X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground%2Fynat%2Fynat_lt.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground%2Fynat%2Fynat_lt.ma;h=0000000000000000000000000000000000000000;hb=888840f6b3a71d3d686b53b702d362ab90ab0038;hp=0ab7bee453552a968341a22389b8e83d35c30a1a;hpb=19b0a814861157ba05f23877d5cd94059f52c2e8;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/ground/ynat/ynat_lt.ma b/matita/matita/contribs/lambdadelta/ground/ynat/ynat_lt.ma deleted file mode 100644 index 0ab7bee45..000000000 --- a/matita/matita/contribs/lambdadelta/ground/ynat/ynat_lt.ma +++ /dev/null @@ -1,271 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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/ynat/ynat_le.ma". - -(* NATURAL NUMBERS WITH INFINITY ********************************************) - -(* strict order relation *) -inductive ylt: relation ynat ≝ -| ylt_inj: ∀m,n. m < n → ylt m n -| ylt_Y : ∀m:nat. ylt m (∞) -. - -interpretation "ynat 'less than'" 'lt x y = (ylt x y). - -(* Basic forward lemmas *****************************************************) - -lemma ylt_fwd_gen: ∀x,y. x < y → ∃m. x = yinj m. -#x #y * -x -y /2 width=2 by ex_intro/ -qed-. - -lemma ylt_fwd_lt_O1: ∀x,y:ynat. x < y → 0 < y. -#x #y #H elim H -x -y /3 width=2 by ylt_inj, ltn_to_ltO/ -qed-. - -(* Basic inversion lemmas ***************************************************) - -fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n → - ∃∃m. m < n & x = yinj m. -#x #y * -x -y -[ #x #y #Hxy #n #Hy elim (le_inv_S1 … Hxy) -Hxy - #m #Hm #H destruct /3 width=3 by le_S_S, ex2_intro/ -| #x #n #Hy destruct -] -qed-. - -lemma ylt_inv_inj2: ∀x,n. x < yinj n → - ∃∃m. m < n & x = yinj m. -/2 width=3 by ylt_inv_inj2_aux/ qed-. - -lemma ylt_inv_inj: ∀m,n. yinj m < yinj n → m < n. -#m #n #H elim (ylt_inv_inj2 … H) -H -#x #Hx #H destruct // -qed-. - -lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥. -#n #H elim (ylt_fwd_gen … H) -H -#y #H destruct -qed-. - -lemma ylt_inv_Y2: ∀x:ynat. x < ∞ → ∃n. x = yinj n. -* /2 width=2 by ex_intro/ -#H elim (ylt_inv_Y1 … H) -qed-. - -lemma ylt_inv_O1: ∀n:ynat. 0 < n → ↑↓n = n. -* // #n #H lapply (ylt_inv_inj … H) -H normalize -/3 width=1 by S_pred, eq_f/ -qed-. - -(* Inversion lemmas on successor ********************************************) - -fact ylt_inv_succ1_aux: ∀x,y:ynat. x < y → ∀m. x = ↑m → m < ↓y ∧ ↑↓y = y. -#x #y * -x -y -[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H - #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy - #m #Hnm #H destruct /3 width=1 by ylt_inj, conj/ -| #x #y #H elim (ysucc_inv_inj_sn … H) -H - #m #H #_ destruct /2 width=1 by ylt_Y, conj/ -] -qed-. - -lemma ylt_inv_succ1: ∀m,y:ynat. ↑m < y → m < ↓y ∧ ↑↓y = y. -/2 