (m := x + t); - [ rewrite Zcompare_plus_compat; assumption - | do 2 rewrite <- (Zplus_comm t); rewrite Zcompare_plus_compat; - assumption ] ]. + intros r x y z t; case r; + [ intros H1 H2; elim (Zcompare_Eq_iff_eq x y); elim (Zcompare_Eq_iff_eq z t); + intros H3 H4 H5 H6; rewrite H3; + [ rewrite H5; + [ elim (Zcompare_Eq_iff_eq (y + t) (y + t)); auto with arith + | auto with arith ] + | auto with arith ] + | intros H1 H2; elim (Zcompare_Gt_Lt_antisym (y + t) (x + z)); intros H3 H4; + apply H3; apply Zcompare_Gt_trans with (m := y + z); + [ rewrite Zcompare_plus_compat; elim (Zcompare_Gt_Lt_antisym t z); + auto with arith + | do 2 rewrite <- (Zplus_comm z); rewrite Zcompare_plus_compat; + elim (Zcompare_Gt_Lt_antisym y x); auto with arith ] + | intros H1 H2; apply Zcompare_Gt_trans with (m := x + t); + [ rewrite Zcompare_plus_compat; assumption + | do 2 rewrite <- (Zplus_comm t); rewrite Zcompare_plus_compat; + assumption ] ]. Qed. Lemma Zcompare_succ_Gt : forall n:Z, (Zsucc n ?= n) = Gt. Proof. -intro x; unfold Zsucc in |- *; pattern x at 2 in |- *; - rewrite <- (Zplus_0_r x); rewrite Zcompare_plus_compat; - reflexivity. + intro x; unfold Zsucc in |- *; pattern x at 2 in |- *; + rewrite <- (Zplus_0_r x); rewrite Zcompare_plus_compat; + reflexivity. Qed. Lemma Zcompare_Gt_not_Lt : forall n m:Z, (n ?= m) = Gt <-> (n ?= m + 1) <> Lt. Proof. -intros x y; split; - [ intro H; elim_compare x (y + 1); - [ intro H1; rewrite H1; discriminate - | intros H1; elim Zcompare_Gt_spec with (1 := H); intros h H2; - absurd ((nat_of_P h > 0)%nat /\ (nat_of_P h < 1)%nat); - [ unfold not in |- *; intros H3; elim H3; intros H4 H5; - absurd (nat_of_P h > 0)%nat; - [ unfold gt in |- *; apply le_not_lt; apply le_S_n; exact H5 - | assumption ] - | split; - [ elim (ZL4 h); intros i H3; rewrite H3; apply gt_Sn_O - | change (nat_of_P h < nat_of_P 1)%nat in |- *; - apply nat_of_P_lt_Lt_compare_morphism; - change ((Zpos h ?= 1) = Lt) in |- *; rewrite <- H2; - rewrite <- (fun m n:Z => Zcompare_plus_compat m n y); - rewrite (Zplus_comm x); rewrite Zplus_assoc; - rewrite Zplus_opp_r; simpl in |- *; exact H1 ] ] - | intros H1; rewrite H1; discriminate ] - | intros H; elim_compare x (y + 1); - [ intros H1; elim (Zcompare_Eq_iff_eq x (y + 1)); intros H2 H3; - rewrite (H2 H1); exact (Zcompare_succ_Gt y) - | intros H1; absurd ((x ?= y + 1) = Lt); assumption - | intros H1; apply Zcompare_Gt_trans with (m := Zsucc y); - [ exact H1 | exact (Zcompare_succ_Gt y) ] ] ]. + intros x y; split; + [ intro H; elim_compare x (y + 1); + [ intro H1; rewrite H1; discriminate + | intros H1; elim Zcompare_Gt_spec with (1 := H); intros h H2; + absurd ((nat_of_P h > 0)%nat /\ (nat_of_P h < 1)%nat); + [ unfold not in |- *; intros H3; elim H3; intros H4 H5; + absurd (nat_of_P h > 0)%nat; + [ unfold gt in |- *; apply le_not_lt; apply le_S_n; exact H5 + | assumption ] + | split; + [ elim (ZL4 h); intros i H3; rewrite H3; apply gt_Sn_O + | change (nat_of_P h < nat_of_P 1)%nat in |- *; + apply nat_of_P_lt_Lt_compare_morphism; + change ((Zpos h ?= 1) = Lt) in |- *; rewrite <- H2; + rewrite <- (fun m n:Z => Zcompare_plus_compat m n y); + rewrite (Zplus_comm x); rewrite Zplus_assoc; + rewrite Zplus_opp_r; simpl in |- *; exact H1 ] ] + | intros H1; rewrite H1; discriminate ] + | intros H; elim_compare x (y + 1); + [ intros H1; elim (Zcompare_Eq_iff_eq x (y + 1)); intros H2 H3; + rewrite (H2 H1); exact (Zcompare_succ_Gt y) + | intros H1; absurd ((x ?