Require Export BinPos.
Require Export BinInt.
Require Import Lt.
Require Import Gt.
Require Import Plus.
Require Import Mult.
Open Local Scope Z_scope.
| Binary Integers (Pierre Crégut, CNET, Lannion, France) |
| 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 ].
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 ] ].
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 ].
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.
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 ].
Qed.
| Transitivity of comparison |
Lemma Zcompare_Gt_trans :
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 ].
Qed.
| 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.
Qed.
Hint Local Resolve Pcompare_refl.
| Comparison first-order specification |
Lemma Zcompare_Gt_spec :
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 ].
Qed.
| 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).
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 ].
Qed.
Hint Local Resolve ZC4.
Lemma weak_Zcompare_Zplus_compatible :
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 ] ] ].
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).
Qed.
Lemma Zplus_compare_compat :
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 ] ].
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.
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) ] ] ].
Qed.
| 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.
Qed.
| Multiplication and comparison |
Lemma Zcompare_mult_compat :
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 ].
Qed.
Reverting x ?= y to trichotomy
|
Lemma rename :
forall (A:Set) (P:A -> Prop) (x:A), (forall y:A, x = y -> P y) -> P x.
Proof.
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.
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 ].
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.
Proof.
intros c1 c2 c3 x y; intros.
rewrite H0; rewrite Zcompare_refl.
assumption.
Qed.
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).
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.
Qed.
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.
Proof.
intros x y; unfold Zle in |- *; elim (x ?= y); auto with arith.
Qed.
Lemma Zlt_compare :
forall n m:Z,
n < m -> match n ?= m with
| Eq => False
| Lt => True
| Gt => False
end.
Proof.
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.
Proof.
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.
Proof.
intros x y; unfold Zgt in |- *; elim (x ?= y); intros;
discriminate || trivial with arith.
Qed.
Lemma Zmult_compare_compat_l :
forall n m p:Z, p > 0 -> (n ?= m) = (p * n ?= p * m).
Proof.
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).
Proof.
intros x y z H; rewrite (Zmult_comm x z); rewrite (Zmult_comm y z);
apply Zmult_compare_compat_l; assumption.
Qed.