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diff --git a/theories7/ZArith/BinInt.v b/theories7/ZArith/BinInt.v deleted file mode 100644 index 9071896b..00000000 --- a/theories7/ZArith/BinInt.v +++ /dev/null @@ -1,1005 +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: BinInt.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*) - -(***********************************************************) -(** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) -(***********************************************************) - -Require Export BinPos. -Require Export Pnat. -Require BinNat. -Require Plus. -Require Mult. -(**********************************************************************) -(** Binary integer numbers *) - -Inductive Z : Set := - ZERO : Z | POS : positive -> Z | NEG : positive -> Z. - -(** Declare Scope Z_scope with Key Z *) -Delimits Scope Z_scope with Z. - -(** Automatically open scope positive_scope for the constructors of Z *) -Bind Scope Z_scope with Z. -Arguments Scope POS [ positive_scope ]. -Arguments Scope NEG [ positive_scope ]. - -(** Subtraction of positive into Z *) - -Definition Zdouble_plus_one [x:Z] := - Cases x of - | ZERO => (POS xH) - | (POS p) => (POS (xI p)) - | (NEG p) => (NEG (double_moins_un p)) - end. - -Definition Zdouble_minus_one [x:Z] := - Cases x of - | ZERO => (NEG xH) - | (NEG p) => (NEG (xI p)) - | (POS p) => (POS (double_moins_un p)) - end. - -Definition Zdouble [x:Z] := - Cases x of - | ZERO => ZERO - | (POS p) => (POS (xO p)) - | (NEG p) => (NEG (xO p)) - end. - -Fixpoint ZPminus [x,y:positive] : Z := - Cases x y of - | (xI x') (xI y') => (Zdouble (ZPminus x' y')) - | (xI x') (xO y') => (Zdouble_plus_one (ZPminus x' y')) - | (xI x') xH => (POS (xO x')) - | (xO x') (xI y') => (Zdouble_minus_one (ZPminus x' y')) - | (xO x') (xO y') => (Zdouble (ZPminus x' y')) - | (xO x') xH => (POS (double_moins_un x')) - | xH (xI y') => (NEG (xO y')) - | xH (xO y') => (NEG (double_moins_un y')) - | xH xH => ZERO - end. - -(** Addition on integers *) - -Definition Zplus := [x,y:Z] - Cases x y of - ZERO y => y - | x ZERO => x - | (POS x') (POS y') => (POS (add x' y')) - | (POS x') (NEG y') => - Cases (compare x' y' EGAL) of - | EGAL => ZERO - | INFERIEUR => (NEG (true_sub y' x')) - | SUPERIEUR => (POS (true_sub x' y')) - end - | (NEG x') (POS y') => - Cases (compare x' y' EGAL) of - | EGAL => ZERO - | INFERIEUR => (POS (true_sub y' x')) - | SUPERIEUR => (NEG (true_sub x' y')) - end - | (NEG x') (NEG y') => (NEG (add x' y')) - end. - -V8Infix "+" Zplus : Z_scope. - -(** Opposite *) - -Definition Zopp := [x:Z] - Cases x of - ZERO => ZERO - | (POS x) => (NEG x) - | (NEG x) => (POS x) - end. - -V8Notation "- x" := (Zopp x) : Z_scope. - -(** Successor on integers *) - -Definition Zs := [x:Z](Zplus x (POS xH)). - -(** Predecessor on integers *) - -Definition Zpred := [x:Z](Zplus x (NEG xH)). - -(** Subtraction on integers *) - -Definition Zminus := [m,n:Z](Zplus m (Zopp n)). - -V8Infix "-" Zminus : Z_scope. - -(** Multiplication on integers *) - -Definition Zmult := [x,y:Z] - Cases x y of - | ZERO _ => ZERO - | _ ZERO => ZERO - | (POS x') (POS y') => (POS (times x' y')) - | (POS x') (NEG y') => (NEG (times x' y')) - | (NEG x') (POS y') => (NEG (times x' y')) - | (NEG x') (NEG y') => (POS (times x' y')) - end. - -V8Infix "*" Zmult : Z_scope. - -(** Comparison of integers *) - -Definition Zcompare := [x,y:Z] - Cases x y of - | ZERO ZERO => EGAL - | ZERO (POS y') => INFERIEUR - | ZERO (NEG y') => SUPERIEUR - | (POS x') ZERO => SUPERIEUR - | (POS x') (POS y') => (compare x' y' EGAL) - | (POS x') (NEG y') => SUPERIEUR - | (NEG x') ZERO => INFERIEUR - | (NEG x') (POS y') => INFERIEUR - | (NEG x') (NEG y') => (Op (compare x' y' EGAL)) - end. - -V8Infix "?