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diff --git a/theories7/NArith/Pnat.v b/theories7/NArith/Pnat.v deleted file mode 100644 index d62661ed..00000000 --- a/theories7/NArith/Pnat.v +++ /dev/null @@ -1,472 +0,0 @@ -(************************************************************************) -(* 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 *) -(************************************************************************) - -(*i $Id: Pnat.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) - -Require BinPos. - -(**********************************************************************) -(** Properties of the injection from binary positive numbers to Peano - natural numbers *) - -(** Original development by Pierre Crégut, CNET, Lannion, France *) - -Require Le. -Require Lt. -Require Gt. -Require Plus. -Require Mult. -Require Minus. - -(** [nat_of_P] is a morphism for addition *) - -Lemma convert_add_un : - (x:positive)(m:nat) - (positive_to_nat (add_un x) m) = (plus m (positive_to_nat x m)). -Proof. -Intro x; NewInduction x as [p IHp|p IHp|]; Simpl; Auto; Intro m; Rewrite IHp; -Rewrite plus_assoc_l; Trivial. -Qed. - -Lemma cvt_add_un : - (p:positive) (convert (add_un p)) = (S (convert p)). -Proof. - Intro; Change (S (convert p)) with (plus (S O) (convert p)); - Unfold convert; Apply convert_add_un. -Qed. - -Theorem convert_add_carry : - (x,y:positive)(m:nat) - (positive_to_nat (add_carry x y) m) = - (plus m (positive_to_nat (add x y) m)). -Proof. -Intro x; NewInduction x as [p IHp|p IHp|]; - Intro y; NewDestruct y; Simpl; Auto with arith; Intro m; [ - Rewrite IHp; Rewrite plus_assoc_l; Trivial with arith -| Rewrite IHp; Rewrite plus_assoc_l; Trivial with arith -| Rewrite convert_add_un; Rewrite plus_assoc_l; Trivial with arith -| Rewrite convert_add_un; Apply plus_assoc_r ]. -Qed. - -Theorem cvt_carry : - (x,y:positive)(convert (add_carry x y)) = (S (convert (add x y))). -Proof. -Intros;Unfold convert; Rewrite convert_add_carry; Simpl; Trivial with arith. -Qed. - -Theorem add_verif : - (x,y:positive)(m:nat) - (positive_to_nat (add x y) m) = - (plus (positive_to_nat x m) (positive_to_nat y m)). -Proof. -Intro x; NewInduction x as [p IHp|p IHp|]; - Intro y; NewDestruct y;Simpl;Auto with arith; [ - Intros m;Rewrite convert_add_carry; Rewrite IHp; - Rewrite plus_assoc_r; Rewrite plus_assoc_r; - Rewrite (plus_permute m (positive_to_nat p (plus m m))); Trivial with arith -| Intros m; Rewrite IHp; Apply plus_assoc_l -| Intros m; Rewrite convert_add_un; - Rewrite (plus_sym (plus m (positive_to_nat p (plus m m)))); - Apply plus_assoc_r -| Intros m; Rewrite IHp; Apply plus_permute -| Intros m; Rewrite convert_add_un; Apply plus_assoc_r ]. -Qed. - -Theorem convert_add: - (x,y:positive) (convert (add x y)) = (plus (convert x) (convert y)). -Proof. -Intros x y; Exact (add_verif x y (S O)). -Qed. - -(** [Pmult_nat] is a morphism for addition *) - -Lemma ZL2: - (y:positive)(m:nat) - (positive_to_nat y (plus m m)) = - (plus (positive_to_nat y m) (positive_to_nat y m)). -Proof. -Intro y; NewInduction y as [p H|p H|]; Intro m; [ - Simpl; Rewrite H; Rewrite plus_assoc_r; - Rewrite (plus_permute m (positive_to_nat p (plus m m))); - Rewrite plus_assoc_r; Auto with arith -| Simpl; Rewrite H; Auto with arith -| Simpl; Trivial with arith ]. -Qed. - -Lemma