X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matitaB%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FBasic-1%2Fleq%2Fprops.ma;fp=matitaB%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FBasic-1%2Fleq%2Fprops.ma;h=0000000000000000000000000000000000000000;hb=88a68a9c334646bc17314d5327cd3b790202acd6;hp=b83fc503e4fe5d920693972058d1c18916d66055;hpb=4904accd80118cb8126e308ae098d87f8651c9f4;p=helm.git diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma deleted file mode 100644 index b83fc503e..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/leq/props.ma +++ /dev/null @@ -1,233 +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/leq/fwd.ma". - -include "Basic-1/aplus/props.ma". - -theorem ahead_inj_snd: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall -(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda -(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def -(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A -(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda -(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) -(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda -(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead -a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in -A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) -\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) -in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a: -A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a: -A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))). -(* COMMENTS -Initial nodes: 259 -END *) - -theorem leq_refl: - \forall (g: G).(\forall (a: A).(leq g a a)) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) -(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A -(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 -a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 -H0))))) a)). -(* COMMENTS -Initial nodes: 87 -END *) - -theorem leq_eq: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 -a2)))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 -a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))). -(* COMMENTS -Initial nodes: 39 -END *) - -theorem leq_sym: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g -a2 a1)))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: -nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) -k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: -(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 -H1 a6 a5 H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 173 -END *) - -theorem leq_trans: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall -(a3: A).((leq g a2 a3) \to (leq g a1 a3)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 -a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g -(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 -\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 -(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus -g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def -(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 -x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 -n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g -(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r -nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) -(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) -x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus -g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 -H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) -(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus -(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g -(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 -x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g -(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: -A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: -A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) -\to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) -a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in -(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: -(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 -(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead -a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) -H5))))))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 869 -END *) - -theorem leq_ahead_false_1: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) -\to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n -n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) -a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: -A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O -n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: -A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow -True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 -n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) -a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H: -((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead -a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq -g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) -(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def -(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0 -H10 P))))) H6))))))) H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 797 -END *) - -theorem leq_ahead_false_2: - \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) -\to (\forall (P: Prop).P)))) -\def - \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: -A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort -n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O -n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 -x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) -(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 -n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) -n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_: -((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead -a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq -g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4: -A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let -H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9 -P))))) H6))))))) H2)))))))))) a2)). -(* COMMENTS -Initial nodes: 797 -END *) -