diff options
Diffstat (limited to 'contrib7/ring')
| -rw-r--r-- | contrib7/ring/ArithRing.v | 81 | ||||
| -rw-r--r-- | contrib7/ring/NArithRing.v | 44 | ||||
| -rw-r--r-- | contrib7/ring/Quote.v | 85 | ||||
| -rw-r--r-- | contrib7/ring/Ring.v | 34 | ||||
| -rw-r--r-- | contrib7/ring/Ring_abstract.v | 699 | ||||
| -rw-r--r-- | contrib7/ring/Ring_normalize.v | 893 | ||||
| -rw-r--r-- | contrib7/ring/Ring_theory.v | 384 | ||||
| -rw-r--r-- | contrib7/ring/Setoid_ring.v | 13 | ||||
| -rw-r--r-- | contrib7/ring/Setoid_ring_normalize.v | 1141 | ||||
| -rw-r--r-- | contrib7/ring/Setoid_ring_theory.v | 429 | ||||
| -rw-r--r-- | contrib7/ring/ZArithRing.v | 35 |
11 files changed, 0 insertions, 3838 deletions
diff --git a/contrib7/ring/ArithRing.v b/contrib7/ring/ArithRing.v deleted file mode 100644 index c2abc4d1..00000000 --- a/contrib7/ring/ArithRing.v +++ /dev/null @@ -1,81 +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 *) -(************************************************************************) - -(* $Id: ArithRing.v,v 1.1.2.1 2004/07/16 19:30:18 herbelin Exp $ *) - -(* Instantiation of the Ring tactic for the naturals of Arith $*) - -Require Export Ring. -Require Export Arith. -Require Eqdep_dec. - -V7only [Import nat_scope.]. -Open Local Scope nat_scope. - -Fixpoint nateq [n,m:nat] : bool := - Cases n m of - | O O => true - | (S n') (S m') => (nateq n' m') - | _ _ => false - end. - -Lemma nateq_prop : (n,m:nat)(Is_true (nateq n m))->n==m. -Proof. - Induction n; Induction m; Intros; Try Contradiction. - Trivial. - Unfold Is_true in H1. - Rewrite (H n1 H1). - Trivial. -Save. - -Hints Resolve nateq_prop eq2eqT : arithring. - -Definition NatTheory : (Semi_Ring_Theory plus mult (1) (0) nateq). - Split; Intros; Auto with arith arithring. - Apply eq2eqT; Apply simpl_plus_l with n:=n. - Apply eqT2eq; Trivial. -Defined. - - -Add Semi Ring nat plus mult (1) (0) nateq NatTheory [O S]. - -Goal (n:nat)(S n)=(plus (S O) n). -Intro; Reflexivity. -Save S_to_plus_one. - -(* Replace all occurrences of (S exp) by (plus (S O) exp), except when - exp is already O and only for those occurrences than can be reached by going - down plus and mult operations *) -Recursive Meta Definition S_to_plus t := - Match t With - | [(S O)] -> '(S O) - | [(S ?1)] -> Let t1 = (S_to_plus ?1) In - '(plus (S O) t1) - | [(plus ?1 ?2)] -> Let t1 = (S_to_plus ?1) - And t2 = (S_to_plus ?2) In - '(plus t1 t2) - | [(mult ?1 ?2)] -> Let t1 = (S_to_plus ?1) - And t2 = (S_to_plus ?2) In - '(mult t1 t2) - | [?] -> 't. - -(* Apply S_to_plus on both sides of an equality *) -Tactic Definition S_to_plus_eq := - Match Context With - | [ |- ?1 = ?2 ] -> - (**) Try (**) - Let t1 = (S_to_plus ?1) - And t2 = (S_to_plus ?2) In - Change t1=t2 - | [ |- ?1 == ?2 ] -> - (**) Try (**) - Let t1 = (S_to_plus ?1) - And t2 = (S_to_plus ?2) In - Change (t1==t2). - -Tactic Definition NatRing := S_to_plus_eq;Ring. diff --git a/contrib7/ring/NArithRing.v b/contrib7/ring/NArithRing.v deleted file mode 100644 index f4548bbb..00000000 --- a/contrib7/ring/NArithRing.v +++ /dev/null @@ -1,44 +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 *) -(************************************************************************) - -(* $Id: NArithRing.v,v 1.1.2.1 2004/07/16 19:30:18 herbelin Exp $ *) - -(* Instantiation of the Ring tactic for the binary natural numbers *) - -Require Export Ring. -Require Export ZArith_base. -Require NArith. -Require Eqdep_dec. - -Definition Neq := [n,m:entier] - Cases (Ncompare n m) of - EGAL => true - | _ => false - end. - -Lemma Neq_prop : (n,m:entier)(Is_true (Neq n m)) -> n=m. - Intros n m H; Unfold Neq in H. - Apply Ncompare_Eq_eq. - NewDestruct (Ncompare n m); [Reflexivity | Contradiction | Contradiction ]. -Save. - -Definition NTheory : (Semi_Ring_Theory Nplus Nmult (Pos xH) Nul Neq). - Split. - Apply Nplus_comm. - Apply Nplus_assoc. - Apply Nmult_comm. - Apply Nmult_assoc. - Apply Nplus_0_l. - Apply Nmult_1_l. - Apply Nmult_0_l. - Apply Nmult_plus_distr_r. - Apply Nplus_reg_l. - Apply Neq_prop. -Save. - -Add Semi Ring entier Nplus Nmult (Pos xH) Nul Neq NTheory [Pos Nul xO xI xH]. diff --git a/contrib7/ring/Quote.v b/contrib7/ring/Quote.v deleted file mode 100644 index 12a51c9f..00000000 --- a/contrib7/ring/Quote.v +++ /dev/null @@ -1,85 +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 *) -(************************************************************************) - -(* $Id: Quote.v,v 1.1.2.1 2004/07/16 19:30:18 herbelin Exp $ *) - -(*********************************************************************** - The "abstract" type index is defined to represent variables. - - index : Set - index_eq : index -> bool - index_eq_prop: (n,m:index)(index_eq n m)=true -> n=m - index_lt : index -> bool - varmap : Type -> Type. - varmap_find : (A:Type)A -> index -> (varmap A) -> A. - - The first arg. of varmap_find is the default value to take - if the object is not found in the varmap. - - index_lt defines a total well-founded order, but we don't prove that. - -***********************************************************************) - -Set Implicit Arguments. - -Section variables_map. - -Variable A : Type. - -Inductive varmap : Type := - Empty_vm : varmap -| Node_vm : A->varmap->varmap->varmap. - -Inductive index : Set := -| Left_idx : index -> index -| Right_idx : index -> index -| End_idx : index -. - -Fixpoint varmap_find [default_value:A; i:index; v:varmap] : A := - Cases i v of - End_idx (Node_vm x _ _) => x - | (Right_idx i1) (Node_vm x v1 v2) => (varmap_find default_value i1 v2) - | (Left_idx i1) (Node_vm x v1 v2) => (varmap_find default_value i1 v1) - | _ _ => default_value - end. - -Fixpoint index_eq [n,m:index] : bool := - Cases n m of - | End_idx End_idx => true - | (Left_idx n') (Left_idx m') => (index_eq n' m') - | (Right_idx n') (Right_idx m') => (index_eq n' m') - | _ _ => false - end. - -Fixpoint index_lt[n,m:index] : bool := - Cases n m of - | End_idx (Left_idx _) => true - | End_idx (Right_idx _) => true - | (Left_idx n') (Right_idx m') => true - | (Right_idx n') (Right_idx m') => (index_lt n' m') - | (Left_idx n') (Left_idx m') => (index_lt n' m') - | _ _ => false - end. - -Lemma index_eq_prop : (n,m:index)(index_eq n m)=true -> n=m. - Induction n; Induction m; Simpl; Intros. - Rewrite (H i0 H1); Reflexivity. - Discriminate. - Discriminate. - Discriminate. - Rewrite (H i0 H1); Reflexivity. - Discriminate. - Discriminate. - Discriminate. - Reflexivity. -Save. - -End variables_map. - -Unset Implicit Arguments. diff --git a/contrib7/ring/Ring.v b/contrib7/ring/Ring.v deleted file mode 100644 index 860dda13..00000000 --- a/contrib7/ring/Ring.v +++ /dev/null @@ -1,34 +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 *) -(************************************************************************) - -(* $Id: Ring.v,v 1.1.2.1 2004/07/16 19:30:18 herbelin Exp $ *) - -Require Export Bool. -Require Export Ring_theory. -Require Export Quote. -Require Export Ring_normalize. -Require Export Ring_abstract. - -(* As an example, we provide an instantation for bool. *) -(* Other instatiations are given in ArithRing and ZArithRing in the - same directory *) - -Definition BoolTheory : (Ring_Theory xorb andb true false [b:bool]b eqb). -Split; Simpl. -NewDestruct n; NewDestruct m; Reflexivity. -NewDestruct n; NewDestruct m; NewDestruct p; Reflexivity. -NewDestruct n; NewDestruct m; Reflexivity. -NewDestruct n; NewDestruct m; NewDestruct p; Reflexivity. -NewDestruct n; Reflexivity. -NewDestruct n; Reflexivity. -NewDestruct n; Reflexivity. -NewDestruct n; NewDestruct m; NewDestruct p; Reflexivity. -NewDestruct x; NewDestruct y; Reflexivity Orelse Simpl; Tauto. -Defined. - -Add Ring bool xorb andb true false [b:bool]b eqb BoolTheory [ true false ]. diff --git a/contrib7/ring/Ring_abstract.v b/contrib7/ring/Ring_abstract.v deleted file mode 100644 index 55bb31da..00000000 --- a/contrib7/ring/Ring_abstract.v +++ /dev/null @@ -1,699 +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 *) -(************************************************************************) - -(* $Id: Ring_abstract.v,v 1.1.2.1 2004/07/16 19:30:18 herbelin Exp $ *) - -Require Ring_theory. -Require Quote. -Require Ring_normalize. - -Section abstract_semi_rings. - -Inductive Type aspolynomial := - ASPvar : index -> aspolynomial -| ASP0 : aspolynomial -| ASP1 : aspolynomial -| ASPplus : aspolynomial -> aspolynomial -> aspolynomial -| ASPmult : aspolynomial -> aspolynomial -> aspolynomial -. - -Inductive abstract_sum : Type := -| Nil_acs : abstract_sum -| Cons_acs : varlist -> abstract_sum -> abstract_sum -. - -Fixpoint abstract_sum_merge [s1:abstract_sum] - : abstract_sum -> abstract_sum := -Cases s1 of -| (Cons_acs l1 t1) => - Fix asm_aux{asm_aux[s2:abstract_sum] : abstract_sum := - Cases s2 of - | (Cons_acs l2 t2) => - if (varlist_lt l1 l2) - then (Cons_acs l1 (abstract_sum_merge t1 s2)) - else (Cons_acs l2 (asm_aux t2)) - | Nil_acs => s1 - end} -| Nil_acs => [s2]s2 -end. - -Fixpoint abstract_varlist_insert [l1:varlist; s2:abstract_sum] - : abstract_sum := - Cases s2 of - | (Cons_acs l2 t2) => - if (varlist_lt l1 l2) - then (Cons_acs l1 s2) - else (Cons_acs l2 (abstract_varlist_insert l1 t2)) - | Nil_acs => (Cons_acs l1 Nil_acs) - end. - -Fixpoint abstract_sum_scalar [l1:varlist; s2:abstract_sum] - : abstract_sum := - Cases s2 of - | (Cons_acs l2 t2) => (abstract_varlist_insert (varlist_merge l1 l2) - (abstract_sum_scalar l1 t2)) - | Nil_acs => Nil_acs - end. - -Fixpoint abstract_sum_prod [s1:abstract_sum] - : abstract_sum -> abstract_sum := - [s2]Cases s1 of - | (Cons_acs l1 t1) => - (abstract_sum_merge (abstract_sum_scalar l1 s2) - (abstract_sum_prod t1 s2)) - | Nil_acs => Nil_acs - end. - -Fixpoint aspolynomial_normalize[p:aspolynomial] : abstract_sum := - Cases p of - | (ASPvar i) => (Cons_acs (Cons_var i Nil_var) Nil_acs) - | ASP1 => (Cons_acs Nil_var Nil_acs) - | ASP0 => Nil_acs - | (ASPplus l r) => (abstract_sum_merge (aspolynomial_normalize l) - (aspolynomial_normalize r)) - | (ASPmult l r) => (abstract_sum_prod (aspolynomial_normalize l) - (aspolynomial_normalize r)) - end. - - - -Variable A : Type. -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aeq : A -> A -> bool. -Variable vm : (varmap A). -Variable T : (Semi_Ring_Theory Aplus Amult Aone Azero Aeq). - -Fixpoint interp_asp [p:aspolynomial] : A := - Cases p of - | (ASPvar i) => (interp_var Azero vm i) - | ASP0 => Azero - | ASP1 => Aone - | (ASPplus l r) => (Aplus (interp_asp l) (interp_asp r)) - | (ASPmult l r) => (Amult (interp_asp l) (interp_asp r)) - end. - -(* Local *) Definition iacs_aux := Fix iacs_aux{iacs_aux [a:A; s:abstract_sum] : A := - Cases s of - | Nil_acs => a - | (Cons_acs l t) => (Aplus a (iacs_aux (interp_vl Amult Aone Azero vm l) t)) - end}. - -Definition interp_acs [s:abstract_sum] : A := - Cases s of - | (Cons_acs l t) => (iacs_aux (interp_vl Amult Aone Azero vm l) t) - | Nil_acs => Azero - end. - -Hint SR_plus_sym_T := Resolve (SR_plus_sym T). -Hint SR_plus_assoc_T := Resolve (SR_plus_assoc T). -Hint SR_plus_assoc2_T := Resolve (SR_plus_assoc2 T). -Hint SR_mult_sym_T := Resolve (SR_mult_sym T). -Hint SR_mult_assoc_T := Resolve (SR_mult_assoc T). -Hint SR_mult_assoc2_T := Resolve (SR_mult_assoc2 T). -Hint SR_plus_zero_left_T := Resolve (SR_plus_zero_left T). -Hint SR_plus_zero_left2_T := Resolve (SR_plus_zero_left2 T). -Hint SR_mult_one_left_T := Resolve (SR_mult_one_left T). -Hint SR_mult_one_left2_T := Resolve (SR_mult_one_left2 T). -Hint SR_mult_zero_left_T := Resolve (SR_mult_zero_left T). -Hint SR_mult_zero_left2_T := Resolve (SR_mult_zero_left2 T). -Hint SR_distr_left_T := Resolve (SR_distr_left T). -Hint SR_distr_left2_T := Resolve (SR_distr_left2 T). -Hint SR_plus_reg_left_T := Resolve (SR_plus_reg_left T). -Hint SR_plus_permute_T := Resolve (SR_plus_permute T). -Hint SR_mult_permute_T := Resolve (SR_mult_permute T). -Hint SR_distr_right_T := Resolve (SR_distr_right T). -Hint SR_distr_right2_T := Resolve (SR_distr_right2 T). -Hint SR_mult_zero_right_T := Resolve (SR_mult_zero_right T). -Hint SR_mult_zero_right2_T := Resolve (SR_mult_zero_right2 T). -Hint SR_plus_zero_right_T := Resolve (SR_plus_zero_right T). -Hint SR_plus_zero_right2_T := Resolve (SR_plus_zero_right2 T). -Hint SR_mult_one_right_T := Resolve (SR_mult_one_right T). -Hint SR_mult_one_right2_T := Resolve (SR_mult_one_right2 T). -Hint SR_plus_reg_right_T := Resolve (SR_plus_reg_right T). -Hints Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) -Hints Immediate T. - -Remark iacs_aux_ok : (x:A)(s:abstract_sum) - (iacs_aux x s)==(Aplus x (interp_acs s)). -Proof. - Induction s; Simpl; Intros. - Trivial. - Reflexivity. -Save. - -Hint rew_iacs_aux : core := Extern 10 (eqT A ? ?) Rewrite iacs_aux_ok. - -Lemma abstract_varlist_insert_ok : (l:varlist)(s:abstract_sum) - (interp_acs (abstract_varlist_insert l s)) - ==(Aplus (interp_vl Amult Aone Azero vm l) (interp_acs s)). - - Induction s. - Trivial. - - Simpl; Intros. - Elim (varlist_lt l v); Simpl. - EAuto. - Rewrite iacs_aux_ok. - Rewrite H; Auto. - -Save. - -Lemma abstract_sum_merge_ok : (x,y:abstract_sum) - (interp_acs (abstract_sum_merge x y)) - ==(Aplus (interp_acs x) (interp_acs y)). - -Proof. - Induction x. - Trivial. - Induction y; Intros. - - Auto. - - Simpl; Elim (varlist_lt v v0); Simpl. - Repeat Rewrite iacs_aux_ok. - Rewrite H; Simpl; Auto. - - Simpl in H0. - Repeat Rewrite iacs_aux_ok. - Rewrite H0. Simpl; Auto. -Save. - -Lemma abstract_sum_scalar_ok : (l:varlist)(s:abstract_sum) - (interp_acs (abstract_sum_scalar l s)) - == (Amult (interp_vl Amult Aone Azero vm l) (interp_acs s)). -Proof. - Induction s. - Simpl; EAuto. - - Simpl; Intros. - Rewrite iacs_aux_ok. - Rewrite abstract_varlist_insert_ok. - Rewrite H. - Rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). - Auto. -Save. - -Lemma