= y + 1) = Lt); assumption + | intros H1; apply Zcompare_Gt_trans with (m := Zsucc y); + [ exact H1 | exact (Zcompare_succ_Gt y) ] ] ]. Qed. -(** Successor and comparison *) +(** * Successor and comparison *) Lemma Zcompare_succ_compat : forall n m:Z, (Zsucc n ?= Zsucc m) = (n ?= m). Proof. -intros n m; unfold Zsucc in |- *; do 2 rewrite (fun t:Z => Zplus_comm t 1); - rewrite Zcompare_plus_compat; auto with arith. + intros n m; unfold Zsucc in |- *; do 2 rewrite (fun t:Z => Zplus_comm t 1); + rewrite Zcompare_plus_compat; auto with arith. Qed. -(** Multiplication and comparison *) +(** * Multiplication and comparison *) Lemma Zcompare_mult_compat : - forall (p:positive) (n m:Z), (Zpos p * n ?= Zpos p * m) = (n ?= m). + forall (p:positive) (n m:Z), (Zpos p * n ?= Zpos p * m) = (n ?= m). Proof. -intros x; induction x as [p H| p H| ]; - [ intros y z; cut (Zpos (xI p) = Zpos p + Zpos p + 1); - [ intros E; rewrite E; do 4 rewrite Zmult_plus_distr_l; - do 2 rewrite Zmult_1_l; apply Zplus_compare_compat; - [ apply Zplus_compare_compat; apply H | trivial with arith ] - | simpl in |- *; rewrite (Pplus_diag p); trivial with arith ] - | intros y z; cut (Zpos (xO p) = Zpos p + Zpos p); - [ intros E; rewrite E; do 2 rewrite Zmult_plus_distr_l; - apply Zplus_compare_compat; apply H - | simpl in |- *; rewrite (Pplus_diag p); trivial with arith ] - | intros y z; do 2 rewrite Zmult_1_l; trivial with arith ]. + intros x; induction x as [p H| p H| ]; + [ intros y z; cut (Zpos (xI p) = Zpos p + Zpos p + 1); + [ intros E; rewrite E; do 4 rewrite Zmult_plus_distr_l; + do 2 rewrite Zmult_1_l; apply Zplus_compare_compat; + [ apply Zplus_compare_compat; apply H | trivial with arith ] + | simpl in |- *; rewrite (Pplus_diag p); trivial with arith ] + | intros y z; cut (Zpos (xO p) = Zpos p + Zpos p); + [ intros E; rewrite E; do 2 rewrite Zmult_plus_distr_l; + apply Zplus_compare_compat; apply H + | simpl in |- *; rewrite (Pplus_diag p); trivial with arith ] + | intros y z; do 2 rewrite Zmult_1_l; trivial with arith ]. Qed. -(** Reverting [x ?= y] to trichotomy *) +(** * Reverting [x ?= y] to trichotomy *) Lemma rename : - forall (A:Type) (P:A -> Prop) (x:A), (forall y:A, x = y -> P y) -> P x. + forall (A:Type) (P:A -> Prop) (x:A), (forall y:A, x = y -> P y) -> P x. Proof. -auto with arith. + auto with arith. Qed. Lemma Zcompare_elim : - forall (c1 c2 c3:Prop) (n m:Z), - (n = m -> c1) -> - (n < m -> c2) -> - (n > m -> c3) -> match n ?= m with - | Eq => c1 - | Lt => c2 - | Gt => c3 - end. + forall (c1 c2 c3:Prop) (n m:Z), + (n = m -> c1) -> + (n < m -> c2) -> + (n > m -> c3) -> match n ?= m with + | Eq => c1 + | Lt => c2 + | Gt => c3 + end. Proof. -intros c1 c2 c3 x y; intros. -apply rename with (x := x ?= y); intro r; elim r; - [ intro; apply H; apply (Zcompare_Eq_eq x y); assumption - | unfold Zlt in H0; assumption - | unfold Zgt in H1; assumption ]. + intros c1 c2 c3 x y; intros. + apply rename with (x := x ?= y); intro r; elim r; + [ intro; apply H; apply (Zcompare_Eq_eq x y); assumption + | unfold Zlt in H0; assumption + | unfold Zgt in H1; assumption ]. Qed. Lemma Zcompare_eq_case : - forall (c1 c2 c3:Prop) (n m:Z), - c1 -> n = m -> match n ?= m with - | Eq => c1 - | Lt => c2 - | Gt => c3 - end. + forall (c1 c2 c3:Prop) (n m:Z), + c1 -> n = m -> match n ?