ZL6: - (p:positive) (positive_to_nat p (S (S O))) = (plus (convert p) (convert p)). -Proof. -Intro p;Change (2) with (plus (S O) (S O)); Rewrite ZL2; Trivial. -Qed. - -(** [nat_of_P] is a morphism for multiplication *) - -Theorem times_convert : - (x,y:positive) (convert (times x y)) = (mult (convert x) (convert y)). -Proof. -Intros x y; NewInduction x as [ x' H | x' H | ]; [ - Change (times (xI x') y) with (add y (xO (times x' y))); Rewrite convert_add; - Unfold 2 3 convert; Simpl; Do 2 Rewrite ZL6; Rewrite H; - Rewrite -> mult_plus_distr; Reflexivity -| Unfold 1 2 convert; Simpl; Do 2 Rewrite ZL6; - Rewrite H; Rewrite mult_plus_distr; Reflexivity -| Simpl; Rewrite <- plus_n_O; Reflexivity ]. -Qed. -V7only [ - Comments "Compatibility with the old version of times and times_convert". - Syntactic Definition times1 := - [x:positive;_:positive->positive;y:positive](times x y). - Syntactic Definition times1_convert := - [x,y:positive;_:positive->positive](times_convert x y). -]. - -(** [nat_of_P] maps to the strictly positive subset of [nat] *) - -Lemma ZL4: (y:positive) (EX h:nat |(convert y)=(S h)). -Proof. -Intro y; NewInduction y as [p H|p H|]; [ - NewDestruct H as [x H1]; Exists (plus (S x) (S x)); - Unfold convert ;Simpl; Change (2) with (plus (1) (1)); Rewrite ZL2; Unfold convert in H1; - Rewrite H1; Auto with arith -| NewDestruct H as [x H2]; Exists (plus x (S x)); Unfold convert; - Simpl; Change (2) with (plus (1) (1)); Rewrite ZL2;Unfold convert in H2; Rewrite H2; Auto with arith -| Exists O ;Auto with arith ]. -Qed. - -(** Extra lemmas on [lt] on Peano natural numbers *) - -Lemma ZL7: - (m,n:nat) (lt m n) -> (lt (plus m m) (plus n n)). -Proof. -Intros m n H; Apply lt_trans with m:=(plus m n); [ - Apply lt_reg_l with 1:=H -| Rewrite (plus_sym m n); Apply lt_reg_l with 1:=H ]. -Qed. - -Lemma ZL8: - (m,n:nat) (lt m n) -> (lt (S (plus m m)) (plus n n)). -Proof. -Intros m n H; Apply le_lt_trans with m:=(plus m n); [ - Change (lt (plus m m) (plus m n)) ; Apply lt_reg_l with 1:=H -| Rewrite (plus_sym m n); Apply lt_reg_l with 1:=H ]. -Qed. - -(** [nat_of_P] is a morphism from [positive] to [nat] for [lt] (expressed - from [compare] on [positive]) - - Part 1: [lt] on [positive] is finer than [lt] on [nat] -*) - -Lemma compare_convert_INFERIEUR : - (x,y:positive) (compare x y EGAL) = INFERIEUR -> - (lt (convert x) (convert y)). -Proof. -Intro x; NewInduction x as [p H|p H|];Intro y; NewDestruct y as [q|q|]; - Intro H2; [ - Unfold convert ;Simpl; Apply lt_n_S; - Do 2 Rewrite ZL6; Apply ZL7; Apply H; Simpl in H2; Assumption -| Unfold convert ;Simpl; Do 2 Rewrite ZL6; - Apply ZL8; Apply H;Simpl in H2; Apply ZLSI;Assumption -| Simpl; Discriminate H2 -| Simpl; Unfold convert ;Simpl;Do 2 Rewrite ZL6; - Elim (ZLII p q H2); [ - Intros H3;Apply lt_S;Apply ZL7; Apply H;Apply H3 - | Intros E;Rewrite E;Apply lt_n_Sn] -| Simpl; Unfold convert ;Simpl;Do 2 Rewrite ZL6; - Apply ZL7;Apply H;Assumption -| Simpl; Discriminate H2 -| Unfold convert ;Simpl; Apply lt_n_S; Rewrite ZL6; - Elim (ZL4 q);Intros h H3; Rewrite H3;Simpl; Apply lt_O_Sn -| Unfold convert ;Simpl; Rewrite ZL6; Elim (ZL4 q);Intros h H3; - Rewrite H3; Simpl; Rewrite <- plus_n_Sm; Apply lt_n_S; Apply lt_O_Sn -| Simpl; Discriminate H2 ]. -Qed. - -(** [nat_of_P] is a morphism from [positive] to [nat] for [gt] (expressed - from [compare] on [positive]) - - Part 1: [gt] on [positive] is finer than [gt] on [nat] -*) - -Lemma compare_convert_SUPERIEUR : - (x,y:positive) (compare x y EGAL)=SUPERIEUR -> (gt (convert x) (convert y)). -Proof. -Unfold gt; Intro x; NewInduction x as [p H|p H|]; - Intro y; NewDestruct y as [q|q|]; Intro H2; [ - Simpl; Unfold convert ;Simpl;Do 2 Rewrite ZL6; - Apply lt_n_S; Apply ZL7; Apply H;Assumption -| Simpl; Unfold convert ;Simpl; Do 2 Rewrite ZL6; - Elim (ZLSS p q H2); [ - Intros H3;Apply lt_S;Apply ZL7;Apply H;Assumption - | Intros E;Rewrite E;Apply lt_n_Sn] -| Unfold convert ;Simpl; Rewrite ZL6;Elim (ZL4 p); - Intros h H3;Rewrite H3;Simpl; Apply lt_n_S; Apply lt_O_Sn -| Simpl;Unfold convert ;Simpl;Do 2 Rewrite ZL6; - Apply ZL8; Apply H; Apply ZLIS; Assumption -| Simpl; Unfold convert ;Simpl;Do 2 Rewrite ZL6; - Apply ZL7;Apply H;Assumption -| Unfold convert ;Simpl; Rewrite ZL6; Elim (ZL4 p); - Intros h H3;Rewrite H3;Simpl; Rewrite <- plus_n_Sm;Apply lt_n_S; - Apply lt_O_Sn -| Simpl; Discriminate H2 -| Simpl; Discriminate H2 -| Simpl; Discriminate H2 ]. -Qed. - -(** [nat_of_P] is a morphism from [positive] to [nat] for [lt] (expressed - from [compare] on [positive]) - - Part 2: [lt] on [nat] is finer than [lt] on [positive] -*) - -Lemma convert_compare_INFERIEUR : - (x,y:positive)(lt (convert x) (convert y)) -> (compare x y EGAL) = INFERIEUR. -Proof. -Intros x y; Unfold gt; Elim (Dcompare (compare x y EGAL)); [ - Intros E; Rewrite (compare_convert_EGAL x y E); - Intros H;Absurd (lt (convert y) (convert y)); [ Apply lt_n_n | Assumption ] -| Intros H;Elim H; [ - Auto - | Intros H1 H2; Absurd (lt (convert x) (convert y)); [ - Apply lt_not_sym; Change (gt (convert x) (convert y)); - Apply compare_convert_SUPERIEUR; Assumption - | Assumption ]]]. -Qed. - -(** [nat_of_P] is a morphism from [positive] to [nat] for [gt] (expressed - from [compare] on [positive]) - - Part 2: [gt] on [nat] is finer than [gt] on [positive] -*) - -Lemma convert_compare_SUPERIEUR : - (x,y:positive)(gt (convert x) (convert y)) -> (compare x y EGAL) = SUPERIEUR. -Proof. -Intros x y; Unfold gt; Elim (Dcompare (compare x y EGAL)); [ - Intros E; Rewrite (compare_convert_EGAL x y E); - Intros H;Absurd (lt (convert y) (convert y)); [ Apply lt_n_n | Assumption ] -| Intros H;Elim H; [ - Intros H1 H2; Absurd (lt (convert y) (convert x)); [ - Apply lt_not_sym; Apply compare_convert_INFERIEUR; Assumption - | Assumption ] - | Auto]]. -Qed. - -(** [nat_of_P] is strictly positive *) - -Lemma compare_positive_to_nat_O : - (p:positive)(m:nat)(le m (positive_to_nat p m)). -NewInduction p; Simpl; Auto with arith. -Intro m; Apply le_trans with (plus m m); Auto with arith. -Qed. - -Lemma compare_convert_O : (p:positive)(lt O (convert p)). -Intro; Unfold convert; Apply lt_le_trans with (S O); Auto with arith. -Apply compare_positive_to_nat_O. -Qed. - -(** Pmult_nat permutes with multiplication *) - -Lemma positive_to_nat_mult : (p:positive) (n,m:nat) - (positive_to_nat p (mult m n))=(mult m (positive_to_nat p n)). -Proof. - Induction p. Intros. Simpl. Rewrite mult_plus_distr_r. Rewrite <- (mult_plus_distr_r m n n). - Rewrite (H (plus n n) m). Reflexivity. - Intros. Simpl. Rewrite <- (mult_plus_distr_r m n n). Apply H. - Trivial. -Qed. - -Lemma