diff options
Diffstat (limited to 'contrib/field')
| -rw-r--r-- | contrib/field/Field.v | 2 | ||||
| -rw-r--r-- | contrib/field/Field_Compl.v | 91 | ||||
| -rw-r--r-- | contrib/field/Field_Tactic.v | 713 | ||||
| -rw-r--r-- | contrib/field/Field_Theory.v | 1015 |
4 files changed, 944 insertions, 877 deletions
diff --git a/contrib/field/Field.v b/contrib/field/Field.v index df2a44f3e7..a7cf332a79 100644 --- a/contrib/field/Field.v +++ b/contrib/field/Field.v @@ -12,4 +12,4 @@ Require Export Field_Compl. Require Export Field_Theory. Require Export Field_Tactic. -(* Command declarations are moved to the ML side *) +(* Command declarations are moved to the ML side *)
\ No newline at end of file diff --git a/contrib/field/Field_Compl.v b/contrib/field/Field_Compl.v index aa3a991470..c90fea1ecc 100644 --- a/contrib/field/Field_Compl.v +++ b/contrib/field/Field_Compl.v @@ -8,55 +8,54 @@ (* $Id$ *) -Inductive listT [A:Type] : Type := - nilT : (listT A) | consT : A->(listT A)->(listT A). - -Fixpoint appT [A:Type][l:(listT A)] : (listT A) -> (listT A) := - [m:(listT A)] - Cases l of - | nilT => m - | (consT a l1) => (consT A a (appT A l1 m)) +Inductive listT (A:Type) : Type := + | nilT : listT A + | consT : A -> listT A -> listT A. + +Fixpoint appT (A:Type) (l m:listT A) {struct l} : listT A := + match l with + | nilT => m + | consT a l1 => consT A a (appT A l1 m) end. -Inductive prodT [A,B:Type] : Type := - pairT: A->B->(prodT A B). +Inductive prodT (A B:Type) : Type := + pairT : A -> B -> prodT A B. Definition assoc_2nd := -Fix assoc_2nd_rec - {assoc_2nd_rec - [A:Type;B:Set;eq_dec:(e1,e2:B){e1=e2}+{~e1=e2};lst:(listT (prodT A B))] - : B->A->A:= - [key:B;default:A] - Cases lst of - | nilT => default - | (consT (pairT v e) l) => - (Cases (eq_dec e key) of - | (left _) => v - | (right _) => (assoc_2nd_rec A B eq_dec l key default) - end) - end}. - -Definition fstT [A,B:Type;c:(prodT A B)] := - Cases c of - | (pairT a _) => a - end. - -Definition sndT [A,B:Type;c:(prodT A B)] := - Cases c of - | (pairT _ a) => a - end. + (fix assoc_2nd_rec (A:Type) (B:Set) + (eq_dec:forall e1 e2:B, {e1 = e2} + {e1 <> e2}) + (lst:listT (prodT A B)) {struct lst} : + B -> A -> A := + fun (key:B) (default:A) => + match lst with + | nilT => default + | consT (pairT v e) l => + match eq_dec e key with + | left _ => v + | right _ => assoc_2nd_rec A B eq_dec l key default + end + end). + +Definition fstT (A B:Type) (c:prodT A B) := match c with + | pairT a _ => a + end. + +Definition sndT (A B:Type) (c:prodT A B) := match c with + | pairT _ a => a + end. Definition mem := -Fix mem {mem [A:Set;eq_dec:(e1,e2:A){e1=e2}+{~e1=e2};a:A;l:(listT A)] : bool := - Cases l of - | nilT => false - | (consT a1 l1) => - Cases (eq_dec a a1) of - | (left _) => true - | (right _) => (mem A eq_dec a l1) - end - end}. - -Inductive option [A:Type] : Type := - | None : (option A) - | Some : (A -> A -> A) -> (option A). + (fix mem (A:Set) (eq_dec:forall e1 e2:A, {e1 = e2} + {e1 <> e2}) + (a:A) (l:listT A) {struct l} : bool := + match l with + | nilT => false + | consT a1 l1 => + match eq_dec a a1 with + | left _ => true + | right _ => mem A eq_dec a l1 + end + end). + +Inductive option (A:Type) : Type := + | None : option A + | Some : (A -> A -> A) -> option A.
\ No newline at end of file diff --git a/contrib/field/Field_Tactic.v b/contrib/field/Field_Tactic.v index cd382bc393..a5b206fb20 100644 --- a/contrib/field/Field_Tactic.v +++ b/contrib/field/Field_Tactic.v @@ -8,102 +8,101 @@ (* $Id$ *) -Require Ring. +Require Import Ring. Require Export Field_Compl. Require Export Field_Theory. (**** Interpretation A --> ExprA ****) -Recursive Tactic Definition MemAssoc var lvar := - Match lvar With - | [(nilT ?)] -> false - | [(consT ? ?1 ?2)] -> - (Match ?1=var With - | [?1=?1] -> true - | _ -> (MemAssoc var ?2)). - -Recursive Tactic Definition SeekVarAux FT lvar trm := - Let AT = Eval Cbv Beta Delta [A] Iota in (A FT) - And AzeroT = Eval Cbv Beta Delta [Azero] Iota in (Azero FT) - And AoneT = Eval Cbv Beta Delta [Aone] Iota in (Aone FT) - And AplusT = Eval Cbv Beta Delta [Aplus] Iota in (Aplus FT) - And AmultT = Eval Cbv Beta Delta [Amult] Iota in (Amult FT) - And AoppT = Eval Cbv Beta Delta [Aopp] Iota in (Aopp FT) - And AinvT = Eval Cbv Beta Delta [Ainv] Iota in (Ainv FT) In - Match trm With - | [(AzeroT)] -> lvar - | [(AoneT)] -> lvar - | [(AplusT ?1 ?2)] -> - Let l1 = (SeekVarAux FT lvar ?1) In - (SeekVarAux FT l1 ?2) - | [(AmultT ?1 ?2)] -> - Let l1 = (SeekVarAux FT lvar ?1) In - (SeekVarAux FT l1 ?2) - | [(AoppT ?1)] -> (SeekVarAux FT lvar ?1) - | [(AinvT ?1)] -> (SeekVarAux FT lvar ?1) - | [?1] -> - Let res = (MemAssoc ?1 lvar) In - Match res With - | [(true)] -> lvar - | [(false)] -> '(consT AT ?1 lvar). - -Tactic Definition SeekVar FT trm := - Let AT = Eval Cbv Beta Delta [A] Iota in (A FT) In - (SeekVarAux FT '(nilT AT) trm). - -Recursive Tactic Definition NumberAux lvar cpt := - Match lvar With - | [(nilT ?1)] -> '(nilT (prodT ?1 nat)) - | [(consT ?1 ?2 ?3)] -> - Let l2 = (NumberAux ?3 '(S cpt)) In - '(consT (prodT ?1 nat) (pairT ?1 nat ?2 cpt) l2). - -Tactic Definition Number lvar := (NumberAux lvar O). - -Tactic Definition BuildVarList FT trm := - Let lvar = (SeekVar FT trm) In - (Number lvar). -V7only [ -(*Used by contrib Maple *) -Tactic Definition build_var_list := BuildVarList. -]. - -Recursive Tactic Definition Assoc elt lst := - Match lst With - | [(nilT ?)] -> Fail - | [(consT (prodT ? nat) (pairT ? nat ?1 ?2) ?3)] -> - Match elt= ?1 With - | [?1= ?1] -> ?2 - | _ -> (Assoc elt ?3). - -Recursive Meta Definition interp_A FT lvar trm := - Let AT = Eval Cbv Beta Delta [A] Iota in (A FT) - And AzeroT = Eval Cbv Beta Delta [Azero] Iota in (Azero FT) - And AoneT = Eval Cbv Beta Delta [Aone] Iota in (Aone FT) - And AplusT = Eval Cbv Beta Delta [Aplus] Iota in (Aplus FT) - And AmultT = Eval Cbv Beta Delta [Amult] Iota in (Amult FT) - And AoppT = Eval Cbv Beta Delta [Aopp] Iota in (Aopp FT) - And AinvT = Eval Cbv Beta Delta [Ainv] Iota in (Ainv FT) In - Match trm With - | [(AzeroT)] -> EAzero - | [(AoneT)] -> EAone - | [(AplusT ?1 ?2)] -> - Let e1 = (interp_A FT lvar ?1) - And e2 = (interp_A FT lvar ?2) In - '(EAplus e1 e2) - | [(AmultT ?1 ?2)] -> - Let e1 = (interp_A FT lvar ?1) - And e2 = (interp_A FT lvar ?2) In - '(EAmult e1 e2) - | [(AoppT ?1)] -> - Let e = (interp_A FT lvar ?1) In - '(EAopp e) - | [(AinvT ?1)] -> - Let e = (interp_A FT lvar ?1) In - '(EAinv e) - | [?1] -> - Let idx = (Assoc ?1 lvar) In - '(EAvar idx). +Ltac mem_assoc var lvar := + match constr:lvar with + | (nilT _) => constr:false + | (consT _ ?X1 ?X2) => + match constr:(X1 = var) with + | (?X1 = ?X1) => constr:true + | _ => mem_assoc var X2 + end + end. + +Ltac seek_var_aux FT lvar trm := + let AT := eval cbv beta iota delta [A] in (A FT) + with AzeroT := eval cbv beta iota delta [Azero] in (Azero FT) + with AoneT := eval cbv beta iota delta [Aone] in (Aone FT) + with AplusT := eval cbv beta iota delta [Aplus] in (Aplus FT) + with AmultT := eval cbv beta iota delta [Amult] in (Amult FT) + with AoppT := eval cbv beta iota delta [Aopp] in (Aopp FT) + with AinvT := eval cbv beta iota delta [Ainv] in (Ainv FT) in + match constr:trm with + | AzeroT => lvar + | AoneT => lvar + | (AplusT ?X1 ?X2) => + let l1 := seek_var_aux FT lvar X1 in + seek_var_aux FT l1 X2 + | (AmultT ?X1 ?X2) => + let l1 := seek_var_aux FT lvar X1 in + seek_var_aux FT l1 X2 + | (AoppT ?X1) => seek_var_aux FT lvar X1 + | (AinvT ?X1) => seek_var_aux FT lvar X1 + | ?X1 => + let res := mem_assoc X1 lvar in + match constr:res with + | true => lvar + | false => constr:(consT AT X1 lvar) + end + end. + +Ltac seek_var FT trm := + let AT := eval cbv beta iota delta [A] in (A FT) in + seek_var_aux FT (nilT AT) trm. + +Ltac number_aux lvar cpt := + match constr:lvar with + | (nilT ?X1) => constr:(nilT (prodT X1 nat)) + | (consT ?X1 ?X2 ?X3) => + let l2 := number_aux X3 (S cpt) in + constr:(consT (prodT X1 nat) (pairT X1 nat X2 cpt) l2) + end. + +Ltac number lvar := number_aux lvar 0. + +Ltac build_varlist FT trm := let lvar := seek_var FT trm in + number lvar. + +Ltac