= m with + | Eq => c1 + | Lt => c2 + | Gt => c3 + end. Proof. -intros c1 c2 c3 x y; intros. -rewrite H0; rewrite Zcompare_refl. -assumption. + intros c1 c2 c3 x y; intros. + rewrite H0; rewrite Zcompare_refl. + assumption. Qed. -(** Decompose an egality between two [?=] relations into 3 implications *) +(** * Decompose an egality between two [?=] relations into 3 implications *) Lemma Zcompare_egal_dec : - forall n m p q:Z, - (n < m -> p < q) -> - ((n ?= m) = Eq -> (p ?= q) = Eq) -> - (n > m -> p > q) -> (n ?= m) = (p ?= q). + forall n m p q:Z, + (n < m -> p < q) -> + ((n ?= m) = Eq -> (p ?= q) = Eq) -> + (n > m -> p > q) -> (n ?= m) = (p ?= q). Proof. -intros x1 y1 x2 y2. -unfold Zgt in |- *; unfold Zlt in |- *; case (x1 ?= y1); case (x2 ?= y2); - auto with arith; symmetry in |- *; auto with arith. + intros x1 y1 x2 y2. + unfold Zgt in |- *; unfold Zlt in |- *; case (x1 ?= y1); case (x2 ?= y2); + auto with arith; symmetry in |- *; auto with arith. Qed. -(** Relating [x ?= y] to [Zle], [Zlt], [Zge] or [Zgt] *) +(** * Relating [x ?= y] to [Zle], [Zlt], [Zge] or [Zgt] *) Lemma Zle_compare : - forall n m:Z, - n <= m -> match n ?= m with - | Eq => True - | Lt => True - | Gt => False - end. + forall n m:Z, + n <= m -> match n ?= m with + | Eq => True + | Lt => True + | Gt => False + end. Proof. -intros x y; unfold Zle in |- *; elim (x ?= y); auto with arith. + intros x y; unfold Zle in |- *; elim (x ?= y); auto with arith. Qed. Lemma Zlt_compare : - forall n m:Z, + forall n m:Z, n < m -> match n ?= m with - | Eq => False - | Lt => True - | Gt => False + | Eq => False + | Lt => True + | Gt => False end. Proof. -intros x y; unfold Zlt in |- *; elim (x ?= y); intros; - discriminate || trivial with arith. + intros x y; unfold Zlt in |- *; elim (x ?= y); intros; + discriminate || trivial with arith. Qed. Lemma Zge_compare : - forall n m:Z, - n >= m -> match n ?= m with - | Eq => True - | Lt => False - | Gt => True - end. + forall n m:Z, + n >= m -> match n ?= m with + | Eq => True + | Lt => False + | Gt => True + end. Proof. -intros x y; unfold Zge in |- *; elim (x ?= y); auto with arith. + intros x y; unfold Zge in |- *; elim (x ?= y); auto with arith. Qed. Lemma Zgt_compare : - forall n m:Z, - n > m -> match n ?= m with - | Eq => False - | Lt => False - | Gt => True - end. + forall n m:Z, + n > m -> match n ?= m with + | Eq => False + | Lt => False + | Gt => True + end. Proof. -intros x y; unfold Zgt in |- *; elim (x ?= y); intros; - discriminate || trivial with arith. + intros x y; unfold Zgt in |- *; elim (x ?= y); intros; + discriminate || trivial with arith. Qed. -(**********************************************************************) -(* Other properties *) - +(*********************) +(** * Other properties *) Lemma Zmult_compare_compat_l : - forall n m p:Z, p > 0 -> (n ?= m) = (p * n ?= p * m). + forall n m p:Z, p > 0 -> (n ?= m) = (p * n ?= p * m). Proof. -intros x y z H; destruct z. + intros x y z H; destruct z. discriminate H. rewrite Zcompare_mult_compat; reflexivity. discriminate H. Qed. Lemma Zmult_compare_compat_r : - forall n m p:Z, p > 0 -> (n ?= m) = (n * p ?= m * p). + forall n m p:Z, p > 0 -> (n ?= m) = (n * p ?= m * p). Proof. -intros x y z H; rewrite (Zmult_comm x z); rewrite (Zmult_comm y z); - apply Zmult_compare_compat_l; assumption. + intros x y z H; rewrite (Zmult_comm x z); rewrite (Zmult_comm y z); + apply Zmult_compare_compat_l; assumption. Qed. |