From 5e72e41f4f86814e56d4b00959ccc56c71042a4c Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Mon, 28 Dec 2020 19:43:58 +0100 Subject: [PATCH] =?utf8?q?arithmetics=20for=20=CE=BB=CE=B4?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit + subtraction for non-negative integers + web site update + apps_2/examples: minor update --- helm/www/lambdadelta/images/bronze-03B4.png | Bin 45625 -> 59369 bytes helm/www/lambdadelta/images/bronze-03BB.png | Bin 45359 -> 58520 bytes .../apps_2/examples/ex_fpbg_refl.ma | 3 +- .../contribs/lambdadelta/ground/arith/nat.txt | 73 ++++++------ .../lambdadelta/ground/arith/nat_iter.ma | 2 +- .../lambdadelta/ground/arith/nat_le.ma | 25 +++- .../lambdadelta/ground/arith/nat_le_minus.ma | 64 ++++++++++ .../ground/arith/nat_le_minus_plus.ma | 109 ++++++++++++++++++ .../lambdadelta/ground/arith/nat_le_plus.ma | 9 +- .../lambdadelta/ground/arith/nat_le_pred.ma | 18 ++- .../lambdadelta/ground/arith/nat_lt.ma | 2 +- .../lambdadelta/ground/arith/nat_lt_minus.ma | 37 ++++++ .../lambdadelta/ground/arith/nat_lt_plus.ma | 6 +- .../lambdadelta/ground/arith/nat_lt_pred.ma | 6 +- .../lambdadelta/ground/arith/nat_minus.ma | 71 ++++++++++++ .../ground/arith/nat_minus_plus.ma | 84 ++++++++++++++ .../lambdadelta/ground/arith/nat_plus.ma | 18 ++- .../lambdadelta/ground/arith/nat_pred.ma | 2 +- .../lambdadelta/ground/arith/nat_pred_succ.ma | 4 +- .../lambdadelta/ground/arith/nat_succ.ma | 4 +- .../lambdadelta/ground/arith/nat_succ_iter.ma | 2 +- .../lambdadelta/ground/arith/pnat_dis.ma | 2 +- .../lambdadelta/ground/arith/pnat_iter.ma | 2 +- .../lambdadelta/ground/arith/pnat_plus.ma | 4 +- .../lambdadelta/ground/web/ground_src.tbl | 9 +- 25 files changed, 481 insertions(+), 75 deletions(-) create mode 100644 matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus.ma create mode 100644 matita/matita/contribs/lambdadelta/ground/arith/nat_le_minus_plus.ma create mode 100644 matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma create mode 100644 matita/matita/contribs/lambdadelta/ground/arith/nat_minus.ma create mode 100644 matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma diff --git a/helm/www/lambdadelta/images/bronze-03B4.png b/helm/www/lambdadelta/images/bronze-03B4.png index d65392f095c4f0cd434dfb640805ee6d0db1cf78..98b99d5b39ea4b6da214cb6ee72c340980802d8c 100644 GIT binary patch literal 59369 zcmbrFrwme(Aq~4*Sn8*IZQr0PIxWYG7AYBM)*% zX9sgDTQhQ3FGn+SGfyjX0KjvlvcoR@v?xb%KngwuZa^Lu~0E3e!pdg!8kI#?$I!_od&< zJLa`ylU+1kWTGzy7tM2fan zg%7=yY|Itup1kk8>6hysz7INiE{EKgllU!}xNptfiviwOA7h)sEUxiYM7`@Dx{QN3 zcUIwf`>z=uMv0F8iw(XfFK?aiJKPTj$XPXpPwA~+dJeONGcsvj$C_s>4*Oq5e^tNG zaftZ@us|@KE_w~cdZ9dEsKrn`Vrs)6eD0?=tk;>&Kf&awn469ywkfIuJmSy~2VaRZqWsw7>ed*WLXLXIOlccDjU_=j;%`fHMH$2 z@3>ZGzfE=CuIOC&-CFb~1=g@1TQ2@Ji9Ou@jh5m6c-zOHZ87-WJlkwx**wQY2L`(vRQXnWzAL&8I7ceyFhB1IyT3f_uB{dw^N<fkE`g=TXc!D 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zK$q3;z50zmSUN*e4s9M4IdyJw?(5j45+X(si3msVDWlvH(G_UcJtL%F<#UES^0YN4 z?28xFpe;8%%qXJD@uOA%XcGKX|C=`i%`;A z8YY?KO0jaDf64hFaZ(3wi-lzr)`cA zfg{*SJS`UZxrOmGG&IOA3U191lF*u}wi+e5JS@vmt9!)hLMmEwYu7Bh)hwU03lP@# zl87@f7kOGqn2=ZzS^DZc7q2HY%;kE{-bL7O$vZBXluk(e4}W+#w(!5Dc}lePrq5_9zJ9=Sn~jJns)1 nGBv!tJ!