=" Zcompare (at level 70, no associativity) : Z_scope. - -Tactic Definition ElimCompare com1 com2:= - Case (Dcompare (Zcompare com1 com2)); [ Idtac | - Let x = FreshId "H" In Intro x; Case x; Clear x ]. - -(** Sign function *) - -Definition Zsgn [z:Z] : Z := - Cases z of - ZERO => ZERO - | (POS p) => (POS xH) - | (NEG p) => (NEG xH) - end. - -(** Direct, easier to handle variants of successor and addition *) - -Definition Zsucc' [x:Z] := - Cases x of - | ZERO => (POS xH) - | (POS x') => (POS (add_un x')) - | (NEG x') => (ZPminus xH x') - end. - -Definition Zpred' [x:Z] := - Cases x of - | ZERO => (NEG xH) - | (POS x') => (ZPminus x' xH) - | (NEG x') => (NEG (add_un x')) - end. - -Definition Zplus' := [x,y:Z] - Cases x y of - ZERO y => y - | x ZERO => x - | (POS x') (POS y') => (POS (add x' y')) - | (POS x') (NEG y') => (ZPminus x' y') - | (NEG x') (POS y') => (ZPminus y' x') - | (NEG x') (NEG y') => (NEG (add x' y')) - end. - -Open Local Scope Z_scope. - -(**********************************************************************) -(** Inductive specification of Z *) - -Theorem Zind : (P:(Z ->Prop)) - (P ZERO) -> ((x:Z)(P x) ->(P (Zsucc' x))) -> ((x:Z)(P x) ->(P (Zpred' x))) -> - (z:Z)(P z). -Proof. -Intros P H0 Hs Hp z; NewDestruct z. - Assumption. - Apply Pind with P:=[p](P (POS p)). - Change (P (Zsucc' ZERO)); Apply Hs; Apply H0. - Intro n; Exact (Hs (POS n)). - Apply Pind with P:=[p](P (NEG p)). - Change (P (Zpred' ZERO)); Apply Hp; Apply H0. - Intro n; Exact (Hp (NEG n)). -Qed. - -(**********************************************************************) -(** Properties of opposite on binary integer numbers *) - -Theorem Zopp_NEG : (x:positive) (Zopp (NEG x)) = (POS x). -Proof. -Reflexivity. -Qed. - -(** [opp] is involutive *) - -Theorem Zopp_Zopp: (x:Z) (Zopp (Zopp x)) = x. -Proof. -Intro x; NewDestruct x; Reflexivity. -Qed. - -(** Injectivity of the opposite *) - -Theorem Zopp_intro : (x,y:Z) (Zopp x) = (Zopp y) -> x = y. -Proof. -Intros x y;Case x;Case y;Simpl;Intros; [ - Trivial | Discriminate H | Discriminate H | Discriminate H -| Simplify_eq H; Intro E; Rewrite E; Trivial -| Discriminate H | Discriminate H | Discriminate H -| Simplify_eq H; Intro E; Rewrite E; Trivial ]. -Qed. - -(**********************************************************************) -(* Properties of the direct definition of successor and predecessor *) - -Lemma Zpred'_succ' : (x:Z)(Zpred' (Zsucc' x))=x. -Proof. -Intro x; NewDestruct x; Simpl. - Reflexivity. -NewDestruct p; Simpl; Try Rewrite double_moins_un_add_un_xI; Reflexivity. -NewDestruct p; Simpl; Try Rewrite is_double_moins_un; Reflexivity. -Qed. - -Lemma Zsucc'_discr : (x:Z)x<>(Zsucc' x). -Proof. -Intro x; NewDestruct x; Simpl. - Discriminate. - Injection; Apply add_un_discr. - NewDestruct p; Simpl. - Discriminate. - Intro H; Symmetry in H; Injection H; Apply double_moins_un_xO_discr. - Discriminate. -Qed. - -(**********************************************************************) -(** Other properties of binary integer numbers *) - -Lemma ZL0 : (S (S O))=(plus (S O) (S O)). -Proof. -Reflexivity. -Qed. - -(**********************************************************************) -(** Properties of the addition on integers *) - -(** zero is left neutral for addition *) - -Theorem Zero_left: (x:Z) (Zplus ZERO x) = x. -Proof. -Intro x; NewDestruct x; Reflexivity. -Qed. - -(** zero is right neutral for addition *) - -Theorem Zero_right: (x:Z) (Zplus x ZERO) = x. -Proof. -Intro x; NewDestruct x; Reflexivity. -Qed. - -(** addition is commutative *) - -Theorem Zplus_sym: (x,y:Z) (Zplus x y) = (Zplus y x). -Proof. -Intro x;NewInduction x as [|p|p];Intro y; NewDestruct y as [|q|q];Simpl;Try Reflexivity. - Rewrite add_sym; Reflexivity. - Rewrite ZC4; NewDestruct (compare q p EGAL); Reflexivity. - Rewrite ZC4; NewDestruct (compare