X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fdemo%2Fpower_derivative.ma;h=adf85249967c8fe1c402b82dcabbb4839679a44c;hb=c99a38b6539be1eb667cced1eed2db3fc75e3162;hp=4f1b44000889a1df1b765d98a29977b8a3b2a432;hpb=5649890273cf8e660bba744e84ce5fee1e5efe69;p=helm.git diff --git a/helm/software/matita/library/demo/power_derivative.ma b/helm/software/matita/library/demo/power_derivative.ma index 4f1b44000..adf852499 100644 --- a/helm/software/matita/library/demo/power_derivative.ma +++ b/helm/software/matita/library/demo/power_derivative.ma @@ -12,8 +12,6 @@ (* *) (**************************************************************************) -set "baseuri" "cic:/matita/demo/power_derivative". - include "nat/plus.ma". include "nat/orders.ma". include "nat/compare.ma". @@ -42,11 +40,7 @@ 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 = +interpretation "Rmult" 'middot x y = (cic:/matita/demo/power_derivative/Rmult.con x y). definition Fplus ≝ @@ -57,7 +51,7 @@ definition Fmult ≝ interpretation "Fplus" 'plus x y = (cic:/matita/demo/power_derivative/Fplus.con x y). -interpretation "Fmult" 'times x y = +interpretation "Fmult" 'middot x y = (cic:/matita/demo/power_derivative/Fmult.con x y). notation "2" with precedence 89 @@ -90,18 +84,18 @@ let rec inj (n:nat) on n : R ≝ ] ]. -coercion cic:/matita/demo/power_derivative/inj.con. +coercion inj. axiom Rplus_Rzero_x: ∀x:R.0+x=x. axiom Rplus_comm: symmetric ? Rplus. axiom Rplus_assoc: associative ? Rplus. -axiom Rmult_Rone_x: ∀x:R.1*x=x. -axiom Rmult_Rzero_x: ∀x:R.0*x=0. +axiom Rmult_Rone_x: ∀x:R.1 · x=x. +axiom Rmult_Rzero_x: ∀x:R.0 · x=0. axiom Rmult_assoc: associative ? Rmult. axiom Rmult_comm: symmetric ? Rmult. axiom Rmult_Rplus_distr: distributive ? Rmult Rplus. -alias symbol "times" = "Rmult". +alias symbol "middot" = "Rmult". alias symbol "plus" = "natural plus". definition monomio ≝ @@ -110,7 +104,7 @@ definition monomio ≝ definition costante : nat → R → R ≝ λa:nat.λx:R.inj a. -coercion cic:/matita/demo/power_derivative/costante.con 1. +coercion costante with 1. axiom f_eq_extensional: ∀f,g:R→R.(∀x:R.f x = g x) → f=g. @@ -247,7 +241,7 @@ 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]. -alias symbol "times" = "Fmult". +alias symbol "middot" = "Fmult". theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n). assume n:nat. @@ -256,7 +250,6 @@ theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n). elim n 0. case O. the thesis becomes (D[x \sup 0] = 0·x \sup (pred 0)). - by _ done. case S (m:nat). by induction hypothesis we know @@ -265,24 +258,24 @@ theorem derivative_power: ∀n:nat. D[x \sup n] = n·x \sup (pred n). (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). - by _ we proved (0 < m ∨ 0=m) (cases). + 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). case left. suppose (0 < m) (m_pos). - by (S_pred m m_pos) we proved (m = 1 + pred m) (H1). - by _ - done. + using (S_pred ? m_pos) we proved (m = 1 + pred m) (H1). + by H1 done. case right. - suppose (0=m) (m_zero). by _ done. + suppose (0=m) (m_zero). + by m_zero, Fmult_zero_f done. conclude (D[x \sup (1+m)]) = (D[x · x \sup m]). = (D[x] · x \sup m + x · D[x \sup m]). - = (x \sup m + x · (m · x \sup (pred m))). + = (x \sup m + x · (m · x \sup (pred m))) timeout=30. = (x \sup m + m · (x \sup (1 + pred m))). = (x \sup m + m · x \sup m). - = ((1+m) · x \sup m) by Fmult_one_f Fmult_commutative Fmult_Fplus_distr