X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matitaB%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FBasic-1%2Fsc3%2Fprops.ma;fp=matitaB%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FBasic-1%2Fsc3%2Fprops.ma;h=0000000000000000000000000000000000000000;hb=88a68a9c334646bc17314d5327cd3b790202acd6;hp=e1d90925157134ce6f8f8b5edfe7a210d3415148;hpb=4904accd80118cb8126e308ae098d87f8651c9f4;p=helm.git diff --git a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma b/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma deleted file mode 100644 index e1d909251..000000000 --- a/matitaB/matita/contribs/LAMBDA-TYPES/Basic-1/sc3/props.ma +++ /dev/null @@ -1,728 +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/sc3/defs.ma". - -include "Basic-1/sn3/lift1.ma". - -include "Basic-1/nf2/lift1.ma". - -include "Basic-1/csuba/arity.ma". - -include "Basic-1/arity/lift1.ma". - -include "Basic-1/arity/aprem.ma". - -include "Basic-1/llt/props.ma". - -include "Basic-1/drop1/getl.ma". - -include "Basic-1/drop1/props.ma". - -include "Basic-1/lift1/props.ma". - -theorem sc3_arity_gen: - \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c -t) \to (arity g c t a))))) -\def - \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(A_ind -(\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c -t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (arity -g c t (ASort n n0)) (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: -(sn3 c t)).H1)) H0))))) (\lambda (a0: A).(\lambda (_: (((sc3 g a0 c t) \to -(arity g c t a0)))).(\lambda (a1: A).(\lambda (_: (((sc3 g a1 c t) \to (arity -g c t a1)))).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H1 in -(land_ind (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat -Appl) w (lift1 is t)))))))) (arity g c t (AHead a0 a1)) (\lambda (H3: (arity -g c t (AHead a0 a1))).(\lambda (_: ((\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat -Appl) w (lift1 is t)))))))))).H3)) H2))))))) a)))). -(* COMMENTS -Initial nodes: 369 -END *) - -theorem sc3_repl: - \forall (g: G).(\forall (a1: A).(\forall (c: C).(\forall (t: T).((sc3 g a1 c -t) \to (\forall (a2: A).((leq g a1 a2) \to (sc3 g a2 c t))))))) -\def - \lambda (g: G).(\lambda (a1: A).(llt_wf_ind (\lambda (a: A).(\forall (c: -C).(\forall (t: T).((sc3 g a c t) \to (\forall (a2: A).((leq g a a2) \to (sc3 -g a2 c t))))))) (\lambda (a2: A).(A_ind (\lambda (a: A).(((\forall (a3: -A).((llt a3 a) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 c t) \to -(\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t))))))))) \to (\forall (c: -C).(\forall (t: T).((sc3 g a c t) \to (\forall (a3: A).((leq g a a3) \to (sc3 -g a3 c t)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (_: ((\forall -(a3: A).((llt a3 (ASort n n0)) \to (\forall (c: C).(\forall (t: T).((sc3 g a3 -c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c t)))))))))).(\lambda -(c: C).(\lambda (t: T).(\lambda (H0: (land (arity g c t (ASort n n0)) (sn3 c -t))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort n n0) a3)).(let H2 \def H0 -in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sc3 g a3 c t) (\lambda -(H3: (arity g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let H_y \def -(arity_repl g c t (ASort n n0) H3 a3 H1) in (let H_x \def (leq_gen_sort1 g n -n0 a3 H1) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort n n0) k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a3 (ASort h2 n2))))) (sc3 g a3 c t) (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A (aplus g (ASort -n n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A a3 (ASort x1 -x0))).(let H8 \def (f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H7) in -(let H9 \def (eq_ind A a3 (\lambda (a: A).(arity g c t a)) H_y (ASort x1 x0) -H8) in (eq_ind_r A (ASort x1 x0) (\lambda (a: A).(sc3 g a c t)) (conj (arity -g c t (ASort x1 x0)) (sn3 c t) H9 H4) a3 H8)))))))) H5)))))) H2)))))))))) -(\lambda (a: A).