q p EGAL); Reflexivity. - Rewrite add_sym; Reflexivity. -Qed. - -(** opposite distributes over addition *) - -Theorem Zopp_Zplus: - (x,y:Z) (Zopp (Zplus x y)) = (Zplus (Zopp x) (Zopp y)). -Proof. -Intro x; NewDestruct x as [|p|p]; Intro y; NewDestruct y as [|q|q]; Simpl; - Reflexivity Orelse NewDestruct (compare p q EGAL); Reflexivity. -Qed. - -(** opposite is inverse for addition *) - -Theorem Zplus_inverse_r: (x:Z) (Zplus x (Zopp x)) = ZERO. -Proof. -Intro x; NewDestruct x as [|p|p]; Simpl; [ - Reflexivity -| Rewrite (convert_compare_EGAL p); Reflexivity -| Rewrite (convert_compare_EGAL p); Reflexivity ]. -Qed. - -Theorem Zplus_inverse_l: (x:Z) (Zplus (Zopp x) x) = ZERO. -Proof. -Intro; Rewrite Zplus_sym; Apply Zplus_inverse_r. -Qed. - -Hints Local Resolve Zero_left Zero_right. - -(** addition is associative *) - -Lemma weak_assoc : - (x,y:positive)(z:Z) (Zplus (POS x) (Zplus (POS y) z))= - (Zplus (Zplus (POS x) (POS y)) z). -Proof. -Intros x y z';Case z'; [ - Auto with arith -| Intros z;Simpl; Rewrite add_assoc;Auto with arith -| Intros z; Simpl; ElimPcompare y z; - Intros E0;Rewrite E0; - ElimPcompare '(add x y) 'z;Intros E1;Rewrite E1; [ - Absurd (compare (add x y) z EGAL)=EGAL; [ (* Case 1 *) - Rewrite convert_compare_SUPERIEUR; [ - Discriminate - | Rewrite convert_add; Rewrite (compare_convert_EGAL y z E0); - Elim (ZL4 x);Intros k E2;Rewrite E2; Simpl; Unfold gt lt; Apply le_n_S; - Apply le_plus_r ] - | Assumption ] - | Absurd (compare (add x y) z EGAL)=INFERIEUR; [ (* Case 2 *) - Rewrite convert_compare_SUPERIEUR; [ - Discriminate - | Rewrite convert_add; Rewrite (compare_convert_EGAL y z E0); - Elim (ZL4 x);Intros k E2;Rewrite E2; Simpl; Unfold gt lt; Apply le_n_S; - Apply le_plus_r] - | Assumption ] - | Rewrite (compare_convert_EGAL y z E0); (* Case 3 *) - Elim (sub_pos_SUPERIEUR (add x z) z);[ - Intros t H; Elim H;Intros H1 H2;Elim H2;Intros H3 H4; - Unfold true_sub; Rewrite H1; Cut x=t; [ - Intros E;Rewrite E;Auto with arith - | Apply simpl_add_r with z:=z; Rewrite <- H3; Rewrite add_sym; Trivial with arith ] - | Pattern 1 z; Rewrite <- (compare_convert_EGAL y z E0); Assumption ] - | Elim (sub_pos_SUPERIEUR z y); [ (* Case 4 *) - Intros k H;Elim H;Intros H1 H2;Elim H2;Intros H3 H4; Unfold 1 true_sub; - Rewrite H1; Cut x=k; [ - Intros E;Rewrite E; Rewrite (convert_compare_EGAL k); Trivial with arith - | Apply simpl_add_r with z:=y; Rewrite (add_sym k y); Rewrite H3; - Apply compare_convert_EGAL; Assumption ] - | Apply ZC2;Assumption] - | Elim (sub_pos_SUPERIEUR z y); [ (* Case 5 *) - Intros k H;Elim H;Intros H1 H2;Elim H2;Intros H3 H4; - Unfold 1 3 5 true_sub; Rewrite H1; - Cut (compare x k EGAL)=INFERIEUR; [ - Intros E2;Rewrite E2; Elim (sub_pos_SUPERIEUR k x); [ - Intros i H5;Elim H5;Intros H6 H7;Elim H7;Intros H8 H9; - Elim (sub_pos_SUPERIEUR z (add x y)); [ - Intros j H10;Elim H10;Intros H11 H12;Elim H12;Intros H13 H14; - Unfold true_sub ;Rewrite H6;Rewrite H11; Cut i=j; [ - Intros E;Rewrite E;Auto with arith - | Apply (simpl_add_l (add x y)); Rewrite H13; - Rewrite (add_sym x y); Rewrite <- add_assoc; Rewrite H8; - Assumption ] - | Apply ZC2; Assumption] - | Apply ZC2;Assumption] - | Apply convert_compare_INFERIEUR; - Apply simpl_lt_plus_l with p:=(convert y); - Do 2 Rewrite <- convert_add; Apply compare_convert_INFERIEUR; - Rewrite H3; Rewrite add_sym; Assumption ] - | Apply ZC2; Assumption ] - | Elim (sub_pos_SUPERIEUR z y); [ (* Case 6 *) - Intros k H;Elim H;Intros H1 H2;Elim H2;Intros H3 H4; - Elim (sub_pos_SUPERIEUR (add x y) z); [ - Intros i H5;Elim H5;Intros H6 H7;Elim H7;Intros H8 H9; - Unfold true_sub; Rewrite H1;Rewrite H6; - Cut (compare x k EGAL)=SUPERIEUR; [ - Intros H10;Elim (sub_pos_SUPERIEUR x k H10); - Intros j H11;Elim H11;Intros H12 H13;Elim H13;Intros H14 H15; - Rewrite