assoc elt lst := + match constr:lst with + | (nilT _) => fail + | (consT (prodT _ nat) (pairT _ nat ?X1 ?X2) ?X3) => + match constr:(elt = X1) with + | (?X1 = ?X1) => constr:X2 + | _ => assoc elt X3 + end + end. + +Ltac interp_A FT lvar trm := + let AT := eval cbv beta iota delta [A] in (A FT) + with AzeroT := eval cbv beta iota delta [Azero] in (Azero FT) + with AoneT := eval cbv beta iota delta [Aone] in (Aone FT) + with AplusT := eval cbv beta iota delta [Aplus] in (Aplus FT) + with AmultT := eval cbv beta iota delta [Amult] in (Amult FT) + with AoppT := eval cbv beta iota delta [Aopp] in (Aopp FT) + with AinvT := eval cbv beta iota delta [Ainv] in (Ainv FT) in + match constr:trm with + | AzeroT => constr:EAzero + | AoneT => constr:EAone + | (AplusT ?X1 ?X2) => + let e1 := interp_A FT lvar X1 with e2 := interp_A FT lvar X2 in + constr:(EAplus e1 e2) + | (AmultT ?X1 ?X2) => + let e1 := interp_A FT lvar X1 with e2 := interp_A FT lvar X2 in + constr:(EAmult e1 e2) + | (AoppT ?X1) => + let e := interp_A FT lvar X1 in + constr:(EAopp e) + | (AinvT ?X1) => let e := interp_A FT lvar X1 in + constr:(EAinv e) + | ?X1 => let idx := assoc X1 lvar in + constr:(EAvar idx) + end. (************************) (* Simplification *) @@ -111,166 +110,190 @@ Recursive Meta Definition interp_A FT lvar trm := (**** Generation of the multiplier ****) -Recursive Tactic Definition Remove e l := - Match l With - | [(nilT ?)] -> l - | [(consT ?1 e ?2)] -> ?2 - | [(consT ?1 ?2 ?3)] -> - Let nl = (Remove e ?3) In - '(consT ?1 ?2 nl). - -Recursive Tactic Definition Union l1 l2 := - Match l1 With - | [(nilT ?)] -> l2 - | [(consT ?1 ?2 ?3)] -> - Let nl2 = (Remove ?2 l2) In - Let nl = (Union ?3 nl2) In - '(consT ?1 ?2 nl). - -Recursive Tactic Definition RawGiveMult trm := - Match trm With - | [(EAinv ?1)] -> '(consT ExprA ?1 (nilT ExprA)) - | [(EAopp ?1)] -> (RawGiveMult ?1) - | [(EAplus ?1 ?2)] -> - Let l1 = (RawGiveMult ?1) - And l2 = (RawGiveMult ?2) In - (Union l1 l2) - | [(EAmult ?1 ?2)] -> - Let l1 = (RawGiveMult ?1) - And l2 = (RawGiveMult ?2) In - Eval Compute in (appT ExprA l1 l2) - | _ -> '(nilT ExprA). - -Tactic Definition GiveMult trm := - Let ltrm = (RawGiveMult trm) In - '(mult_of_list ltrm). +Ltac remove e l := + match constr:l with + | (nilT _) => l + | (consT ?X1 e ?X2) => constr:X2 + | (consT ?X1 ?X2 ?X3) => let nl := remove e X3 in + constr:(consT X1 X2 nl) + end. + +Ltac union l1 l2 := + match constr:l1 with + | (nilT _) => l2 + | (consT ?X1 ?X2 ?X3) => + let nl2 := remove X2 l2 in + let nl := union X3 nl2 in + constr:(consT X1 X2 nl) + end. + +Ltac raw_give_mult trm := + match constr:trm with + | (EAinv ?X1) => constr:(consT ExprA X1 (nilT ExprA)) + | (EAopp ?X1) => raw_give_mult X1 + | (EAplus ?X1 ?X2) => + let l1 := raw_give_mult X1 with l2 := raw_give_mult X2 in + union l1 l2 + | (EAmult ?X1 ?X2) => + let l1 := raw_give_mult X1 with l2 := raw_give_mult X2 in + eval compute in (appT ExprA l1 l2) + | _ => constr:(nilT ExprA) + end. + +Ltac give_mult trm := + let ltrm := raw_give_mult trm in + constr:(mult_of_list ltrm). (**** Associativity ****) -Tactic Definition ApplyAssoc FT lvar trm := - Let t=Eval Compute in (assoc trm) In - Match t=trm With - | [ ?1=?1 ] -> Idtac - | _ -> Rewrite <- (assoc_correct FT trm); Change (assoc trm) with t. +Ltac apply_assoc FT lvar trm := + let t := eval compute in (assoc trm) in + match constr:(t = trm) with + | (?X1 = ?X1) => idtac + | _ => + rewrite <- (assoc_correct FT trm); change (assoc trm) with t in |- * + end. (**** Distribution *****) -Tactic Definition ApplyDistrib FT lvar trm := - Let t=Eval Compute in (distrib trm) In - Match t=trm With - | [ ?1=?1 ] -> Idtac - | _ -> Rewrite <- (distrib_correct FT trm); Change (distrib trm) with t. +Ltac apply_distrib FT lvar trm := + let t := eval compute in (distrib trm) in + match constr:(t = trm) with + | (?X1 = ?X1) => idtac + | _ => + rewrite <- (distrib_correct FT trm); + change (distrib trm) with t in |- * + end. (**** Multiplication by the inverse product ****) -Tactic Definition GrepMult := - Match Context With - | [ id: ~(interp_ExprA ? ? ?)= ? |- ?] -> id. - -Tactic Definition WeakReduce := - Match Context With - | [|-[(interp_ExprA ?1 ?2 ?)]] -> - Cbv Beta Delta [interp_ExprA assoc_2nd eq_nat_dec mult_of_list ?1 ?2 A - Azero Aone Aplus Amult Aopp Ainv] Zeta Iota. - -Tactic Definition Multiply mul := - Match Context With - | [|-(interp_ExprA ?1 ?2 ?3)=(interp_ExprA ?1 ?2 ?4)] -> - Let AzeroT = Eval Cbv Beta Delta [Azero ?1] Iota in (Azero ?1) In - Cut ~(interp_ExprA ?1 ?2 mul)=AzeroT; - [Intro; - Let id = GrepMult In - Apply (mult_eq ?1 ?3 ?4 mul ?2 id) - |WeakReduce; - Let AoneT = Eval Cbv Beta Delta [Aone ?1] Iota in (Aone ?1) - And AmultT = Eval Cbv Beta Delta [Amult ?1] Iota in (Amult ?1) In - Try (Match Context With - | [|-[(AmultT ? AoneT)]] -> Rewrite (AmultT_1r ?1));Clear ?1 ?2]. - -Tactic Definition ApplyMultiply FT lvar trm := - Let t=Eval Compute in (multiply trm) In - Match t=trm With - | [ ?1=?1 ] -> Idtac - | _ -> Rewrite <- (multiply_correct FT trm); Change (multiply trm) with t. +Ltac grep_mult := match goal with + | id:(interp_ExprA _ _ _ <> _) |- _ => id + end. + +Ltac weak_reduce := + match goal with + | |- context [(interp_ExprA ?X1 ?X2 _)] => + cbv beta iota zeta + delta [interp_ExprA assoc_2nd eq_nat_dec mult_of_list X1 X2 A Azero + Aone Aplus Amult Aopp Ainv] in |- * + end. + +Ltac multiply mul := + match goal with + | |- (interp_ExprA ?X1 ?X2 ?X3 = interp_ExprA ?X1 ?X2 ?X4) => + let AzeroT := eval cbv beta iota delta [Azero X1] in (Azero X1) in + (cut (interp_ExprA X1 X2 mul <> AzeroT); + [ intro; let id := grep_mult in + apply (mult_eq X1 X3 X4 mul X2 id) + | weak_reduce; + let AoneT := eval cbv beta iota delta [Aone X1] in (Aone X1) + with AmultT := eval cbv beta iota delta [Amult X1] in (Amult X1) in + (try + match goal with + | |- context [(AmultT _ AoneT)] => rewrite (AmultT_1r X1) + end; clear X1 X2) ]) + end. + +Ltac apply_multiply FT lvar trm := + let t := eval compute in (multiply trm) in + match constr:(t = trm) with + | (?X1 = ?X1) => idtac + | _ => + rewrite <- (multiply_correct FT trm); + change (multiply trm) with t in |- * + end. (**** Permutations and simplification ****) -Tactic Definition ApplyInverse mul FT lvar trm := - Let t=Eval Compute in (inverse_simplif mul trm) In - Match t=trm With - | [ ?1=?1 ] -> Idtac - | _ -> Rewrite <- (inverse_correct FT trm mul); - [Change (inverse_simplif mul trm) with t|Assumption]. +Ltac apply_inverse mul FT lvar trm := + let t := eval compute in (inverse_simplif mul trm) in + match constr:(t = trm) with + | (?X1 = ?X1) => idtac + | _ => + rewrite <- (inverse_correct FT trm mul); + [ change (inverse_simplif mul trm) with t in |- * | assumption ] + end. (**** Inverse test ****) -Tactic Definition StrongFail tac := First [tac|Fail 2]. - -Recursive Tactic Definition InverseTestAux FT trm := - Let AplusT = Eval Cbv Beta Delta [Aplus] Iota in (Aplus FT) - And AmultT = Eval Cbv Beta Delta [Amult] Iota in (Amult FT) - And AoppT = Eval Cbv Beta Delta [Aopp] Iota in (Aopp FT) - And AinvT = Eval Cbv Beta Delta [Ainv] Iota in (Ainv FT) In - Match trm With - | [(AinvT ?)] -> Fail 1 - | [(AoppT ?1)] -> StrongFail ((InverseTestAux FT ?1);Idtac) - | [(AplusT ?1 ?2)] -> - StrongFail ((InverseTestAux FT ?1);(InverseTestAux FT ?2)) - | [(AmultT ?1 ?2)] -> - StrongFail ((InverseTestAux FT ?1);(InverseTestAux FT ?2)) - | _ -> Idtac. - -Tactic Definition InverseTest FT := - Let AplusT = Eval Cbv Beta Delta [Aplus] Iota in (Aplus FT) In - Match Context With - | [|- ?1=?2] -> (InverseTestAux FT '(AplusT ?1 ?2)). +Ltac strong_fail tac := first [ tac | fail 2 ]. + +Ltac inverse_test_aux FT trm := + let AplusT := eval cbv beta iota delta [Aplus] in (Aplus FT) + with AmultT := eval cbv beta iota delta [Amult] in (Amult FT) + with AoppT := eval cbv beta iota delta [Aopp] in (Aopp FT) + with AinvT := eval cbv beta iota delta [Ainv] in (Ainv FT) in + match constr:trm with + | (AinvT _) => fail 1 + | (AoppT ?X1) => + strong_fail ltac:(inverse_test_aux FT X1; idtac) + | (AplusT ?X1 ?X2) => + strong_fail ltac:(inverse_test_aux FT X1; inverse_test_aux FT X2) + | (AmultT ?X1 ?X2) => + strong_fail ltac:(inverse_test_aux FT X1; inverse_test_aux FT X2) + | _ => idtac + end. + +Ltac inverse_test FT := + let AplusT := eval cbv beta iota delta [Aplus] in (Aplus FT) in + match goal with + | |- (?X1 = ?X2) => inverse_test_aux FT (AplusT X1 X2) + end. (**** Field itself ****) -Tactic Definition ApplySimplif sfun := - (Match Context With - | [|- (interp_ExprA ?1 ?2 ?3)=(interp_ExprA ? ? ?)] -> - (sfun ?1 ?2 ?3)); - (Match Context With - | [|- (interp_ExprA ? ? ?)