abstract_sum_prod_ok : (x,y:abstract_sum) - (interp_acs (abstract_sum_prod x y)) - == (Amult (interp_acs x) (interp_acs y)). - -Proof. - Induction x. - Intros; Simpl; EAuto. - - NewDestruct y; Intros. - - Simpl; Rewrite H; EAuto. - - Unfold abstract_sum_prod; Fold abstract_sum_prod. - Rewrite abstract_sum_merge_ok. - Rewrite abstract_sum_scalar_ok. - Rewrite H; Simpl; Auto. -Save. - -Theorem aspolynomial_normalize_ok : (x:aspolynomial) - (interp_asp x)==(interp_acs (aspolynomial_normalize x)). -Proof. - Induction x; Simpl; Intros; Trivial. - Rewrite abstract_sum_merge_ok. - Rewrite H; Rewrite H0; EAuto. - Rewrite abstract_sum_prod_ok. - Rewrite H; Rewrite H0; EAuto. -Save. - -End abstract_semi_rings. - -Section abstract_rings. - -(* In abstract polynomials there is no constants other - than 0 and 1. An abstract ring is a ring whose operations plus, - and mult are not functions but constructors. In other words, - when c1 and c2 are closed, (plus c1 c2) doesn't reduce to a closed - term. "closed" mean here "without plus and mult". *) - -(* this section is not parametrized by a (semi-)ring. - Nevertheless, they are two different types for semi-rings and rings - and there will be 2 correction theorems *) - -Inductive Type apolynomial := - APvar : index -> apolynomial -| AP0 : apolynomial -| AP1 : apolynomial -| APplus : apolynomial -> apolynomial -> apolynomial -| APmult : apolynomial -> apolynomial -> apolynomial -| APopp : apolynomial -> apolynomial -. - -(* A canonical "abstract" sum is a list of varlist with the sign "+" or "-". - Invariant : the list is sorted and there is no varlist is present - with both signs. +x +x +x -x is forbidden => the canonical form is +x+x *) - -Inductive signed_sum : Type := -| Nil_varlist : signed_sum -| Plus_varlist : varlist -> signed_sum -> signed_sum -| Minus_varlist : varlist -> signed_sum -> signed_sum -. - -Fixpoint signed_sum_merge [s1:signed_sum] - : signed_sum -> signed_sum := -Cases s1 of -| (Plus_varlist l1 t1) => - Fix ssm_aux{ssm_aux[s2:signed_sum] : signed_sum := - Cases s2 of - | (Plus_varlist l2 t2) => - if (varlist_lt l1 l2) - then (Plus_varlist l1 (signed_sum_merge t1 s2)) - else (Plus_varlist l2 (ssm_aux t2)) - | (Minus_varlist l2 t2) => - if (varlist_eq l1 l2) - then (signed_sum_merge t1 t2) - else if (varlist_lt l1 l2) - then (Plus_varlist l1 (signed_sum_merge t1 s2)) - else (Minus_varlist l2 (ssm_aux t2)) - | Nil_varlist => s1 - end} -| (Minus_varlist l1 t1) => - Fix ssm_aux2{ssm_aux2[s2:signed_sum] : signed_sum := - Cases s2 of - | (Plus_varlist l2 t2) => - if (varlist_eq l1 l2) - then (signed_sum_merge t1 t2) - else if (varlist_lt l1 l2) - then (Minus_varlist l1 (signed_sum_merge t1 s2)) - else (Plus_varlist l2 (ssm_aux2 t2)) - | (Minus_varlist l2 t2) => - if (varlist_lt l1 l2) - then (Minus_varlist l1 (signed_sum_merge t1 s2)) - else (Minus_varlist l2 (ssm_aux2 t2)) - | Nil_varlist => s1 - end} -| Nil_varlist => [s2]s2 -end. - -Fixpoint plus_varlist_insert [l1:varlist; s2:signed_sum] - : signed_sum := - Cases s2 of - | (Plus_varlist l2 t2) => - if (varlist_lt l1 l2) - then (Plus_varlist l1 s2) - else (Plus_varlist l2 (plus_varlist_insert l1 t2)) - | (Minus_varlist l2 t2) => - if (varlist_eq l1 l2) - then t2 - else if (varlist_lt l1 l2) - then (Plus_varlist l1 s2) - else (Minus_varlist l2 (plus_varlist_insert l1 t2)) - | Nil_varlist => (Plus_varlist l1 Nil_varlist) - end. - -Fixpoint minus_varlist_insert [l1:varlist; s2:signed_sum] - : signed_sum := - Cases s2 of - | (Plus_varlist l2 t2) => - if (varlist_eq l1 l2) - then t2 - else if (varlist_lt l1 l2) - then (Minus_varlist l1 s2) - else (Plus_varlist l2 (minus_varlist_insert l1 t2)) - | (Minus_varlist l2 t2) => - if (varlist_lt l1 l2) - then (Minus_varlist l1 s2) - else (Minus_varlist l2 (minus_varlist_insert l1 t2)) - | Nil_varlist => (Minus_varlist l1 Nil_varlist) - end. - -Fixpoint signed_sum_opp [s:signed_sum] : signed_sum := - Cases s of - | (Plus_varlist l2 t2) => (Minus_varlist l2 (signed_sum_opp t2)) - | (Minus_varlist l2 t2) => (Plus_varlist l2 (signed_sum_opp t2)) - | Nil_varlist => Nil_varlist - end. - - -Fixpoint plus_sum_scalar [l1:varlist; s2:signed_sum] - : signed_sum := - Cases s2 of - | (Plus_varlist l2 t2) => (plus_varlist_insert (varlist_merge l1 l2) - (plus_sum_scalar l1 t2)) - | (Minus_varlist l2 t2) => (minus_varlist_insert (varlist_merge l1 l2) - (plus_sum_scalar l1 t2)) - | Nil_varlist => Nil_varlist - end. - -Fixpoint minus_sum_scalar [l1:varlist; s2:signed_sum] - : signed_sum := - Cases s2 of - | (Plus_varlist l2 t2) => (minus_varlist_insert (varlist_merge l1 l2) - (minus_sum_scalar l1 t2)) - | (Minus_varlist l2 t2) => (plus_varlist_insert (varlist_merge l1 l2) - (minus_sum_scalar l1 t2)) - | Nil_varlist => Nil_varlist - end. - -Fixpoint signed_sum_prod [s1:signed_sum] - : signed_sum -> signed_sum := - [s2]Cases s1 of - | (Plus_varlist l1 t1) => - (signed_sum_merge (plus_sum_scalar l1 s2) - (signed_sum_prod t1 s2)) - | (Minus_varlist l1 t1) => - (signed_sum_merge (minus_sum_scalar l1 s2) - (signed_sum_prod t1 s2)) - | Nil_varlist => Nil_varlist - end. - -Fixpoint apolynomial_normalize[p:apolynomial] : signed_sum := - Cases p of - | (APvar i) => (Plus_varlist (Cons_var i Nil_var) Nil_varlist) - | AP1 => (Plus_varlist Nil_var Nil_varlist) - | AP0 => Nil_varlist - | (APplus l r) => (signed_sum_merge (apolynomial_normalize l) - (apolynomial_normalize r)) - | (APmult l r) => (signed_sum_prod (apolynomial_normalize l) - (apolynomial_normalize r)) - | (APopp q) => (signed_sum_opp (apolynomial_normalize q)) - end. - - -Variable A : Type. -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aopp :A -> A. -Variable Aeq : A -> A -> bool. -Variable vm : (varmap A). -Variable T : (Ring_Theory Aplus Amult Aone Azero Aopp Aeq). - -(* Local *) Definition isacs_aux := Fix isacs_aux{isacs_aux [a:A; s:signed_sum] : A := - Cases s of - | Nil_varlist => a - | (Plus_varlist l t) => - (Aplus a (isacs_aux (interp_vl Amult Aone Azero vm l) t)) - | (Minus_varlist l t) => - (Aplus a (isacs_aux (Aopp (interp_vl Amult Aone Azero vm l)) t)) - end}. - -Definition interp_sacs [s:signed_sum] : A := - Cases s of - | (Plus_varlist l t) => (isacs_aux (interp_vl Amult Aone Azero vm l) t) - | (Minus_varlist l t) => - (isacs_aux (Aopp (interp_vl Amult Aone Azero vm l)) t) - | Nil_varlist => Azero - end. - -Fixpoint interp_ap [p:apolynomial] : A := - Cases p of - | (APvar i) => (interp_var Azero vm i) - | AP0 => Azero - | AP1 => Aone - | (APplus l r) => (Aplus (interp_ap l) (interp_ap r)) - | (APmult l r) => (Amult (interp_ap l) (interp_ap r)) - | (APopp q) => (Aopp (interp_ap q)) - end. - -Hint Th_plus_sym_T := Resolve (Th_plus_sym T). -Hint Th_plus_assoc_T := Resolve (Th_plus_assoc T). -Hint Th_plus_assoc2_T := Resolve (Th_plus_assoc2 T). -Hint Th_mult_sym_T := Resolve (Th_mult_sym T). -Hint Th_mult_assoc_T := Resolve (Th_mult_assoc T). -Hint Th_mult_assoc2_T := Resolve (Th_mult_assoc2 T). -Hint Th_plus_zero_left_T := Resolve (Th_plus_zero_left T). -Hint Th_plus_zero_left2_T := Resolve (Th_plus_zero_left2 T). -Hint Th_mult_one_left_T := Resolve (Th_mult_one_left T). -Hint Th_mult_one_left2_T := Resolve (Th_mult_one_left2 T). -Hint Th_mult_zero_left_T := Resolve (Th_mult_zero_left T). -Hint Th_mult_zero_left2_T := Resolve (Th_mult_zero_left2 T). -Hint Th_distr_left_T := Resolve (Th_distr_left T). -Hint Th_distr_left2_T := Resolve (Th_distr_left2 T). -Hint Th_plus_reg_left_T := Resolve (Th_plus_reg_left T). -Hint Th_plus_permute_T := Resolve (Th_plus_permute T). -Hint Th_mult_permute_T := Resolve (Th_mult_permute T). -Hint Th_distr_right_T := Resolve (Th_distr_right T). -Hint Th_distr_right2_T := Resolve (Th_distr_right2 T). -Hint Th_mult_zero_right2_T := Resolve (Th_mult_zero_right2 T). -Hint Th_plus_zero_right_T := Resolve (Th_plus_zero_right T). -Hint Th_plus_zero_right2_T := Resolve (Th_plus_zero_right2 T). -Hint Th_mult_one_right_T := Resolve (Th_mult_one_right T). -Hint Th_mult_one_right2_T := Resolve (Th_mult_one_right2 T). -Hint Th_plus_reg_right_T := Resolve (Th_plus_reg_right T). -Hints Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) -Hints Immediate T. - -Lemma isacs_aux_ok : (x:A)(s:signed_sum) - (isacs_aux x s)==(Aplus x (interp_sacs s)). -Proof. - Induction s; Simpl; Intros. - Trivial. - Reflexivity. - Reflexivity. -Save. - -Hint rew_isacs_aux : core := Extern 10 (eqT A ? ?) Rewrite isacs_aux_ok. - -Tactic Definition Solve1 v v0 H H0 := - Simpl; Elim (varlist_lt v v0); Simpl; Rewrite isacs_aux_ok; - [Rewrite H; Simpl; Auto - |Simpl in H0; Rewrite H0; Auto ]. - -Lemma signed_sum_merge_ok : (x,y:signed_sum) - (interp_sacs (signed_sum_merge x y)) - ==(Aplus (interp_sacs x) (interp_sacs y)). - - Induction x. - Intro; Simpl; Auto. - - Induction y; Intros. - - Auto. - - Solve1 v v0 H H0. - - Simpl; Generalize (varlist_eq_prop v v0). - Elim (varlist_eq v v0); Simpl. - - Intro Heq; Rewrite (Heq I). - Rewrite H. - Repeat Rewrite isacs_aux_ok. - Rewrite (Th_plus_permute T). - Repeat Rewrite (Th_plus_assoc T). - Rewrite (Th_plus_sym T (Aopp (interp_vl Amult Aone Azero vm v0)) - (interp_vl Amult Aone Azero vm v0)). - Rewrite (Th_opp_def T). - Rewrite (Th_plus_zero_left T). - Reflexivity. - - Solve1 v v0 H H0. - - Induction y; Intros. - - Auto. - - Simpl; Generalize (varlist_eq_prop v v0). - Elim (varlist_eq v v0); Simpl. - - Intro Heq; Rewrite (Heq I). - Rewrite H. - Repeat Rewrite isacs_aux_ok. - Rewrite (Th_plus_permute T). - Repeat Rewrite (Th_plus_assoc T). - Rewrite (Th_opp_def T). - Rewrite (Th_plus_zero_left T). - Reflexivity. - - Solve1 v v0 H H0. - - Solve1 v v0 H H0. - -Save. - -Tactic Definition Solve2 l v H := - Elim (varlist_lt l v); Simpl; Rewrite isacs_aux_ok; - [ Auto - | Rewrite H; Auto ]. - -Lemma plus_varlist_insert_ok : (l:varlist)(s:signed_sum) - (interp_sacs (plus_varlist_insert l s)) - == (Aplus (interp_vl Amult Aone Azero vm l) (interp_sacs s)). -Proof. - - Induction s. - Trivial. - - Simpl; Intros. - Solve2 l v H. - - Simpl; Intros. - Generalize (varlist_eq_prop l v). - Elim (varlist_eq l v); Simpl. - - Intro Heq; Rewrite (Heq I). - Repeat Rewrite isacs_aux_ok. - Repeat Rewrite (Th_plus_assoc T). - Rewrite (Th_opp_def T). - Rewrite (Th_plus_zero_left T). - Reflexivity. - - Solve2 l v H. - -Save. - -Lemma minus_varlist_insert_ok : (l:varlist)(s:signed_sum) - (interp_sacs (minus_varlist_insert l s)) - == (Aplus (Aopp (interp_vl Amult Aone Azero vm l)) (interp_sacs s)). -Proof. - - Induction s. - Trivial. - - Simpl; Intros. - Generalize (varlist_eq_prop l v). - Elim (varlist_eq l v); Simpl. - - Intro Heq; Rewrite (Heq I). - Repeat Rewrite isacs_aux_ok. - Repeat Rewrite (Th_plus_assoc T). - Rewrite (Th_plus_sym T (Aopp (interp_vl Amult Aone Azero vm v)) - (interp_vl Amult Aone Azero vm v)). - Rewrite (Th_opp_def T). - Auto. - - Simpl; Intros. - Solve2 l v H. - - Simpl; Intros; Solve2 l v H. - -Save. - -Lemma signed_sum_opp_ok : (s:signed_sum) - (interp_sacs (signed_sum_opp s)) - == (Aopp (interp_sacs s)). -Proof. - - Induction s; Simpl; Intros. - - Symmetry; Apply (Th_opp_zero T). - - Repeat Rewrite isacs_aux_ok. - Rewrite H. - Rewrite (Th_plus_opp_opp T). - Reflexivity. - - Repeat Rewrite isacs_aux_ok. - Rewrite H. - Rewrite <- (Th_plus_opp_opp T). - Rewrite (Th_opp_opp T). - Reflexivity. - -Save. - -Lemma plus_sum_scalar_ok : (l:varlist)(s:signed_sum) - (interp_sacs (plus_sum_scalar l s)) - == (Amult (interp_vl Amult Aone Azero vm l) (interp_sacs s)). -Proof. - - Induction s. - Trivial. - - Simpl; Intros. - Rewrite plus_varlist_insert_ok. - Rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). - Repeat Rewrite isacs_aux_ok. - Rewrite H. - Auto. - - Simpl; Intros. - Rewrite minus_varlist_insert_ok. - Repeat Rewrite isacs_aux_ok. - Rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). - Rewrite H. - Rewrite (Th_distr_right T). - Rewrite <- (Th_opp_mult_right T). - Reflexivity. - -Save. - -Lemma minus_sum_scalar_ok : (l:varlist)(s:signed_sum) - (interp_sacs (minus_sum_scalar l s)) - == (Aopp (Amult (interp_vl Amult Aone Azero vm l) (interp_sacs s))). -Proof. - - Induction s; Simpl; Intros. - - Rewrite (Th_mult_zero_right T); Symmetry; Apply (Th_opp_zero T). - - Simpl; Intros. - Rewrite minus_varlist_insert_ok. - Rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). - Repeat Rewrite isacs_aux_ok. - Rewrite H. - Rewrite (Th_distr_right T). - Rewrite (Th_plus_opp_opp T). - Reflexivity. - - Simpl; Intros. - Rewrite plus_varlist_insert_ok. - Repeat Rewrite isacs_aux_ok. - Rewrite (varlist_merge_ok A Aplus Amult Aone Azero Aeq vm T). - Rewrite H. - Rewrite (Th_distr_right T). - Rewrite <- (Th_opp_mult_right T). - Rewrite <- (Th_plus_opp_opp T). - Rewrite (Th_opp_opp T). - Reflexivity. - -Save. - -Lemma signed_sum_prod_ok : (x,y:signed_sum) - (interp_sacs (signed_sum_prod x y)) == - (Amult (interp_sacs x) (interp_sacs y)). -Proof. - - Induction x. - - Simpl; EAuto 1. - - Intros; Simpl. - Rewrite signed_sum_merge_ok. - Rewrite plus_sum_scalar_ok. - Repeat Rewrite isacs_aux_ok. - Rewrite H. - Auto. - - Intros; Simpl. - Repeat Rewrite isacs_aux_ok. - Rewrite signed_sum_merge_ok. - Rewrite minus_sum_scalar_ok. - Rewrite H. - Rewrite (Th_distr_left T). - Rewrite (Th_opp_mult_left T). - Reflexivity. - -Save. - -Theorem apolynomial_normalize_ok : (p:apolynomial) - (interp_sacs (apolynomial_normalize p))==(interp_ap p). -Proof. - Induction p; Simpl; Auto 1. - Intros. - Rewrite signed_sum_merge_ok. - Rewrite H; Rewrite H0; Reflexivity. - Intros. - Rewrite signed_sum_prod_ok. - Rewrite H; Rewrite H0; Reflexivity. - Intros. - Rewrite signed_sum_opp_ok. - Rewrite H; Reflexivity. -Save. - -End abstract_rings. diff --git a/contrib7/ring/Ring_normalize.v b/contrib7/ring/Ring_normalize.v deleted file mode 100644 index 1dbd9d56..00000000 --- a/contrib7/ring/Ring_normalize.v +++ /dev/null @@ -1,893 +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 *) -(************************************************************************) - -(* $Id: Ring_normalize.v,v 1.1.2.1 2004/07/16 19:30:18 herbelin Exp $ *) - -Require Ring_theory. -Require Quote. - -Set Implicit Arguments. - -Lemma index_eq_prop: (n,m:index)(Is_true (index_eq n m)) -> n=m. -Proof. - Intros. - Apply Quote.index_eq_prop. - Generalize H. - Case (index_eq n m); Simpl; Trivial; Intros. - Contradiction. -Save. - -Section semi_rings. - -Variable A : Type. -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aeq : A -> A -> bool. - -(* Section definitions. *) - - -(******************************************) -(* Normal abtract Polynomials *) -(******************************************) -(* DEFINITIONS : -- A varlist is a sorted product of one or more variables : x, x*y*z -- A monom is a constant, a varlist or the product of a constant by a varlist - variables. 