H10; Rewrite H12; Cut i=j; [ - Intros H16;Rewrite H16;Auto with arith - | Apply (simpl_add_l (add z k)); Rewrite <- (add_assoc z k j); - Rewrite H14; Rewrite (add_sym z k); Rewrite <- add_assoc; - Rewrite H8; Rewrite (add_sym x y); Rewrite add_assoc; - Rewrite (add_sym k y); Rewrite H3; Trivial with arith] - | Apply convert_compare_SUPERIEUR; Unfold lt gt; - Apply simpl_lt_plus_l with p:=(convert y); - Do 2 Rewrite <- convert_add; Apply compare_convert_INFERIEUR; - Rewrite H3; Rewrite add_sym; Apply ZC1; Assumption ] - | Assumption ] - | Apply ZC2;Assumption ] - | Absurd (compare (add x y) z EGAL)=EGAL; [ (* Case 7 *) - Rewrite convert_compare_SUPERIEUR; [ - Discriminate - | Rewrite convert_add; Unfold gt;Apply lt_le_trans with m:=(convert y);[ - Apply compare_convert_INFERIEUR; Apply ZC1; Assumption - | Apply le_plus_r]] - | Assumption ] - | Absurd (compare (add x y) z EGAL)=INFERIEUR; [ (* Case 8 *) - Rewrite convert_compare_SUPERIEUR; [ - Discriminate - | Unfold gt; Apply lt_le_trans with m:=(convert y);[ - Exact (compare_convert_SUPERIEUR y z E0) - | Rewrite convert_add; Apply le_plus_r]] - | Assumption ] - | Elim sub_pos_SUPERIEUR with 1:=E0;Intros k H1; (* Case 9 *) - Elim sub_pos_SUPERIEUR with 1:=E1; Intros i H2;Elim H1;Intros H3 H4; - Elim H4;Intros H5 H6; Elim H2;Intros H7 H8;Elim H8;Intros H9 H10; - Unfold true_sub ;Rewrite H3;Rewrite H7; Cut (add x k)=i; [ - Intros E;Rewrite E;Auto with arith - | Apply (simpl_add_l z);Rewrite (add_sym x k); - Rewrite add_assoc; Rewrite H5;Rewrite H9; - Rewrite add_sym; Trivial with arith ]]]. -Qed. - -Hints Local Resolve weak_assoc. - -Theorem Zplus_assoc : - (n,m,p:Z) (Zplus n (Zplus m p))= (Zplus (Zplus n m) p). -Proof. -Intros x y z;Case x;Case y;Case z;Auto with arith; Intros; [ - Rewrite (Zplus_sym (NEG p0)); Rewrite weak_assoc; - Rewrite (Zplus_sym (Zplus (POS p1) (NEG p0))); Rewrite weak_assoc; - Rewrite (Zplus_sym (POS p1)); Trivial with arith -| Apply Zopp_intro; Do 4 Rewrite Zopp_Zplus; - Do 2 Rewrite Zopp_NEG; Rewrite Zplus_sym; Rewrite <- weak_assoc; - Rewrite (Zplus_sym (Zopp (POS p1))); - Rewrite (Zplus_sym (Zplus (POS p0) (Zopp (POS p1)))); - Rewrite (weak_assoc p); Rewrite weak_assoc; Rewrite (Zplus_sym (POS p0)); - Trivial with arith -| Rewrite Zplus_sym; Rewrite (Zplus_sym (POS p0) (POS p)); - Rewrite <- weak_assoc; Rewrite Zplus_sym; Rewrite (Zplus_sym (POS p0)); - Trivial with arith -| Apply Zopp_intro; Do 4 Rewrite Zopp_Zplus; - Do 2 Rewrite Zopp_NEG; Rewrite (Zplus_sym (Zopp (POS p0))); - Rewrite weak_assoc; Rewrite (Zplus_sym (Zplus (POS p1) (Zopp (POS p0)))); - Rewrite weak_assoc;Rewrite (Zplus_sym (POS p)); Trivial with arith -| Apply Zopp_intro; Do 4 Rewrite Zopp_Zplus; Do 2 Rewrite Zopp_NEG; - Apply weak_assoc -| Apply Zopp_intro; Do 4 Rewrite Zopp_Zplus; Do 2 Rewrite Zopp_NEG; - Apply weak_assoc] -. -Qed. - -V7only [Notation Zplus_assoc_l := Zplus_assoc.]. - -Lemma Zplus_assoc_r : (n,m,p:Z)(Zplus (Zplus n m) p) =(Zplus n (Zplus m p)). -Proof. -Intros; Symmetry; Apply Zplus_assoc. -Qed. - -(** Associativity mixed with commutativity *) - -Theorem Zplus_permute : (n,m,p:Z) (Zplus n (Zplus m p))=(Zplus m (Zplus n p)). -Proof. -Intros n m p; -Rewrite Zplus_sym;Rewrite <- Zplus_assoc; Rewrite (Zplus_sym p n); Trivial with arith. -Qed. - -(** addition simplifies *) - -Theorem Zsimpl_plus_l : (n,m,p:Z)(Zplus n m)=(Zplus n p)->m=p. -Intros n m p H; Cut (Zplus (Zopp n) (Zplus n m))=(Zplus (Zopp n) (Zplus n p));[ - Do 2 Rewrite -> Zplus_assoc; Rewrite -> (Zplus_sym (Zopp n) n); - Rewrite -> Zplus_inverse_r;Simpl; Trivial with arith -| Rewrite -> H; Trivial with arith ]. -Qed. - -(** addition and successor permutes *) - -Lemma Zplus_S_n: (x,y:Z) (Zplus (Zs x) y) = (Zs (Zplus x y)). -Proof. -Intros x y; Unfold Zs; Rewrite (Zplus_sym (Zplus x y)); Rewrite