=(interp_ExprA ?1 ?2 ?3)] -> - (sfun ?1 ?2 ?3)). - -Tactic Definition Unfolds FT := - (Match Eval Cbv Beta Delta [Aminus] Iota in (Aminus FT) With - | [(Some ? ?1)] -> Unfold ?1 - | _ -> Idtac); - (Match Eval Cbv Beta Delta [Adiv] Iota in (Adiv FT) With - | [(Some ? ?1)] -> Unfold ?1 - | _ -> Idtac). - -Tactic Definition Reduce FT := - Let AzeroT = Eval Cbv Beta Delta [Azero] Iota in (Azero FT) - And AoneT = Eval Cbv Beta Delta [Aone] Iota in (Aone FT) - And AplusT = Eval Cbv Beta Delta [Aplus] Iota in (Aplus FT) - And AmultT = Eval Cbv Beta Delta [Amult] Iota in (Amult FT) - And AoppT = Eval Cbv Beta Delta [Aopp] Iota in (Aopp FT) - And AinvT = Eval Cbv Beta Delta [Ainv] Iota in (Ainv FT) In - Cbv Beta Delta -[AzeroT AoneT AplusT AmultT AoppT AinvT] Zeta Iota - Orelse Compute. - -Recursive Tactic Definition Field_Gen_Aux FT := - Let AplusT = Eval Cbv Beta Delta [Aplus] Iota in (Aplus FT) In - Match Context With - | [|- ?1=?2] -> - Let lvar = (BuildVarList FT '(AplusT ?1 ?2)) In - Let trm1 = (interp_A FT lvar ?1) - And trm2 = (interp_A FT lvar ?2) In - Let mul = (GiveMult '(EAplus trm1 trm2)) In - Cut [ft:=FT][vm:=lvar](interp_ExprA ft vm trm1)=(interp_ExprA ft vm trm2); - [Compute;Auto - |Intros ft vm;(ApplySimplif ApplyDistrib);(ApplySimplif ApplyAssoc); - (Multiply mul);[(ApplySimplif ApplyMultiply); - (ApplySimplif (ApplyInverse mul)); - (Let id = GrepMult In Clear id);WeakReduce;Clear ft vm; - First [(InverseTest FT);Ring|(Field_Gen_Aux FT)]|Idtac]]. - -Tactic Definition Field_Gen FT := - Unfolds FT;((InverseTest FT);Ring) Orelse (Field_Gen_Aux FT). -V7only [Tactic Definition field_gen := Field_Gen.]. +Ltac apply_simplif sfun := + match goal with + | |- (interp_ExprA ?X1 ?X2 ?X3 = interp_ExprA _ _ _) => + sfun X1 X2 X3 + end; + match goal with + | |- (interp_ExprA _ _ _ = interp_ExprA ?X1 ?X2 ?X3) => + sfun X1 X2 X3 + end. + +Ltac unfolds FT := + match eval cbv beta iota delta [Aminus] in (Aminus FT) with + | (Some _ ?X1) => unfold X1 in |- * + | _ => idtac + end; + match eval cbv beta iota delta [Adiv] in (Adiv FT) with + | (Some _ ?X1) => unfold X1 in |- * + | _ => idtac + end. + +Ltac reduce FT := + let AzeroT := eval cbv beta iota delta [Azero] in (Azero FT) + with AoneT := eval cbv beta iota delta [Aone] in (Aone FT) + with AplusT := eval cbv beta iota delta [Aplus] in (Aplus FT) + with AmultT := eval cbv beta iota delta [Amult] in (Amult FT) + with AoppT := eval cbv beta iota delta [Aopp] in (Aopp FT) + with AinvT := eval cbv beta iota delta [Ainv] in (Ainv FT) in + (cbv beta iota zeta delta -[AzeroT AoneT AplusT AmultT AoppT AinvT] in |- * || + compute in |- *). + +Ltac field_gen_aux FT := + let AplusT := eval cbv beta iota delta [Aplus] in (Aplus FT) in + match goal with + | |- (?X1 = ?X2) => + let lvar := build_varlist FT (AplusT X1 X2) in + let trm1 := interp_A FT lvar X1 with trm2 := interp_A FT lvar X2 in + let mul := give_mult (EAplus trm1 trm2) in + (cut + (let ft := FT in + let vm := lvar in interp_ExprA ft vm trm1 = interp_ExprA ft vm trm2); + [ compute in |- *; auto + | intros ft vm; apply_simplif apply_distrib; + apply_simplif apply_assoc; multiply mul; + [ apply_simplif apply_multiply; + apply_simplif ltac:(apply_inverse mul); + let id := grep_mult in + clear id; weak_reduce; clear ft vm; first + [ inverse_test FT; ring | field_gen_aux FT ] + | idtac ] ]) + end. + +Ltac field_gen FT := unfolds FT; (inverse_test FT; ring) || field_gen_aux FT. (*****************************) (* Term Simplification *) @@ -278,120 +301,132 @@ V7only [Tactic Definition field_gen := Field_Gen.]. (**** Minus and division expansions ****) -Meta Definition InitExp FT trm := - Let e = - (Match Eval Cbv Beta Delta [Aminus] Iota in (Aminus FT) With - | [(Some ? ?1)] -> Eval Cbv Beta Delta [?1] in trm - | _ -> trm) In - Match Eval Cbv Beta Delta [Adiv] Iota in (Adiv FT) With - | [(Some ? ?1)] -> Eval Cbv Beta Delta [?1] in e - | _ -> e. -V7only [ -(*Used by contrib Maple *) -Tactic Definition init_exp := InitExp. -]. +Ltac init_exp FT trm := + let e := + (match eval cbv beta iota delta [Aminus] in (Aminus FT) with + | (Some _ ?X1) => eval cbv beta delta [X1] in trm + | _ => trm + end) in + match eval cbv beta iota delta [Adiv] in (Adiv FT) with + | (Some _ ?X1) => eval cbv beta delta [X1] in e + | _ => e + end. (**** Inverses simplification ****) -Recursive Meta Definition SimplInv trm:= - Match trm With - | [(EAplus ?1 ?2)] -> - Let e1 = (SimplInv ?1) - And e2 = (SimplInv ?2) In - '(EAplus e1 e2) - | [(EAmult ?1 ?2)] -> - Let e1 = (SimplInv ?1) - And e2 = (SimplInv ?2) In - '(EAmult e1 e2) - | [(EAopp ?1)] -> Let e = (SimplInv ?1) In '(EAopp e) - | [(EAinv ?1)] -> (SimplInvAux ?1) - | [?1] -> ?1 -And SimplInvAux trm := - Match trm With - | [(EAinv ?1)] -> (SimplInv ?1) - | [(EAmult ?1 ?2)] -> - Let e1 = (SimplInv '(EAinv ?1)) - And e2 = (SimplInv '(EAinv ?2)) In - '(EAmult e1 e2) - | [?1] -> Let e = (SimplInv ?1) In '(EAinv e). +Ltac simpl_inv trm := + match constr:trm with + | (EAplus ?X1 ?X2) => + let e1 := simpl_inv X1 with e2 := simpl_inv X2 in + constr:(EAplus e1 e2) + | (EAmult ?X1 ?X2) => + let e1 := simpl_inv X1 with e2 := simpl_inv X2 in + constr:(EAmult e1 e2) + | (EAopp ?X1) => let e := simpl_inv X1 in + constr:(EAopp e) + | (EAinv ?X1) => SimplInvAux X1 + | ?X1 => constr:X1 + end + with SimplInvAux trm := + match constr:trm with + | (EAinv ?X1) => simpl_inv X1 + | (EAmult ?X1 ?X2) => + let e1 := simpl_inv (EAinv X1) with e2 := simpl_inv (EAinv X2) in + constr:(EAmult e1 e2) + | ?X1 => let e := simpl_inv X1 in + constr:(EAinv e) + end. (**** Monom simplification ****) -Recursive Meta Definition Map fcn lst := - Match lst With - | [(nilT ?)] -> lst - | [(consT ?1 ?2 ?3)] -> - Let r = (fcn ?2) - And t = (Map fcn ?3) In - '(consT ?1 r t). - -Recursive Meta Definition BuildMonomAux lst trm := - Match lst With - | [(nilT ?)] -> Eval Compute in (assoc trm) - | [(consT ? ?1 ?2)] -> BuildMonomAux ?2 '(EAmult trm ?1). - -Recursive Meta Definition BuildMonom lnum lden := - Let ildn = (Map (Fun e -> '(EAinv e)) lden) In - Let ltot = Eval Compute in (appT ExprA lnum ildn) In - Let trm = (BuildMonomAux ltot EAone) In - Match trm With - | [(EAmult ? ?1)] -> ?1 - | [?1] -> ?1. - -Recursive Meta Definition SimplMonomAux lnum lden trm := - Match trm With - | [(EAmult (EAinv ?1) ?2)] -> - Let mma = (MemAssoc ?1 lnum) In - (Match mma With - | [true] -> - Let newlnum = (Remove ?1 lnum) In SimplMonomAux newlnum lden ?2 - | [false] -> SimplMonomAux lnum '(consT ExprA ?1 lden) ?2) - | [(EAmult ?1 ?2)] -> - Let mma = (MemAssoc ?1 lden) In - (Match mma With - | [true] -> - Let newlden = (Remove ?1 lden) In SimplMonomAux lnum newlden ?2 - | [false] -> SimplMonomAux '(consT ExprA ?1 lnum) lden ?2) - | [(EAinv ?1)] -> - Let mma = (MemAssoc ?1 lnum) In - (Match mma With - | [true] -> - Let newlnum = (Remove ?1 lnum) In BuildMonom newlnum lden - | [false] -> BuildMonom lnum '(consT ExprA ?1 lden)) - | [?1] -> - Let mma = (MemAssoc ?1 lden) In - (Match mma With - | [true] -> - Let newlden = (Remove ?1 lden) In BuildMonom lnum newlden - | [false] -> BuildMonom '(consT ExprA ?1 lnum) lden). - -Meta Definition SimplMonom trm := - SimplMonomAux '(nilT ExprA) '(nilT ExprA) trm. - -Recursive Meta Definition SimplAllMonoms trm := - Match trm With - | [(EAplus ?1 ?2)] -> - Let e1 = (SimplMonom ?1) - And e2 = (SimplAllMonoms ?2) In - '(EAplus e1 e2) - | [?1] -> SimplMonom ?1. +Ltac map_tactic fcn lst := + match constr:lst with + | (nilT _) => lst + | (consT ?X1 ?X2 ?X3) => + let r := fcn X2 with t := map_tactic fcn X3 in + constr:(consT