(\lambda (_: ((((\forall (a3: A).((llt a3 a) \to (\forall (c: -C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to -(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a c t) \to -(\forall (a3: A).((leq g a a3) \to (sc3 g a3 c t))))))))).(\lambda (a0: -A).(\lambda (H0: ((((\forall (a3: A).((llt a3 a0) \to (\forall (c: -C).(\forall (t: T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to -(sc3 g a4 c t))))))))) \to (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) -\to (\forall (a3: A).((leq g a0 a3) \to (sc3 g a3 c t))))))))).(\lambda (H1: -((\forall (a3: A).((llt a3 (AHead a a0)) \to (\forall (c: C).(\forall (t: -T).((sc3 g a3 c t) \to (\forall (a4: A).((leq g a3 a4) \to (sc3 g a4 c -t)))))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H2: (land (arity g c t -(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is -t)))))))))).(\lambda (a3: A).(\lambda (H3: (leq g (AHead a a0) a3)).(let H4 -\def H2 in (land_ind (arity g c t (AHead a a0)) (\forall (d: C).(\forall (w: -T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w (lift1 is t)))))))) (sc3 g a3 c t) (\lambda (H5: (arity -g c t (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat -Appl) w (lift1 is t)))))))))).(let H_x \def (leq_gen_head1 g a a0 a3 H3) in -(let H7 \def H_x in (ex3_2_ind A A (\lambda (a4: A).(\lambda (_: A).(leq g a -a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a0 a5))) (\lambda (a4: -A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))) (sc3 g a3 c t) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H8: (leq g a x0)).(\lambda (H9: (leq g a0 -x1)).(\lambda (H10: (eq A a3 (AHead x0 x1))).(let H11 \def (f_equal A A -(\lambda (e: A).e) a3 (AHead x0 x1) H10) in (eq_ind_r A (AHead x0 x1) -(\lambda (a4: A).(sc3 g a4 c t)) (conj (arity g c t (AHead x0 x1)) (\forall -(d: C).(\forall (w: T).((sc3 g x0 d w) \to (\forall (is: PList).((drop1 is d -c) \to (sc3 g x1 d (THead (Flat Appl) w (lift1 is t)))))))) (arity_repl g c t -(AHead a a0) H5 (AHead x0 x1) (leq_head g a x0 H8 a0 x1 H9)) (\lambda (d: -C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is: -PList).(\lambda (H13: (drop1 is d c)).(H0 (\lambda (a4: A).(\lambda (H14: -(llt a4 a0)).(\lambda (c0: C).(\lambda (t0: T).(\lambda (H15: (sc3 g a4 c0 -t0)).(\lambda (a5: A).(\lambda (H16: (leq g a4 a5)).(H1 a4 (llt_trans a4 a0 -(AHead a a0) H14 (llt_head_dx a a0)) c0 t0 H15 a5 H16)))))))) d (THead (Flat -Appl) w (lift1 is t)) (H6 d w (H1 x0 (llt_repl g a x0 H8 (AHead a a0) -(llt_head_sx a a0)) d w H12 a (leq_sym g a x0 H8)) is H13) x1 H9))))))) a3 -H11))))))) H7))))) H4)))))))))))) a2)) a1)). -(* COMMENTS -Initial nodes: 1359 -END *) - -theorem sc3_lift: - \forall (g: G).(\forall (a: A).(\forall (e: C).(\forall (t: T).((sc3 g a e -t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) -\to (sc3 g a c (lift h d t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (e: -C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d t)))))))))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (e: C).(\lambda (t: T).(\lambda -(H: (land (arity g e t (ASort n n0)) (sn3 e t))).(\lambda (c: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(let H1 \def H in -(land_ind (arity g e t (ASort n n0)) (sn3 e t) (land (arity g c (lift h d t) -(ASort n n0)) (sn3 c (lift h d t))) (\lambda (H2: (arity g e t (ASort n -n0))).(\lambda (H3: (sn3 e t)).(conj (arity g c (lift h d t) (ASort n n0)) -(sn3 c (lift h d t)) (arity_lift g e t (ASort n n0) H2 c h d H0) (sn3_lift e -t H3 c h d H0)))) H1))))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (e: -C).(\forall (t: T).((sc3 g a0 e t) \to (\forall (c: C).(\forall (h: -nat).(\forall (d: nat).((drop h d c e) \to (sc3 g a0 c (lift h d -t))))))))))).(\lambda (a1: A).(\lambda (_: ((\forall (e: C).(\forall (t: -T).((sc3 g a1 e t) \to (\forall (c: C).(\forall (h: nat).