%Xyr7;BH0TKe)7gs*lP*)g(zb3e)YXYs&af(nminus_plus_sn_refl_sn m o) +/2 width=1 by nle_minus_sn_bi/ +qed. + +(*** le_inv_plus_l *) +lemma nle_minus_dx_full (o) (m) (n): m + o ≤ n → ∧∧ m ≤ n - o & o ≤ n. +/3 width=3 by nle_minus_dx, nle_trans, conj/ qed-. + +(* Inversions with nminus and nplus *****************************************) + +(*** plus_minus_m_m *) +lemma nplus_minus_sn_refl_sn (m) (n): m ≤ n → n = n - m + m. +#m #n #H elim H -n // +#n #Hn #IH <(nminus_succ_sn … Hn) -Hn +(nplus_minus_sn_refl_sn … Hon) +/2 width=1 by nle_plus_bi_dx/ +qed-. + +(* Destructions with nminus and nplus ***************************************) + +(*** plus_minus *) +lemma nminus_plus_comm_23 (o) (m) (n): + m ≤ n → n - m + o = n + o - m. +#o #m #n #H elim H -n // +#n #Hn #IH <(nminus_succ_sn … Hn) +nplus_assoc +(nplus_minus_sn_refl_sn … Hm) in ⊢ (??%?); // +qed-. diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma index 513ea5257..37a0efbb2 100644 --- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_plus.ma @@ -15,9 +15,9 @@ include "ground/arith/nat_plus.ma". include "ground/arith/nat_le.ma". -(* NON-NEGATIVE INTEGERS ****************************************************) +(* ORDER FOR NON-NEGATIVE INTEGERS ****************************************************) -(* Basic constructions with plus ********************************************) +(* Constructions with nplus ***********************************************************) (*** monotonic_le_plus_l *) lemma nle_plus_bi_dx (m) (n1) (n2): n1 ≤ n2 → n1 + m ≤ n2 + m. @@ -46,14 +46,14 @@ lemma nle_plus_dx_dx (m) (n) (o): n + o ≤ m → n ≤ m. lemma nle_plus_dx_sn (m) (n) (o): n ≤ m → n ≤ o + m. /2 width=3 by nle_trans/ qed. -(* Main constructions with plus *********************************************) +(* Main constructions with nplus ********************************************) (*** le_plus *) theorem nle_plus_bi (m1) (m2) (n1) (n2): m1 ≤ m2 → n1 ≤ n2 → m1 + n1 ≤ m2 + n2. /3 width=3 by nle_plus_bi_dx, nle_plus_bi_sn, nle_trans/ qed. -(* Basic inversions with plus ***********************************************) +(* Inversions with nplus ****************************************************) (*** plus_le_0 *) lemma nle_inv_plus_zero (m) (n): m + n ≤ 𝟎 → ∧∧ 𝟎 = m & 𝟎 = n. @@ -67,4 +67,3 @@ qed-. (*** le_plus_to_le *) lemma nle_inv_plus_bi_sn (o) (m) (n): o + n ≤ o + m → n ≤ m. /2 width=2 by nle_inv_plus_bi_dx/ qed-. - diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma index f1824e22e..da1de97f5 100644 --- a/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_le_pred.ma @@ -15,9 +15,9 @@ include "ground/arith/nat_pred_succ.ma". include "ground/arith/nat_le.ma". -(* NON-NEGATIVE INTEGERS ****************************************************) +(* ORDER FOR NON-NEGATIVE INTEGERS ******************************************) -(* Basic constructions with pred ********************************************) +(* Constructions with npred *************************************************) (*** le_pred_n *) lemma nle_pred_sn_refl (m): ↓m ≤ m. @@ -26,6 +26,18 @@ qed. (*** monotonic_pred *) lemma nle_pred_bi (m) (n): m ≤ n → ↓m ≤ ↓n. -#m #n #H elim H -n +#m #n #H elim H -n // /2 width=3 by nle_trans/ qed. + +lemma nle_pred_sn (m) (n): m ≤ ↑n → ↓m ≤ n. +#m #n elim m -m // +/2 width=1 by nle_pred_bi/ +qed-. + +(* Destructions with npred **************************************************) + +lemma nle_inv_pred_sn (m) (n): ↓m ≤ n → m ≤ ↑n. +#m #n elim m -m +/2 width=1 by