X1 r t) + end. + +Ltac build_monom_aux lst trm := + match constr:lst with + | (nilT _) => eval compute in (assoc trm) + | (consT _ ?X1 ?X2) => build_monom_aux X2 (EAmult trm X1) + end. + +Ltac build_monom lnum lden := + let ildn := map_tactic ltac:(fun e => constr:(EAinv e)) lden in + let ltot := eval compute in (appT ExprA lnum ildn) in + let trm := build_monom_aux ltot EAone in + match constr:trm with + | (EAmult _ ?X1) => constr:X1 + | ?X1 => constr:X1 + end. + +Ltac simpl_monom_aux lnum lden trm := + match constr:trm with + | (EAmult (EAinv ?X1) ?X2) => + let mma := mem_assoc X1 lnum in + match constr:mma with + | true => + let newlnum := remove X1 lnum in + simpl_monom_aux newlnum lden X2 + | false => simpl_monom_aux lnum (consT ExprA X1 lden) X2 + end + | (EAmult ?X1 ?X2) => + let mma := mem_assoc X1 lden in + match constr:mma with + | true => + let newlden := remove X1 lden in + simpl_monom_aux lnum newlden X2 + | false => simpl_monom_aux (consT ExprA X1 lnum) lden X2 + end + | (EAinv ?X1) => + let mma := mem_assoc X1 lnum in + match constr:mma with + | true => + let newlnum := remove X1 lnum in + build_monom newlnum lden + | false => build_monom lnum (consT ExprA X1 lden) + end + | ?X1 => + let mma := mem_assoc X1 lden in + match constr:mma with + | true => + let newlden := remove X1 lden in + build_monom lnum newlden + | false => build_monom (consT ExprA X1 lnum) lden + end + end. + +Ltac simpl_monom trm := simpl_monom_aux (nilT ExprA) (nilT ExprA) trm. + +Ltac simpl_all_monomials trm := + match constr:trm with + | (EAplus ?X1 ?X2) => + let e1 := simpl_monom X1 with e2 := simpl_all_monomials X2 in + constr:(EAplus e1 e2) + | ?X1 => simpl_monom X1 + end. (**** Associativity and distribution ****) -Meta Definition AssocDistrib trm := Eval Compute in (assoc (distrib trm)). +Ltac assoc_distrib trm := eval compute in (assoc (distrib trm)). (**** The tactic Field_Term ****) -Tactic Definition EvalWeakReduce trm := - Eval Cbv Beta Delta [interp_ExprA assoc_2nd eq_nat_dec mult_of_list A Azero - Aone Aplus Amult Aopp Ainv] Zeta Iota in trm. - -Tactic Definition Field_Term FT exp := - Let newexp = (InitExp FT exp) In - Let lvar = (BuildVarList FT newexp) In - Let trm = (interp_A FT lvar newexp) In - Let tma = Eval Compute in (assoc trm) In - Let tsmp = (SimplAllMonoms (AssocDistrib (SimplAllMonoms - (SimplInv tma)))) In - Let trep = (EvalWeakReduce '(interp_ExprA FT lvar tsmp)) In - Replace exp with trep;[Ring trep|Field_Gen FT]. +Ltac eval_weak_reduce trm := + eval + cbv beta iota zeta + delta [interp_ExprA assoc_2nd eq_nat_dec mult_of_list A Azero Aone Aplus + Amult Aopp Ainv] in trm. + +Ltac field_term FT exp := + let newexp := init_exp FT exp in + let lvar := build_varlist FT newexp in + let trm := interp_A FT lvar newexp in + let tma := eval compute in (assoc trm) in + let tsmp := + simpl_all_monomials + ltac:(assoc_distrib ltac:(simpl_all_monomials ltac:(simpl_inv tma))) in + let trep := eval_weak_reduce (interp_ExprA FT lvar tsmp) in + (replace exp with trep; [ ring trep | field_gen FT ]).
\ No newline at end of file diff --git a/contrib/field/Field_Theory.v b/contrib/field/Field_Theory.v index e2710260f9..35f1125e41 100644 --- a/contrib/field/Field_Theory.v +++ b/contrib/field/Field_Theory.v @@ -8,86 +8,87 @@ (* $Id$ *) -Require Peano_dec. -Require Ring. -Require Field_Compl. - -Record Field_Theory : Type := -{ A : Type; - Aplus : A -> A -> A; - Amult : A -> A -> A; - Aone : A; - Azero : A; - Aopp : A -> A; - Aeq : A -> A -> bool; - Ainv : A -> A; - Aminus : (option A); - Adiv : (option A); - RT : (Ring_Theory Aplus Amult Aone Azero Aopp Aeq); - Th_inv_def : (n:A)~(n=Azero)->(Amult (Ainv n) n)=Aone -}. +Require Import Peano_dec. +Require Import Ring. +Require Import Field_Compl. + +Record Field_Theory : Type := + {A : Type; + Aplus : A -> A -> A; + Amult : A -> A -> A; + Aone : A; + Azero : A; + Aopp : A -> A; + Aeq : A -> A -> bool; + Ainv : A -> A; + Aminus : option A; + Adiv : option A; + RT : Ring_Theory Aplus Amult Aone Azero Aopp Aeq; + Th_inv_def : forall n:A, n <> Azero -> Amult (Ainv n) n = Aone}. (* The reflexion structure *) Inductive ExprA : Set := -| EAzero : ExprA -| EAone : ExprA -| EAplus : ExprA -> ExprA -> ExprA -| EAmult : ExprA -> ExprA -> ExprA -| EAopp : ExprA -> ExprA -| EAinv : ExprA -> ExprA -| EAvar : nat -> ExprA. + | EAzero : ExprA + | EAone : ExprA + | EAplus : ExprA -> ExprA -> ExprA + | EAmult : ExprA -> ExprA -> ExprA + | EAopp : ExprA -> ExprA + | EAinv : ExprA -> ExprA + | EAvar : nat -> ExprA. (**** Decidability of equality ****) -Lemma eqExprA_O:(e1,e2:ExprA){e1=e2}+{~e1=e2}. -Proof. - Double Induction e1 e2;Try Intros; - Try (Left;Reflexivity) Orelse Try (Right;Discriminate). - Elim (H1 e0);Intro y;Elim (H2 e);Intro y0; - Try (Left; Rewrite y; Rewrite y0;Auto) - Orelse (Right;Red;Intro;Inversion H3;Auto). - Elim (H1 e0);Intro y;Elim (H2 e);Intro y0; - Try (Left; Rewrite y; Rewrite y0;Auto) - Orelse (Right;Red;Intro;Inversion H3;Auto). - Elim (H0 e);Intro y. - Left; Rewrite y; Auto. - Right;Red; Intro;Inversion H1;Auto. - Elim (H0 e);Intro y. - Left; Rewrite y; Auto. - Right;Red; Intro;Inversion H1;Auto. - Elim (eq_nat_dec n n0);Intro y. - Left; Rewrite y; Auto. - Right;Red;Intro;Inversion H;Auto. +Lemma eqExprA_O : forall e1 e2:ExprA, {e1 = e2} + {e1 <> e2}. +Proof. + double induction e1 e2; try intros; + try (left; reflexivity) || (try (right; discriminate)). + elim (H1 e0); intro y; elim (H2 e); intro y0; + try + (left; rewrite y; rewrite y0; auto) || + (right; red in |- *; intro; inversion H3; auto). + elim (H1 e0); intro y; elim (H2 e); intro y0; + try + (left; rewrite y; rewrite y0; auto) || + (right; red in |- *; intro; inversion H3; auto). + elim (H0 e); intro y. + left; rewrite y; auto. + right; red in |- *; intro; inversion H1; auto. + elim (H0 e); intro y. + left; rewrite y; auto. + right; red in |- *; intro; inversion H1; auto. + elim (eq_nat_dec n n0); intro y. + left; rewrite y; auto. + right; red in |- *; intro; inversion H; auto. Defined. -Definition eq_nat_dec := Eval Compute in Peano_dec.eq_nat_dec. -Definition eqExprA := Eval Compute in eqExprA_O. +Definition eq_nat_dec := Eval compute in eq_nat_dec. +Definition eqExprA := Eval compute in eqExprA_O. (**** Generation of the multiplier ****) -Fixpoint mult_of_list [e:(listT ExprA)]: ExprA := - Cases e of +Fixpoint mult_of_list (e:listT ExprA) : ExprA := + match e with | nilT => EAone - | (consT e1 l1) => (EAmult e1 (mult_of_list l1)) + | consT e1 l1 => EAmult e1 (mult_of_list l1) end. Section Theory_of_fields. Variable T : Field_Theory. -Local AT := (A T). -Local AplusT := (Aplus T). -Local AmultT := (Amult T). -Local AoneT := (Aone T). -Local AzeroT := (Azero T). -Local AoppT := (Aopp T). -Local AeqT := (Aeq T). -Local AinvT := (Ainv T). -Local RTT := (RT T). -Local Th_inv_defT := (Th_inv_def T). +Let AT := A T. +Let AplusT := Aplus T. +Let AmultT := Amult T. +Let AoneT := Aone T. +Let AzeroT := Azero T. +Let AoppT := Aopp T. +Let AeqT := Aeq T. +Let AinvT := Ainv T. +Let RTT := RT T. +Let Th_inv_defT := Th_inv_def T. -Add Abstract Ring (A T) (Aplus T) (Amult T) (Aone T) (Azero T) (Aopp T) - (Aeq T) (RT T). +Add Abstract Ring (A T) (Aplus T) (Amult T) (Aone T) ( + Azero T) (Aopp T) (Aeq T) (RT T). Add Abstract Ring AT AplusT AmultT AoneT AzeroT AoppT AeqT RTT. @@ -95,93 +96,94 @@ Add Abstract Ring AT AplusT AmultT AoneT AzeroT AoppT AeqT RTT. (* Lemmas to be used *) (***************************) -Lemma AplusT_sym:(r1,r2:AT)(AplusT r1 r2)=(AplusT r2 r1). +Lemma AplusT_sym : forall r1 r2:AT, AplusT r1 r2 = AplusT r2 r1. Proof. - Intros;Ring. -Save. + intros; ring. +Qed. -Lemma AplusT_assoc:(r1,r2,r3:AT)(AplusT (AplusT r1 r2) r3)= - (AplusT r1 (AplusT r2 r3)). +Lemma AplusT_assoc : + forall r1 r2 r3:AT, AplusT (AplusT r1 r2) r3 = AplusT r1 (AplusT r2 r3). Proof. - Intros;Ring. -Save. + intros; ring. +Qed. -Lemma AmultT_sym:(r1,r2:AT)(AmultT r1 r2)=(AmultT r2 r1). +Lemma AmultT_sym : forall r1 r2:AT, AmultT r1 r2 = AmultT r2 r1. Proof. - Intros;Ring. -Save. + intros; ring. +Qed. -Lemma AmultT_assoc:(r1,r2,r3:AT)(AmultT (AmultT