Zplus_assoc; -Rewrite (Zplus_sym (POS xH)); Trivial with arith. -Qed. - -Lemma Zplus_n_Sm : (n,m:Z) (Zs (Zplus n m))=(Zplus n (Zs m)). -Proof. -Intros n m; Unfold Zs; Rewrite Zplus_assoc; Trivial with arith. -Qed. - -Lemma Zplus_Snm_nSm : (n,m:Z)(Zplus (Zs n) m)=(Zplus n (Zs m)). -Proof. -Unfold Zs ;Intros n m; Rewrite <- Zplus_assoc; Rewrite (Zplus_sym (POS xH)); -Trivial with arith. -Qed. - -(** Misc properties, usually redundant or non natural *) - -Lemma Zplus_n_O : (n:Z) n=(Zplus n ZERO). -Proof. -Symmetry; Apply Zero_right. -Qed. - -Lemma Zplus_unit_left : (n,m:Z) (Zplus n ZERO)=m -> n=m. -Proof. -Intros n m; Rewrite Zero_right; Intro; Assumption. -Qed. - -Lemma Zplus_unit_right : (n,m:Z) n=(Zplus m ZERO) -> n=m. -Proof. -Intros n m; Rewrite Zero_right; Intro; Assumption. -Qed. - -Lemma Zplus_simpl : (x,y,z,t:Z) x=y -> z=t -> (Zplus x z)=(Zplus y t). -Proof. -Intros; Rewrite H; Rewrite H0; Reflexivity. -Qed. - -Lemma Zplus_Zopp_expand : (x,y,z:Z) - (Zplus x (Zopp y))=(Zplus (Zplus x (Zopp z)) (Zplus z (Zopp y))). -Proof. -Intros x y z. -Rewrite <- (Zplus_assoc x). -Rewrite (Zplus_assoc (Zopp z)). -Rewrite Zplus_inverse_l. -Reflexivity. -Qed. - -(**********************************************************************) -(** Properties of successor and predecessor on binary integer numbers *) - -Theorem Zn_Sn : (x:Z) ~ x=(Zs x). -Proof. -Intros n;Cut ~ZERO=(POS xH);[ - Unfold not ;Intros H1 H2;Apply H1;Apply (Zsimpl_plus_l n);Rewrite Zero_right; - Exact H2 -| Discriminate ]. -Qed. - -Theorem add_un_Zs : (x:positive) (POS (add_un x)) = (Zs (POS x)). -Proof. -Intro; Rewrite -> ZL12; Unfold Zs; Simpl; Trivial with arith. -Qed. - -(** successor and predecessor are inverse functions *) - -Theorem Zs_pred : (n:Z) n=(Zs (Zpred n)). -Proof. -Intros n; Unfold Zs Zpred ;Rewrite <- Zplus_assoc; Simpl; Rewrite Zero_right; -Trivial with arith. -Qed. - -Hints Immediate Zs_pred : zarith. - -Theorem Zpred_Sn : (x:Z) x=(Zpred (Zs x)). -Proof. -Intros m; Unfold Zpred Zs; Rewrite <- Zplus_assoc; Simpl; -Rewrite Zplus_sym; Auto with arith. -Qed. - -Theorem Zeq_add_S : (n,m:Z) (Zs n)=(Zs m) -> n=m. -Proof. -Intros n m H. -Change (Zplus (Zplus (NEG xH) (POS xH)) n)= - (Zplus (Zplus (NEG xH) (POS xH)) m); -Do 2 Rewrite <- Zplus_assoc; Do 2 Rewrite (Zplus_sym (POS xH)); -Unfold Zs in H;Rewrite H; Trivial with arith. -Qed. - -(** Misc properties, usually redundant or non natural *) - -Lemma Zeq_S : (n,m:Z) n=m -> (Zs n)=(Zs m). -Proof. -Intros n m H; Rewrite H; Reflexivity. -Qed. - -Lemma Znot_eq_S : (n,m:Z) ~(n=m) -> ~((Zs n)=(Zs m)). -Proof. -Unfold not ;Intros n m H1 H2;Apply H1;Apply Zeq_add_S; Assumption. -Qed. - -(**********************************************************************) -(** Properties of subtraction on binary integer numbers *) - -Lemma Zminus_0_r : (x:Z) (Zminus x ZERO)=x. -Proof. -Intro; Unfold Zminus; Simpl;Rewrite Zero_right; Trivial with arith. -Qed. - -Lemma Zminus_n_O : (x:Z) x=(Zminus x ZERO). -Proof. -Intro; Symmetry; Apply Zminus_0_r. -Qed. - -Lemma Zminus_diag : (n:Z)(Zminus n n)=ZERO. -Proof. -Intro; Unfold Zminus; Rewrite Zplus_inverse_r; Trivial with arith. -Qed. - -Lemma Zminus_n_n : (n:Z)(ZERO=(Zminus n n)). -Proof. -Intro; Symmetry; Apply Zminus_diag. -Qed. - -Lemma Zplus_minus : (x,y,z:Z)(x=(Zplus y z))->(z=(Zminus x y)). -Proof. -Intros n m p H;Unfold Zminus;Apply (Zsimpl_plus_l m); -Rewrite (Zplus_sym m (Zplus n (Zopp m))); Rewrite <- Zplus_assoc; -Rewrite Zplus_inverse_l; Rewrite Zero_right; Rewrite H; Trivial with arith. -Qed. - -Lemma Zminus_plus : (x,y:Z)(Zminus (Zplus x y) x)=y. -Proof. -Intros n m;Unfold Zminus ;Rewrite -> (Zplus_sym n m);Rewrite <- Zplus_assoc; -Rewrite -> Zplus_inverse_r; Apply Zero_right. -Qed. - -Lemma Zle_plus_minus : (n,m:Z) (Zplus n (Zminus m n))=m. -Proof. -Unfold