positive_to_nat_2 : (p:positive) - (positive_to_nat p (2))=(mult (2) (positive_to_nat p (1))). -Proof. - Intros. Rewrite <- positive_to_nat_mult. Reflexivity. -Qed. - -Lemma positive_to_nat_4 : (p:positive) - (positive_to_nat p (4))=(mult (2) (positive_to_nat p (2))). -Proof. - Intros. Rewrite <- positive_to_nat_mult. Reflexivity. -Qed. - -(** Mapping of xH, xO and xI through [nat_of_P] *) - -Lemma convert_xH : (convert xH)=(1). -Proof. - Reflexivity. -Qed. - -Lemma convert_xO : (p:positive) (convert (xO p))=(mult (2) (convert p)). -Proof. - Induction p. Unfold convert. Simpl. Intros. Rewrite positive_to_nat_2. - Rewrite positive_to_nat_4. Rewrite H. Simpl. Rewrite <- plus_Snm_nSm. Reflexivity. - Unfold convert. Simpl. Intros. Rewrite positive_to_nat_2. Rewrite positive_to_nat_4. - Rewrite H. Reflexivity. - Reflexivity. -Qed. - -Lemma convert_xI : (p:positive) (convert (xI p))=(S (mult (2) (convert p))). -Proof. - Induction p. Unfold convert. Simpl. Intro p0. Intro. Rewrite positive_to_nat_2. - Rewrite positive_to_nat_4; Injection H; Intro H1; Rewrite H1; Rewrite <- plus_Snm_nSm; Reflexivity. - Unfold convert. Simpl. Intros. Rewrite positive_to_nat_2. Rewrite positive_to_nat_4. - Injection H; Intro H1; Rewrite H1; Reflexivity. - Reflexivity. -Qed. - -(**********************************************************************) -(** Properties of the shifted injection from Peano natural numbers to - binary positive numbers *) - -(** Composition of [P_of_succ_nat] and [nat_of_P] is successor on [nat] *) - -Theorem bij1 : (m:nat) (convert (anti_convert m)) = (S m). -Proof. -Intro m; NewInduction m as [|n H]; [ - Reflexivity -| Simpl; Rewrite cvt_add_un; Rewrite H; Auto ]. -Qed. - -(** Miscellaneous lemmas on [P_of_succ_nat] *) - -Lemma ZL3: (x:nat) (add_un (anti_convert (plus x x))) = (xO (anti_convert x)). -Proof. -Intro x; NewInduction x as [|n H]; [ - Simpl; Auto with arith -| Simpl; Rewrite plus_sym; Simpl; Rewrite H; Rewrite ZL1;Auto with arith]. -Qed. - -Lemma ZL5: (x:nat) (anti_convert (plus (S x) (S x))) = (xI (anti_convert x)). -Proof. -Intro x; NewInduction x as [|n H];Simpl; [ - Auto with arith -| Rewrite <- plus_n_Sm; Simpl; Simpl in H; Rewrite H; Auto with arith]. -Qed. - -(** Composition of [nat_of_P] and [P_of_succ_nat] is successor on [positive] *) - -Theorem bij2 : (x:positive) (anti_convert (convert x)) = (add_un x). -Proof. -Intro x; NewInduction x as [p H|p H|]; [ - Simpl; Rewrite <- H; Change (2) with (plus (1) (1)); - Rewrite ZL2; Elim (ZL4 p); - Unfold convert; Intros n H1;Rewrite H1; Rewrite ZL3; Auto with arith -| Unfold convert ;Simpl; Change (2) with (plus (1) (1)); - Rewrite ZL2; - Rewrite <- (sub_add_one - (anti_convert - (plus (positive_to_nat p (S O)) (positive_to_nat p (S O))))); - Rewrite <- (sub_add_one (xI p)); - Simpl;Rewrite <- H;Elim (ZL4 p); Unfold convert ;Intros n H1;Rewrite H1; - Rewrite ZL5; Simpl; Trivial with arith -| Unfold convert; Simpl; Auto with arith ]. -Qed. - -(** Composition of [nat_of_P], [P_of_succ_nat] and [Ppred] is identity - on [positive] *) - -Theorem bij3: (x:positive)(sub_un (anti_convert (convert x))) = x. -Proof. -Intros x; Rewrite bij2; Rewrite sub_add_one; Trivial with arith. -Qed. - -(**********************************************************************) -(** Extra properties of the