r1 r2) r3)= - (AmultT r1 (AmultT r2 r3)). +Lemma AmultT_assoc : + forall r1 r2 r3:AT, AmultT (AmultT r1 r2) r3 = AmultT r1 (AmultT r2 r3). Proof. - Intros;Ring. -Save. + intros; ring. +Qed. -Lemma AplusT_Ol:(r:AT)(AplusT AzeroT r)=r. +Lemma AplusT_Ol : forall r:AT, AplusT AzeroT r = r. Proof. - Intros;Ring. -Save. + intros; ring. +Qed. -Lemma AmultT_1l:(r:AT)(AmultT AoneT r)=r. +Lemma AmultT_1l : forall r:AT, AmultT AoneT r = r. Proof. - Intros;Ring. -Save. + intros; ring. +Qed. -Lemma AplusT_AoppT_r:(r:AT)(AplusT r (AoppT r))=AzeroT. +Lemma AplusT_AoppT_r : forall r:AT, AplusT r (AoppT r) = AzeroT. Proof. - Intros;Ring. -Save. + intros; ring. +Qed. -Lemma AmultT_AplusT_distr:(r1,r2,r3:AT)(AmultT r1 (AplusT r2 r3))= - (AplusT (AmultT r1 r2) (AmultT r1 r3)). +Lemma AmultT_AplusT_distr : + forall r1 r2 r3:AT, + AmultT r1 (AplusT r2 r3) = AplusT (AmultT r1 r2) (AmultT r1 r3). Proof. - Intros;Ring. -Save. + intros; ring. +Qed. -Lemma r_AplusT_plus:(r,r1,r2:AT)(AplusT r r1)=(AplusT r r2)->r1=r2. +Lemma r_AplusT_plus : forall r r1 r2:AT, AplusT r r1 = AplusT r r2 -> r1 = r2. Proof. - Intros; Transitivity (AplusT (AplusT (AoppT r) r) r1). - Ring. - Transitivity (AplusT (AplusT (AoppT r) r) r2). - Repeat Rewrite -> AplusT_assoc; Rewrite <- H; Reflexivity. - Ring. -Save. + intros; transitivity (AplusT (AplusT (AoppT r) r) r1). + ring. + transitivity (AplusT (AplusT (AoppT r) r) r2). + repeat rewrite AplusT_assoc; rewrite <- H; reflexivity. + ring. +Qed. -Lemma r_AmultT_mult: - (r,r1,r2:AT)(AmultT r r1)=(AmultT r r2)->~r=AzeroT->r1=r2. +Lemma r_AmultT_mult : + forall r r1 r2:AT, AmultT r r1 = AmultT r r2 -> r <> AzeroT -> r1 = r2. Proof. - Intros; Transitivity (AmultT (AmultT (AinvT r) r) r1). - Rewrite Th_inv_defT;[Symmetry; Apply AmultT_1l;Auto|Auto]. - Transitivity (AmultT (AmultT (AinvT r) r) r2). - Repeat Rewrite AmultT_assoc; Rewrite H; Trivial. - Rewrite Th_inv_defT;[Apply AmultT_1l;Auto|Auto]. -Save. + intros; transitivity (AmultT (AmultT (AinvT r) r) r1). + rewrite Th_inv_defT; [ symmetry in |- *; apply AmultT_1l; auto | auto ]. + transitivity (AmultT (AmultT (AinvT r) r) r2). + repeat rewrite AmultT_assoc; rewrite H; trivial. + rewrite Th_inv_defT; [ apply AmultT_1l; auto | auto ]. +Qed. -Lemma AmultT_Or:(r:AT) (AmultT r AzeroT)=AzeroT. +Lemma AmultT_Or : forall r:AT, AmultT r AzeroT = AzeroT. Proof. - Intro; Ring. -Save. + intro; ring. +Qed. -Lemma AmultT_Ol:(r:AT)(AmultT AzeroT r)=AzeroT. +Lemma AmultT_Ol : forall r:AT, AmultT AzeroT r = AzeroT. Proof. - Intro; Ring. -Save. + intro; ring. +Qed. -Lemma AmultT_1r:(r:AT)(AmultT r AoneT)=r. +Lemma AmultT_1r : forall r:AT, AmultT r AoneT = r. Proof. - Intro; Ring. -Save. + intro; ring. +Qed. -Lemma AinvT_r:(r:AT)~r=AzeroT->(AmultT r (AinvT r))=AoneT. +Lemma AinvT_r : forall r:AT, r <> AzeroT -> AmultT r (AinvT r) = AoneT. Proof. - Intros; Rewrite -> AmultT_sym; Apply Th_inv_defT; Auto. -Save. + intros; rewrite AmultT_sym; apply Th_inv_defT; auto. +Qed. -Lemma without_div_O_contr: - (r1,r2:AT)~(AmultT r1 r2)=AzeroT ->~r1=AzeroT/\~r2=AzeroT. +Lemma Rmult_neq_0_reg : + forall r1 r2:AT, AmultT r1 r2 <> AzeroT -> r1 <> AzeroT /\ r2 <> AzeroT. Proof. - Intros r1 r2 H; Split; Red; Intro; Apply H; Rewrite H0; Ring. -Save. + intros r1 r2 H; split; red in |- *; intro; apply H; rewrite H0; ring. +Qed. (************************) (* Interpretation *) @@ -189,15 +191,16 @@ Save. (**** ExprA --> A ****) -Fixpoint interp_ExprA [lvar:(listT (prodT AT nat));e:ExprA] : AT := - Cases e of - | EAzero => AzeroT - | EAone => AoneT - | (EAplus e1 e2) => (AplusT (interp_ExprA lvar e1) (interp_ExprA lvar e2)) - | (EAmult e1 e2) => (AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2)) - | (EAopp e) => ((Aopp T) (interp_ExprA lvar e)) - | (EAinv e) => ((Ainv T) (interp_ExprA lvar e)) - | (EAvar n) => (assoc_2nd AT nat eq_nat_dec lvar n AzeroT) +Fixpoint interp_ExprA (lvar:listT (prodT AT nat)) (e:ExprA) {struct e} : + AT := + match e with + | EAzero => AzeroT + | EAone => AoneT + | EAplus e1 e2 => AplusT (interp_ExprA lvar e1) (interp_ExprA lvar e2) + | EAmult e1 e2 => AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2) + | EAopp e => Aopp T (interp_ExprA lvar e) + | EAinv e => Ainv T (interp_ExprA lvar e) + | EAvar n => assoc_2nd AT nat eq_nat_dec lvar n AzeroT end. (************************) @@ -207,406 +210,436 @@ Fixpoint interp_ExprA [lvar:(listT (prodT AT nat));e:ExprA] : AT := (**** Associativity ****) Definition merge_mult := - Fix merge_mult {merge_mult [e1:ExprA] : ExprA -> ExprA := - [e2:ExprA]Cases e1 of - | (EAmult t1 t2) => - Cases t2 of - | (EAmult t2 t3) => (EAmult t1 (EAmult t2 (merge_mult t3 e2))) - | _ => (EAmult t1 (EAmult t2 e2)) - end - | _ => (EAmult e1 e2) - end}. - -Fixpoint assoc_mult [e:ExprA] : ExprA := - Cases e of - | (EAmult e1 e3) => - Cases e1 of - | (EAmult e1 e2) => - (merge_mult (merge_mult (assoc_mult e1) (assoc_mult e2)) - (assoc_mult e3)) - | _ => (EAmult e1 (assoc_mult e3)) - end + (fix merge_mult (e1:ExprA) : ExprA -> ExprA := + fun e2:ExprA => + match e1 with + | EAmult t1 t2 => + match t2 with + | EAmult t2 t3 => EAmult t1 (EAmult t2 (merge_mult t3 e2)) + | _ => EAmult t1 (EAmult t2 e2) + end + | _ => EAmult e1 e2 + end). + +Fixpoint assoc_mult (e:ExprA) : ExprA := + match e with + | EAmult e1 e3 => + match e1 with + | EAmult e1 e2 => + merge_mult (merge_mult (assoc_mult e1) (assoc_mult e2)) + (assoc_mult e3) + | _ => EAmult e1 (assoc_mult e3) + end | _ => e end. Definition merge_plus := - Fix merge_plus {merge_plus [e1:ExprA]:ExprA->ExprA:= - [e2:ExprA]Cases e1 of - | (EAplus t1 t2) => - Cases t2 of - | (EAplus t2 t3) => (EAplus t1 (EAplus t2 (merge_plus t3 e2))) - | _ => (EAplus t1 (EAplus t2 e2)) - end - | _ => (EAplus e1 e2) - end}. - -Fixpoint assoc [e:ExprA] : ExprA := - Cases e of - | (EAplus e1 e3) => - Cases e1 of - | (EAplus e1 e2) => - (merge_plus (merge_plus (assoc e1) (assoc e2)) (assoc e3)) - | _ => (EAplus (assoc_mult e1) (assoc e3)) - end - | _ => (assoc_mult e) + (fix merge_plus (e1:ExprA) : ExprA -> ExprA := + fun e2:ExprA => + match e1 with + | EAplus t1 t2 => + match t2 with + | EAplus t2 t3 => EAplus t1 (EAplus t2 (merge_plus t3 e2)) + | _ => EAplus t1 (EAplus t2 e2) + end + | _ => EAplus e1 e2 + end). + +Fixpoint assoc (e:ExprA) : ExprA := + match e with + | EAplus e1 e3 => + match e1 with + | EAplus e1 e2 => + merge_plus (merge_plus (assoc e1) (assoc e2)) (assoc e3) + | _ => EAplus (assoc_mult e1) (assoc e3) + end + | _ => assoc_mult e end. -Lemma merge_mult_correct1: - (e1,e2,e3:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (merge_mult (EAmult e1 e2) e3))= - (interp_ExprA lvar (EAmult e1 (merge_mult e2 e3))). -Proof. -Intros e1 e2;Generalize e1;Generalize e2;Clear e1 e2. -Induction e2;Auto;Intros. -Unfold 1 merge_mult;Fold merge_mult; - Unfold 2 interp_ExprA;Fold interp_ExprA; - Rewrite (H0 e e3 lvar); - Unfold 1 interp_ExprA;Fold interp_ExprA; - Unfold 5 interp_ExprA;Fold interp_ExprA;Auto. -Save. - -Lemma merge_mult_correct: - (e1,e2:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (merge_mult e1 e2))= - (interp_ExprA lvar (EAmult e1 e2)). -Proof. -Induction e1;Auto;Intros. -Elim e0;Try (Intros;Simpl;Ring). -Unfold interp_ExprA in H2;Fold interp_ExprA in H2; - Cut (AmultT (interp_ExprA lvar e2) (AmultT (interp_ExprA lvar e4) - (AmultT (interp_ExprA lvar e) (interp_ExprA lvar e3))))= - (AmultT (AmultT (AmultT (interp_ExprA lvar e) (interp_ExprA lvar e4)) - (interp_ExprA lvar e2)) (interp_ExprA lvar e3)). -Intro H3;Rewrite H3;Rewrite <-H2; - Rewrite merge_mult_correct1;Simpl;Ring. -Ring. -Save. - -Lemma assoc_mult_correct1:(e1,e2:ExprA)(lvar:(listT (prodT AT nat))) - (AmultT (interp_ExprA lvar (assoc_mult e1)) - (interp_ExprA lvar (assoc_mult e2)))= - (interp_ExprA lvar (assoc_mult (EAmult e1 e2))). -Proof. -Induction e1;Auto;Intros. -Rewrite <-(H e0 lvar);Simpl;Rewrite merge_mult_correct;Simpl; - Rewrite merge_mult_correct;Simpl;Auto. -Save. - -Lemma assoc_mult_correct: - (e:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (assoc_mult e))=(interp_ExprA lvar e). -Proof. -Induction e;Auto;Intros. -Elim e0;Intros. -Intros;Simpl;Ring. -Simpl;Rewrite (AmultT_1l (interp_ExprA lvar (assoc_mult e1))); - Rewrite (AmultT_1l (interp_ExprA lvar e1)); Apply H0. -Simpl;Rewrite (H0 lvar);Auto. -Simpl;Rewrite merge_mult_correct;Simpl;Rewrite merge_mult_correct;Simpl; - Rewrite AmultT_assoc;Rewrite assoc_mult_correct1;Rewrite H2;Simpl; - Rewrite <-assoc_mult_correct1 in H1; - Unfold 3 interp_ExprA in H1;Fold interp_ExprA in H1; - Rewrite (H0 lvar) in H1; - Rewrite (AmultT_sym (interp_ExprA lvar e3) (interp_ExprA lvar e1)); - Rewrite <-AmultT_assoc;Rewrite H1;Rewrite AmultT_assoc;Ring. -Simpl;Rewrite (H0 lvar);Auto. -Simpl;Rewrite (H0 lvar);Auto. -Simpl;Rewrite (H0 lvar);Auto. -Save. - -Lemma merge_plus_correct1: - (e1,e2,e3:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (merge_plus (EAplus e1 e2) e3))= - (interp_ExprA lvar (EAplus e1 (merge_plus e2 e3))). -Proof. -Intros e1 e2;Generalize e1;Generalize e2;Clear e1 e2. -Induction e2;Auto;Intros. -Unfold 1 merge_plus;Fold merge_plus; - Unfold 2 interp_ExprA;Fold interp_ExprA; - Rewrite (H0 e e3 lvar); - Unfold 1 interp_ExprA;Fold interp_ExprA; - Unfold 5 interp_ExprA;Fold interp_ExprA;Auto. -Save. - -Lemma merge_plus_correct: - (e1,e2:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (merge_plus e1 e2))= - (interp_ExprA lvar (EAplus e1 e2)). -Proof. -Induction e1;Auto;Intros. -Elim e0;Try Intros;Try (Simpl;Ring). -Unfold interp_ExprA in H2;Fold interp_ExprA in H2; - Cut (AplusT (interp_ExprA lvar e2) (AplusT (interp_ExprA lvar e4) - (AplusT (interp_ExprA lvar e) (interp_ExprA lvar e3))))= - (AplusT (AplusT (AplusT (interp_ExprA lvar e) (interp_ExprA lvar e4)) - (interp_ExprA lvar e2)) (interp_ExprA lvar e3)). -Intro H3;Rewrite H3;Rewrite <-H2;Rewrite merge_plus_correct1;Simpl;Ring. -Ring. -Save. - -Lemma assoc_plus_correct:(e1,e2:ExprA)(lvar:(listT (prodT AT nat))) - (AplusT (interp_ExprA lvar (assoc e1)) (interp_ExprA lvar (assoc e2)))= - (interp_ExprA lvar (assoc (EAplus e1 e2))). -Proof. -Induction e1;Auto;Intros. -Rewrite <-(H e0 lvar);Simpl;Rewrite merge_plus_correct;Simpl; - Rewrite merge_plus_correct;Simpl;Auto. -Save. - -Lemma assoc_correct: - (e:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (assoc e))=(interp_ExprA lvar e). -Proof. -Induction e;Auto;Intros. -Elim e0;Intros. -Simpl;Rewrite (H0 lvar);Auto. -Simpl;Rewrite (H0 lvar);Auto. -Simpl;Rewrite merge_plus_correct;Simpl;Rewrite merge_plus_correct; - Simpl;Rewrite AplusT_assoc;Rewrite assoc_plus_correct;Rewrite H2; - Simpl;Apply (r_AplusT_plus (interp_ExprA lvar (assoc e1)) - (AplusT (interp_ExprA lvar (assoc e2)) - (AplusT (interp_ExprA lvar e3) (interp_ExprA lvar e1))) - (AplusT (AplusT (interp_ExprA lvar e2) (interp_ExprA lvar e3)) - (interp_ExprA lvar e1)));Rewrite <-AplusT_assoc; - Rewrite (AplusT_sym (interp_ExprA lvar (assoc e1)) - (interp_ExprA lvar (assoc e2))); - Rewrite assoc_plus_correct;Rewrite H1;Simpl;Rewrite (H0 lvar); - Rewrite <-(AplusT_assoc (AplusT (interp_ExprA lvar e2) - (interp_ExprA lvar e1)) - (interp_ExprA lvar e3) (interp_ExprA lvar e1)); - Rewrite (AplusT_assoc (interp_ExprA lvar e2) (interp_ExprA lvar e1) - (interp_ExprA lvar e3)); - Rewrite (AplusT_sym (interp_ExprA lvar e1) (interp_ExprA lvar e3)); - Rewrite <-(AplusT_assoc (interp_ExprA lvar e2) (interp_ExprA lvar e3) - (interp_ExprA lvar e1));Apply AplusT_sym. -Unfold assoc;Fold assoc;Unfold interp_ExprA;Fold interp_ExprA; - Rewrite assoc_mult_correct;Rewrite (H0 lvar);Simpl;Auto. -Simpl;Rewrite (H0 lvar);Auto. -Simpl;Rewrite (H0 lvar);Auto. -Simpl;Rewrite (H0 lvar);Auto. -Unfold assoc;Fold assoc;Unfold interp_ExprA;Fold interp_ExprA; - Rewrite assoc_mult_correct;Simpl;Auto. -Save. +Lemma merge_mult_correct1 : + forall (e1 e2 e3:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (merge_mult (EAmult e1 e2) e3) = + interp_ExprA lvar (EAmult e1 (merge_mult e2 e3)). +Proof. +intros e1 e2; generalize e1; generalize e2; clear e1 e2. +simple induction e2; auto; intros. +unfold merge_mult at 1 in |- *; fold merge_mult in |- *; + unfold interp_ExprA at 2 in |- *; fold interp_ExprA in |- *; + rewrite (H0 e e3 lvar); unfold interp_ExprA at 1 in |- *; + fold interp_ExprA in |- *; unfold interp_ExprA at 5 in |- *; + fold interp_ExprA in |- *; auto. +Qed. + +Lemma merge_mult_correct : + forall (e1 e2:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (merge_mult e1 e2) = interp_ExprA lvar (EAmult e1 e2). +Proof. +simple induction e1; auto; intros. +elim e0; try (intros; simpl in |- *; ring). +unfold interp_ExprA in H2; fold interp_ExprA in H2; + cut + (AmultT (interp_ExprA lvar e2) + (AmultT (interp_ExprA lvar e4) + (AmultT (interp_ExprA lvar e) (interp_ExprA lvar e3))) = + AmultT + (AmultT (AmultT (interp_ExprA lvar e) (interp_ExprA lvar e4)) + (interp_ExprA lvar e2)) (interp_ExprA lvar e3)). +intro H3; rewrite H3; rewrite <- H2; rewrite merge_mult_correct1; + simpl in |- *; ring. +ring. +Qed. + +Lemma assoc_mult_correct1 : + forall (e1 e2:ExprA) (lvar:listT (prodT AT nat)), + AmultT (interp_ExprA lvar (assoc_mult e1)) + (interp_ExprA lvar (assoc_mult e2)) = + interp_ExprA lvar (assoc_mult (EAmult e1 e2)). +Proof. +simple induction e1; auto; intros. +rewrite <- (H e0 lvar); simpl in |- *; rewrite merge_mult_correct; + simpl in |- *; rewrite merge_mult_correct; simpl in |- *; + auto. +Qed. + +Lemma assoc_mult_correct : + forall (e:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (assoc_mult e) = interp_ExprA lvar e. +Proof. +simple induction e; auto; intros. +elim e0; intros. +intros; simpl in |- *; ring. +simpl in |- *; rewrite (AmultT_1l (interp_ExprA lvar (assoc_mult e1))); + rewrite (AmultT_1l (interp_ExprA lvar e1)); apply H0. +simpl in |- *; rewrite (H0 lvar); auto. +simpl in |- *; rewrite merge_mult_correct; simpl in |- *; + rewrite merge_mult_correct; simpl in |- *; rewrite AmultT_assoc; + rewrite assoc_mult_correct1; rewrite H2; simpl in |- *; + rewrite <- assoc_mult_correct1 in H1; unfold interp_ExprA at 3 in H1; + fold interp_ExprA in H1; rewrite (H0 lvar) in H1; + rewrite (AmultT_sym (interp_ExprA lvar e3) (interp_ExprA lvar e1)); + rewrite <- AmultT_assoc; rewrite H1; rewrite AmultT_assoc; + ring. +simpl in |- *; rewrite (H0 lvar); auto. +simpl in |- *; rewrite (H0 lvar); auto. +simpl in |- *; rewrite (H0 lvar); auto. +Qed. + +Lemma merge_plus_correct1 : + forall (e1 e2 e3:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (merge_plus (EAplus e1 e2) e3) = + interp_ExprA lvar (EAplus e1 (merge_plus e2 e3)). +Proof. +intros e1 e2; generalize e1; generalize e2; clear e1 e2. +simple induction e2; auto; intros. +unfold merge_plus at 1 in |- *; fold merge_plus in |- *; + unfold interp_ExprA at 2 in |- *; fold interp_ExprA in |- *; + rewrite (H0 e e3 lvar); unfold interp_ExprA at 1 in |- *; + fold interp_ExprA in |- *; unfold interp_ExprA at 5 in |- *; + fold interp_ExprA in |- *; auto. +Qed. + +Lemma merge_plus_correct : + forall (e1 e2:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (merge_plus e1 e2) = interp_ExprA lvar (EAplus e1 e2). +Proof. +simple induction e1; auto; intros. +elim e0; try intros; try (simpl in |- *; ring). +unfold interp_ExprA in H2; fold interp_ExprA in H2; + cut + (AplusT (interp_ExprA lvar e2) + (AplusT (interp_ExprA lvar e4) + (AplusT (interp_ExprA lvar e) (interp_ExprA lvar e3))) = + AplusT + (AplusT (AplusT (interp_ExprA lvar e) (interp_ExprA lvar e4)) + (interp_ExprA lvar e2)) (interp_ExprA lvar e3)). +intro H3; rewrite H3; rewrite <- H2; rewrite merge_plus_correct1; + simpl in |- *; ring. +ring. +Qed. + +Lemma assoc_plus_correct : + forall (e1 e2:ExprA) (lvar:listT (prodT AT nat)), + AplusT (interp_ExprA lvar (assoc e1)) (interp_ExprA lvar (assoc e2)) = + interp_ExprA lvar (assoc (EAplus e1 e2)). +Proof. +simple induction e1; auto; intros. +rewrite <- (H e0 lvar); simpl in |- *; rewrite merge_plus_correct; + simpl in |- *; rewrite merge_plus_correct; simpl in |- *; + auto. +Qed. + +Lemma assoc_correct : + forall (e:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (assoc e) = interp_ExprA lvar e. +Proof. +simple induction e; auto; intros. +elim e0; intros. +simpl in |- *; rewrite (H0 lvar); auto. +simpl in |- *; rewrite (H0 