Zminus; Intros n m; Rewrite Zplus_permute; Rewrite Zplus_inverse_r; -Apply Zero_right. -Qed. - -Lemma Zminus_Sn_m : (n,m:Z)((Zs (Zminus n m))=(Zminus (Zs n) m)). -Proof. -Intros n m;Unfold Zminus Zs; Rewrite (Zplus_sym n (Zopp m)); -Rewrite <- Zplus_assoc;Apply Zplus_sym. -Qed. - -Lemma Zminus_plus_simpl_l : - (x,y,z:Z)(Zminus (Zplus z x) (Zplus z y))=(Zminus x y). -Proof. -Intros n m p;Unfold Zminus; Rewrite Zopp_Zplus; Rewrite Zplus_assoc; -Rewrite (Zplus_sym p); Rewrite <- (Zplus_assoc n p); Rewrite Zplus_inverse_r; -Rewrite Zero_right; Trivial with arith. -Qed. - -Lemma Zminus_plus_simpl : - (x,y,z:Z)((Zminus x y)=(Zminus (Zplus z x) (Zplus z y))). -Proof. -Intros; Symmetry; Apply Zminus_plus_simpl_l. -Qed. - -Lemma Zminus_Zplus_compatible : - (x,y,z:Z) (Zminus (Zplus x z) (Zplus y z)) = (Zminus x y). -Intros x y n. -Unfold Zminus. -Rewrite -> Zopp_Zplus. -Rewrite -> (Zplus_sym (Zopp y) (Zopp n)). -Rewrite -> Zplus_assoc. -Rewrite <- (Zplus_assoc x n (Zopp n)). -Rewrite -> (Zplus_inverse_r n). -Rewrite <- Zplus_n_O. -Reflexivity. -Qed. - -(** Misc redundant properties *) - -V7only [Set Implicit Arguments.]. - -Lemma Zeq_Zminus : (x,y:Z)x=y -> (Zminus x y)=ZERO. -Proof. -Intros x y H; Rewrite H; Symmetry; Apply Zminus_n_n. -Qed. - -Lemma Zminus_Zeq : (x,y:Z)(Zminus x y)=ZERO -> x=y. -Proof. -Intros x y H; Rewrite <- (Zle_plus_minus y x); Rewrite H; Apply Zero_right. -Qed. - -V7only [Unset Implicit Arguments.]. - -(**********************************************************************) -(** Properties of multiplication on binary integer numbers *) - -(** One is neutral for multiplication *) - -Theorem Zmult_1_n : (n:Z)(Zmult (POS xH) n)=n. -Proof. -Intro x; NewDestruct x; Reflexivity. -Qed. -V7only [Notation Zmult_one := Zmult_1_n.]. - -Theorem Zmult_n_1 : (n:Z)(Zmult n (POS xH))=n. -Proof. -Intro x; NewDestruct x; Simpl; Try Rewrite times_x_1; Reflexivity. -Qed. - -(** Zero property of multiplication *) - -Theorem Zero_mult_left: (x:Z) (Zmult ZERO x) = ZERO. -Proof. -Intro x; NewDestruct x; Reflexivity. -Qed. - -Theorem Zero_mult_right: (x:Z) (Zmult x ZERO) = ZERO. -Proof. -Intro x; NewDestruct x; Reflexivity. -Qed. - -Hints Local Resolve Zero_mult_left Zero_mult_right. - -Lemma Zmult_n_O : (n:Z) ZERO=(Zmult n ZERO). -Proof. -Intro x; NewDestruct x; Reflexivity. -Qed. - -(** Commutativity of multiplication *) - -Theorem Zmult_sym : (x,y:Z) (Zmult x y) = (Zmult y x). -Proof. -Intros x y; NewDestruct x as [|p|p]; NewDestruct y as [|q|q]; Simpl; - Try Rewrite (times_sym p q); Reflexivity. -Qed. - -(** Associativity of multiplication *) - -Theorem Zmult_assoc : - (x,y,z:Z) (Zmult x (Zmult y z))= (Zmult (Zmult x y) z). -Proof. -Intros x y z; NewDestruct x; NewDestruct y; NewDestruct z; Simpl; - Try Rewrite times_assoc; Reflexivity. -Qed. -V7only [Notation Zmult_assoc_l := Zmult_assoc.]. - -Lemma Zmult_assoc_r : (n,m,p:Z)((Zmult (Zmult n m) p) = (Zmult n (Zmult m p))). -Proof. -Intros n m p; Rewrite Zmult_assoc; Trivial with arith. -Qed. - -(** Associativity mixed with commutativity *) - -Theorem Zmult_permute : (n,m,p:Z)(Zmult n (Zmult m p)) = (Zmult m (Zmult n p)). -Proof. -Intros x y z; Rewrite -> (Zmult_assoc y x z); Rewrite -> (Zmult_sym y x). -Apply Zmult_assoc. -Qed. - -(** Z is integral *) - -Theorem Zmult_eq: (x,y:Z) ~(x=ZERO) -> (Zmult y x) = ZERO -> y = ZERO. -Proof. -Intros x y; NewDestruct x as [|p|p]. - Intro H; Absurd ZERO=ZERO; Trivial. - Intros _ H; NewDestruct y as [|q|q]; Reflexivity Orelse Discriminate. - Intros _ H; NewDestruct y as [|q|q]; Reflexivity Orelse Discriminate. -Qed. - -V7only [Set Implicit Arguments.]. - -Theorem Zmult_zero : (x,y:Z)(Zmult x y)=ZERO -> x=ZERO \/ y=ZERO. -Proof. -Intros x y; NewDestruct x; NewDestruct y; Auto; Simpl; Intro H; Discriminate H. -Qed. - -V7only [Unset Implicit Arguments.]