nle_succ_bi/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt.ma index 1df0f857d..f7a868b53 100644 --- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt.ma +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt.ma @@ -14,7 +14,7 @@ include "ground/arith/nat_le.ma". -(* NON-NEGATIVE INTEGERS ****************************************************) +(* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************) (*** lt *) definition nlt: relation2 nat nat ≝ diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma new file mode 100644 index 000000000..34a16e01f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_minus.ma @@ -0,0 +1,37 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground/arith/nat_le_minus.ma". +include "ground/arith/nat_lt_pred.ma". + +(* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************) + +(* Rewrites with nminus *****************************************************) + +(*** minus_pred_pred *) +lemma nminus_pred_bi (m) (n): 𝟎 < m → 𝟎 < n → n - m = ↓n - ↓m. +#m #n #Hm #Hn +>(nlt_inv_zero_sn … Hm) in ⊢ (??%?); -Hm +>(nlt_inv_zero_sn … Hn) in ⊢ (??%?); -Hn +// +qed-. + +(* Constructions with nminus ************************************************) + +(*** monotonic_lt_minus_l *) +lemma nlt_minus_sn_bi (o) (m) (n): o ≤ m → m < n → m - o < n - o. +#o #m #n #Hom #Hmn +lapply (nle_minus_sn_bi … o Hmn) -Hmn +<(nminus_succ_sn … Hom) // +qed. diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_plus.ma index e9920e928..05776ef20 100644 --- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_plus.ma +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_plus.ma @@ -15,9 +15,9 @@ include "ground/arith/nat_le_plus.ma". include "ground/arith/nat_lt.ma". -(* NON-NEGATIVE INTEGERS ****************************************************) +(* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************) -(* Basic constructions with plus ********************************************) +(* Constructions with nplus *************************************************) (*** monotonic_lt_plus_l *) lemma nlt_plus_bi_dx (m) (n1) (n2): n1 < n2 → n1 + m < n2 + m. @@ -35,7 +35,7 @@ qed. lemma lt_plus_Sn_r: ∀a,x,n. a < a + x + ↑n. /2 width=1/ qed-. -(* Basic inversions with plus ***********************************************) +(* Inversions with nplus ****************************************************) (*** lt_plus_to_lt_l *) lemma nlt_inv_plus_bi_dx (m) (n1) (n2): n1 + m < n2 + m → n1 < n2. diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma index 16e5bb239..2b3502259 100644 --- a/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_lt_pred.ma @@ -15,14 +15,14 @@ include "ground/arith/nat_pred_succ.ma". include "ground/arith/nat_lt.ma". -(* NON-NEGATIVE INTEGERS ****************************************************) +(* STRICT ORDER FOR NON-NEGATIVE INTEGERS ***********************************) -(* Basic constructions with pred ********************************************) +(* Constructions with npred *************************************************) lemma nlt_zero_sn (m): m = ↑↓m → 𝟎 < m. // qed. -(* Basic inversions with pred ***********************************************) +(* Inversions with npred ****************************************************) (*** S_pred *) lemma nlt_inv_zero_sn (m): 𝟎 < m → m = ↑↓m. diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_minus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_minus.ma