X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2A%2Fynat%2Fynat_lt.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2A%2Fynat%2Fynat_lt.ma;h=850fae157577dec01cf733a534c3274bb8686fa6;hb=d2545ffd201b1aa49887313791386add78fa8603;hp=0000000000000000000000000000000000000000;hpb=57ae1762497a5f3ea75740e2908e04adb8642cc2;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma new file mode 100644 index 000000000..850fae157 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_lt.ma @@ -0,0 +1,182 @@ +(**************************************************************************) +(* ___ *) +(* ||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_2A/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_le_succ: ∀x,y. x < y → ⫯x ≤ y. +#x #y * -x -y /2 width=1 by yle_inj/ +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_O1: ∀n. 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. 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. ⫯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 yle **************************************) + +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: ∀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-. + +(* Basic properties *********************************************************) + +lemma ylt_O: ∀x. ⫯⫰(yinj x) = yinj x → 0 < x. +* /2 width=1 by/ normalize +#H destruct +qed. + +(* Properties on predecessor ************************************************) + +lemma ylt_pred: ∀m,n. 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: ∀n. 0 < ⫯n. +* /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. + +(* Properties on order ******************************************************) + +lemma yle_split_eq: ∀m:ynat. ∀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_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-. + +(* 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-.