X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matitaB%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FBasic-1%2Faplus%2Fprops.ma;fp=matitaB%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FBasic-1%2Faplus%2Fprops.ma;h=0000000000000000000000000000000000000000;hb=88a68a9c334646bc17314d5327cd3b790202acd6;hp=94bb9a069a40f695aa638ce0643baaf992a94e02;hpb=4904accd80118cb8126e308ae098d87f8651c9f4;p=helm.git diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma deleted file mode 100644 index 94bb9a069..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/aplus/props.ma +++ /dev/null @@ -1,282 +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 *) -(* *) -(**************************************************************************) - -(* This file was automatically generated: do not edit *********************) - -include "Basic-1/aplus/defs.ma". - -include "Basic-1/next_plus/props.ma". - -theorem aplus_reg_r: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall -(h2: nat).((eq A (aplus g a1 h1) (aplus g a2 h2)) \to (\forall (h: nat).(eq A -(aplus g a1 (plus h h1)) (aplus g a2 (plus h h2))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 -(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))). -(* COMMENTS -Initial nodes: 143 -END *) - -theorem aplus_assoc: - \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A -(aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2)))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: -nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n -h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: -nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus -g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A -(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) -(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g -(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O -n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda -(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g -(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) -(plus_n_Sm n n0)))) h2)))) h1))). -(* COMMENTS -Initial nodes: 361 -END *) - -theorem aplus_asucc: - \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) -h) (asucc g (aplus g a h))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a -(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) -(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) -(aplus_assoc g a (S O) h)))). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem aplus_sort_O_S_simpl: - \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O -n) (S k)) (aplus g (ASort O (next g n)) k)))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc -g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) -(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) -k)) (aplus_asucc g k (ASort O n))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem aplus_sort_S_S_simpl: - \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A -(aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k))))) -\def - \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind -A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g -(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g -(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))). -(* COMMENTS -Initial nodes: 97 -END *) - -theorem aplus_asort_O_simpl: - \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O -n) h) (ASort O (next_plus g n h))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda -(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall -(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 -n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) -(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat -(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next -g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) -(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n -(ASort O n0)))))) h)). -(* COMMENTS -Initial nodes: 229 -END *) - -theorem aplus_asort_le_simpl: - \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h -k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n)))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: -nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort -(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O -k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) -(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: -nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A -(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda -(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat -O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) -h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S -x))).(\lambda (_: (le h0 x)).(let H3 \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) H1) in (False_ind (eq A (asucc g (aplus -g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0)))) -(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda -(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort -(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n -n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g -h0 (ASort (S n) n0))))))) k)))) h)). -(* COMMENTS -Initial nodes: 484 -END *) - -theorem aplus_asort_simpl: - \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A -(aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k))))))) -\def - \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: -nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus -g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) -(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) -(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) -(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus -h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a -(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O -(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k -h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A -(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) -(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h -(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k) -(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus -h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h -(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort -(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n -(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) -n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) -(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) -(aplus_asort_le_simpl g h k n H))))))). -(* COMMENTS -Initial nodes: 587 -END *) - -theorem aplus_ahead_simpl: - \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h)))))) -\def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: -A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 -n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) -(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: -A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda -(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g -(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) -(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n -a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) -h)). -(* COMMENTS -Initial nodes: 239 -END *) - -theorem aplus_asucc_false: - \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) -h) a) \to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: -nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A -(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) -\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) -(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 -\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 -(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat -(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1) -\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1: -nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0) -\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0) -(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1: -nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in -(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g -n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n -(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda -(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort -O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to -P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let -H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort -(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) -(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow -n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat) -\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow -(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in -minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S -n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) -\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def -(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0 -n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus -g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 -h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 -n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: -A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to -(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: -nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: -Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc -g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g -(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 -(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) -in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n: -nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc -g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow -a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 -P)))))))))) a)). -(* COMMENTS -Initial nodes: 977 -END *) - -theorem aplus_inj: - \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A -(aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2))))) -\def - \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A -(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: -(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: -A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: -(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g -a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g -n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq -nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq -A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: -A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A -(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) -n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) -(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a -n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: -(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def -(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus -g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def -(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g -a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat -nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)). -(* COMMENTS -Initial nodes: 599 -END *) -