From 45d8e093b732768f58794dc3169e2980257dc7d3 Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Thu, 30 Nov 2006 22:09:47 +0000 Subject: [PATCH] Notation \middot used everywhere in place of *. --- .../matita/library/demo/power_derivative.ma | 69 ++++++++++--------- 1 file changed, 37 insertions(+), 32 deletions(-) diff --git a/helm/software/matita/library/demo/power_derivative.ma b/helm/software/matita/library/demo/power_derivative.ma index 879f55d7e..24656c1d3 100644 --- a/helm/software/matita/library/demo/power_derivative.ma +++ b/helm/software/matita/library/demo/power_derivative.ma @@ -41,6 +41,11 @@ interpretation "None" 'one = interpretation "Rplus" 'plus x y = (cic:/matita/demo/power_derivative/Rplus.con x y). + +notation "hvbox(a break \middot b)" + left associative with precedence 55 +for @{ 'times $a $b }. + interpretation "Rmult" 'times x y = (cic:/matita/demo/power_derivative/Rmult.con x y). @@ -48,7 +53,7 @@ definition Fplus ≝ λf,g:R→R.λx:R.f x + g x. definition Fmult ≝ - λf,g:R→R.λx:R.f x * g x. + λf,g:R→R.λx:R.f x · g x. interpretation "Fplus" 'plus x y = (cic:/matita/demo/power_derivative/Fplus.con x y). @@ -69,7 +74,7 @@ interpretation "Ntwo" 'two = let rec Rpower (x:R) (n:nat) on n ≝ match n with [ O ⇒ 1 - | S n ⇒ x * (Rpower x n) + | S n ⇒ x · (Rpower x n) ]. interpretation "Rpower" 'exp x n = @@ -110,7 +115,7 @@ coercion cic:/matita/demo/power_derivative/costante.con 1. axiom f_eq_extensional: ∀f,g:R→R.(∀x:R.f x = g x) → f=g. -lemma Fmult_one_f: ∀f:R→R.1*f=f. +lemma Fmult_one_f: ∀f:R→R.1·f=f. intro; unfold Fmult; simplify; @@ -119,7 +124,7 @@ lemma Fmult_one_f: ∀f:R→R.1*f=f. auto. qed. -lemma Fmult_zero_f: ∀f:R→R.0*f=0. +lemma Fmult_zero_f: ∀f:R→R.0·f=0. intro; unfold Fmult; simplify; @@ -159,7 +164,7 @@ lemma Fmult_Fplus_distr: distributive ? Fmult Fplus. qed. lemma monomio_product: - ∀n,m.monomio (n+m) = monomio n * monomio m. + ∀n,m.monomio (n+m) = monomio n · monomio m. intros; unfold monomio; unfold Fmult; @@ -172,10 +177,10 @@ lemma monomio_product: | simplify; apply f_eq_extensional; intro; - cut (x\sup (n1+m) = x \sup n1 * x \sup m); + cut (x\sup (n1+m) = x \sup n1 · x \sup m); [ rewrite > Hcut; auto - | change in ⊢ (? ? % ?) with ((λx:R.(x)\sup(n1+m)) x); + | change in ⊢ (? ? % ?) with ((λx:R.x\sup(n1+m)) x); rewrite > H; reflexivity ] @@ -240,28 +245,28 @@ interpretation "Rmonomio" 'monomio n = axiom derivative_x0: D[x \sup 0] = 0. axiom derivative_x1: D[x] = 1. -axiom derivative_mult: ∀f,g:R→R. D[f*g] = D[f]*g + f*D[g]. +axiom derivative_mult: ∀f,g:R→R. D[f·g] = D[f]·g + f·D[g]. alias symbol "times" = "Fmult". -theorem derivative_power: ∀n:nat. D[x \sup n] = n*x \sup (pred n). +theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n). assume n:nat. we proceed by induction on n to prove - (D[x \sup n] = n*x \sup (pred n)). + (D[x \sup n] = n · x \sup (pred n)). case O. - the thesis becomes (D[x \sup 0] = 0*x \sup (pred 0)). + the thesis becomes (D[x \sup 0] = 0·x \sup (pred 0)). by _ done. case S (m:nat). by induction hypothesis we know - (D[x \sup m] = m*x \sup (pred m)) (H). + (D[x \sup m] = m·x \sup (pred m)) (H). the thesis becomes - (D[x \sup (1+m)] = (1+m)*x \sup m). + (D[x \sup (1+m)] = (1+m) · x \sup m). we need to prove - (m * (x \sup (1+ pred m)) = m * x \sup m) (Ppred). + (m · (x \sup (1+ pred m)) = m · x \sup m) (Ppred). by _ we proved (0 < m ∨ 0=m) (cases). we proceed by induction on cases - to prove (m * (x \sup (1+ pred m)) = m * x \sup m). + to prove (m · (x \sup (1+ pred m)) = m · x \sup m). case left. suppose (0 < m) (m_pos). by (S_pred m m_pos) we proved (m = 1 + pred m) (H1). @@ -271,13 +276,13 @@ theorem derivative_power: ∀n:nat. D[x \sup n] = n*x \sup (pred n). suppose (0=m) (m_zero). by _ done. conclude (D[x \sup (1+m)]) - = (D[x * x \sup m]) by _. - = (D[x] * x \sup m + x * D[x \sup m]) by _. - = (x \sup m + x * (m * x \sup (pred m))) by _. + = (D[x · x \sup m]) by _. + = (D[x] · x \sup m + x · D[x \sup m]) by _. + = (x \sup m + x · (m · x \sup (pred m))) by _. clear H. - = (x \sup m + m * (x \sup (1 + pred m))) by _. - = (x \sup m + m * x \sup m) by _. - = ((1+m)*x \sup m) by _ (timeout=30) + = (x \sup m + m · (x \sup (1 + pred m))) by _. + = (x \sup m + m · x \sup m) by _. + = ((1+m) · x \sup m) by _ (timeout=30) done. qed. @@ -299,26 +304,26 @@ for @{ 'derivative $p}. interpretation "Rderivative" 'derivative f = (cic:/matita/demo/power_derivative/derivative.con f). -theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n)*x \sup n. +theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n) · x \sup n. assume n:nat. we proceed by induction on n to prove - (D[x \sup (1+n)] = (1+n)*x \sup n). + (D[x \sup (1+n)] = (1+n) · x \sup n). case O. - the thesis becomes (D[x \sup 1] = 1*x \sup 0). + the thesis becomes (D[x \sup 1] = 1 · x \sup 0). by _ done. case S (m:nat). by induction hypothesis we know - (D[x \sup (1+m)] = (1+m)*x \sup m) (H). + (D[x \sup (1+m)] = (1+m) · x \sup m) (H). the thesis becomes - (D[x \sup (2+m)] = (2+m)*x \sup (1+m)). + (D[x \sup (2+m)] = (2+m) · x \sup (1+m)). conclude (D[x \sup (2+m)]) - = (D[x \sup 1 * x \sup (1+m)]) by _. - = (D[x \sup 1] * x \sup (1+m) + x * D[x \sup (1+m)]) by _. - = (x \sup (1+m) + x * (costante (1+m) * x \sup m)) by _. + = (D[x \sup 1 · x \sup (1+m)]) by _. + = (D[x \sup 1] · x \sup (1+m) + x · D[x \sup (1+m)]) by _. + = (x \sup (1+m) + x · (costante (1+m) · x \sup m)) by _. clear H. - = (x \sup (1+m) + costante (1+m) * x \sup (1+m)) by _. - = (x \sup (1+m) * (costante (2 + m))) by _ + = (x \sup (1+m) + costante (1+m) · x \sup (1+m)) by _. + = (x \sup (1+m) · (costante (2 + m))) by _ done. -qed. +qed. \ No newline at end of file -- 2.39.2