. - -Lemma Zmult_1_inversion_l : - (x,y:Z) (Zmult x y)=(POS xH) -> x=(POS xH) \/ x=(NEG xH). -Proof. -Intros x y; NewDestruct x as [|p|p]; Intro; [ Discriminate | Left | Right ]; - (NewDestruct y as [|q|q]; Try Discriminate; - Simpl in H; Injection H; Clear H; Intro H; - Rewrite times_one_inversion_l with 1:=H; Reflexivity). -Qed. - -(** Multiplication and Opposite *) - -Theorem Zopp_Zmult_l : (x,y:Z)(Zopp (Zmult x y)) = (Zmult (Zopp x) y). -Proof. -Intros x y; NewDestruct x; NewDestruct y; Reflexivity. -Qed. - -Theorem Zopp_Zmult_r : (x,y:Z)(Zopp (Zmult x y)) = (Zmult x (Zopp y)). -Intros x y; Rewrite (Zmult_sym x y); Rewrite Zopp_Zmult_l; Apply Zmult_sym. -Qed. - -Lemma Zopp_Zmult: (x,y:Z) (Zmult (Zopp x) y) = (Zopp (Zmult x y)). -Proof. -Intros x y; Symmetry; Apply Zopp_Zmult_l. -Qed. - -Theorem Zmult_Zopp_left : (x,y:Z)(Zmult (Zopp x) y) = (Zmult x (Zopp y)). -Intros x y; Rewrite Zopp_Zmult; Rewrite Zopp_Zmult_r; Trivial with arith. -Qed. - -Theorem Zmult_Zopp_Zopp: (x,y:Z) (Zmult (Zopp x) (Zopp y)) = (Zmult x y). -Proof. -Intros x y; NewDestruct x; NewDestruct y; Reflexivity. -Qed. - -Theorem Zopp_one : (x:Z)(Zopp x)=(Zmult x (NEG xH)). -Intro x; NewInduction x; Intros; Rewrite Zmult_sym; Auto with arith. -Qed. - -(** Distributivity of multiplication over addition *) - -Lemma weak_Zmult_plus_distr_r: - (x:positive)(y,z:Z) - (Zmult (POS x) (Zplus y z)) = (Zplus (Zmult (POS x) y) (Zmult (POS x) z)). -Proof. -Intros x y' z';Case y';Case z';Auto with arith;Intros y z; - (Simpl; Rewrite times_add_distr; Trivial with arith) -Orelse - (Simpl; ElimPcompare z y; Intros E0;Rewrite E0; [ - Rewrite (compare_convert_EGAL z y E0); - Rewrite (convert_compare_EGAL (times x y)); Trivial with arith - | Cut (compare (times x z) (times x y) EGAL)=INFERIEUR; [ - Intros E;Rewrite E; Rewrite times_true_sub_distr; [ - Trivial with arith - | Apply ZC2;Assumption ] - | Apply convert_compare_INFERIEUR;Do 2 Rewrite times_convert; - Elim (ZL4 x);Intros h H1;Rewrite H1;Apply lt_mult_left; - Exact (compare_convert_INFERIEUR z y E0)] - | Cut (compare (times x z) (times x y) EGAL)=SUPERIEUR; [ - Intros E;Rewrite E; Rewrite times_true_sub_distr; Auto with arith - | Apply convert_compare_SUPERIEUR; Unfold gt; Do 2 Rewrite times_convert; - Elim (ZL4 x);Intros h H1;Rewrite H1;Apply lt_mult_left; - Exact (compare_convert_SUPERIEUR z y E0) ]]). -Qed. - -Theorem Zmult_plus_distr_r: - (x,y,z:Z) (Zmult x (Zplus y z)) = (Zplus (Zmult x y) (Zmult x z)). -Proof. -Intros x y z; Case x; [ - Auto with arith -| Intros x';Apply weak_Zmult_plus_distr_r -| Intros p; Apply Zopp_intro; Rewrite Zopp_Zplus; - Do 3 Rewrite <- Zopp_Zmult; Rewrite Zopp_NEG; - Apply weak_Zmult_plus_distr_r ]. -Qed. - -Theorem Zmult_plus_distr_l : - (n,m,p:Z)((Zmult (Zplus n m) p)=(Zplus (Zmult n p) (Zmult m p))). -Proof. -Intros n m p;Rewrite Zmult_sym;Rewrite Zmult_plus_distr_r; -Do 2 Rewrite -> (Zmult_sym p); Trivial with arith. -Qed. - -(** Distributivity of multiplication over subtraction *) - -Lemma Zmult_Zminus_distr_l : - (x,y,z:Z)((Zmult (Zminus x y) z)=(Zminus (Zmult x z) (Zmult y z))). -Proof. -Intros x y z; Unfold Zminus. -Rewrite <- Zopp_Zmult. -Apply Zmult_plus_distr_l. -Qed. - -V7only [Notation Zmult_minus_distr := Zmult_Zminus_distr_l.]. - -Lemma Zmult_Zminus_distr_r : - (x,y,z:Z)(Zmult z (Zminus x y)) = (Zminus (Zmult z x) (Zmult z y)). -Proof. -Intros x y z; Rewrite (Zmult_sym z (Zminus x y)). -Rewrite (Zmult_sym z x). -Rewrite (Zmult_sym z y). -Apply Zmult_Zminus_distr_l. -Qed. - -(** Simplification of multiplication for non-zero integers *) -V7only [Set Implicit Arguments.]