From 9762655f6853bacf4786dab274afa4c33d3e32f3 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Sat, 15 Nov 2008 13:18:05 +0000 Subject: [PATCH 1/1] housekeeping --- .../matita/contribs/didactic/algebra.ma | 101 ------------------ .../software/matita/contribs/didactic/depends | 9 -- .../matita/contribs/didactic/depends.png | Bin 41036 -> 0 bytes helm/software/matita/contribs/didactic/root | 1 - 4 files changed, 111 deletions(-) delete mode 100644 helm/software/matita/contribs/didactic/algebra.ma delete mode 100644 helm/software/matita/contribs/didactic/depends delete mode 100644 helm/software/matita/contribs/didactic/depends.png delete mode 100644 helm/software/matita/contribs/didactic/root diff --git a/helm/software/matita/contribs/didactic/algebra.ma b/helm/software/matita/contribs/didactic/algebra.ma deleted file mode 100644 index 17c30e341..000000000 --- a/helm/software/matita/contribs/didactic/algebra.ma +++ /dev/null @@ -1,101 +0,0 @@ - -include "nat/times.ma". -include "nat/compare.ma". - -definition if_then_else ≝ λT:Type.λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f]. -notation > "'if' term 19 e 'then' term 19 t 'else' term 90 f" non associative with precedence 90 for @{ 'if_then_else $e $t $f }. -notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 90 for @{ 'if_then_else $e $t $f }. -interpretation "Formula if_then_else" 'if_then_else e t f = (if_then_else _ e t f). - -inductive F : Type ≝ -| EZero : F -| EOne : F -| EPlus : F → F → F -| ETimes : F → F → F -| X : ℕ → F -. - -let rec val (E : F) (v : ℕ → ℕ) on E : ℕ := - match E with - [ X i ⇒ v i - | EPlus e1 e2 ⇒ (val e1 v) + (val e2 v) - | ETimes e1 e2 ⇒ (val e1 v) * (val e2 v) - | EZero ⇒ 0 - | EOne ⇒ 1 - ] - . - -definition v201 ≝ λx. - if (eqb x 0) then 2 - else if (eqb x 1) then 0 - else if (eqb x 2) then 1 - else 0 -. - -eval normalize on - (val - (EPlus (X 3) - (ETimes EOne (X 2))) v201). - -let rec subst (E : F) (G : F) (i : ℕ) on E : F := - match E with - [ X j => if (eqb i j) then G else (X j) - | EPlus e1 e2 => EPlus (subst e1 G i) (subst e2 G i) - | ETimes e1 e2 => ETimes (subst e1 G i) (subst e2 G i) - | EZero => EZero - | EOne => EOne - ] - . - -eval normalize on - (subst - (EPlus (X 0) (ETimes (X 1) (X 2))) - (EOne) - 1) - . - -definition equiv := - λe1,e2.∀v. val e1 v = val e2 v. - -theorem sostituzione : - ∀E1,E2,G,i. - equiv E1 E2 → equiv (subst G E1 i) (subst G E2 i). -assume E1 : F. -assume E2 : F. -assume G : F. -assume i : ℕ. -suppose (equiv E1 E2) (H). -we proceed by induction on G to prove - (equiv (subst G E1 i) (subst G E2 i)). -case EZero. - the thesis becomes (equiv (subst EZero E1 i) (subst EZero E2 i)). - the thesis becomes (equiv EZero (subst EZero E2 i)). - the thesis becomes (equiv EZero EZero). - the thesis becomes (∀v.val EZero v = val EZero v). - assume v : (ℕ → ℕ). - the thesis becomes (0 = val EZero v). - the thesis becomes (0 = 0). - done. -case EOne. - the thesis becomes (∀v.val EOne v = val EOne v). - assume v : (ℕ → ℕ). - the thesis becomes (1 = 1). - done. -case EPlus. - assume e1 : F. - by induction hypothesis we know (equiv (subst e1 E1 i) (subst e1 E2 i)) (H1). - assume e2 : F. - by induction hypothesis we know (equiv (subst e2 E1 i) (subst e2 E2 i)) (H2). - by H1 we proved (∀v.val (subst e1 E1 i) v = val (subst e1 E2 i) v) (H11). - by H2 we proved (∀v.val (subst e2 E1 i) v = val (subst e2 E2 i) v) (H22). - the thesis becomes - (∀v. val (subst e1 E1 i) v + val (subst e2 E1 i) v = - val (subst e1 E2 i) v + val (subst e2 E2 i) v). - assume v : (ℕ → ℕ). - conclude - (val (subst e1 E1 i) v + val (subst e2 E1 i) v) = - (val (subst e1 E1 i) v + val (subst e2 E2 i) v) by H11. - conclude - (val (subst e1 E1 i) v + val (subst e2 E1 i) v) = - (val (subst e1 E1 i) v + val (subst e2 E1 i) v) by H22. - done. diff --git a/helm/software/matita/contribs/didactic/depends b/helm/software/matita/contribs/didactic/depends deleted file mode 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