2*x*y, x*y*z, 3 are monoms : 2*3, x*3*y, 4*x*3 are NOT. -- A canonical sum is either a monom or an ordered sum of monoms - (the order on monoms is defined later) -- A normal polynomial it either a constant or a canonical sum or a constant - plus a canonical sum -*) - -(* varlist is isomorphic to (list var), but we built a special inductive - for efficiency *) -Inductive varlist : Type := -| Nil_var : varlist -| Cons_var : index -> varlist -> varlist -. - -Inductive canonical_sum : Type := -| Nil_monom : canonical_sum -| Cons_monom : A -> varlist -> canonical_sum -> canonical_sum -| Cons_varlist : varlist -> canonical_sum -> canonical_sum -. - -(* Order on monoms *) - -(* That's the lexicographic order on varlist, extended by : - - A constant is less than every monom - - The relation between two varlist is preserved by multiplication by a - constant. - - Examples : - 3 < x < y - x*y < x*y*y*z - 2*x*y < x*y*y*z - x*y < 54*x*y*y*z - 4*x*y < 59*x*y*y*z -*) - -Fixpoint varlist_eq [x,y:varlist] : bool := - Cases x y of - | Nil_var Nil_var => true - | (Cons_var i xrest) (Cons_var j yrest) => - (andb (index_eq i j) (varlist_eq xrest yrest)) - | _ _ => false - end. - -Fixpoint varlist_lt [x,y:varlist] : bool := - Cases x y of - | Nil_var (Cons_var _ _) => true - | (Cons_var i xrest) (Cons_var j yrest) => - if (index_lt i j) then true - else (andb (index_eq i j) (varlist_lt xrest yrest)) - | _ _ => false - end. - -(* merges two variables lists *) -Fixpoint varlist_merge [l1:varlist] : varlist -> varlist := - Cases l1 of - | (Cons_var v1 t1) => - Fix vm_aux {vm_aux [l2:varlist] : varlist := - Cases l2 of - | (Cons_var v2 t2) => - if (index_lt v1 v2) - then (Cons_var v1 (varlist_merge t1 l2)) - else (Cons_var v2 (vm_aux t2)) - | Nil_var => l1 - end} - | Nil_var => [l2]l2 - end. - -(* returns the sum of two canonical sums *) -Fixpoint canonical_sum_merge [s1:canonical_sum] - : canonical_sum -> canonical_sum := -Cases s1 of -| (Cons_monom c1 l1 t1) => - Fix csm_aux{csm_aux[s2:canonical_sum] : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 c2) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 (canonical_sum_merge t1 s2)) - else (Cons_monom c2 l2 (csm_aux t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 Aone) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 (canonical_sum_merge t1 s2)) - else (Cons_varlist l2 (csm_aux t2)) - | Nil_monom => s1 - end} -| (Cons_varlist l1 t1) => - Fix csm_aux2{csm_aux2[s2:canonical_sum] : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone c2) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 (canonical_sum_merge t1 s2)) - else (Cons_monom c2 l2 (csm_aux2 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone Aone) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 (canonical_sum_merge t1 s2)) - else (Cons_varlist l2 (csm_aux2 t2)) - | Nil_monom => s1 - end} -| Nil_monom => [s2]s2 -end. - -(* Insertion of a monom in a canonical sum *) -Fixpoint monom_insert [c1:A; l1:varlist; s2 : canonical_sum] - : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 c2) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 s2) - else (Cons_monom c2 l2 (monom_insert c1 l1 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 Aone) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 s2) - else (Cons_varlist l2 (monom_insert c1 l1 t2)) - | Nil_monom => (Cons_monom c1 l1 Nil_monom) - end. - -Fixpoint varlist_insert [l1:varlist; s2:canonical_sum] - : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone c2) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 s2) - else (Cons_monom c2 l2 (varlist_insert l1 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone Aone) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 s2) - else (Cons_varlist l2 (varlist_insert l1 t2)) - | Nil_monom => (Cons_varlist l1 Nil_monom) - end. - -(* Computes c0*s *) -Fixpoint canonical_sum_scalar [c0:A; s:canonical_sum] : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (Cons_monom (Amult c0 c) l (canonical_sum_scalar c0 t)) - | (Cons_varlist l t) => - (Cons_monom c0 l (canonical_sum_scalar c0 t)) - | Nil_monom => Nil_monom - end. - -(* Computes l0*s *) -Fixpoint canonical_sum_scalar2 [l0:varlist; s:canonical_sum] - : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (monom_insert c (varlist_merge l0 l) (canonical_sum_scalar2 l0 t)) - | (Cons_varlist l t) => - (varlist_insert (varlist_merge l0 l) (canonical_sum_scalar2 l0 t)) - | Nil_monom => Nil_monom - end. - -(* Computes c0*l0*s *) -Fixpoint canonical_sum_scalar3 [c0:A;l0:varlist; s:canonical_sum] - : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (monom_insert (Amult c0 c) (varlist_merge l0 l) - (canonical_sum_scalar3 c0 l0 t)) - | (Cons_varlist l t) => - (monom_insert c0 (varlist_merge l0 l) - (canonical_sum_scalar3 c0 l0 t)) - | Nil_monom => Nil_monom - end. - -(* returns the product of two canonical sums *) -Fixpoint canonical_sum_prod [s1:canonical_sum] - : canonical_sum -> canonical_sum := - [s2]Cases s1 of - | (Cons_monom c1 l1 t1) => - (canonical_sum_merge (canonical_sum_scalar3 c1 l1 s2) - (canonical_sum_prod t1 s2)) - | (Cons_varlist l1 t1) => - (canonical_sum_merge (canonical_sum_scalar2 l1 s2) - (canonical_sum_prod t1 s2)) - | Nil_monom => Nil_monom - end. - -(* The type to represent concrete semi-ring polynomials *) -Inductive Type spolynomial := - SPvar : index -> spolynomial -| SPconst : A -> spolynomial -| SPplus : spolynomial -> spolynomial -> spolynomial -| SPmult : spolynomial -> spolynomial -> spolynomial. - -Fixpoint spolynomial_normalize[p:spolynomial] : canonical_sum := - Cases p of - | (SPvar i) => (Cons_varlist (Cons_var i Nil_var) Nil_monom) - | (SPconst c) => (Cons_monom c Nil_var Nil_monom) - | (SPplus l r) => (canonical_sum_merge (spolynomial_normalize l) - (spolynomial_normalize r)) - | (SPmult l r) => (canonical_sum_prod (spolynomial_normalize l) - (spolynomial_normalize r)) - end. - -(* Deletion of useless 0 and 1 in canonical sums *) -Fixpoint canonical_sum_simplify [ s:canonical_sum] : canonical_sum := - Cases s of - | (Cons_monom c l t) => - if (Aeq c Azero) - then (canonical_sum_simplify t) - else if (Aeq c Aone) - then (Cons_varlist l (canonical_sum_simplify t)) - else (Cons_monom c l (canonical_sum_simplify t)) - | (Cons_varlist l t) => (Cons_varlist l (canonical_sum_simplify t)) - | Nil_monom => Nil_monom - end. - -Definition spolynomial_simplify := - [x:spolynomial](canonical_sum_simplify (spolynomial_normalize x)). - -(* End definitions. *) - -(* Section interpretation. *) - -(*** Here a variable map is defined and the interpetation of a spolynom - acording to a certain variables map. Once again the choosen definition - is generic and could be changed ****) - -Variable vm : (varmap A). - -(* Interpretation of list of variables - * [x1; ... ; xn ] is interpreted as (find v x1)* ... *(find v xn) - * The unbound variables are mapped to 0. Normally this case sould - * never occur. Since we want only to prove correctness theorems, which form - * is : for any varmap and any spolynom ... this is a safe and pain-saving - * choice *) -Definition interp_var [i:index] := (varmap_find Azero i vm). - -(* Local *) Definition ivl_aux := Fix ivl_aux {ivl_aux[x:index; t:varlist] : A := - Cases t of - | Nil_var => (interp_var x) - | (Cons_var x' t') => (Amult (interp_var x) (ivl_aux x' t')) - end}. - -Definition interp_vl := [l:varlist] - Cases l of - | Nil_var => Aone - | (Cons_var x t) => (ivl_aux x t) - end. - -(* Local *) Definition interp_m := [c:A][l:varlist] - Cases l of - | Nil_var => c - | (Cons_var x t) => - (Amult c (ivl_aux x t)) - end. - -(* Local *) Definition ics_aux := Fix ics_aux{ics_aux[a:A; s:canonical_sum] : A := - Cases s of - | Nil_monom => a - | (Cons_varlist l t) => (Aplus a (ics_aux (interp_vl l) t)) - | (Cons_monom c l t) => (Aplus a (ics_aux (interp_m c l) t)) - end}. - -(* Interpretation of a canonical sum *) -Definition interp_cs : canonical_sum -> A := - [s]Cases s of - | Nil_monom => Azero - | (Cons_varlist l t) => - (ics_aux (interp_vl l) t) - | (Cons_monom c l t) => - (ics_aux (interp_m c l) t) - end. - -Fixpoint interp_sp [p:spolynomial] : A := - Cases p of - (SPconst c) => c - | (SPvar i) => (interp_var i) - | (SPplus p1 p2) => (Aplus (interp_sp p1) (interp_sp p2)) - | (SPmult p1 p2) => (Amult (interp_sp p1) (interp_sp p2)) - end. - - -(* End interpretation. *) - -Unset Implicit Arguments. - -(* Section properties. *) - -Variable T : (Semi_Ring_Theory Aplus Amult Aone Azero Aeq). - -Hint SR_plus_sym_T := Resolve (SR_plus_sym T). -Hint SR_plus_assoc_T := Resolve (SR_plus_assoc T). -Hint SR_plus_assoc2_T := Resolve (SR_plus_assoc2 T). -Hint SR_mult_sym_T := Resolve (SR_mult_sym T). -Hint SR_mult_assoc_T := Resolve (SR_mult_assoc T). -Hint SR_mult_assoc2_T := Resolve (SR_mult_assoc2 T). -Hint SR_plus_zero_left_T := Resolve (SR_plus_zero_left T). -Hint SR_plus_zero_left2_T := Resolve (SR_plus_zero_left2 T). -Hint SR_mult_one_left_T := Resolve (SR_mult_one_left T). -Hint SR_mult_one_left2_T := Resolve (SR_mult_one_left2 T). -Hint SR_mult_zero_left_T := Resolve (SR_mult_zero_left T). -Hint SR_mult_zero_left2_T := Resolve (SR_mult_zero_left2 T). -Hint SR_distr_left_T := Resolve (SR_distr_left T). -Hint SR_distr_left2_T := Resolve (SR_distr_left2 T). -Hint SR_plus_reg_left_T := Resolve (SR_plus_reg_left T). -Hint SR_plus_permute_T := Resolve (SR_plus_permute T). -Hint SR_mult_permute_T := Resolve (SR_mult_permute T). -Hint SR_distr_right_T := Resolve (SR_distr_right T). -Hint SR_distr_right2_T := Resolve (SR_distr_right2 T). -Hint SR_mult_zero_right_T := Resolve (SR_mult_zero_right T). -Hint SR_mult_zero_right2_T := Resolve (SR_mult_zero_right2 T). -Hint SR_plus_zero_right_T := Resolve (SR_plus_zero_right T). -Hint SR_plus_zero_right2_T := Resolve (SR_plus_zero_right2 T). -Hint SR_mult_one_right_T := Resolve (SR_mult_one_right T). -Hint SR_mult_one_right2_T := Resolve (SR_mult_one_right2 T). -Hint SR_plus_reg_right_T := Resolve (SR_plus_reg_right T). -Hints Resolve refl_equal sym_equal trans_equal. -(* Hints Resolve refl_eqT sym_eqT trans_eqT. *) -Hints Immediate T. - -Lemma varlist_eq_prop : (x,y:varlist) - (Is_true (varlist_eq x y))->x==y. -Proof. - Induction x; Induction y; Contradiction Orelse Try Reflexivity. - Simpl; Intros. - Generalize (andb_prop2 ? ? H1); Intros; Elim H2; Intros. - Rewrite (index_eq_prop H3); Rewrite (H v0 H4); Reflexivity. -Save. - -Remark ivl_aux_ok : (v:varlist)(i:index) - (ivl_aux i v)==(Amult (interp_var i) (interp_vl v)). -Proof. - Induction v; Simpl; Intros. - Trivial. - Rewrite H; Trivial. -Save. - -Lemma varlist_merge_ok : (x,y:varlist) - (interp_vl (varlist_merge x y)) - ==(Amult (interp_vl x) (interp_vl y)). -Proof. - Induction x. - Simpl; Trivial. - Induction y. - Simpl; Trivial. - Simpl; Intros. - Elim (index_lt i i0); Simpl; Intros. - - Repeat Rewrite ivl_aux_ok. - Rewrite H. Simpl. - Rewrite ivl_aux_ok. - EAuto. - - Repeat Rewrite ivl_aux_ok. - Rewrite H0. - Rewrite ivl_aux_ok. - EAuto. -Save. - -Remark ics_aux_ok : (x:A)(s:canonical_sum) - (ics_aux x s)==(Aplus x (interp_cs s)). -Proof. - Induction s; Simpl; Intros. - Trivial. - Reflexivity. - Reflexivity. -Save. - -Remark interp_m_ok : (x:A)(l:varlist) - (interp_m x l)==(Amult x (interp_vl l)). -Proof. - NewDestruct l. - Simpl; Trivial. - Reflexivity. -Save. - -Lemma canonical_sum_merge_ok : (x,y:canonical_sum) - (interp_cs (canonical_sum_merge x y)) - ==(Aplus (interp_cs x) (interp_cs y)). - -Induction x; Simpl. -Trivial. - -Induction y; Simpl; Intros. -(* monom and nil *) -EAuto. - -(* monom and monom *) -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl; Repeat Rewrite ics_aux_ok; Rewrite H. -Repeat Rewrite interp_m_ok. -Rewrite (SR_distr_left T). -Repeat Rewrite <- (SR_plus_assoc T). -Apply congr_eqT with f:=(Aplus (Amult a (interp_vl v0))). -Trivial. - -Elim (varlist_lt v v0); Simpl. -Repeat Rewrite ics_aux_ok. -Rewrite H; Simpl; Rewrite ics_aux_ok; EAuto. - -Rewrite ics_aux_ok; Rewrite H0; Repeat Rewrite ics_aux_ok; Simpl; EAuto. - -(* monom and varlist *) -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl; Repeat Rewrite ics_aux_ok; Rewrite H. -Repeat Rewrite interp_m_ok. -Rewrite (SR_distr_left T). -Repeat Rewrite <- (SR_plus_assoc T). -Apply congr_eqT with f:=(Aplus (Amult a (interp_vl v0))). -Rewrite (SR_mult_one_left T). -Trivial. - -Elim (varlist_lt v v0); Simpl. -Repeat Rewrite ics_aux_ok. -Rewrite H; Simpl; Rewrite ics_aux_ok; EAuto. -Rewrite ics_aux_ok; Rewrite H0; Repeat Rewrite ics_aux_ok; Simpl; EAuto. - -Induction y; Simpl; Intros. -(* varlist and nil *) -Trivial. - -(* varlist and monom *) -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl; Repeat Rewrite ics_aux_ok; Rewrite H. -Repeat Rewrite interp_m_ok. -Rewrite (SR_distr_left T). -Repeat Rewrite <- (SR_plus_assoc T). -Rewrite (SR_mult_one_left T). -Apply congr_eqT with f:=(Aplus (interp_vl