. - -Lemma Zmult_reg_left : (x,y,z:Z) z<>ZERO -> (Zmult z x)=(Zmult z y) -> x=y. -Proof. -Intros x y z H H0. -Generalize (Zeq_Zminus H0). -Intro. -Apply Zminus_Zeq. -Rewrite <- Zmult_Zminus_distr_r in H1. -Clear H0; NewDestruct (Zmult_zero H1). -Contradiction. -Trivial. -Qed. - -Lemma Zmult_reg_right : (x,y,z:Z) z<>ZERO -> (Zmult x z)=(Zmult y z) -> x=y. -Proof. -Intros x y z Hz. -Rewrite (Zmult_sym x z). -Rewrite (Zmult_sym y z). -Intro; Apply Zmult_reg_left with z; Assumption. -Qed. -V7only [Unset Implicit Arguments.]. - -(** Addition and multiplication by 2 *) - -Lemma Zplus_Zmult_2 : (x:Z) (Zplus x x) = (Zmult x (POS (xO xH))). -Proof. -Intros x; Pattern 1 2 x ; Rewrite <- (Zmult_n_1 x); -Rewrite <- Zmult_plus_distr_r; Reflexivity. -Qed. - -(** Multiplication and successor *) - -Lemma Zmult_succ_r : (n,m:Z) (Zmult n (Zs m))=(Zplus (Zmult n m) n). -Proof. -Intros n m;Unfold Zs; Rewrite Zmult_plus_distr_r; -Rewrite (Zmult_sym n (POS xH));Rewrite Zmult_one; Trivial with arith. -Qed. - -Lemma Zmult_n_Sm : (n,m:Z) (Zplus (Zmult n m) n)=(Zmult n (Zs m)). -Proof. -Intros; Symmetry; Apply Zmult_succ_r. -Qed. - -Lemma Zmult_succ_l : (n,m:Z) (Zmult (Zs n) m)=(Zplus (Zmult n m) m). -Proof. -Intros n m; Unfold Zs; Rewrite Zmult_plus_distr_l; Rewrite Zmult_1_n; -Trivial with arith. -Qed. - -Lemma Zmult_Sm_n : (n,m:Z) (Zplus (Zmult n m) m)=(Zmult (Zs n) m). -Proof. -Intros; Symmetry; Apply Zmult_succ_l. -Qed. - -(** Misc redundant properties *) - -Lemma Z_eq_mult: - (x,y:Z) y = ZERO -> (Zmult y x) = ZERO. -Intros x y H; Rewrite H; Auto with arith. -Qed. - -(**********************************************************************) -(** Relating binary positive numbers and binary integers *) - -Lemma POS_xI : (p:positive) (POS (xI p))=(Zplus (Zmult (POS (xO xH)) (POS p)) (POS xH)). -Proof. -Intro; Apply refl_equal. -Qed. - -Lemma POS_xO : (p:positive) (POS (xO p))=(Zmult (POS (xO xH)) (POS p)). -Proof. -Intro; Apply refl_equal. -Qed. - -Lemma NEG_xI : (p:positive) (NEG (xI p))=(Zminus (Zmult (POS (xO xH)) (NEG p)) (POS xH)). -Proof. -Intro; Apply refl_equal. -Qed. - -Lemma NEG_xO : (p:positive) (NEG (xO p))=(Zmult (POS (xO xH)) (NEG p)). -Proof. -Reflexivity. -Qed. - -Lemma POS_add : (p,p':positive)(POS (add p p'))=(Zplus (POS p) (POS p')). -Proof. -Intros p p'; NewDestruct p; NewDestruct p'; Reflexivity. -Qed. - -Lemma NEG_add : (p,p':positive)(NEG (add p p'))=(Zplus (NEG p) (NEG p')). -Proof. -Intros p p'; NewDestruct p; NewDestruct p'; Reflexivity. -Qed. - -(**********************************************************************) -(** Order relations *) - -Definition Zlt := [x,y:Z](Zcompare x y) = INFERIEUR. -Definition Zgt := [x,y:Z](Zcompare x y) = SUPERIEUR. -Definition Zle := [x,y:Z]~(Zcompare x y) = SUPERIEUR. -Definition Zge := [x,y:Z]~(Zcompare x y) = INFERIEUR. -Definition Zne := [x,y:Z] ~(x=y). - -V8Infix "<=" Zle : Z_scope. -V8Infix "<" Zlt : Z_scope. -V8Infix ">=" Zge : Z_scope. -V8Infix ">" Zgt : Z_scope. - -V8Notation "x <= y <= z" := (Zle x y)/\(Zle y z) :Z_scope. -V8Notation "x <= y < z" := (Zle x y)/\(Zlt y z) :Z_scope. -V8Notation "x < y < z" := (Zlt x y)/\(Zlt y z) :Z_scope. -V8Notation "x < y <= z" := (Zlt x y)/\(Zle y z) :Z_scope. - -(**********************************************************************) -(** Absolute value on integers *) - -Definition absolu [x:Z] : nat := - Cases x of - ZERO => O - | (POS p) => (convert p) - | (NEG p) => (convert p) - end. - -Definition Zabs [z:Z] : Z := - Cases z of - ZERO => ZERO - | (POS p) => (POS p) - | (NEG p) => (POS p) - end. - -(**********************************************************************) -(** From [nat] to [Z] *) - -Definition inject_nat := - [x:nat]Cases x of - O => ZERO - | (S y) => (POS (anti_convert y)) - end. - -Require BinNat. - -Definition entier_of_Z := - [z:Z]Cases z of ZERO => Nul | (POS p) => (Pos p) | (NEG p) => (Pos p) end. - -Definition Z_of_entier := - [x:entier]Cases x of Nul => ZERO | (Pos p) => (POS p) end. |