injection from binary positive numbers to Peano - natural numbers *) - -(** [nat_of_P] is a morphism for subtraction on positive numbers *) - -Theorem true_sub_convert: - (x,y:positive) (compare x y EGAL) = SUPERIEUR -> - (convert (true_sub x y)) = (minus (convert x) (convert y)). -Proof. -Intros x y H; Apply plus_reg_l with (convert y); -Rewrite le_plus_minus_r; [ - Rewrite <- convert_add; Rewrite sub_add; Auto with arith -| Apply lt_le_weak; Exact (compare_convert_SUPERIEUR x y H)]. -Qed. - -(** [nat_of_P] is injective *) - -Lemma convert_intro : (x,y:positive)(convert x)=(convert y) -> x=y. -Proof. -Intros x y H;Rewrite <- (bij3 x);Rewrite <- (bij3 y); Rewrite H; Trivial with arith. -Qed. - -Lemma ZL16: (p,q:positive)(lt (minus (convert p) (convert q)) (convert p)). -Proof. -Intros p q; Elim (ZL4 p);Elim (ZL4 q); Intros h H1 i H2; -Rewrite H1;Rewrite H2; Simpl;Unfold lt; Apply le_n_S; Apply le_minus. -Qed. - -Lemma ZL17: (p,q:positive)(lt (convert p) (convert (add p q))). -Proof. -Intros p q; Rewrite convert_add;Unfold lt;Elim (ZL4 q); Intros k H;Rewrite H; -Rewrite plus_sym;Simpl; Apply le_n_S; Apply le_plus_r. -Qed. - -(** Comparison and subtraction *) - -Lemma compare_true_sub_right : - (p,q,z:positive) - (compare q p EGAL)=INFERIEUR-> - (compare z p EGAL)=SUPERIEUR-> - (compare z q EGAL)=SUPERIEUR-> - (compare (true_sub z p) (true_sub z q) EGAL)=INFERIEUR. -Proof. -Intros; Apply convert_compare_INFERIEUR; Rewrite true_sub_convert; [ - Rewrite true_sub_convert; [ - Apply simpl_lt_plus_l with p:=(convert q); Rewrite le_plus_minus_r; [ - Rewrite plus_sym; Apply simpl_lt_plus_l with p:=(convert p); - Rewrite plus_assoc_l; Rewrite le_plus_minus_r; [ - Rewrite (plus_sym (convert p)); Apply lt_reg_l; - Apply compare_convert_INFERIEUR; Assumption - | Apply lt_le_weak; Apply compare_convert_INFERIEUR; - Apply ZC1; Assumption ] - | Apply lt_le_weak;Apply compare_convert_INFERIEUR; - Apply ZC1; Assumption ] - | Assumption ] - | Assumption ]. -Qed. - -Lemma compare_true_sub_left : - (p,q,z:positive) - (compare q p EGAL)=INFERIEUR-> - (compare p z EGAL)=SUPERIEUR-> - (compare q z EGAL)=SUPERIEUR-> - (compare (true_sub q z) (true_sub p z) EGAL)=INFERIEUR. -Proof. -Intros p q z; Intros; - Apply convert_compare_INFERIEUR; Rewrite true_sub_convert; [ - Rewrite true_sub_convert; [ - Unfold gt; Apply simpl_lt_plus_l with p:=(convert z); - Rewrite le_plus_minus_r; [ - Rewrite le_plus_minus_r; [ - Apply compare_convert_INFERIEUR;Assumption - | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Apply ZC1;Assumption] - | Apply lt_le_weak; Apply compare_convert_INFERIEUR;Apply ZC1; Assumption] - | Assumption] -| Assumption]. -Qed. - -(** Distributivity of multiplication over subtraction *) - -Theorem times_true_sub_distr: - (x,y,z:positive) (compare y z EGAL) = SUPERIEUR -> - (times x (true_sub y z)) = (true_sub (times x y) (times x z)). -Proof. -Intros x y z H; Apply convert_intro; -Rewrite times_convert; Rewrite true_sub_convert; [ - Rewrite true_sub_convert; [ - Do 2 Rewrite times_convert; - Do 3 Rewrite (mult_sym (convert x));Apply mult_minus_distr - | Apply convert_compare_SUPERIEUR; Do 2 Rewrite times_convert; - Unfold gt; Elim (ZL4 x);Intros h H1;Rewrite H1; Apply lt_mult_left; - Exact (compare_convert_SUPERIEUR y z H) ] -| Assumption ]. -Qed. - |