v0)). -Trivial. - -Elim (varlist_lt v v0); Simpl. -Repeat Rewrite ics_aux_ok. -Rewrite H; Simpl; Rewrite ics_aux_ok; EAuto. -Rewrite ics_aux_ok; Rewrite H0; Repeat Rewrite ics_aux_ok; Simpl; EAuto. - -(* varlist and varlist *) -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl; Repeat Rewrite ics_aux_ok; Rewrite H. -Repeat Rewrite interp_m_ok. -Rewrite (SR_distr_left T). -Repeat Rewrite <- (SR_plus_assoc T). -Rewrite (SR_mult_one_left T). -Apply congr_eqT with f:=(Aplus (interp_vl v0)). -Trivial. - -Elim (varlist_lt v v0); Simpl. -Repeat Rewrite ics_aux_ok. -Rewrite H; Simpl; Rewrite ics_aux_ok; EAuto. -Rewrite ics_aux_ok; Rewrite H0; Repeat Rewrite ics_aux_ok; Simpl; EAuto. -Save. - -Lemma monom_insert_ok: (a:A)(l:varlist)(s:canonical_sum) - (interp_cs (monom_insert a l s)) - == (Aplus (Amult a (interp_vl l)) (interp_cs s)). -Intros; Generalize s; Induction s0. - -Simpl; Rewrite interp_m_ok; Trivial. - -Simpl; Intros. -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl; Rewrite interp_m_ok; - Repeat Rewrite ics_aux_ok; Rewrite interp_m_ok; - Rewrite (SR_distr_left T); EAuto. -Elim (varlist_lt l v); Simpl; -[ Repeat Rewrite interp_m_ok; Rewrite ics_aux_ok; EAuto -| Repeat Rewrite interp_m_ok; Rewrite ics_aux_ok; - Rewrite H; Rewrite ics_aux_ok; EAuto]. - -Simpl; Intros. -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl; Rewrite interp_m_ok; - Repeat Rewrite ics_aux_ok; - Rewrite (SR_distr_left T); Rewrite (SR_mult_one_left T); EAuto. -Elim (varlist_lt l v); Simpl; -[ Repeat Rewrite interp_m_ok; Rewrite ics_aux_ok; EAuto -| Repeat Rewrite interp_m_ok; Rewrite ics_aux_ok; - Rewrite H; Rewrite ics_aux_ok; EAuto]. -Save. - -Lemma varlist_insert_ok : - (l:varlist)(s:canonical_sum) - (interp_cs (varlist_insert l s)) - == (Aplus (interp_vl l) (interp_cs s)). -Intros; Generalize s; Induction s0. - -Simpl; Trivial. - -Simpl; Intros. -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl; Rewrite interp_m_ok; - Repeat Rewrite ics_aux_ok; Rewrite interp_m_ok; - Rewrite (SR_distr_left T); Rewrite (SR_mult_one_left T); EAuto. -Elim (varlist_lt l v); Simpl; -[ Repeat Rewrite interp_m_ok; Rewrite ics_aux_ok; EAuto -| Repeat Rewrite interp_m_ok; Rewrite ics_aux_ok; - Rewrite H; Rewrite ics_aux_ok; EAuto]. - -Simpl; Intros. -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl; Rewrite interp_m_ok; - Repeat Rewrite ics_aux_ok; - Rewrite (SR_distr_left T); Rewrite (SR_mult_one_left T); EAuto. -Elim (varlist_lt l v); Simpl; -[ Repeat Rewrite interp_m_ok; Rewrite ics_aux_ok; EAuto -| Repeat Rewrite interp_m_ok; Rewrite ics_aux_ok; - Rewrite H; Rewrite ics_aux_ok; EAuto]. -Save. - -Lemma canonical_sum_scalar_ok : (a:A)(s:canonical_sum) - (interp_cs (canonical_sum_scalar a s)) - ==(Amult a (interp_cs s)). -Induction s. -Simpl; EAuto. - -Simpl; Intros. -Repeat Rewrite ics_aux_ok. -Repeat Rewrite interp_m_ok. -Rewrite H. -Rewrite (SR_distr_right T). -Repeat Rewrite <- (SR_mult_assoc T). -Reflexivity. - -Simpl; Intros. -Repeat Rewrite ics_aux_ok. -Repeat Rewrite interp_m_ok. -Rewrite H. -Rewrite (SR_distr_right T). -Repeat Rewrite <- (SR_mult_assoc T). -Reflexivity. -Save. - -Lemma canonical_sum_scalar2_ok : (l:varlist; s:canonical_sum) - (interp_cs (canonical_sum_scalar2 l s)) - ==(Amult (interp_vl l) (interp_cs s)). -Induction s. -Simpl; Trivial. - -Simpl; Intros. -Rewrite monom_insert_ok. -Repeat Rewrite ics_aux_ok. -Repeat Rewrite interp_m_ok. -Rewrite H. -Rewrite varlist_merge_ok. -Repeat Rewrite (SR_distr_right T). -Repeat Rewrite <- (SR_mult_assoc T). -Repeat Rewrite <- (SR_plus_assoc T). -Rewrite (SR_mult_permute T a (interp_vl l) (interp_vl v)). -Reflexivity. - -Simpl; Intros. -Rewrite varlist_insert_ok. -Repeat Rewrite ics_aux_ok. -Repeat Rewrite interp_m_ok. -Rewrite H. -Rewrite varlist_merge_ok. -Repeat Rewrite (SR_distr_right T). -Repeat Rewrite <- (SR_mult_assoc T). -Repeat Rewrite <- (SR_plus_assoc T). -Reflexivity. -Save. - -Lemma canonical_sum_scalar3_ok : (c:A; l:varlist; s:canonical_sum) - (interp_cs (canonical_sum_scalar3 c l s)) - ==(Amult c (Amult (interp_vl l) (interp_cs s))). -Induction s. -Simpl; Repeat Rewrite (SR_mult_zero_right T); Reflexivity. - -Simpl; Intros. -Rewrite monom_insert_ok. -Repeat Rewrite ics_aux_ok. -Repeat Rewrite interp_m_ok. -Rewrite H. -Rewrite varlist_merge_ok. -Repeat Rewrite (SR_distr_right T). -Repeat Rewrite <- (SR_mult_assoc T). -Repeat Rewrite <- (SR_plus_assoc T). -Rewrite (SR_mult_permute T a (interp_vl l) (interp_vl v)). -Reflexivity. - -Simpl; Intros. -Rewrite monom_insert_ok. -Repeat Rewrite ics_aux_ok. -Repeat Rewrite interp_m_ok. -Rewrite H. -Rewrite varlist_merge_ok. -Repeat Rewrite (SR_distr_right T). -Repeat Rewrite <- (SR_mult_assoc T). -Repeat Rewrite <- (SR_plus_assoc T). -Rewrite (SR_mult_permute T c (interp_vl l) (interp_vl v)). -Reflexivity. -Save. - -Lemma canonical_sum_prod_ok : (x,y:canonical_sum) - (interp_cs (canonical_sum_prod x y)) - ==(Amult (interp_cs x) (interp_cs y)). -Induction x; Simpl; Intros. -Trivial. - -Rewrite canonical_sum_merge_ok. -Rewrite canonical_sum_scalar3_ok. -Rewrite ics_aux_ok. -Rewrite interp_m_ok. -Rewrite H. -Rewrite (SR_mult_assoc T a (interp_vl v) (interp_cs y)). -Symmetry. -EAuto. - -Rewrite canonical_sum_merge_ok. -Rewrite canonical_sum_scalar2_ok. -Rewrite ics_aux_ok. -Rewrite H. -Trivial. -Save. - -Theorem spolynomial_normalize_ok : (p:spolynomial) - (interp_cs (spolynomial_normalize p)) == (interp_sp p). -Induction p; Simpl; Intros. - -Reflexivity. -Reflexivity. - -Rewrite canonical_sum_merge_ok. -Rewrite H; Rewrite H0. -Reflexivity. - -Rewrite canonical_sum_prod_ok. -Rewrite H; Rewrite H0. -Reflexivity. -Save. - -Lemma canonical_sum_simplify_ok : (s:canonical_sum) - (interp_cs (canonical_sum_simplify s)) == (interp_cs s). -Induction s. - -Reflexivity. - -(* cons_monom *) -Simpl; Intros. -Generalize (SR_eq_prop T 8!a 9!Azero). -Elim (Aeq a Azero). -Intro Heq; Rewrite (Heq I). -Rewrite H. -Rewrite ics_aux_ok. -Rewrite interp_m_ok. -Rewrite (SR_mult_zero_left T). -Trivial. - -Intros; Simpl. -Generalize (SR_eq_prop T 8!a 9!Aone). -Elim (Aeq a Aone). -Intro Heq; Rewrite (Heq I). -Simpl. -Repeat Rewrite ics_aux_ok. -Rewrite interp_m_ok. -Rewrite H. -Rewrite (SR_mult_one_left T). -Reflexivity. - -Simpl. -Repeat Rewrite ics_aux_ok. -Rewrite interp_m_ok. -Rewrite H. -Reflexivity. - -(* cons_varlist *) -Simpl; Intros. -Repeat Rewrite ics_aux_ok. -Rewrite H. -Reflexivity. - -Save. - -Theorem spolynomial_simplify_ok : (p:spolynomial) - (interp_cs (spolynomial_simplify p)) == (interp_sp p). -Intro. -Unfold spolynomial_simplify. -Rewrite canonical_sum_simplify_ok. -Apply spolynomial_normalize_ok. -Save. - -(* End properties. *) -End semi_rings. - -Implicits Cons_varlist. -Implicits Cons_monom. -Implicits SPconst. -Implicits SPplus. -Implicits SPmult. - -Section rings. - -(* Here the coercion between Ring and Semi-Ring will be useful *) - -Set Implicit Arguments. - -Variable A : Type. -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aopp : A -> A. -Variable Aeq : A -> A -> bool. -Variable vm : (varmap A). -Variable T : (Ring_Theory Aplus Amult Aone Azero Aopp Aeq). - -Hint Th_plus_sym_T := Resolve (Th_plus_sym T). -Hint Th_plus_assoc_T := Resolve (Th_plus_assoc T). -Hint Th_plus_assoc2_T := Resolve (Th_plus_assoc2 T). -Hint Th_mult_sym_T := Resolve (Th_mult_sym T). -Hint Th_mult_assoc_T := Resolve (Th_mult_assoc T). -Hint Th_mult_assoc2_T := Resolve (Th_mult_assoc2 T). -Hint Th_plus_zero_left_T := Resolve (Th_plus_zero_left T). -Hint Th_plus_zero_left2_T := Resolve (Th_plus_zero_left2 T). -Hint Th_mult_one_left_T := Resolve (Th_mult_one_left T). -Hint Th_mult_one_left2_T := Resolve (Th_mult_one_left2 T). -Hint Th_mult_zero_left_T := Resolve (Th_mult_zero_left T). -Hint Th_mult_zero_left2_T := Resolve (Th_mult_zero_left2 T). -Hint Th_distr_left_T := Resolve (Th_distr_left T). -Hint Th_distr_left2_T := Resolve (Th_distr_left2 T). -Hint Th_plus_reg_left_T := Resolve (Th_plus_reg_left T). -Hint Th_plus_permute_T := Resolve (Th_plus_permute T). -Hint Th_mult_permute_T := Resolve (Th_mult_permute T). -Hint Th_distr_right_T := Resolve (Th_distr_right T). -Hint Th_distr_right2_T := Resolve (Th_distr_right2 T). -Hint Th_mult_zero_right_T := Resolve (Th_mult_zero_right T). -Hint Th_mult_zero_right2_T := Resolve (Th_mult_zero_right2 T). -Hint Th_plus_zero_right_T := Resolve (Th_plus_zero_right T). -Hint Th_plus_zero_right2_T := Resolve (Th_plus_zero_right2 T). -Hint Th_mult_one_right_T := Resolve (Th_mult_one_right T). -Hint Th_mult_one_right2_T := Resolve (Th_mult_one_right2 T). -Hint Th_plus_reg_right_T := Resolve (Th_plus_reg_right T). -Hints Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) -Hints Immediate T. - -(*** Definitions *) - -Inductive Type polynomial := - Pvar : index -> polynomial -| Pconst : A -> polynomial -| Pplus : polynomial -> polynomial -> polynomial -| Pmult : polynomial -> polynomial -> polynomial -| Popp : polynomial -> polynomial. - -Fixpoint polynomial_normalize [x:polynomial] : (canonical_sum A) := - Cases x of - (Pplus l r) => (canonical_sum_merge Aplus Aone - (polynomial_normalize l) - (polynomial_normalize r)) - | (Pmult l r) => (canonical_sum_prod Aplus Amult Aone - (polynomial_normalize l) - (polynomial_normalize r)) - | (Pconst c) => (Cons_monom c Nil_var (Nil_monom A)) - | (Pvar i) => (Cons_varlist (Cons_var i Nil_var) (Nil_monom A)) - | (Popp p) => (canonical_sum_scalar3 Aplus Amult Aone - (Aopp Aone) Nil_var - (polynomial_normalize p)) - end. - -Definition polynomial_simplify := - [x:polynomial](canonical_sum_simplify Aone Azero Aeq - (polynomial_normalize x)). - -Fixpoint spolynomial_of [x:polynomial] : (spolynomial A) := - Cases x of - (Pplus l r) => (SPplus (spolynomial_of l) (spolynomial_of r)) - | (Pmult l r) => (SPmult (spolynomial_of l) (spolynomial_of r)) - | (Pconst c) => (SPconst c) - | (Pvar i) => (SPvar A i) - | (Popp p) => (SPmult (SPconst (Aopp Aone)) (spolynomial_of p)) - end. - -(*** Interpretation *) - -Fixpoint interp_p [p:polynomial] : A := - Cases p of - (Pconst c) => c - | (Pvar i) => (varmap_find Azero i vm) - | (Pplus p1 p2) => (Aplus (interp_p p1) (interp_p p2)) - | (Pmult p1 p2) => (Amult (interp_p p1) (interp_p p2)) - | (Popp p1) => (Aopp (interp_p p1)) - end. - -(*** Properties *) - -Unset Implicit Arguments. - -Lemma spolynomial_of_ok : (p:polynomial) - (interp_p p)==(interp_sp Aplus Amult Azero vm (spolynomial_of p)). -Induction p; Reflexivity Orelse (Simpl; Intros). -Rewrite H; Rewrite H0; Reflexivity. -Rewrite H; Rewrite H0; Reflexivity. -Rewrite H. -Rewrite (Th_opp_mult_left2 T). -Rewrite (Th_mult_one_left T). -Reflexivity. -Save. - -Theorem polynomial_normalize_ok : (p:polynomial) - (polynomial_normalize p) - ==(spolynomial_normalize Aplus Amult Aone (spolynomial_of p)). -Induction p; Reflexivity Orelse (Simpl; Intros). -Rewrite H; Rewrite H0; Reflexivity. -Rewrite H; Rewrite H0; Reflexivity. -Rewrite H; Simpl. -Elim (canonical_sum_scalar3 Aplus Amult Aone (Aopp Aone) Nil_var - (spolynomial_normalize Aplus Amult Aone (spolynomial_of p0))); -[ Reflexivity -| Simpl; Intros; Rewrite H0; Reflexivity -| Simpl; Intros; Rewrite H0; Reflexivity ]. -Save. - -Theorem polynomial_simplify_ok : (p:polynomial) - (interp_cs Aplus Amult Aone Azero vm (polynomial_simplify p)) - ==(interp_p p). -Intro. -Unfold polynomial_simplify. -Rewrite spolynomial_of_ok. -Rewrite polynomial_normalize_ok. -Rewrite (canonical_sum_simplify_ok A Aplus Amult Aone Azero Aeq vm T). -Rewrite (spolynomial_normalize_ok A Aplus Amult Aone Azero Aeq vm T). -Reflexivity. -Save. - -End rings. - -V8Infix "+" Pplus : ring_scope. -V8Infix "*" Pmult : ring_scope. -V8Notation "- x" := (Popp x) : ring_scope. -V8Notation "[ x ]" := (Pvar x) (at level 1) : ring_scope. - -Delimits Scope ring_scope with ring. diff --git a/contrib7/ring/Ring_theory.v b/contrib7/ring/Ring_theory.v deleted file mode 100644 index 85fb7f6c..00000000 --- a/contrib7/ring/Ring_theory.v +++ /dev/null @@ -1,384 +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 *) -(************************************************************************) - -(* $Id: Ring_theory.v,v 1.1.2.1 2004/07/16 19:30:19 herbelin Exp $ *) - -Require Export Bool. - -Set Implicit Arguments. - -Section Theory_of_semi_rings. - -Variable A : Type. -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -(* There is also a "weakly decidable" equality on A. That means - that if (A_eq x y)=true then x=y but x=y can arise when - (A_eq x y)=false. On an abstract ring the function [x,y:A]false - is a good choice. The proof of A_eq_prop is in this case easy. *) -Variable Aeq : A -> A -> bool. - -Infix 4 "+" Aplus V8only 50 (left associativity). -Infix 4 "*" Amult V8only 40 (left associativity). -Notation "0" := Azero. -Notation "1" := Aone. - -Record Semi_Ring_Theory : Prop := -{ SR_plus_sym : (n,m:A) n + m == m + n; - SR_plus_assoc : (n,m,p:A) n + (m + p) == (n + m) + p; - SR_mult_sym : (n,m:A) n*m == m*n; - SR_mult_assoc : (n,m,p:A) n*(m*p) == (n*m)*p; - SR_plus_zero_left :(n:A) 0 + n == n; - SR_mult_one_left : (n:A) 1*n == n; - SR_mult_zero_left : (n:A) 0*n == 0; - SR_distr_left : (n,m,p:A) (n + m)*p == n*p + m*p; - SR_plus_reg_left : (n,m,p:A) n + m == n + p -> m==p; - SR_eq_prop : (x,y:A) (Is_true (Aeq x y)) -> x==y -}. - -Variable T : Semi_Ring_Theory. - -Local plus_sym := (SR_plus_sym T). -Local plus_assoc := (SR_plus_assoc T). -Local mult_sym := ( SR_mult_sym T). -Local mult_assoc := (SR_mult_assoc T). -Local plus_zero_left := (SR_plus_zero_left T). -Local mult_one_left := (SR_mult_one_left T). -Local mult_zero_left := (SR_mult_zero_left T). -Local distr_left := (SR_distr_left T). -Local plus_reg_left := (SR_plus_reg_left T). - -Hints Resolve plus_sym plus_assoc mult_sym mult_assoc - plus_zero_left mult_one_left mult_zero_left distr_left - plus_reg_left. - -(* Lemmas whose form