new file mode 100644 index 000000000..8129ad55c --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_minus.ma @@ -0,0 +1,71 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground/arith/nat_succ_iter.ma". +include "ground/arith/nat_pred_succ.ma". + +(* SUBTRACTION FOR NON-NEGATIVE INTEGERS ************************************) + +(*** minus *) +definition nminus: nat → nat → nat ≝ + λm,n. npred^n m. + +interpretation + "minus (positive integers" + 'minus m n = (nminus m n). + +(* Basic rewrites ***********************************************************) + +(*** minus_n_O *) +lemma nminus_zero_dx (m): m = m - 𝟎. +// qed. + +lemma nminus_pred_sn (m) (n): ↓(m - n) = ↓m - n. +#m #n @(niter_appl … npred) +qed. + +(*** eq_minus_S_pred *) +lemma nminus_succ_dx (m) (n): ↓(m - n) = m - ↑n. +#m #n @(niter_succ … npred) +qed. + +(*** minus_O_n *) +lemma nminus_zero_sn (n): 𝟎 = 𝟎 - n. +#n elim n -n // +qed. + +(*** minus_S_S *) +lemma nminus_succ_bi (m) (n): m - n = ↑m - ↑n. +#m #n elim n -n // +qed. + +(* Advanced rewrites ********************************************************) + +lemma nminus_succ_dx_pred_sn (m) (n): ↓m - n = m - ↑n. +// qed-. + +(*** minus_n_n *) +lemma nminus_refl (m): 𝟎 = m - m. +#m elim m -m // +qed. + +(*** minus_Sn_n *) +lemma nminus_succ_sn_refl (m): ninj (𝟏) = ↑m - m. +#m elim m -m // +qed. + +(*** minus_minus_comm *) +lemma nminus_minus_comm (o) (m) (n): o - m - n = o - n - m. +#o #m #n elim n -n // +qed-. diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma new file mode 100644 index 000000000..47e88de49 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_minus_plus.ma @@ -0,0 +1,84 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "ground/arith/nat_plus.ma". +include "ground/arith/nat_minus.ma". + +(* SUBTRACTION FOR NON-NEGATIVE INTEGERS ************************************) + +(* Rewrites with nplus ******************************************************) + +(*** minus_plus_m_m *) +lemma nminus_plus_sn_refl_sn (m) (n): m = m + n - n. +#m #n elim n -n // +#n #IH nplus_succ_shift #Ho + elim (IH … Ho) -IH -Ho * #_ #H + elim (eq_inv_nzero_succ … H) + ] +] +qed-. + +(*** discr_minus_x_xy *) +lemma eq_inv_nminus_refl_sn (m) (n): m = m - n → ∨∨ 𝟎 = m | 𝟎 = n. +#m #n #Hmn +elim (eq_inv_plus_nminus_refl_sn … (𝟎) Hmn) -Hmn * #H1 #H2 +/2 width=1 by or_introl, or_intror/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma index e7702b80d..75349d2da 100644 --- a/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma +++ b/matita/matita/contribs/lambdadelta/ground/arith/nat_plus.ma @@ -14,7 +14,7 @@ include "ground/arith/nat_succ_iter.ma". -(* NON-NEGATIVE INTEGERS ****************************************************) +(* ADDITION FOR NON-NEGATIVE INTEGERS ***************************************) (*** plus *) definition nplus: nat → nat → nat ≝ @@ -33,7 +33,7 @@ lemma nplus_zero_dx (m): m = m + 𝟎. lemma nplus_one_dx (n): ↑n = n + 𝟏. // qed. -(* Semigroup properties *****************************************************) +(* Advanved rewrites (semigroup properties) *********************************) (*** plus_n_Sm *) lemma nplus_succ_dx (m) (n): ↑(m+n) = m + ↑n. @@ -54,15 +54,20 @@ lemma nplus_comm: commutative … nplus. #m @(nat_ind … m) -m // qed-. -lemma nplus_one_sn (n): ↑n = 𝟏 + n. -#n