width=3 by ylt_inv_succ1_aux/ qed-. - -lemma ylt_inv_succ: ∀m,n. ↑m < ↑n → m < n. -#m #n #H elim (ylt_inv_succ1 … H) -H // -qed-. - -(* Forward lemmas on successor **********************************************) - -fact ylt_fwd_succ2_aux: ∀x,y. x < y → ∀n. y = ↑n → x ≤ n. -#x #y * -x -y -[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H - #n #H1 #H2 destruct /3 width=1 by yle_inj, le_S_S_to_le/ -| #x #n #H lapply (ysucc_inv_Y_sn … H) -H // -] -qed-. - -lemma ylt_fwd_succ2: ∀m,n. m < ↑n → m ≤ n. -/2 width=3 by ylt_fwd_succ2_aux/ qed-. - -(* inversion and forward lemmas on order ************************************) - -lemma ylt_fwd_le_succ1: ∀m,n. m < n → ↑m ≤ n. -#m #n * -m -n /2 width=1 by yle_inj/ -qed-. - -lemma ylt_fwd_le_pred2: ∀x,y:ynat. x < y → x ≤ ↓y. -#x #y #H elim H -x -y /3 width=1 by yle_inj, monotonic_pred/ -qed-. - -lemma ylt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n. -#m #n * -m -n /3 width=1 by lt_to_le, yle_inj/ -qed-. - -lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥. -#m #n * -m -n -[ #m #n #Hmn #H lapply (yle_inv_inj … H) -H - #H elim (lt_refl_false n) /2 width=3 by le_to_lt_to_lt/ -| #m #H lapply (yle_inv_Y1 … H) -H - #H destruct -] -qed-. - -lemma ylt_inv_le: ∀x,y. x < y → x < ∞ ∧ ↑x ≤ y. -#x #y #H elim H -x -y /3 width=1 by yle_inj, conj/ -qed-. - -(* Basic properties *********************************************************) - -lemma ylt_O1: ∀x:ynat. ↑↓x = x → 0 < x. -* // * /2 width=1 by ylt_inj/ normalize -#H destruct -qed. - -lemma yle_inv_succ_sn_lt (x:ynat) (y:ynat): - ↑x ≤ y → ∧∧ x ≤ ↓y & 0 < y. -#x #y #H elim (yle_inv_succ1 … H) -H /3 width=2 by ylt_O1, conj/ -qed-. - -(* Properties on predecessor ************************************************) - -lemma ylt_pred: ∀m,n:ynat. m < n → 0 < m → ↓m < ↓n. -#m #n * -m -n -/4 width=1 by ylt_inv_inj, ylt_inj, monotonic_lt_pred/ -qed. - -(* Properties on successor **************************************************) - -lemma ylt_O_succ: ∀x:ynat. 0 < ↑x. -* /2 width=1 by ylt_inj/ -qed. - -lemma ylt_succ: ∀m,n. m < n → ↑m < ↑n. -#m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/ -qed. - -lemma ylt_succ_Y: ∀x. x < ∞ → ↑x < ∞. -* /2 width=1 by/ qed. - -lemma yle_succ1_inj: ∀x. ∀y:ynat. ↑yinj x ≤ y → x < y. -#x * /3 width=1 by yle_inv_inj, ylt_inj/ -qed. - -lemma ylt_succ2_refl: ∀x,y:ynat. x < y → x < ↑x. -#x #y #H elim (ylt_fwd_gen … H) -y /2 width=1 by ylt_inj/ -qed. - -(* Properties on order ******************************************************) - -lemma yle_split_eq: ∀m,n:ynat. m ≤ n → m < n ∨ m = n. -#m #n * -m -n -[ #m #n #Hmn elim (le_to_or_lt_eq … Hmn) -Hmn - /3 width=1 by or_introl, ylt_inj/ -| * /2 width=1 by or_introl, ylt_Y/ -] -qed-. - -lemma ylt_split: ∀m,n:ynat. m < n ∨ n ≤ m. -#m #n elim (yle_split m n) /2 width=1 by or_intror/ -#H elim (yle_split_eq … H) -H /2 width=1 by or_introl, or_intror/ -qed-. - -lemma ylt_split_eq: ∀m,n:ynat. ∨∨ m < n | n = m | n < m. -#m #n elim (ylt_split m n) /2 width=1 by or3_intro0/ -#H elim (yle_split_eq … H) -H /2 width=1 by or3_intro1, or3_intro2/ -qed-. - -lemma ylt_yle_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y ≤ z → x < y → x < z. -#x #y #z * -y -z -[ #y #z #Hyz #H elim (ylt_inv_inj2 … H) -H - #m #Hm #H destruct /3 width=3 by ylt_inj, lt_to_le_to_lt/ -| #y * // -] -qed-. - -lemma yle_ylt_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y < z → x ≤ y → x < z. -#x #y #z * -y -z -[ #y #z #Hyz #H elim (yle_inv_inj2 … H) -H - #m #Hm #H destruct /3 width=3 by ylt_inj, le_to_lt_to_lt/ -| #y #H elim (yle_inv_inj2 … H) -H // -] -qed-. - -lemma le_ylt_trans (x) (y) (z): x ≤ y → yinj y < z → yinj x < z. -/3 width=3 by yle_ylt_trans, yle_inj/ -qed-. - -lemma yle_inv_succ1_lt: ∀x,y:ynat. ↑x ≤ y → 0 < y ∧ x ≤ ↓y. -#x #y #H elim (yle_inv_succ1 … H) -H /3 width=1 by ylt_O1, conj/ -qed-. - -lemma yle_lt: ∀x,y. x < ∞ → ↑x ≤ y → x < y. -#x * // #y #H elim (ylt_inv_Y2 … H) -H #n #H destruct -/3 width=1 by ylt_inj, yle_inv_inj/ -qed-. - -(* Main properties **********************************************************) - -theorem ylt_trans: Transitive … ylt. -#x #y * -x -y -[ #x #y #Hxy * // - #z #H lapply (ylt_inv_inj … H) -H - /3 width=3 by transitive_lt, ylt_inj/ (**) (* full auto too slow *) -| #x #z #H elim (ylt_yle_false … H) // -] -qed-. - -lemma lt_ylt_trans (x) (y) (z): x < y → yinj y < z → yinj x < z. -/3 width=3 by ylt_trans, ylt_inj/ -qed-. - -(* Elimination principles ***************************************************) - -fact ynat_ind_lt_le_aux: ∀R:predicate ynat. - (∀y. (∀x. x < y → R x) → R y) → - ∀y:nat. ∀x. x ≤ y → R x. -#R #IH #y elim y -y -[ #x #H >(yle_inv_O2 … H) -x - @IH -IH #x #H elim (ylt_yle_false … H) -H // -| /5 width=3 by ylt_yle_trans, ylt_fwd_succ2/ -] -qed-. - -fact ynat_ind_lt_aux: ∀R:predicate ynat. - (∀y. (∀x. x < y → R x) → R y) → - ∀y:nat. R y. -/4 width=2 by ynat_ind_lt_le_aux/ qed-. - -lemma ynat_ind_lt: ∀R:predicate ynat. - (∀y. (∀x. x < y → R x) → R y) → - ∀y. R y. -#R #IH * /4 width=1 by ynat_ind_lt_aux/ -@IH #x #H elim (ylt_inv_Y2 … H) -H -#n #H destruct /4 width=1 by ynat_ind_lt_aux/ -qed-. - -fact ynat_f_ind_aux: ∀A. ∀f:A→ynat. ∀R:predicate A. - (∀x. (∀a. f a < x → R a) → ∀a. f a = x → R a) → - ∀x,a. f a = x → R a. -#A #f #R #IH #x @(ynat_ind_lt … x) -x -/3 width=3 by/ -qed-. - -lemma ynat_f_ind: ∀A. ∀f:A→ynat. ∀R:predicate A. - (∀x. (∀a. f a < x → R a) → ∀a. f a = x → R a) → ∀a. R a. -#A #f #R #IH #a -@(ynat_f_ind_aux … IH) -IH [2: // | skip ] -qed-.