is x=y are also provided in form y=x because Auto does - not symmetry *) -Lemma SR_mult_assoc2 : (n,m,p:A) (n * m) * p == n * (m * p). -Symmetry; EAuto. Qed. - -Lemma SR_plus_assoc2 : (n,m,p:A) (n + m) + p == n + (m + p). -Symmetry; EAuto. Qed. - -Lemma SR_plus_zero_left2 : (n:A) n == 0 + n. -Symmetry; EAuto. Qed. - -Lemma SR_mult_one_left2 : (n:A) n == 1*n. -Symmetry; EAuto. Qed. - -Lemma SR_mult_zero_left2 : (n:A) 0 == 0*n. -Symmetry; EAuto. Qed. - -Lemma SR_distr_left2 : (n,m,p:A) n*p + m*p == (n + m)*p. -Symmetry; EAuto. Qed. - -Lemma SR_plus_permute : (n,m,p:A) n + (m + p) == m + (n + p). -Intros. -Rewrite -> plus_assoc. -Elim (plus_sym m n). -Rewrite <- plus_assoc. -Reflexivity. -Qed. - -Lemma SR_mult_permute : (n,m,p:A) n*(m*p) == m*(n*p). -Intros. -Rewrite -> mult_assoc. -Elim (mult_sym m n). -Rewrite <- mult_assoc. -Reflexivity. -Qed. - -Hints Resolve SR_plus_permute SR_mult_permute. - -Lemma SR_distr_right : (n,m,p:A) n*(m + p) == (n*m) + (n*p). -Intros. -Repeat Rewrite -> (mult_sym n). -EAuto. -Qed. - -Lemma SR_distr_right2 : (n,m,p:A) (n*m) + (n*p) == n*(m + p). -Symmetry; Apply SR_distr_right. Qed. - -Lemma SR_mult_zero_right : (n:A) n*0 == 0. -Intro; Rewrite mult_sym; EAuto. -Qed. - -Lemma SR_mult_zero_right2 : (n:A) 0 == n*0. -Intro; Rewrite mult_sym; EAuto. -Qed. - -Lemma SR_plus_zero_right :(n:A) n + 0 == n. -Intro; Rewrite plus_sym; EAuto. -Qed. -Lemma SR_plus_zero_right2 :(n:A) n == n + 0. -Intro; Rewrite plus_sym; EAuto. -Qed. - -Lemma SR_mult_one_right : (n:A) n*1 == n. -Intro; Elim mult_sym; Auto. -Qed. - -Lemma SR_mult_one_right2 : (n:A) n == n*1. -Intro; Elim mult_sym; Auto. -Qed. - -Lemma SR_plus_reg_right : (n,m,p:A) m + n == p + n -> m==p. -Intros n m p; Rewrite (plus_sym m n); Rewrite (plus_sym p n); EAuto. -Qed. - -End Theory_of_semi_rings. - -Section Theory_of_rings. - -Variable A : Type. - -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aopp : A -> A. -Variable Aeq : A -> A -> bool. - -Infix 4 "+" Aplus V8only 50 (left associativity). -Infix 4 "*" Amult V8only 40 (left associativity). -Notation "0" := Azero. -Notation "1" := Aone. -Notation "- x" := (Aopp x) (at level 0) V8only. - -Record Ring_Theory : Prop := -{ Th_plus_sym : (n,m:A) n + m == m + n; - Th_plus_assoc : (n,m,p:A) n + (m + p) == (n + m) + p; - Th_mult_sym : (n,m:A) n*m == m*n; - Th_mult_assoc : (n,m,p:A) n*(m*p) == (n*m)*p; - Th_plus_zero_left :(n:A) 0 + n == n; - Th_mult_one_left : (n:A) 1*n == n; - Th_opp_def : (n:A) n + (-n) == 0; - Th_distr_left : (n,m,p:A) (n + m)*p == n*p + m*p; - Th_eq_prop : (x,y:A) (Is_true (Aeq x y)) -> x==y -}. - -Variable T : Ring_Theory. - -Local plus_sym := (Th_plus_sym T). -Local plus_assoc := (Th_plus_assoc T). -Local mult_sym := ( Th_mult_sym T). -Local mult_assoc := (Th_mult_assoc T). -Local plus_zero_left := (Th_plus_zero_left T). -Local mult_one_left := (Th_mult_one_left T). -Local opp_def := (Th_opp_def T). -Local distr_left := (Th_distr_left T). - -Hints Resolve plus_sym plus_assoc mult_sym mult_assoc - plus_zero_left mult_one_left opp_def distr_left. - -(* Lemmas whose form is x=y are also provided in form y=x because Auto does - not symmetry *) -Lemma Th_mult_assoc2 : (n,m,p:A) (n * m) * p == n * (m * p). -Symmetry; EAuto. Qed. - -Lemma Th_plus_assoc2 : (n,m,p:A) (n + m) + p == n + (m + p). -Symmetry; EAuto. Qed. - -Lemma Th_plus_zero_left2 : (n:A) n == 0 + n. -Symmetry; EAuto. Qed. - -Lemma Th_mult_one_left2 : (n:A) n == 1*n. -Symmetry; EAuto. Qed. - -Lemma Th_distr_left2 : (n,m,p:A) n*p + m*p == (n + m)*p. -Symmetry; EAuto. Qed. - -Lemma Th_opp_def2 : (n:A) 0 == n + (-n). -Symmetry; EAuto. Qed. - -Lemma Th_plus_permute : (n,m,p:A) n + (m + p) == m + (n + p). -Intros. -Rewrite -> plus_assoc. -Elim (plus_sym m n). -Rewrite <- plus_assoc. -Reflexivity. -Qed. - -Lemma Th_mult_permute : (n,m,p:A) n*(m*p) == m*(n*p). -Intros. -Rewrite -> mult_assoc. -Elim (mult_sym m n). -Rewrite <- mult_assoc. -Reflexivity. -Qed. - -Hints Resolve Th_plus_permute Th_mult_permute. - -Lemma aux1 : (a:A) a + a == a -> a == 0. -Intros. -Generalize (opp_def a). -Pattern 1 a. -Rewrite <- H. -Rewrite <- plus_assoc. -Rewrite -> opp_def. -Elim plus_sym. -Rewrite plus_zero_left. -Trivial. -Qed. - -Lemma Th_mult_zero_left :(n:A) 0*n == 0. -Intros. -Apply aux1. -Rewrite <- distr_left. -Rewrite plus_zero_left. -Reflexivity. -Qed. -Hints Resolve Th_mult_zero_left. - -Lemma Th_mult_zero_left2 : (n:A) 0 == 0*n. -Symmetry; EAuto. Qed. - -Lemma aux2 : (x,y,z:A) x+y==0 -> x+z==0 -> y==z. -Intros. -Rewrite <- (plus_zero_left y). -Elim H0. -Elim plus_assoc. -Elim (plus_sym y z). -Rewrite -> plus_assoc. -Rewrite -> H. -Rewrite plus_zero_left. -Reflexivity. -Qed. - -Lemma Th_opp_mult_left : (x,y:A) -(x*y) == (-x)*y. -Intros. -Apply (aux2 1!x*y); -[ Apply opp_def -| Rewrite <- distr_left; - Rewrite -> opp_def; - Auto]. -Qed. -Hints Resolve Th_opp_mult_left. - -Lemma Th_opp_mult_left2 : (x,y:A) (-x)*y == -(x*y). -Symmetry; EAuto. Qed. - -Lemma Th_mult_zero_right : (n:A) n*0 == 0. -Intro; Elim mult_sym; EAuto. -Qed. - -Lemma Th_mult_zero_right2 : (n:A) 0 == n*0. -Intro; Elim mult_sym; EAuto. -Qed. - -Lemma Th_plus_zero_right :(n:A) n + 0 == n. -Intro; Rewrite plus_sym; EAuto. -Qed. - -Lemma Th_plus_zero_right2 :(n:A) n == n + 0. -Intro; Rewrite plus_sym; EAuto. -Qed. - -Lemma Th_mult_one_right : (n:A) n*1 == n. -Intro;Elim mult_sym; EAuto. -Qed. - -Lemma Th_mult_one_right2 : (n:A) n == n*1. -Intro;Elim mult_sym; EAuto. -Qed. - -Lemma Th_opp_mult_right : (x,y:A) -(x*y) == x*(-y). -Intros; Do 2 Rewrite -> (mult_sym x); Auto. -Qed. - -Lemma Th_opp_mult_right2 : (x,y:A) x*(-y) == -(x*y). -Intros; Do 2 Rewrite -> (mult_sym x); Auto. -Qed. - -Lemma Th_plus_opp_opp : (x,y:A) (-x) + (-y) == -(x+y). -Intros. -Apply (aux2 1! x + y); -[ Elim plus_assoc; - Rewrite -> (Th_plus_permute y (-x)); Rewrite -> plus_assoc; - Rewrite -> opp_def; Rewrite plus_zero_left; Auto -| Auto ]. -Qed. - -Lemma Th_plus_permute_opp: (n,m,p:A) (-m)+(n+p) == n+((-m)+p). -EAuto. Qed. - -Lemma Th_opp_opp : (n:A) -(-n) == n. -Intro; Apply (aux2 1! -n); - [ Auto | Elim plus_sym; Auto ]. -Qed. -Hints Resolve Th_opp_opp. - -Lemma Th_opp_opp2 : (n:A) n == -(-n). -Symmetry; EAuto. Qed. - -Lemma Th_mult_opp_opp : (x,y:A) (-x)*(-y) == x*y. -Intros; Rewrite <- Th_opp_mult_left; Rewrite <- Th_opp_mult_right; Auto. -Qed. - -Lemma Th_mult_opp_opp2 : (x,y:A) x*y == (-x)*(-y). -Symmetry; Apply Th_mult_opp_opp. Qed. - -Lemma Th_opp_zero : -0 == 0. -Rewrite <- (plus_zero_left (-0)). -Auto. Qed. - -Lemma Th_plus_reg_left : (n,m,p:A) n + m == n + p -> m==p. -Intros; Generalize (congr_eqT ? ? [z] (-n)+z ? ? H). -Repeat Rewrite plus_assoc. -Rewrite (plus_sym (-n) n). -Rewrite opp_def. -Repeat Rewrite Th_plus_zero_left; EAuto. -Qed. - -Lemma Th_plus_reg_right : (n,m,p:A) m + n == p + n -> m==p. -Intros. -EApply Th_plus_reg_left with n. -Rewrite (plus_sym n m). -Rewrite (plus_sym n p). -Auto. -Qed. - -Lemma Th_distr_right : (n,m,p:A) n*(m + p) == (n*m) + (n*p). -Intros. -Repeat Rewrite -> (mult_sym n). -EAuto. -Qed. - -Lemma Th_distr_right2 : (n,m,p:A) (n*m) + (n*p) == n*(m + p). -Symmetry; Apply Th_distr_right. -Qed. - -End Theory_of_rings. - -Hints Resolve Th_mult_zero_left Th_plus_reg_left : core. - -Unset Implicit Arguments. - -Definition Semi_Ring_Theory_of : - (A:Type)(Aplus : A -> A -> A)(Amult : A -> A -> A)(Aone : A) - (Azero : A)(Aopp : A -> A)(Aeq : A -> A -> bool) - (Ring_Theory Aplus Amult Aone Azero Aopp Aeq) - ->(Semi_Ring_Theory Aplus Amult Aone Azero Aeq). -Intros until 1; Case H. -Split; Intros; Simpl; EAuto. -Defined. - -(* Every ring can be viewed as a semi-ring : this property will be used - in Abstract_polynom. *) -Coercion Semi_Ring_Theory_of : Ring_Theory >-> Semi_Ring_Theory. - - -Section product_ring. - -End product_ring. - -Section power_ring. - -End power_ring. diff --git a/contrib7/ring/Setoid_ring.v b/contrib7/ring/Setoid_ring.v deleted file mode 100644 index 222104e5..00000000 --- a/contrib7/ring/Setoid_ring.v +++ /dev/null @@ -1,13 +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 *) -(************************************************************************) - -(* $Id: Setoid_ring.v,v 1.1.2.1 2004/07/16 19:30:19 herbelin Exp $ *) - -Require Export Setoid_ring_theory. -Require Export Quote. -Require Export Setoid_ring_normalize. diff --git a/contrib7/ring/Setoid_ring_normalize.v b/contrib7/ring/Setoid_ring_normalize.v deleted file mode 100644 index b6b79dae..00000000 --- a/contrib7/ring/Setoid_ring_normalize.v +++ /dev/null @@ -1,1141 +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 *) -(************************************************************************) - -(* $Id: Setoid_ring_normalize.v,v 1.1.2.1 2004/07/16 19:30:19 herbelin Exp $ *) - -Require Setoid_ring_theory. -Require Quote. - -Set Implicit Arguments. - -Lemma index_eq_prop: (n,m:index)(Is_true (index_eq n m)) -> n=m. -Proof. - Induction n; Induction m; Simpl; Try (Reflexivity Orelse Contradiction). - Intros; Rewrite (H i0); Trivial. - Intros; Rewrite (H i0); Trivial. -Save. - -Section setoid. - -Variable A : Type. -Variable Aequiv : A -> A -> Prop. -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aopp : A -> A. -Variable Aeq : A -> A -> bool. - -Variable S : (Setoid_Theory A Aequiv). - -Add Setoid A Aequiv S. - -Variable plus_morph : (a,a0,a1,a2:A) - (Aequiv a a0)->(Aequiv a1 a2)->(Aequiv (Aplus a a1) (Aplus a0 a2)). -Variable mult_morph : (a,a0,a1,a2:A) - (Aequiv a a0)->(Aequiv a1 a2)->(Aequiv (Amult a a1) (Amult a0 a2)). -Variable opp_morph : (a,a0:A) - (Aequiv a a0)->(Aequiv (Aopp a) (Aopp a0)). - -Add Morphism Aplus : Aplus_ext. -Exact plus_morph. -Save. - -Add Morphism Amult : Amult_ext. -Exact mult_morph. -Save. - -Add Morphism Aopp : Aopp_ext. -Exact opp_morph. -Save. - -Local equiv_refl := (Seq_refl A Aequiv S). -Local equiv_sym := (Seq_sym A Aequiv S). -Local equiv_trans := (Seq_trans A Aequiv S). - -Hints Resolve equiv_refl equiv_trans. -Hints Immediate equiv_sym. - -Section semi_setoid_rings. - -(* Section definitions. *) - - -(******************************************) -(* Normal abtract Polynomials *) -(******************************************) -(* DEFINITIONS : -- A varlist is a sorted product of one or more variables : x, x*y*z -- A monom is a constant, a varlist or the product of a constant by a varlist - variables. 2*x*y, x*y*z, 3 are monoms : 2*3, x*3*y, 4*x*3 are NOT. -- A canonical sum is either a monom or an ordered sum of monoms - (the order on monoms is defined later) -- A normal polynomial it either a constant or a canonical sum or a constant - plus a canonical sum -*) - -(* varlist is isomorphic to (list var), but we built a special inductive - for efficiency *) -Inductive varlist : Type := -| Nil_var : varlist -| Cons_var : index -> varlist -> varlist -. - -Inductive canonical_sum : Type := -| Nil_monom : canonical_sum -| Cons_monom : A -> varlist -> canonical_sum -> canonical_sum -| Cons_varlist : varlist -> canonical_sum -> canonical_sum -. - -(* Order on monoms *) - -(* That's the lexicographic order on varlist, extended by : - - A constant is less than every monom - - The relation between two varlist is preserved by multiplication by a - constant. - - Examples : - 3 < x < y - x*y < x*y*y*z - 2*x*y < x*y*y*z - x*y < 54*x*y*y*z - 4*x*y < 59*x*y*y*z -*) - -Fixpoint varlist_eq [x,y:varlist] : bool := - Cases x y of - | Nil_var Nil_var => true - | (Cons_var i xrest) (Cons_var j yrest) => - (andb (index_eq i j) (varlist_eq xrest yrest)) - | _ _ => false - end. - -Fixpoint varlist_lt [x,y:varlist] : bool := - Cases x y of - | Nil_var (Cons_var _ _) => true - | (Cons_var i xrest) (Cons_var j yrest) => - if (index_lt i j) then true - else (andb (index_eq i j) (varlist_lt xrest yrest)) - | _ _ => false - end. - -(* merges two variables lists *) -Fixpoint varlist_merge [l1:varlist] : varlist -> varlist := - Cases l1 of - | (Cons_var v1 t1) => - Fix vm_aux {vm_aux [l2:varlist] : varlist := - Cases l2 of - | (Cons_var v2 t2) => - if (index_lt v1 v2) - then (Cons_var v1 (varlist_merge t1 l2)) - else (Cons_var v2 (vm_aux t2)) - | Nil_var => l1 - end} - | Nil_var => [l2]l2 - end. - -(* returns the sum of two canonical sums *) -Fixpoint canonical_sum_merge [s1:canonical_sum] - : canonical_sum -> canonical_sum := -Cases s1 of -| (Cons_monom c1 l1 t1) => - Fix csm_aux{csm_aux[s2:canonical_sum] : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 c2) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 (canonical_sum_merge t1 s2)) - else (Cons_monom c2 l2 (csm_aux t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 Aone) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 (canonical_sum_merge t1 s2)) - else (Cons_varlist l2 (csm_aux t2)) - | Nil_monom => s1 - end} -| (Cons_varlist l1 t1) => - Fix csm_aux2{csm_aux2[s2:canonical_sum] : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone c2) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 (canonical_sum_merge t1 s2)) - else (Cons_monom c2 l2 (csm_aux2 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone Aone) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 (canonical_sum_merge t1 s2)) - else (Cons_varlist l2 (csm_aux2 t2)) - | Nil_monom => s1 - end} -| Nil_monom => [s2]s2 -end. - -(* Insertion of a monom in a canonical sum *) -Fixpoint monom_insert [c1:A; l1:varlist; s2 : canonical_sum] - : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 c2) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 s2) - else (Cons_monom c2 l2 (monom_insert c1 l1 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 Aone) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 s2) - else (Cons_varlist l2 (monom_insert c1 l1 t2)) - | Nil_monom => (Cons_monom c1 l1 Nil_monom) - end. - -Fixpoint varlist_insert [l1:varlist; s2:canonical_sum] - : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone c2) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 s2) - else (Cons_monom c2 l2 (varlist_insert l1 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone Aone) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 s2) - else (Cons_varlist l2 (varlist_insert l1 t2)) - | Nil_monom => (Cons_varlist l1 Nil_monom) - end. - -(* Computes c0*s *) -Fixpoint canonical_sum_scalar [c0:A; s:canonical_sum] : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (Cons_monom (Amult c0 c) l (canonical_sum_scalar c0 t)) - | (Cons_varlist l t) => - (Cons_monom c0 l (canonical_sum_scalar c0 t)) - | Nil_monom => Nil_monom - end. - -(* Computes l0*s *) -Fixpoint canonical_sum_scalar2 [l0:varlist; s:canonical_sum] - : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (monom_insert c (varlist_merge l0 l) (canonical_sum_scalar2 l0 t)) - | (Cons_varlist l t) => - (varlist_insert (varlist_merge l0 l) (canonical_sum_scalar2 l0 t)) - | Nil_monom => Nil_monom - end. - -(* Computes c0*l0*s *) -Fixpoint canonical_sum_scalar3 [c0:A;l0:varlist; s:canonical_sum] - : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (monom_insert (Amult c0 c) (varlist_merge l0 l) - (canonical_sum_scalar3 c0 l0 t)) - | (Cons_varlist l t) => - (monom_insert c0 (varlist_merge l0 l) - (canonical_sum_scalar3 c0 l0 t)) - | Nil_monom => Nil_monom - end. - -(* returns the product of two canonical sums *) -Fixpoint canonical_sum_prod [s1:canonical_sum] - : canonical_sum -> canonical_sum := - [s2]Cases s1 of - | (Cons_monom c1 l1 t1) => - (canonical_sum_merge (canonical_sum_scalar3 c1 l1 s2) - (canonical_sum_prod t1 s2)) - | (Cons_varlist l1 t1) => - (canonical_sum_merge (canonical_sum_scalar2 l1 s2) - (canonical_sum_prod t1 s2)) - | Nil_monom => Nil_monom - end. - -(* The type to represent concrete semi-setoid-ring polynomials *) - -Inductive Type setspolynomial := - SetSPvar : index -> setspolynomial -| SetSPconst : A -> setspolynomial -| SetSPplus : setspolynomial -> setspolynomial -> setspolynomial -| SetSPmult : setspolynomial -> setspolynomial -> setspolynomial. - -Fixpoint setspolynomial_normalize [p:setspolynomial] : canonical_sum := - Cases p of - | (SetSPplus l r) => (canonical_sum_merge (setspolynomial_normalize l) (setspolynomial_normalize r)) - | (SetSPmult l r) => (canonical_sum_prod (setspolynomial_normalize l) (setspolynomial_normalize r)) - | (SetSPconst c) => (Cons_monom c Nil_var Nil_monom) - | (SetSPvar i) => (Cons_varlist (Cons_var i Nil_var) Nil_monom) - end. - -Fixpoint canonical_sum_simplify [ s:canonical_sum] : canonical_sum := - Cases s of - | (Cons_monom c l t) => - if (Aeq c Azero) - then (canonical_sum_simplify t) - else if (Aeq c Aone) - then (Cons_varlist l (canonical_sum_simplify t)) - else (Cons_monom c l (canonical_sum_simplify t)) - | (Cons_varlist l t) => (Cons_varlist l (canonical_sum_simplify t)) - | Nil_monom => Nil_monom - end. - -Definition setspolynomial_simplify := - [x:setspolynomial] (canonical_sum_simplify (setspolynomial_normalize x)). - -Variable vm : (varmap A). - -Definition interp_var [i:index] := (varmap_find Azero i vm). - -Definition ivl_aux := Fix ivl_aux {ivl_aux[x:index; t:varlist] : A := - Cases t of - | Nil_var => (interp_var x) - | (Cons_var x' t') => (Amult (interp_var x) (ivl_aux x' t')) - end}. - -Definition interp_vl := [l:varlist] - Cases l of - | Nil_var => Aone - | (Cons_var x t) => (ivl_aux x t) - end. - -Definition interp_m := [c:A][l:varlist] - Cases l of - | Nil_var => c - | (Cons_var x t) => - (Amult c (ivl_aux x t)) - end. - -Definition ics_aux := Fix ics_aux{ics_aux[a:A; s:canonical_sum] : A := - Cases s of - | Nil_monom => a - | (Cons_varlist l t) => (Aplus a (ics_aux (interp_vl l) t)) - | (Cons_monom c l t) => (Aplus a (ics_aux (interp_m c l) t)) - end}. - -Definition interp_setcs : canonical_sum -> A := - [s]Cases s of - | Nil_monom => Azero - | (Cons_varlist l t) => - (ics_aux (interp_vl l) t) - | (Cons_monom c l t) => - (ics_aux (interp_m c l) t) - end. - -Fixpoint interp_setsp [p:setspolynomial] : A := - Cases p of - | (SetSPconst c) => c - | (SetSPvar i) => (interp_var i) - | (SetSPplus p1 p2) => (Aplus (interp_setsp p1) (interp_setsp p2)) - | (SetSPmult p1 p2) => (Amult (interp_setsp p1) (interp_setsp p2)) - end. - -(* End interpretation. *) - -Unset Implicit Arguments. - -(* Section properties. *) - -Variable T : (Semi_Setoid_Ring_Theory Aequiv Aplus Amult Aone Azero Aeq). - -Hint SSR_plus_sym_T := Resolve (SSR_plus_sym T). -Hint SSR_plus_assoc_T := Resolve (SSR_plus_assoc T). -Hint SSR_plus_assoc2_T := Resolve (SSR_plus_assoc2 S T). -Hint SSR_mult_sym_T := Resolve (SSR_mult_sym T). -Hint SSR_mult_assoc_T := Resolve (SSR_mult_assoc T). -Hint SSR_mult_assoc2_T := Resolve (SSR_mult_assoc2 S T). -Hint SSR_plus_zero_left_T := Resolve (SSR_plus_zero_left T). -Hint SSR_plus_zero_left2_T := Resolve (SSR_plus_zero_left2 S T). -Hint SSR_mult_one_left_T := Resolve (SSR_mult_one_left T). -Hint SSR_mult_one_left2_T := Resolve (SSR_mult_one_left2 S T). -Hint SSR_mult_zero_left_T := Resolve (SSR_mult_zero_left T). -Hint SSR_mult_zero_left2_T := Resolve (SSR_mult_zero_left2 S T). -Hint SSR_distr_left_T := Resolve (SSR_distr_left T). -Hint SSR_distr_left2_T := Resolve (SSR_distr_left2 S T). -Hint SSR_plus_reg_left_T := Resolve (SSR_plus_reg_left T). -Hint SSR_plus_permute_T := Resolve (SSR_plus_permute S plus_morph T). -Hint SSR_mult_permute_T := Resolve (SSR_mult_permute S mult_morph T). -Hint SSR_distr_right_T := Resolve (SSR_distr_right S plus_morph T). -Hint SSR_distr_right2_T := Resolve (SSR_distr_right2 S plus_morph T). -Hint SSR_mult_zero_right_T := Resolve (SSR_mult_zero_right S T). -Hint SSR_mult_zero_right2_T := Resolve (SSR_mult_zero_right2 S T). -Hint SSR_plus_zero_right_T := Resolve (SSR_plus_zero_right S T). -Hint SSR_plus_zero_right2_T := Resolve (SSR_plus_zero_right2 S T). -Hint SSR_mult_one_right_T := Resolve (SSR_mult_one_right S T). -Hint SSR_mult_one_right2_T := Resolve (SSR_mult_one_right2 S T). -Hint SSR_plus_reg_right_T := Resolve (SSR_plus_reg_right S T). -Hints Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) -Hints Immediate T. - -Lemma varlist_eq_prop : (x,y:varlist) - (Is_true (varlist_eq x y))->x==y. -Proof. - Induction x; Induction y; Contradiction Orelse Try Reflexivity. - Simpl; Intros. - Generalize (andb_prop2 ? ? H1); Intros; Elim H2; Intros. - Rewrite (index_eq_prop H3); Rewrite (H v0 H4); Reflexivity. -Save. - -Remark ivl_aux_ok : (v:varlist)(i:index) - (Aequiv (ivl_aux i v) (Amult (interp_var i) (interp_vl v))). -Proof. - Induction v; Simpl; Intros. - Trivial. - Rewrite (H i); Trivial. -Save. - -Lemma varlist_merge_ok : (x,y:varlist) - (Aequiv (interp_vl (varlist_merge x y)) (Amult (interp_vl x) (interp_vl y))). -Proof. - Induction x. - Simpl; Trivial. - Induction y. - Simpl; Trivial. - Simpl; Intros. - Elim (index_lt i i0); Simpl; Intros. - - Rewrite (ivl_aux_ok v i). - Rewrite (ivl_aux_ok v0 i0). - Rewrite (ivl_aux_ok (varlist_merge v (Cons_var i0 v0)) i). - Rewrite (H (Cons_var i0 v0)). - Simpl. - Rewrite (ivl_aux_ok v0 i0). - EAuto. - - Rewrite (ivl_aux_ok v i). - Rewrite (ivl_aux_ok v0 i0). - Rewrite (ivl_aux_ok - (Fix vm_aux - {vm_aux [l2:varlist] : varlist := - Cases (l2) of - Nil_var => (Cons_var i v) - | (Cons_var v2 t2) => - (if (index_lt i v2) - then (Cons_var i (varlist_merge v l2)) - else (Cons_var v2 (vm_aux t2))) - end} v0) i0). - Rewrite H0. - Rewrite (ivl_aux_ok v i). - EAuto. -Save. - -Remark ics_aux_ok : (x:A)(s:canonical_sum) - (Aequiv (ics_aux x s) (Aplus x (interp_setcs s))). -Proof. - Induction s; Simpl; Intros;Trivial. -Save. - -Remark interp_m_ok : (x:A)(l:varlist) - (Aequiv (interp_m x l) (Amult x (interp_vl l))). -Proof. - NewDestruct l;Trivial. -Save. - -Hint ivl_aux_ok_ := Resolve ivl_aux_ok. -Hint ics_aux_ok_ := Resolve ics_aux_ok. -Hint interp_m_ok_ := Resolve interp_m_ok. - -(* Hints Resolve ivl_aux_ok ics_aux_ok interp_m_ok. *) - -Lemma canonical_sum_merge_ok : (x,y:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_merge x y)) - (Aplus (interp_setcs x) (interp_setcs y))). -Proof. -Induction x; Simpl. -Trivial. - -Induction y; Simpl; Intros. -EAuto. - -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl. -Rewrite (ics_aux_ok (interp_m a v0) c). -Rewrite (ics_aux_ok (interp_m a0 v0) c0). -Rewrite (ics_aux_ok (interp_m (Aplus a a0) v0) - (canonical_sum_merge c c0)). -Rewrite (H c0). -Rewrite (interp_m_ok (Aplus a a0) v0). -Rewrite (interp_m_ok a v0). -Rewrite (interp_m_ok a0 v0). -Setoid_replace (Amult (Aplus a a0) (interp_vl v0)) - with (Aplus (Amult a (interp_vl v0)) (Amult a0 (interp_vl v0))). -Setoid_replace (Aplus - (Aplus (Amult a (interp_vl v0)) - (Amult a0 (interp_vl v0))) - (Aplus (interp_setcs c) (interp_setcs c0))) - with (Aplus (Amult a (interp_vl v0)) - (Aplus (Amult a0 (interp_vl v0)) - (Aplus (interp_setcs c) (interp_setcs c0)))). -Setoid_replace (Aplus (Aplus (Amult a (interp_vl v0)) (interp_setcs c)) - (Aplus (Amult a0 (interp_vl v0)) (interp_setcs c0))) - with (Aplus (Amult a (interp_vl v0)) - (Aplus (interp_setcs c) - (Aplus (Amult a0 (interp_vl v0)) (interp_setcs c0)))). -Auto. - -Elim (varlist_lt v v0); Simpl. -Intro. -Rewrite (ics_aux_ok (interp_m a v) - (canonical_sum_merge c (Cons_monom a0 v0 c0))). -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (ics_aux_ok (interp_m a0 v0) c0). -Rewrite (H (Cons_monom a0 v0 c0)); Simpl. -Rewrite (ics_aux_ok (interp_m a0 v0) c0); Auto. - -Intro. -Rewrite (ics_aux_ok (interp_m a0 v0) - (Fix csm_aux - {csm_aux [s2:canonical_sum] : canonical_sum := - Cases (s2) of - Nil_monom => (Cons_monom a v c) - | (Cons_monom c2 l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus a c2) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_monom a v - (canonical_sum_merge c s2)) - else (Cons_monom c2 l2 (csm_aux t2)))) - | (Cons_varlist l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus a Aone) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_monom a v - (canonical_sum_merge c s2)) - else (Cons_varlist l2 (csm_aux t2)))) - end} c0)). -Rewrite H0. -Rewrite (ics_aux_ok (interp_m a v) c); -Rewrite (ics_aux_ok (interp_m a0 v0) c0); Simpl; Auto. - -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus a Aone) v0) - (canonical_sum_merge c c0)); -Rewrite (ics_aux_ok (interp_m a v0) c); -Rewrite (ics_aux_ok (interp_vl v0) c0). -Rewrite (H c0). -Rewrite (interp_m_ok (Aplus a Aone) v0). -Rewrite (interp_m_ok a v0). -Setoid_replace (Amult (Aplus a Aone) (interp_vl v0)) - with (Aplus (Amult a (interp_vl v0)) (Amult Aone (interp_vl v0))). -Setoid_replace (Aplus - (Aplus (Amult a (interp_vl v0)) - (Amult Aone (interp_vl v0))) - (Aplus (interp_setcs c) (interp_setcs c0))) - with (Aplus (Amult a (interp_vl v0)) - (Aplus (Amult Aone (interp_vl v0)) - (Aplus (interp_setcs c) (interp_setcs c0)))). -Setoid_replace (Aplus (Aplus (Amult a (interp_vl v0)) (interp_setcs c)) - (Aplus (interp_vl v0) (interp_setcs c0))) - with (Aplus (Amult a (interp_vl v0)) - (Aplus (interp_setcs c) (Aplus (interp_vl v0) (interp_setcs c0)))). -Setoid_replace (Amult Aone (interp_vl v0)) with (interp_vl v0). -Auto. - -Elim (varlist_lt v v0); Simpl. -Intro. -Rewrite (ics_aux_ok (interp_m a v) - (canonical_sum_merge c (Cons_varlist v0 c0))); -Rewrite (ics_aux_ok (interp_m a v) c); -Rewrite (ics_aux_ok (interp_vl v0) c0). -Rewrite (H (Cons_varlist v0 c0)); Simpl. -Rewrite (ics_aux_ok (interp_vl v0) c0). -Auto. - -Intro. -Rewrite (ics_aux_ok (interp_vl v0) - (Fix csm_aux - {csm_aux [s2:canonical_sum] : canonical_sum := - Cases (s2) of - Nil_monom => (Cons_monom a v c) - | (Cons_monom c2 l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus a c2) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_monom a v - (canonical_sum_merge c s2)) - else (Cons_monom c2 l2 (csm_aux t2)))) - | (Cons_varlist l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus a Aone) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_monom a v - (canonical_sum_merge c s2)) - else (Cons_varlist l2 (csm_aux t2)))) - end} c0)); Rewrite H0. -Rewrite (ics_aux_ok (interp_m a v) c); -Rewrite (ics_aux_ok (interp_vl v0) c0); Simpl. -Auto. - -Induction y; Simpl; Intros. -Trivial. - -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus Aone a) v0) - (canonical_sum_merge c c0)); -Rewrite (ics_aux_ok (interp_vl v0) c); -Rewrite (ics_aux_ok (interp_m a v0) c0); Rewrite ( -H c0). -Rewrite (interp_m_ok (Aplus Aone a) v0); -Rewrite (interp_m_ok a v0). -Setoid_replace (Amult (Aplus Aone a) (interp_vl v0)) - with (Aplus (Amult Aone (interp_vl v0)) (Amult a (interp_vl v0))); -Setoid_replace (Aplus - (Aplus (Amult Aone (interp_vl v0)) - (Amult a (interp_vl v0))) - (Aplus (interp_setcs c) (interp_setcs c0))) - with (Aplus (Amult Aone (interp_vl v0)) - (Aplus (Amult a (interp_vl v0)) - (Aplus (interp_setcs c) (interp_setcs c0)))); -Setoid_replace (Aplus (Aplus (interp_vl v0) (interp_setcs c)) - (Aplus (Amult a (interp_vl v0)) (interp_setcs c0))) - with (Aplus (interp_vl v0) - (Aplus (interp_setcs c) - (Aplus (Amult a (interp_vl v0)) (interp_setcs c0)))). -Auto. - -Elim (varlist_lt v v0); Simpl; Intros. -Rewrite (ics_aux_ok (interp_vl v) - (canonical_sum_merge c (Cons_monom a v0 c0))); -Rewrite (ics_aux_ok (interp_vl v) c); -Rewrite (ics_aux_ok (interp_m a v0) c0). -Rewrite (H (Cons_monom a v0 c0)); Simpl. -Rewrite (ics_aux_ok (interp_m a v0) c0); Auto. - -Rewrite (ics_aux_ok (interp_m a v0) - (Fix csm_aux2 - {csm_aux2 [s2:canonical_sum] : canonical_sum := - Cases (s2) of - Nil_monom => (Cons_varlist v c) - | (Cons_monom c2 l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus Aone c2) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_varlist v - (canonical_sum_merge c s2)) - else (Cons_monom c2 l2 (csm_aux2 t2)))) - | (Cons_varlist l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus Aone Aone) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_varlist