X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fex0%2Fprops.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fex0%2Fprops.ma;h=148879216ca30fefbabaee3675b8da399f34e3ee;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hp=0000000000000000000000000000000000000000;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1;p=helm.git diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma new file mode 100644 index 000000000..148879216 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/ex0/props.ma @@ -0,0 +1,192 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "LambdaDelta-1/ex0/defs.ma". + +include "LambdaDelta-1/leq/defs.ma". + +include "LambdaDelta-1/aplus/props.ma". + +theorem aplus_gz_le: + \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A +(aplus gz (ASort h n) k) (ASort O (plus (minus k h) n)))))) +\def + \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0: +nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h) +n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y +\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n) +(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0: +nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A +(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h: +nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A +(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O +\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n: +nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n)) +k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O +(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0 +n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n))) +(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0: +nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S +(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0)) +(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz +(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n: +nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz +(aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S +k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda +(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz +(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) +k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: +A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) +k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S +n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k). + +theorem aplus_gz_ge: + \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A +(aplus gz (ASort h n) k) (ASort (minus h k) n))))) +\def + \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h: +nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0) +n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0: +nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O) +(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 +h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda +(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz +(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le +(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0: +nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) +(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0 +x)).(let H2 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) +I (S x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O +n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0) +n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) +n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in +(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus +gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) +k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus +gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc +gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)). + +theorem next_plus_gz: + \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) +\def + \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat +(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0: +nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat +S (next_plus gz n n0) (plus n0 n) H))) h)). + +theorem leqz_leq: + \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) +\def + \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz +(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda +(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda +(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k +h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2 +(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def +(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort +h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1 +(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) +(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2) +(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def +(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with +[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat) +on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O | +(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow +(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2 +k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A +return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) +\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in +(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n: +nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n: +nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal +nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1)) +(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2 +k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a +(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 +(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort +h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus +(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat +(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2) +n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind +A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return +(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]) +| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in +(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1 +k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k +h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A +a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1)) +(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) +k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort +(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in +(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k) +n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A +(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k))) +(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k))) +n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort +n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow +False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1 +n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A +(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) +k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 +\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O +(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le +k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A +return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) +\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: +nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in +plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus +(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in +(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3: +A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: +A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 +a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). + +theorem leq_leqz: + \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) +\def + \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind +(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda +(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus +h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A +(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1))) +(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A +(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2))) +(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus +(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus +h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2)) +(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n: +nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus +h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq +A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz +n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1 +h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n: +nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus +h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal +nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2 +n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2) +(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2) +(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2) +(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus +(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2)) +(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1 +h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0: +A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0 +a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda +(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). +