lvar); auto. +simpl in |- *; rewrite merge_plus_correct; simpl in |- *; + rewrite merge_plus_correct; simpl in |- *; rewrite AplusT_assoc; + rewrite assoc_plus_correct; rewrite H2; simpl in |- *; + apply + (r_AplusT_plus (interp_ExprA lvar (assoc e1)) + (AplusT (interp_ExprA lvar (assoc e2)) + (AplusT (interp_ExprA lvar e3) (interp_ExprA lvar e1))) + (AplusT (AplusT (interp_ExprA lvar e2) (interp_ExprA lvar e3)) + (interp_ExprA lvar e1))); rewrite <- AplusT_assoc; + rewrite + (AplusT_sym (interp_ExprA lvar (assoc e1)) (interp_ExprA lvar (assoc e2))) + ; rewrite assoc_plus_correct; rewrite H1; simpl in |- *; + rewrite (H0 lvar); + rewrite <- + (AplusT_assoc (AplusT (interp_ExprA lvar e2) (interp_ExprA lvar e1)) + (interp_ExprA lvar e3) (interp_ExprA lvar e1)) + ; + rewrite + (AplusT_assoc (interp_ExprA lvar e2) (interp_ExprA lvar e1) + (interp_ExprA lvar e3)); + rewrite (AplusT_sym (interp_ExprA lvar e1) (interp_ExprA lvar e3)); + rewrite <- + (AplusT_assoc (interp_ExprA lvar e2) (interp_ExprA lvar e3) + (interp_ExprA lvar e1)); apply AplusT_sym. +unfold assoc in |- *; fold assoc in |- *; unfold interp_ExprA in |- *; + fold interp_ExprA in |- *; rewrite assoc_mult_correct; + rewrite (H0 lvar); simpl in |- *; auto. +simpl in |- *; rewrite (H0 lvar); auto. +simpl in |- *; rewrite (H0 lvar); auto. +simpl in |- *; rewrite (H0 lvar); auto. +unfold assoc in |- *; fold assoc in |- *; unfold interp_ExprA in |- *; + fold interp_ExprA in |- *; rewrite assoc_mult_correct; + simpl in |- *; auto. +Qed. (**** Distribution *****) -Fixpoint distrib_EAopp [e:ExprA] : ExprA := - Cases e of - | (EAplus e1 e2) => (EAplus (distrib_EAopp e1) (distrib_EAopp e2)) - | (EAmult e1 e2) => (EAmult (distrib_EAopp e1) (distrib_EAopp e2)) - | (EAopp e) => (EAmult (EAopp EAone) (distrib_EAopp e)) +Fixpoint distrib_EAopp (e:ExprA) : ExprA := + match e with + | EAplus e1 e2 => EAplus (distrib_EAopp e1) (distrib_EAopp e2) + | EAmult e1 e2 => EAmult (distrib_EAopp e1) (distrib_EAopp e2) + | EAopp e => EAmult (EAopp EAone) (distrib_EAopp e) | e => e end. Definition distrib_mult_right := - Fix distrib_mult_right {distrib_mult_right [e1:ExprA]:ExprA->ExprA:= - [e2:ExprA]Cases e1 of - | (EAplus t1 t2) => - (EAplus (distrib_mult_right t1 e2) (distrib_mult_right t2 e2)) - | _ => (EAmult e1 e2) - end}. - -Fixpoint distrib_mult_left [e1:ExprA] : ExprA->ExprA := - [e2:ExprA] - Cases e1 of - | (EAplus t1 t2) => - (EAplus (distrib_mult_left t1 e2) (distrib_mult_left t2 e2)) - | _ => (distrib_mult_right e2 e1) + (fix distrib_mult_right (e1:ExprA) : ExprA -> ExprA := + fun e2:ExprA => + match e1 with + | EAplus t1 t2 => + EAplus (distrib_mult_right t1 e2) (distrib_mult_right t2 e2) + | _ => EAmult e1 e2 + end). + +Fixpoint distrib_mult_left (e1 e2:ExprA) {struct e1} : ExprA := + match e1 with + | EAplus t1 t2 => + EAplus (distrib_mult_left t1 e2) (distrib_mult_left t2 e2) + | _ => distrib_mult_right e2 e1 end. -Fixpoint distrib_main [e:ExprA] : ExprA := - Cases e of - | (EAmult e1 e2) => (distrib_mult_left (distrib_main e1) (distrib_main e2)) - | (EAplus e1 e2) => (EAplus (distrib_main e1) (distrib_main e2)) - | (EAopp e) => (EAopp (distrib_main e)) +Fixpoint distrib_main (e:ExprA) : ExprA := + match e with + | EAmult e1 e2 => distrib_mult_left (distrib_main e1) (distrib_main e2) + | EAplus e1 e2 => EAplus (distrib_main e1) (distrib_main e2) + | EAopp e => EAopp (distrib_main e) | _ => e end. -Definition distrib [e:ExprA] : ExprA := (distrib_main (distrib_EAopp e)). - -Lemma distrib_mult_right_correct: - (e1,e2:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (distrib_mult_right e1 e2))= - (AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2)). -Proof. -Induction e1;Try Intros;Simpl;Auto. -Rewrite AmultT_sym;Rewrite AmultT_AplusT_distr; - Rewrite (H e2 lvar);Rewrite (H0 e2 lvar);Ring. -Save. - -Lemma distrib_mult_left_correct: - (e1,e2:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (distrib_mult_left e1 e2))= - (AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2)). -Proof. -Induction e1;Try Intros;Simpl. -Rewrite AmultT_Ol;Rewrite distrib_mult_right_correct;Simpl;Apply AmultT_Or. -Rewrite distrib_mult_right_correct;Simpl; - Apply AmultT_sym. -Rewrite AmultT_sym; - Rewrite (AmultT_AplusT_distr (interp_ExprA lvar e2) (interp_ExprA lvar e) - (interp_ExprA lvar e0)); - Rewrite (AmultT_sym (interp_ExprA lvar e2) (interp_ExprA lvar e)); - Rewrite (AmultT_sym (interp_ExprA lvar e2) (interp_ExprA lvar e0)); - Rewrite (H e2 lvar);Rewrite (H0 e2 lvar);Auto. -Rewrite distrib_mult_right_correct;Simpl;Apply AmultT_sym. -Rewrite distrib_mult_right_correct;Simpl;Apply AmultT_sym. -Rewrite distrib_mult_right_correct;Simpl;Apply AmultT_sym. -Rewrite distrib_mult_right_correct;Simpl;Apply AmultT_sym. -Save. - -Lemma distrib_correct: - (e:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (distrib e))=(interp_ExprA lvar e). -Proof. -Induction e;Intros;Auto. -Simpl;Rewrite <- (H lvar);Rewrite <- (H0 lvar); Unfold distrib;Simpl;Auto. -Simpl;Rewrite <- (H lvar);Rewrite <- (H0 lvar); Unfold distrib;Simpl; - Apply distrib_mult_left_correct. -Simpl;Fold AoppT;Rewrite <- (H lvar);Unfold distrib;Simpl; - Rewrite distrib_mult_right_correct; - Simpl;Fold AoppT;Ring. -Save. +Definition distrib (e:ExprA) : ExprA := distrib_main (distrib_EAopp e). + +Lemma distrib_mult_right_correct : + forall (e1 e2:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (distrib_mult_right e1 e2) = + AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2). +Proof. +simple induction e1; try intros; simpl in |- *; auto. +rewrite AmultT_sym; rewrite AmultT_AplusT_distr; rewrite (H e2 lvar); + rewrite (H0 e2 lvar); ring. +Qed. + +Lemma distrib_mult_left_correct : + forall (e1 e2:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (distrib_mult_left e1 e2) = + AmultT (interp_ExprA lvar e1) (interp_ExprA lvar e2). +Proof. +simple induction e1; try intros; simpl in |- *. +rewrite AmultT_Ol; rewrite distrib_mult_right_correct; simpl in |- *; + apply AmultT_Or. +rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_sym. +rewrite AmultT_sym; + rewrite + (AmultT_AplusT_distr (interp_ExprA lvar e2) (interp_ExprA lvar e) + (interp_ExprA lvar e0)); + rewrite (AmultT_sym (interp_ExprA lvar e2) (interp_ExprA lvar e)); + rewrite (AmultT_sym (interp_ExprA lvar e2) (interp_ExprA lvar e0)); + rewrite (H e2 lvar); rewrite (H0 e2 lvar); auto. +rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_sym. +rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_sym. +rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_sym. +rewrite distrib_mult_right_correct; simpl in |- *; apply AmultT_sym. +Qed. + +Lemma distrib_correct : + forall (e:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (distrib e) = interp_ExprA lvar e. +Proof. +simple induction e; intros; auto. +simpl in |- *; rewrite <- (H lvar); rewrite <- (H0 lvar); + unfold distrib in |- *; simpl in |- *; auto. +simpl in |- *; rewrite <- (H lvar); rewrite <- (H0 lvar); + unfold distrib in |- *; simpl in |- *; apply distrib_mult_left_correct. +simpl in |- *; fold AoppT in |- *; rewrite <- (H lvar); + unfold distrib in |- *; simpl in |- *; rewrite distrib_mult_right_correct; + simpl in |- *; fold AoppT in |- *; ring. +Qed. (**** Multiplication by the inverse product ****) -Lemma mult_eq: - (e1,e2,a:ExprA)(lvar:(listT (prodT AT nat))) - ~((interp_ExprA lvar a)=AzeroT)-> - (interp_ExprA lvar (EAmult a e1))=(interp_ExprA lvar (EAmult a e2))-> - (interp_ExprA lvar e1)=(interp_ExprA lvar e2). -Proof. - Simpl;Intros; - Apply (r_AmultT_mult (interp_ExprA lvar a) (interp_ExprA lvar e1) - (interp_ExprA lvar e2));Assumption. -Save. - -Fixpoint multiply_aux [a,e:ExprA] : ExprA := - Cases e of - | (EAplus e1 e2) => - (EAplus (EAmult a e1) (multiply_aux a e2)) - | _ => (EAmult a e) +Lemma mult_eq : + forall (e1 e2 a:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar a <> AzeroT -> + interp_ExprA lvar (EAmult a e1) = interp_ExprA lvar (EAmult a e2) -> + interp_ExprA lvar e1 = interp_ExprA lvar e2. +Proof. + simpl in |- *; intros; + apply + (r_AmultT_mult (interp_ExprA lvar a) (interp_ExprA lvar