v - (canonical_sum_merge c s2)) - else (Cons_varlist l2 (csm_aux2 t2)))) - end} c0)); Rewrite H0. -Rewrite (ics_aux_ok (interp_vl v) c); -Rewrite (ics_aux_ok (interp_m a v0) c0); Simpl; Auto. - -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0); Intros. -Rewrite (H1 I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus Aone Aone) v0) - (canonical_sum_merge c c0)); -Rewrite (ics_aux_ok (interp_vl v0) c); -Rewrite (ics_aux_ok (interp_vl v0) c0); Rewrite ( -H c0). -Rewrite (interp_m_ok (Aplus Aone Aone) v0). -Setoid_replace (Amult (Aplus Aone Aone) (interp_vl v0)) - with (Aplus (Amult Aone (interp_vl v0)) (Amult Aone (interp_vl v0))); -Setoid_replace (Aplus - (Aplus (Amult Aone (interp_vl v0)) - (Amult Aone (interp_vl v0))) - (Aplus (interp_setcs c) (interp_setcs c0))) - with (Aplus (Amult Aone (interp_vl v0)) - (Aplus (Amult Aone (interp_vl v0)) - (Aplus (interp_setcs c) (interp_setcs c0)))); -Setoid_replace (Aplus (Aplus (interp_vl v0) (interp_setcs c)) - (Aplus (interp_vl v0) (interp_setcs c0))) - with (Aplus (interp_vl v0) - (Aplus (interp_setcs c) (Aplus (interp_vl v0) (interp_setcs c0)))). -Setoid_replace (Amult Aone (interp_vl v0)) with (interp_vl v0); Auto. - -Elim (varlist_lt v v0); Simpl. -Rewrite (ics_aux_ok (interp_vl v) - (canonical_sum_merge c (Cons_varlist v0 c0))); -Rewrite (ics_aux_ok (interp_vl v) c); -Rewrite (ics_aux_ok (interp_vl v0) c0); -Rewrite (H (Cons_varlist v0 c0)); Simpl. -Rewrite (ics_aux_ok (interp_vl v0) c0); Auto. - -Rewrite (ics_aux_ok (interp_vl v0) - (Fix csm_aux2 - {csm_aux2 [s2:canonical_sum] : canonical_sum := - Cases (s2) of - Nil_monom => (Cons_varlist v c) - | (Cons_monom c2 l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus Aone c2) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_varlist v - (canonical_sum_merge c s2)) - else (Cons_monom c2 l2 (csm_aux2 t2)))) - | (Cons_varlist l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus Aone Aone) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_varlist v - (canonical_sum_merge c s2)) - else (Cons_varlist l2 (csm_aux2 t2)))) - end} c0)); Rewrite H0. -Rewrite (ics_aux_ok (interp_vl v) c); -Rewrite (ics_aux_ok (interp_vl v0) c0); Simpl; Auto. -Save. - -Lemma monom_insert_ok: (a:A)(l:varlist)(s:canonical_sum) - (Aequiv (interp_setcs (monom_insert a l s)) - (Aplus (Amult a (interp_vl l)) (interp_setcs s))). -Proof. -Induction s; Intros. -Simpl; Rewrite (interp_m_ok a l); Trivial. - -Simpl; Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus a a0) v) c); -Rewrite (ics_aux_ok (interp_m a0 v) c). -Rewrite (interp_m_ok (Aplus a a0) v); -Rewrite (interp_m_ok a0 v). -Setoid_replace (Amult (Aplus a a0) (interp_vl v)) - with (Aplus (Amult a (interp_vl v)) (Amult a0 (interp_vl v))). -Auto. - -Elim (varlist_lt l v); Simpl; Intros. -Rewrite (ics_aux_ok (interp_m a0 v) c). -Rewrite (interp_m_ok a0 v); Rewrite (interp_m_ok a l). -Auto. - -Rewrite (ics_aux_ok (interp_m a0 v) (monom_insert a l c)); -Rewrite (ics_aux_ok (interp_m a0 v) c); Rewrite H. -Auto. - -Simpl. -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus a Aone) v) c); -Rewrite (ics_aux_ok (interp_vl v) c). -Rewrite (interp_m_ok (Aplus a Aone) v). -Setoid_replace (Amult (Aplus a Aone) (interp_vl v)) - with (Aplus (Amult a (interp_vl v)) (Amult Aone (interp_vl v))). -Setoid_replace (Amult Aone (interp_vl v)) with (interp_vl v). -Auto. - -Elim (varlist_lt l v); Simpl; Intros; Auto. -Rewrite (ics_aux_ok (interp_vl v) (monom_insert a l c)); -Rewrite H. -Rewrite (ics_aux_ok (interp_vl v) c); Auto. -Save. - -Lemma varlist_insert_ok : - (l:varlist)(s:canonical_sum) - (Aequiv (interp_setcs (varlist_insert l s)) - (Aplus (interp_vl l) (interp_setcs s))). -Proof. -Induction s; Simpl; Intros. -Trivial. - -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus Aone a) v) c); -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok (Aplus Aone a) v); -Rewrite (interp_m_ok a v). -Setoid_replace (Amult (Aplus Aone a) (interp_vl v)) - with (Aplus (Amult Aone (interp_vl v)) (Amult a (interp_vl v))). -Setoid_replace (Amult Aone (interp_vl v)) with (interp_vl v); Auto. - -Elim (varlist_lt l v); Simpl; Intros; Auto. -Rewrite (ics_aux_ok (interp_m a v) (varlist_insert l c)); -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite H; Auto. - -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus Aone Aone) v) c); -Rewrite (ics_aux_ok (interp_vl v) c). -Rewrite (interp_m_ok (Aplus Aone Aone) v). -Setoid_replace (Amult (Aplus Aone Aone) (interp_vl v)) - with (Aplus (Amult Aone (interp_vl v)) (Amult Aone (interp_vl v))). -Setoid_replace (Amult Aone (interp_vl v)) with (interp_vl v); Auto. - -Elim (varlist_lt l v); Simpl; Intros; Auto. -Rewrite (ics_aux_ok (interp_vl v) (varlist_insert l c)). -Rewrite H. -Rewrite (ics_aux_ok (interp_vl v) c); Auto. -Save. - -Lemma canonical_sum_scalar_ok : (a:A)(s:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_scalar a s)) (Amult a (interp_setcs s))). -Proof. -Induction s; Simpl; Intros. -Trivial. - -Rewrite (ics_aux_ok (interp_m (Amult a a0) v) - (canonical_sum_scalar a c)); -Rewrite (ics_aux_ok (interp_m a0 v) c). -Rewrite (interp_m_ok (Amult a a0) v); -Rewrite (interp_m_ok a0 v). -Rewrite H. -Setoid_replace (Amult a (Aplus (Amult a0 (interp_vl v)) (interp_setcs c))) - with (Aplus (Amult a (Amult a0 (interp_vl v))) (Amult a (interp_setcs c))). -Auto. - -Rewrite (ics_aux_ok (interp_m a v) (canonical_sum_scalar a c)); -Rewrite (ics_aux_ok (interp_vl v) c); Rewrite H. -Rewrite (interp_m_ok a v). -Auto. -Save. - -Lemma canonical_sum_scalar2_ok : (l:varlist; s:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_scalar2 l s)) (Amult (interp_vl l) (interp_setcs s))). -Proof. -Induction s; Simpl; Intros; Auto. -Rewrite (monom_insert_ok a (varlist_merge l v) - (canonical_sum_scalar2 l c)). -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite H. -Rewrite (varlist_merge_ok l v). -Setoid_replace (Amult (interp_vl l) - (Aplus (Amult a (interp_vl v)) (interp_setcs c))) - with (Aplus (Amult (interp_vl l) (Amult a (interp_vl v))) - (Amult (interp_vl l) (interp_setcs c))). -Auto. - -Rewrite (varlist_insert_ok (varlist_merge l v) - (canonical_sum_scalar2 l c)). -Rewrite (ics_aux_ok (interp_vl v) c). -Rewrite H. -Rewrite (varlist_merge_ok l v). -Auto. -Save. - -Lemma canonical_sum_scalar3_ok : (c:A; l:varlist; s:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_scalar3 c l s)) (Amult c (Amult (interp_vl l) (interp_setcs s)))). -Proof. -Induction s; Simpl; Intros. -Rewrite (SSR_mult_zero_right S T (interp_vl l)). -Auto. - -Rewrite (monom_insert_ok (Amult c a) (varlist_merge l v) - (canonical_sum_scalar3 c l c0)). -Rewrite (ics_aux_ok (interp_m a v) c0). -Rewrite (interp_m_ok a v). -Rewrite H. -Rewrite (varlist_merge_ok l v). -Setoid_replace (Amult (interp_vl l) - (Aplus (Amult a (interp_vl v)) (interp_setcs c0))) - with (Aplus (Amult (interp_vl l) (Amult a (interp_vl v))) - (Amult (interp_vl l) (interp_setcs c0))). -Setoid_replace (Amult c - (Aplus (Amult (interp_vl l) (Amult a (interp_vl v))) - (Amult (interp_vl l) (interp_setcs c0)))) - with (Aplus (Amult c (Amult (interp_vl l) (Amult a (interp_vl v)))) - (Amult c (Amult (interp_vl l) (interp_setcs c0)))). -Setoid_replace (Amult (Amult c a) (Amult (interp_vl l) (interp_vl v))) - with (Amult c (Amult a (Amult (interp_vl l) (interp_vl v)))). -Auto. - -Rewrite (monom_insert_ok c (varlist_merge l v) - (canonical_sum_scalar3 c l c0)). -Rewrite (ics_aux_ok (interp_vl v) c0). -Rewrite H. -Rewrite (varlist_merge_ok l v). -Setoid_replace (Aplus (Amult c (Amult (interp_vl l) (interp_vl v))) - (Amult c (Amult (interp_vl l) (interp_setcs c0)))) - with (Amult c - (Aplus (Amult (interp_vl l) (interp_vl v)) - (Amult (interp_vl l) (interp_setcs c0)))). -Auto. -Save. - -Lemma canonical_sum_prod_ok : (x,y:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_prod x y)) (Amult (interp_setcs x) (interp_setcs y))). -Proof. -Induction x; Simpl; Intros. -Trivial. - -Rewrite (canonical_sum_merge_ok (canonical_sum_scalar3 a v y) - (canonical_sum_prod c y)). -Rewrite (canonical_sum_scalar3_ok a v y). -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite (H y). -Setoid_replace (Amult a (Amult (interp_vl v) (interp_setcs y))) - with (Amult (Amult a (interp_vl v)) (interp_setcs y)). -Setoid_replace (Amult (Aplus (Amult a (interp_vl v)) (interp_setcs c)) - (interp_setcs y)) - with (Aplus (Amult (Amult a (interp_vl v)) (interp_setcs y)) - (Amult (interp_setcs c) (interp_setcs y))). -Trivial. - -Rewrite (canonical_sum_merge_ok (canonical_sum_scalar2 v y) - (canonical_sum_prod c y)). -Rewrite (canonical_sum_scalar2_ok v y). -Rewrite (ics_aux_ok (interp_vl v) c). -Rewrite (H y). -Trivial. -Save. - -Theorem setspolynomial_normalize_ok : (p:setspolynomial) - (Aequiv (interp_setcs (setspolynomial_normalize p)) (interp_setsp p)). -Proof. -Induction p; Simpl; Intros; Trivial. -Rewrite (canonical_sum_merge_ok (setspolynomial_normalize s) - (setspolynomial_normalize s0)). -Rewrite H; Rewrite H0; Trivial. - -Rewrite (canonical_sum_prod_ok (setspolynomial_normalize s) - (setspolynomial_normalize s0)). -Rewrite H; Rewrite H0; Trivial. -Save. - -Lemma canonical_sum_simplify_ok : (s:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_simplify s)) (interp_setcs s)). -Proof. -Induction s; Simpl; Intros. -Trivial. - -Generalize (SSR_eq_prop T 9!a 10!Azero). -Elim (Aeq a Azero). -Simpl. -Intros. -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite (H0 I). -Setoid_replace (Amult Azero (interp_vl v)) with Azero. -Rewrite H. -Trivial. - -Intros; Simpl. -Generalize (SSR_eq_prop T 9!a 10!Aone). -Elim (Aeq a Aone). -Intros. -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite (H1 I). -Simpl. -Rewrite (ics_aux_ok (interp_vl v) (canonical_sum_simplify c)). -Rewrite H. -Auto. - -Simpl. -Intros. -Rewrite (ics_aux_ok (interp_m a v) (canonical_sum_simplify c)). -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite H; Trivial. - -Rewrite (ics_aux_ok (interp_vl v) (canonical_sum_simplify c)). -Rewrite H. -Auto. -Save. - -Theorem setspolynomial_simplify_ok : (p:setspolynomial) - (Aequiv (interp_setcs (setspolynomial_simplify p)) (interp_setsp p)). -Proof. -Intro. -Unfold setspolynomial_simplify. -Rewrite (canonical_sum_simplify_ok (setspolynomial_normalize p)). -Exact (setspolynomial_normalize_ok p). -Save. - -End semi_setoid_rings. - -Implicits Cons_varlist. -Implicits Cons_monom. -Implicits SetSPconst. -Implicits SetSPplus. -Implicits SetSPmult. - - - -Section setoid_rings. - -Set Implicit Arguments. - -Variable vm : (varmap A). -Variable T : (Setoid_Ring_Theory Aequiv Aplus Amult Aone Azero Aopp Aeq). - -Hint STh_plus_sym_T := Resolve (STh_plus_sym T). -Hint STh_plus_assoc_T := Resolve (STh_plus_assoc T). -Hint STh_plus_assoc2_T := Resolve (STh_plus_assoc2 S T). -Hint STh_mult_sym_T := Resolve (STh_mult_sym T). -Hint STh_mult_assoc_T := Resolve (STh_mult_assoc T). -Hint STh_mult_assoc2_T := Resolve (STh_mult_assoc2 S T). -Hint STh_plus_zero_left_T := Resolve (STh_plus_zero_left T). -Hint STh_plus_zero_left2_T := Resolve (STh_plus_zero_left2 S T). -Hint STh_mult_one_left_T := Resolve (STh_mult_one_left T). -Hint STh_mult_one_left2_T := Resolve (STh_mult_one_left2 S T). -Hint STh_mult_zero_left_T := Resolve (STh_mult_zero_left S plus_morph mult_morph T). -Hint STh_mult_zero_left2_T := Resolve (STh_mult_zero_left2 S plus_morph mult_morph T). -Hint STh_distr_left_T := Resolve (STh_distr_left T). -Hint STh_distr_left2_T := Resolve (STh_distr_left2 S T). -Hint STh_plus_reg_left_T := Resolve (STh_plus_reg_left S plus_morph T). -Hint STh_plus_permute_T := Resolve (STh_plus_permute S plus_morph T). -Hint STh_mult_permute_T := Resolve (STh_mult_permute S mult_morph T). -Hint STh_distr_right_T := Resolve (STh_distr_right S plus_morph T). -Hint STh_distr_right2_T := Resolve (STh_distr_right2 S plus_morph T). -Hint STh_mult_zero_right_T := Resolve (STh_mult_zero_right S plus_morph mult_morph T). -Hint STh_mult_zero_right2_T := Resolve (STh_mult_zero_right2 S plus_morph mult_morph T). -Hint STh_plus_zero_right_T := Resolve (STh_plus_zero_right S T). -Hint STh_plus_zero_right2_T := Resolve (STh_plus_zero_right2 S T). -Hint STh_mult_one_right_T := Resolve (STh_mult_one_right S T). -Hint STh_mult_one_right2_T := Resolve (STh_mult_one_right2 S T). -Hint STh_plus_reg_right_T := Resolve (STh_plus_reg_right S plus_morph T). -Hints Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) -Hints Immediate T. - - -(*** Definitions *) - -Inductive Type setpolynomial := - SetPvar : index -> setpolynomial -| SetPconst : A -> setpolynomial -| SetPplus : setpolynomial -> setpolynomial -> setpolynomial -| SetPmult : setpolynomial -> setpolynomial -> setpolynomial -| SetPopp : setpolynomial -> setpolynomial. - -Fixpoint setpolynomial_normalize [x:setpolynomial] : canonical_sum := - Cases x of - | (SetPplus l r) => (canonical_sum_merge - (setpolynomial_normalize l) - (setpolynomial_normalize r)) - | (SetPmult l r) => (canonical_sum_prod - (setpolynomial_normalize l) - (setpolynomial_normalize r)) - | (SetPconst c) => (Cons_monom c Nil_var Nil_monom) - | (SetPvar i) => (Cons_varlist (Cons_var i Nil_var) Nil_monom) - | (SetPopp p) => (canonical_sum_scalar3 - (Aopp Aone) Nil_var - (setpolynomial_normalize p)) - end. - -Definition setpolynomial_simplify := - [x:setpolynomial](canonical_sum_simplify (setpolynomial_normalize x)). - -Fixpoint setspolynomial_of [x:setpolynomial] : setspolynomial := - Cases x of - | (SetPplus l r) => (SetSPplus (setspolynomial_of l) (setspolynomial_of r)) - | (SetPmult l r) => (SetSPmult (setspolynomial_of l) (setspolynomial_of r)) - | (SetPconst c) => (SetSPconst c) - | (SetPvar i) => (SetSPvar i) - | (SetPopp p) => (SetSPmult (SetSPconst (Aopp Aone)) (setspolynomial_of p)) - end. - -(*** Interpretation *) - -Fixpoint interp_setp [p:setpolynomial] : A := - Cases p of - | (SetPconst c) => c - | (SetPvar i) => (varmap_find Azero i vm) - | (SetPplus p1 p2) => (Aplus (interp_setp p1) (interp_setp p2)) - | (SetPmult p1 p2) => (Amult (interp_setp p1) (interp_setp p2)) - | (SetPopp p1) => (Aopp (interp_setp p1)) - end. - -(*** Properties *) - -Unset Implicit Arguments. - -Lemma setspolynomial_of_ok : (p:setpolynomial) - (Aequiv (interp_setp p) (interp_setsp vm (setspolynomial_of p))). -Induction p; Trivial; Simpl; Intros. -Rewrite H; Rewrite H0; Trivial. -Rewrite H; Rewrite H0; Trivial. -Rewrite H. -Rewrite (STh_opp_mult_left2 S plus_morph mult_morph T Aone - (interp_setsp vm (setspolynomial_of s))). -Rewrite (STh_mult_one_left T - (interp_setsp vm (setspolynomial_of s))). -Trivial. -Save. - -Theorem setpolynomial_normalize_ok : (p:setpolynomial) - (setpolynomial_normalize p) - ==(setspolynomial_normalize (setspolynomial_of p)). -Induction p; Trivial; Simpl; Intros. -Rewrite H; Rewrite H0; Reflexivity. -Rewrite H; Rewrite H0; Reflexivity. -Rewrite H; Simpl. -Elim (canonical_sum_scalar3 (Aopp Aone) Nil_var - (setspolynomial_normalize (setspolynomial_of s))); - [ Reflexivity - | Simpl; Intros; Rewrite H0; Reflexivity - | Simpl; Intros; Rewrite H0; Reflexivity ]. -Save. - -Theorem setpolynomial_simplify_ok : (p:setpolynomial) - (Aequiv (interp_setcs vm (setpolynomial_simplify p)) (interp_setp p)). -Intro. -Unfold setpolynomial_simplify. -Rewrite (setspolynomial_of_ok p). -Rewrite setpolynomial_normalize_ok. -Rewrite (canonical_sum_simplify_ok vm - (Semi_Setoid_Ring_Theory_of A Aequiv S Aplus Amult Aone Azero Aopp - Aeq plus_morph mult_morph T) - (setspolynomial_normalize (setspolynomial_of p))). -Rewrite (setspolynomial_normalize_ok vm - (Semi_Setoid_Ring_Theory_of A Aequiv S Aplus Amult Aone Azero Aopp - Aeq plus_morph mult_morph T) (setspolynomial_of p)). -Trivial. -Save. - -End setoid_rings. - -End setoid. diff --git a/contrib7/ring/Setoid_ring_theory.v b/contrib7/ring/Setoid_ring_theory.v deleted file mode 100644 index 13afc5ee..00000000 --- a/contrib7/ring/Setoid_ring_theory.v +++ /dev/null @@ -1,429 +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 *) -(************************************************************************) - -(* $Id: Setoid_ring_theory.v,v 1.1.2.1 2004/07/16 19:30:19 herbelin Exp $ *) - -Require Export Bool. -Require Export Setoid. - -Set Implicit Arguments. - -Section Setoid_rings. - -Variable A : Type. -Variable Aequiv : A -> A -> Prop. - -Infix Local "==" Aequiv (at level 5, no associativity). - -Variable S : (Setoid_Theory A Aequiv). - -Add Setoid A Aequiv S. - -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aopp : A -> A. -Variable Aeq : A -> A -> bool. - -Infix 4 "+" Aplus V8only 50 (left associativity). -Infix 4 "*" Amult V8only 40 (left associativity). -Notation "0" := Azero. -Notation "1" := Aone. -Notation "- x" := (Aopp x) (at level 0) V8only. - -Variable plus_morph : (a,a0,a1,a2:A) a == a0 -> a1 == a2 -> a+a1 == a0+a2. -Variable mult_morph : (a,a0,a1,a2:A) a == a0 -> a1 == a2 -> a*a1 == a0*a2. -Variable opp_morph : (a,a0:A) a == a0 -> -a == -a0. - -Add Morphism Aplus : Aplus_ext. -Exact plus_morph. -Save. - -Add Morphism Amult : Amult_ext. -Exact mult_morph. -Save. - -Add Morphism Aopp : Aopp_ext. -Exact opp_morph. -Save. - -Section Theory_of_semi_setoid_rings. - -Record Semi_Setoid_Ring_Theory : Prop := -{ SSR_plus_sym : (n,m:A) n + m == m + n; - SSR_plus_assoc : (n,m,p:A) n + (m + p) == (n + m) + p; - SSR_mult_sym : (n,m:A) n*m == m*n; - SSR_mult_assoc : (n,m,p:A) n*(m*p) == (n*m)*p; - SSR_plus_zero_left :(n:A) 0 + n == n; - SSR_mult_one_left : (n:A) 1*n == n; - SSR_mult_zero_left : (n:A) 0*n == 0; - SSR_distr_left : (n,m,p:A) (n + m)*p == n*p + m*p; - SSR_plus_reg_left : (n,m,p:A)n + m == n + p -> m == p; - SSR_eq_prop : (x,y:A) (Is_true (Aeq x y)) -> x == y -}. - -Variable T : Semi_Setoid_Ring_Theory. - -Local plus_sym := (SSR_plus_sym T). -Local plus_assoc := (SSR_plus_assoc T). -Local mult_sym := ( SSR_mult_sym T). -Local mult_assoc := (SSR_mult_assoc T). -Local plus_zero_left := (SSR_plus_zero_left T). -Local mult_one_left := (SSR_mult_one_left T). -Local mult_zero_left := (SSR_mult_zero_left T). -Local distr_left := (SSR_distr_left T). -Local plus_reg_left := (SSR_plus_reg_left T). -Local equiv_refl := (Seq_refl A Aequiv S). -Local equiv_sym := (Seq_sym A Aequiv S). -Local equiv_trans := (Seq_trans A Aequiv S). - -Hints Resolve plus_sym plus_assoc mult_sym mult_assoc - plus_zero_left mult_one_left mult_zero_left distr_left - plus_reg_left equiv_refl (*equiv_sym*). -Hints Immediate equiv_sym. - -(* Lemmas whose form is x=y are also provided in form y=x because - Auto does not symmetry *) -Lemma SSR_mult_assoc2 : (n,m,p:A) (n * m) * p == n * (m * p). -Auto. Save. - -Lemma SSR_plus_assoc2 : (n,m,p:A) (n + m) + p == n + (m + p). -Auto. Save. - -Lemma SSR_plus_zero_left2 : (n:A) n == 0 + n. -Auto. Save. - -Lemma SSR_mult_one_left2 : (n:A) n == 1*n. -Auto. Save. - -Lemma SSR_mult_zero_left2 : (n:A) 0 == 0*n. -Auto. Save. - -Lemma SSR_distr_left2 : (n,m,p:A) n*p + m*p == (n + m)*p. -Auto. Save. - -Lemma SSR_plus_permute : (n,m,p:A) n+(m+p) == m+(n+p). -Intros. -Rewrite (plus_assoc n m p). -Rewrite (plus_sym n m). -Rewrite <- (plus_assoc m n p). -Trivial. -Save. - -Lemma SSR_mult_permute : (n,m,p:A) n*(m*p) == m*(n*p). -Intros. -Rewrite (mult_assoc n m p). -Rewrite (mult_sym n m). -Rewrite <- (mult_assoc m n p). -Trivial. -Save. - -Hints Resolve SSR_plus_permute SSR_mult_permute. - -Lemma SSR_distr_right : (n,m,p:A) n*(m+p) == (n*m) + (n*p). -Intros. -Rewrite (mult_sym n (Aplus m p)). -Rewrite (mult_sym n m). -Rewrite (mult_sym n p). -Auto. -Save. - -Lemma SSR_distr_right2 : (n,m,p:A) (n*m) + (n*p) == n*(m + p). -Intros. -Apply equiv_sym. -Apply SSR_distr_right. -Save. - -Lemma SSR_mult_zero_right : (n:A) n*0 == 0. -Intro; Rewrite (mult_sym n Azero); Auto. -Save. - -Lemma SSR_mult_zero_right2 : (n:A) 0 == n*0. -Intro; Rewrite (mult_sym n Azero); Auto. -Save. - -Lemma SSR_plus_zero_right :(n:A) n + 0 == n. -Intro; Rewrite (plus_sym n Azero); Auto. -Save. - -Lemma SSR_plus_zero_right2 :(n:A) n == n + 0. -Intro; Rewrite (plus_sym n Azero); Auto. -Save. - -Lemma SSR_mult_one_right : (n:A) n*1 == n. -Intro; Rewrite (mult_sym n Aone); Auto. -Save. - -Lemma SSR_mult_one_right2 : (n:A) n == n*1. -Intro; Rewrite (mult_sym n Aone); Auto. -Save. - -Lemma SSR_plus_reg_right : (n,m,p:A) m+n == p+n -> m==p. -Intros n m p; Rewrite (plus_sym m n); Rewrite (plus_sym p n). -Intro; Apply plus_reg_left with n; Trivial. -Save. - -End Theory_of_semi_setoid_rings. - -Section Theory_of_setoid_rings. - -Record Setoid_Ring_Theory : Prop := -{ STh_plus_sym : (n,m:A) n + m == m + n; - STh_plus_assoc : (n,m,p:A) n + (m + p) == (n + m) + p; - STh_mult_sym : (n,m:A) n*m == m*n; - STh_mult_assoc : (n,m,p:A) n*(m*p) == (n*m)*p; - STh_plus_zero_left :(n:A) 0 + n == n; - STh_mult_one_left : (n:A) 1*n == n; - STh_opp_def : (n:A) n + (-n) == 0; - STh_distr_left : (n,m,p:A) (n + m)*p == n*p + m*p; - STh_eq_prop : (x,y:A) (Is_true (Aeq x y)) -> x == y -}. - -Variable T : Setoid_Ring_Theory. - -Local plus_sym := (STh_plus_sym T). -Local plus_assoc := (STh_plus_assoc T). -Local mult_sym := (STh_mult_sym T). -Local mult_assoc := (STh_mult_assoc T). -Local plus_zero_left := (STh_plus_zero_left T). -Local mult_one_left := (STh_mult_one_left T). -Local opp_def := (STh_opp_def T). -Local distr_left := (STh_distr_left T). -Local equiv_refl := (Seq_refl A Aequiv S). -Local equiv_sym := (Seq_sym A Aequiv S). -Local equiv_trans := (Seq_trans A Aequiv S). - -Hints Resolve plus_sym plus_assoc mult_sym mult_assoc - plus_zero_left mult_one_left opp_def distr_left - equiv_refl equiv_sym. - -(* Lemmas whose form is x=y are also provided in form y=x because Auto does - not symmetry *) - -Lemma STh_mult_assoc2 : (n,m,p:A) (n * m) * p == n * (m * p). -Auto. Save. - -Lemma STh_plus_assoc2 : (n,m,p:A) (n + m) + p == n + (m + p). -Auto. Save. - -Lemma STh_plus_zero_left2 : (n:A) n == 0 + n. -Auto. Save. - -Lemma STh_mult_one_left2 : (n:A) n == 1*n. -Auto. Save. - -Lemma STh_distr_left2 : (n,m,p:A) n*p + m*p == (n + m)*p. -Auto. Save. - -Lemma STh_opp_def2 : (n:A) 0 == n + (-n). -Auto. Save. - -Lemma STh_plus_permute : (n,m,p:A) n + (m + p) == m + (n + p). -Intros. -Rewrite (plus_assoc n m p). -Rewrite (plus_sym n m). -Rewrite <- (plus_assoc m n p). -Trivial. -Save. - -Lemma STh_mult_permute : (n,m,p:A) n*(m*p) == m*(n*p). -Intros. -Rewrite (mult_assoc n m p). -Rewrite (mult_sym n m). -Rewrite <- (mult_assoc m n p). -Trivial. -Save. - -Hints Resolve STh_plus_permute STh_mult_permute. - -Lemma Saux1 : (a:A) a + a == a -> a == 0. -Intros. -Rewrite <- (plus_zero_left a). -Rewrite (plus_sym Azero a). -Setoid_replace (Aplus a Azero) with (Aplus a (Aplus a (Aopp a))); Auto. -Rewrite (plus_assoc a a (Aopp a)). -Rewrite H. -Apply opp_def. -Save. - -Lemma STh_mult_zero_left :(n:A) 0*n == 0. -Intros. -Apply Saux1. -Rewrite <- (distr_left Azero Azero n). -Rewrite (plus_zero_left Azero). -Trivial. -Save. -Hints Resolve STh_mult_zero_left. - -Lemma STh_mult_zero_left2 : (n:A) 0 == 0*n. -Auto. -Save. - -Lemma Saux2 : (x,y,z:A) x+y==0 -> x+z==0 -> y == z. -Intros. -Rewrite <- (plus_zero_left y). -Rewrite <- H0. -Rewrite <- (plus_assoc x z y). -Rewrite (plus_sym z y). -Rewrite (plus_assoc x y z). -Rewrite H. -Auto. -Save. - -Lemma STh_opp_mult_left : (x,y:A) -(x*y) == (-x)*y. -Intros. -Apply Saux2 with (Amult x y); Auto. -Rewrite <- (distr_left x (Aopp x) y). -Rewrite (opp_def x). -Auto. -Save. -Hints Resolve STh_opp_mult_left. - -Lemma STh_opp_mult_left2 : (x,y:A) (-x)*y == -(x*y) . -Auto. -Save. - -Lemma STh_mult_zero_right : (n:A) n*0 == 0. -Intro; Rewrite (mult_sym n Azero); Auto. -Save. - -Lemma STh_mult_zero_right2 : (n:A) 0 == n*0. -Intro; Rewrite (mult_sym n Azero); Auto. -Save. - -Lemma STh_plus_zero_right :(n:A) n + 0 == n. -Intro; Rewrite (plus_sym n Azero); Auto. -Save. - -Lemma STh_plus_zero_right2 :(n:A) n == n + 0. -Intro; Rewrite (plus_sym n Azero); Auto. -Save. - -Lemma STh_mult_one_right : (n:A) n*1 == n. -Intro; Rewrite (mult_sym n Aone); Auto. -Save. - -Lemma STh_mult_one_right2 : (n:A) n == n*1. -Intro; Rewrite (mult_sym n Aone); Auto. -Save. - -Lemma STh_opp_mult_right : (x,y:A) -(x*y) == x*(-y). -Intros. -Rewrite (mult_sym x y). -Rewrite (mult_sym x (Aopp y)). -Auto. -Save. - -Lemma STh_opp_mult_right2 : (x,y:A) x*(-y) == -(x*y). -Intros. -Rewrite (mult_sym x y). -Rewrite (mult_sym x (Aopp y)). -Auto. -Save. - -Lemma STh_plus_opp_opp : (x,y:A) (-x) + (-y) == -(x+y). -Intros. -Apply Saux2 with (Aplus x y); Auto. -Rewrite (STh_plus_permute (Aplus x y) (Aopp x) (Aopp y)). -Rewrite <- (plus_assoc x y (Aopp y)). -Rewrite (opp_def y); Rewrite (STh_plus_zero_right x). -Rewrite (STh_opp_def2 x); Trivial. -Save. - -Lemma STh_plus_permute_opp: (n,m,p:A) (-m)+(n+p) == n+((-m)+p). -Auto. -Save. - -Lemma STh_opp_opp : (n:A) -(-n) == n. -Intro. -Apply Saux2 with (Aopp n); Auto. -Rewrite (plus_sym (Aopp n) n); Auto. -Save. -Hints Resolve STh_opp_opp. - -Lemma STh_opp_opp2 : (n:A) n == -(-n). -Auto. -Save. - -Lemma STh_mult_opp_opp : (x,y:A) (-x)*(-y) == x*y. -Intros. -Rewrite (STh_opp_mult_left2 x (Aopp y)). -Rewrite (STh_opp_mult_right2 x y). -Trivial. -Save. - -Lemma STh_mult_opp_opp2 : (x,y:A) x*y == (-x)*(-y). -Intros. -Apply equiv_sym. -Apply STh_mult_opp_opp. -Save. - -Lemma STh_opp_zero : -0 == 0. -Rewrite <- (plus_zero_left (Aopp Azero)). -Trivial. -Save. - -Lemma STh_plus_reg_left : (n,m,p:A) n+m == n+p -> m==p. -Intros. -Rewrite <- (plus_zero_left m). -Rewrite <- (plus_zero_left p). -Rewrite <- (opp_def n). -Rewrite (plus_sym n (Aopp n)). -Rewrite <- (plus_assoc (Aopp n) n m). -Rewrite <- (plus_assoc (Aopp n) n p). -Auto. -Save. - -Lemma STh_plus_reg_right : (n,m,p:A) m+n == p+n -> m==p. -Intros. -Apply STh_plus_reg_left with n. -Rewrite (plus_sym n m); Rewrite (plus_sym n p); -Assumption. -Save. - -Lemma STh_distr_right : (n,m,p:A) n*(m+p) == (n*m)+(n*p). -Intros. -Rewrite (mult_sym n (Aplus m p)). -Rewrite (mult_sym n m). -Rewrite (mult_sym n p). -Trivial. -Save. - -Lemma STh_distr_right2 : (n,m,p:A) (n*m)+(n*p) == n*(m+p). -Intros. -Apply equiv_sym. -Apply STh_distr_right. -Save. - -End Theory_of_setoid_rings. - -Hints Resolve STh_mult_zero_left STh_plus_reg_left : core. - -Unset Implicit Arguments. - -Definition Semi_Setoid_Ring_Theory_of : - Setoid_Ring_Theory -> Semi_Setoid_Ring_Theory. -Intros until 1; Case H. -Split; Intros; Simpl; EAuto. -Defined. - -Coercion Semi_Setoid_Ring_Theory_of : - Setoid_Ring_Theory >-> Semi_Setoid_Ring_Theory. - - - -Section product_ring. - -End product_ring. - -Section power_ring. - -End power_ring. - -End Setoid_rings. diff --git a/contrib7/ring/ZArithRing.v b/contrib7/ring/ZArithRing.v deleted file mode 100644 index fc7ef29f..00000000 --- a/contrib7/ring/ZArithRing.v +++ /dev/null @@ -1,35 +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 *) -(************************************************************************) - -(* $Id: ZArithRing.v,v 1.1.2.1 2004/07/16 19:30:19 herbelin Exp $ *) - -(* Instantiation of the Ring tactic for the binary integers of ZArith *) - -Require Export ArithRing. -Require Export ZArith_base. -Require Eqdep_dec. - -Definition Zeq := [x,y:Z] - Cases `x ?= y ` of - EGAL => true - | _ => false - end. - -Lemma Zeq_prop : (x,y:Z)(Is_true (Zeq x y)) -> x==y. - Intros x y H; Unfold Zeq in H. - Apply Zcompare_EGAL_eq. - NewDestruct (Zcompare x y); [Reflexivity | Contradiction | Contradiction ]. -Save. - -Definition ZTheory : (Ring_Theory Zplus Zmult `1` `0` Zopp Zeq). - Split; Intros; Apply eq2eqT; EAuto with zarith. - Apply eqT2eq; Apply Zeq_prop; Assumption. -Save. - -(* NatConstants and NatTheory are defined in Ring_theory.v *) -Add Ring Z Zplus Zmult `1` `0` Zopp Zeq ZTheory [POS NEG ZERO xO xI xH]. |