costante_sum + = ((1+m) · x \sup m) timeout=30 by Fmult_one_f, Fmult_commutative, Fmult_Fplus_distr, costante_sum done. qed. @@ -297,8 +290,7 @@ interpretation "Rderivative" 'derivative \eta.f = (cic:/matita/demo/power_derivative/derivative.con f). *) -notation "hvbox(\frac 'd' ('d' 'x') break p)" - right associative with precedence 90 +notation "hvbox(\frac 'd' ('d' 'x') break p)" with precedence 90 for @{ 'derivative $p}. interpretation "Rderivative" 'derivative f = @@ -310,7 +302,6 @@ theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n) · x \sup n. (D[x \sup (1+n)] = (1+n) · x \sup n).*) elim n 0. case O. the thesis becomes (D[x \sup 1] = 1 · x \sup 0). - by _ done. case S (m:nat). by induction hypothesis we know @@ -323,75 +314,7 @@ theorem derivative_power': ∀n:nat. D[x \sup (1+n)] = (1+n) · x \sup n. = (D[x] · x \sup (1+m) + x · D[x \sup (1+m)]). = (x \sup (1+m) + x · (costante (1+m) · x \sup m)). = (x \sup (1+m) + costante (1+m) · x \sup (1+m)). - - - - conclude (x \sup (1+m) + costante (1+m) · x \sup (1+m)) - = (costante 1 · x \sup (1+m) + costante (1+m) ·(x) \sup (1+m)) proof. - by (Fmult_one_f ((x)\sup(1+m))) - we proved (costante 1·(x)\sup(1+m)=(x)\sup(1+m)) (previous). - by (eq_OF_eq ? ? (λfoo:(R→R).foo+costante (1+m)·(x)\sup(1+m)) (costante 1 - ·(x)\sup(1 - +m)) ((x)\sup(1 - +m)) previous) - done. - = ((x)\sup(1+m)·costante 1+costante (1+m)·(x)\sup(1+m)) proof. - by (Fmult_commutative (costante 1) ((x)\sup(1+m))) - we proved (costante 1·(x)\sup(1+m)=(x)\sup(1+m)·costante 1) (previous). - by (eq_f ? ? (λfoo:(R→R).foo+costante (1+m)·(x)\sup(1+m)) (costante 1 - ·(x)\sup(1+m)) ((x)\sup(1 - +m) - ·costante - 1) previous) - done. - = ((x)\sup(1+m)·costante 1+(x)\sup(1+m)·costante (1+m)) proof. - by (Fmult_commutative ((x)\sup(1+m)) (costante (1+m))) - we proved ((x)\sup(1+m)·costante (1+m)=costante (1+m)·(x)\sup(1+m)) - - (previous) - . - by (eq_OF_eq ? ? (λfoo:(R→R).(x)\sup(1+m)·costante 1+foo) ((x)\sup(1+m) - ·costante - (1+m)) (costante - (1 - +m) - ·(x)\sup(1 - +m)) previous) - done. - = ((x)\sup(1+m)·(costante 1+costante (1+m))) proof. - by (Fmult_Fplus_distr ((x)\sup(1+m)) (costante 1) (costante (1+m))) - we proved - ((x)\sup(1+m)·(costante 1+costante (1+m)) - =(x)\sup(1+m)·costante 1+(x)\sup(1+m)·costante (1+m)) - - (previous) - . - by (sym_eq ? ((x)\sup(1+m)·(costante 1+costante (1+m))) ((x)\sup(1+m) - ·costante 1 - +(x)\sup(1+m) - ·costante (1+m)) previous) - done. - = ((costante 1+costante (1+m))·(x)\sup(1+m)) - exact (Fmult_commutative ((x)\sup(1+m)) (costante 1+costante (1+m))). - = (costante (1+(1+m))·(x)\sup(1+m)) proof. - by (costante_sum 1 (1+m)) - we proved (costante 1+costante (1+m)=costante (1+(1+m))) (previous). - by (eq_f ? ? (λfoo:(R→R).foo·(x)\sup(1+m)) (costante 1+costante (1+m)) (costante - (1 - +(1 - +m))) previous) - done. - = (costante (1+1+m)·(x)\sup(1+m)) proof. - by (assoc_plus 1 1 m) - we proved (1+1+m=1+(1+m)) (previous). - by (eq_OF_eq ? ? (λfoo:nat.costante foo·(x)\sup(1+m)) ? ? previous) - done. - = (costante (2+m)·(x)\sup(1+m)) proof done. - by (plus_n_SO 1) - we proved (2=1+1) (previous). - by (eq_OF_eq ? ? (λfoo:nat.costante (foo+m)·(x)\sup(1+m)) ? ? previous) - done. - - -(* end auto($Revision: 8206 $) proof: TIME=0.06 SIZE=100 DEPTH=100 *) done. + = ((2+m) · x \sup (1+m)) timeout=30 by Fmult_one_f, Fmult_commutative, + Fmult_Fplus_distr, assoc_plus, plus_n_SO, costante_sum + done. qed.