e1) + (interp_ExprA lvar e2)); assumption. +Qed. + +Fixpoint multiply_aux (a e:ExprA) {struct e} : ExprA := + match e with + | EAplus e1 e2 => EAplus (EAmult a e1) (multiply_aux a e2) + | _ => EAmult a e end. -Definition multiply [e:ExprA] : ExprA := - Cases e of - | (EAmult a e1) => (multiply_aux a e1) +Definition multiply (e:ExprA) : ExprA := + match e with + | EAmult a e1 => multiply_aux a e1 | _ => e end. -Lemma multiply_aux_correct: - (a,e:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (multiply_aux a e))= - (AmultT (interp_ExprA lvar a) (interp_ExprA lvar e)). +Lemma multiply_aux_correct : + forall (a e:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (multiply_aux a e) = + AmultT (interp_ExprA lvar a) (interp_ExprA lvar e). Proof. -Induction e;Simpl;Intros;Try (Rewrite merge_mult_correct);Auto. - Simpl;Rewrite (H0 lvar);Ring. -Save. +simple induction e; simpl in |- *; intros; try rewrite merge_mult_correct; + auto. + simpl in |- *; rewrite (H0 lvar); ring. +Qed. -Lemma multiply_correct: - (e:ExprA)(lvar:(listT (prodT AT nat))) - (interp_ExprA lvar (multiply e))=(interp_ExprA lvar e). +Lemma multiply_correct : + forall (e:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar (multiply e) = interp_ExprA lvar e. Proof. - Induction e;Simpl;Auto. - Intros;Apply multiply_aux_correct. -Save. + simple induction e; simpl in |- *; auto. + intros; apply multiply_aux_correct. +Qed. (**** Permutations and simplification ****) -Fixpoint monom_remove [a,m:ExprA] : ExprA := - Cases m of - | (EAmult m0 m1) => - (Cases (eqExprA m0 (EAinv a)) of - | (left _) => m1 - | (right _) => (EAmult m0 (monom_remove a m1)) - end) +Fixpoint monom_remove (a m:ExprA) {struct m} : ExprA := + match m with + | EAmult m0 m1 => + match eqExprA m0 (EAinv a) with + | left _ => m1 + | right _ => EAmult m0 (monom_remove a m1) + end | _ => - (Cases (eqExprA m (EAinv a)) of - | (left _) => EAone - | (right _) => (EAmult a m) - end) + match eqExprA m (EAinv a) with + | left _ => EAone + | right _ => EAmult a m + end end. -Definition monom_simplif_rem := - Fix monom_simplif_rem {monom_simplif_rem/1:ExprA->ExprA->ExprA:= - [a,m:ExprA] - Cases a of - | (EAmult a0 a1) => (monom_simplif_rem a1 (monom_remove a0 m)) - | _ => (monom_remove a m) - end}. - -Definition monom_simplif [a,m:ExprA] : ExprA := - Cases m of - | (EAmult a' m') => - (Cases (eqExprA a a') of - | (left _) => (monom_simplif_rem a m') - | (right _) => m - end) +Definition monom_simplif_rem := + (fix monom_simplif_rem (a:ExprA) : ExprA -> ExprA := + fun m:ExprA => + match a with + | EAmult a0 a1 => monom_simplif_rem a1 (monom_remove a0 m) + | _ => monom_remove a m + end). + +Definition monom_simplif (a m:ExprA) : ExprA := + match m with + | EAmult a' m' => + match eqExprA a a' with + | left _ => monom_simplif_rem a m' + | right _ => m + end | _ => m end. -Fixpoint inverse_simplif [a,e:ExprA] : ExprA := - Cases e of - | (EAplus e1 e2) => (EAplus (monom_simplif a e1) (inverse_simplif a e2)) - | _ => (monom_simplif a e) +Fixpoint inverse_simplif (a e:ExprA) {struct e} : ExprA := + match e with + | EAplus e1 e2 => EAplus (monom_simplif a e1) (inverse_simplif a e2) + | _ => monom_simplif a e end. -Lemma monom_remove_correct:(e,a:ExprA) - (lvar:(listT (prodT AT nat)))~((interp_ExprA lvar a)=AzeroT)-> - (interp_ExprA lvar (monom_remove a e))= - (AmultT (interp_ExprA lvar a) (interp_ExprA lvar e)). -Proof. -Induction e; Intros. -Simpl;Case (eqExprA EAzero (EAinv a));Intros;[Inversion e0|Simpl;Trivial]. -Simpl;Case (eqExprA EAone (EAinv a));Intros;[Inversion e0|Simpl;Trivial]. -Simpl;Case (eqExprA (EAplus e0 e1) (EAinv a));Intros;[Inversion e2| - Simpl;Trivial]. -Simpl;Case (eqExprA e0 (EAinv a));Intros. -Rewrite e2;Simpl;Fold AinvT. -Rewrite <-(AmultT_assoc (interp_ExprA lvar a) (AinvT (interp_ExprA lvar a)) - (interp_ExprA lvar e1)); - Rewrite AinvT_r;[Ring|Assumption]. -Simpl;Rewrite H0;Auto; Ring. -Simpl;Fold AoppT;Case (eqExprA (EAopp e0) (EAinv a));Intros;[Inversion e1| - Simpl;Trivial]. -Unfold monom_remove;Case (eqExprA (EAinv e0) (EAinv a));Intros. -Case (eqExprA e0 a);Intros. -Rewrite e2;Simpl;Fold AinvT;Rewrite AinvT_r;Auto. -Inversion e1;Simpl;ElimType False;Auto. -Simpl;Trivial. -Unfold monom_remove;Case (eqExprA (EAvar n) (EAinv a));Intros; - [Inversion e0|Simpl;Trivial]. -Save. - -Lemma monom_simplif_rem_correct:(a,e:ExprA) - (lvar:(listT (prodT AT nat)))~((interp_ExprA lvar a)=AzeroT)-> - (interp_ExprA lvar (monom_simplif_rem a e))= - (AmultT (interp_ExprA lvar a) (interp_ExprA lvar e)). -Proof. -Induction a;Simpl;Intros; Try Rewrite monom_remove_correct;Auto. -Elim (without_div_O_contr (interp_ExprA lvar e) - (interp_ExprA lvar e0) H1);Intros. -Rewrite (H0 (monom_remove e e1) lvar H3);Rewrite monom_remove_correct;Auto. -Ring. -Save. - -Lemma monom_simplif_correct:(e,a:ExprA) - (lvar:(listT (prodT AT nat)))~((interp_ExprA lvar a)=AzeroT)-> - (interp_ExprA lvar (monom_simplif a e))=(interp_ExprA lvar e). -Proof. -Induction e;Intros;Auto. -Simpl;Case (eqExprA a e0);Intros. -Rewrite <-e2;Apply monom_simplif_rem_correct;Auto. -Simpl;Trivial. -Save. - -Lemma inverse_correct: - (e,a:ExprA)(lvar:(listT (prodT AT nat)))~((interp_ExprA lvar a)=AzeroT)-> - (interp_ExprA lvar (inverse_simplif a e))=(interp_ExprA lvar e). -Proof. -Induction e;Intros;Auto. -Simpl;Rewrite (H0 a lvar H1); Rewrite monom_simplif_correct ; Auto. -Unfold inverse_simplif;Rewrite monom_simplif_correct ; Auto. -Save. - -End Theory_of_fields. +Lemma monom_remove_correct : + forall (e a:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar a <> AzeroT -> + interp_ExprA lvar (monom_remove a e) = + AmultT (interp_ExprA lvar a) (interp_ExprA lvar e). +Proof. +simple induction e; intros. +simpl in |- *; case (eqExprA EAzero (EAinv a)); intros; + [ inversion e0 | simpl in |- *; trivial ]. +simpl in |- *; case (eqExprA EAone (EAinv a)); intros; + [ inversion e0 | simpl in |- *; trivial ]. +simpl in |- *; case (eqExprA (EAplus e0 e1) (EAinv a)); intros; + [ inversion e2 | simpl in |- *; trivial ]. +simpl in |- *; case (eqExprA e0 (EAinv a)); intros. +rewrite e2; simpl in |- *; fold AinvT in |- *. +rewrite <- + (AmultT_assoc (interp_ExprA lvar a) (AinvT (interp_ExprA lvar a)) + (interp_ExprA lvar e1)); rewrite AinvT_r; [ ring | assumption ]. +simpl in |- *; rewrite H0; auto; ring. +simpl in |- *; fold AoppT in |- *; case (eqExprA (EAopp e0) (EAinv a)); + intros; [ inversion e1 | simpl in |- *; trivial ]. +unfold monom_remove in |- *; case (eqExprA (EAinv e0) (EAinv a)); intros. +case (eqExprA e0 a); intros. +rewrite e2; simpl in |- *; fold AinvT in |- *; rewrite AinvT_r; auto. +inversion e1; simpl in |- *; elimtype False; auto. +simpl in |- *; trivial. +unfold monom_remove in |- *; case (eqExprA (EAvar n) (EAinv a)); intros; + [ inversion e0 | simpl in |- *; trivial ]. +Qed. + +Lemma monom_simplif_rem_correct : + forall (a e:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar a <> AzeroT -> + interp_ExprA lvar (monom_simplif_rem a e) = + AmultT (interp_ExprA lvar a) (interp_ExprA lvar e). +Proof. +simple induction a; simpl in |- *; intros; try rewrite monom_remove_correct; + auto. +elim (Rmult_neq_0_reg (interp_ExprA lvar e) (interp_ExprA lvar e0) H1); + intros. +rewrite (H0 (monom_remove e e1) lvar H3); rewrite monom_remove_correct; auto. +ring. +Qed. + +Lemma monom_simplif_correct : + forall (e a:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar a <> AzeroT -> + interp_ExprA lvar (monom_simplif a e) = interp_ExprA lvar e. +Proof. +simple induction e; intros; auto. +simpl in |- *; case (eqExprA a e0); intros. +rewrite <- e2; apply monom_simplif_rem_correct; auto. +simpl in |- *; trivial. +Qed. + +Lemma inverse_correct : + forall (e a:ExprA) (lvar:listT (prodT AT nat)), + interp_ExprA lvar a <> AzeroT -> + interp_ExprA lvar (inverse_simplif a e) = interp_ExprA lvar e. +Proof. +simple induction e; intros; auto. +simpl in |- *; rewrite (H0 a lvar H1); rewrite monom_simplif_correct; auto. +unfold inverse_simplif in |- *; rewrite monom_simplif_correct; auto. +Qed. + +End Theory_of_fields.
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