(\forall (d: -nat).((drop h d c e) \to (sc3 g a1 c (lift h d t))))))))))).(\lambda (e: -C).(\lambda (t: T).(\lambda (H1: (land (arity g e t (AHead a0 a1)) (\forall -(d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d -e) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t)))))))))).(\lambda (c: -C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d c e)).(let H3 -\def H1 in (land_ind (arity g e t (AHead a0 a1)) (\forall (d0: C).(\forall -(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) \to (sc3 g -a1 d0 (THead (Flat Appl) w (lift1 is t)))))))) (land (arity g c (lift h d t) -(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall -(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(lift h d t)))))))))) (\lambda (H4: (arity g e t (AHead a0 a1))).(\lambda -(H5: ((\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 e) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -t)))))))))).(conj (arity g c (lift h d t) (AHead a0 a1)) (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (lift h d t))))))))) -(arity_lift g e t (AHead a0 a1) H4 c h d H2) (\lambda (d0: C).(\lambda (w: -T).(\lambda (H6: (sc3 g a0 d0 w)).(\lambda (is: PList).(\lambda (H7: (drop1 -is d0 c)).(let H_y \def (H5 d0 w H6 (PConsTail is h d)) in (eq_ind T (lift1 -(PConsTail is h d) t) (\lambda (t0: T).(sc3 g a1 d0 (THead (Flat Appl) w -t0))) (H_y (drop1_cons_tail c e h d H2 is d0 H7)) (lift1 is (lift h d t)) -(lift1_cons_tail t h d is))))))))))) H3))))))))))))) a)). -(* COMMENTS -Initial nodes: 849 -END *) - -theorem sc3_lift1: - \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: -PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) -\to (sc3 g a c (lift1 hds t))))))))) -\def - \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: -PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g -a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: -C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c -e)).(let H_y \def (drop1_gen_pnil c e H0) in (eq_ind_r C e (\lambda (c0: -C).(sc3 g a c0 t)) H c H_y)))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 -g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: -C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n -n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H1) in (let H2 \def H_x -in (ex2_ind C (\lambda (c2: C).(drop n n0 c c2)) (\lambda (c2: C).(drop1 p c2 -e)) (sc3 g a c (lift n n0 (lift1 p t))) (\lambda (x: C).(\lambda (H3: (drop n -n0 c x)).(\lambda (H4: (drop1 p x e)).(sc3_lift g a x (lift1 p t) (H x t H0 -H4) c n n0 H3)))) H2))))))))))) hds)))). -(* COMMENTS -Initial nodes: 289 -END *) - -theorem sc3_abbr: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: -nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads -(Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to -(sc3 g a c (THeads (Flat Appl) vs (TLRef i))))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: -TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: -C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c -(CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs (TLRef -i))))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: -TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (v: T).(\lambda (c: -C).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs (lift (S i) O v)) -(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (lift (S i) O v))))).(\lambda -(H0: (getl i c (CHead d (Bind Abbr) v))).(let H1 \def H in (land_ind (arity g -c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0)) (sn3 c (THeads (Flat -Appl) vs (lift (S i) O v))) (land (arity g c (THeads (Flat Appl) vs (TLRef -i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))) (\lambda (H2: -(arity g c (THeads (Flat Appl) vs (lift (S i) O v)) (ASort n n0))).(\lambda -(H3: (sn3 c (THeads (Flat Appl) vs (lift (S i) O v)))).(conj (arity g c -(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs -(TLRef i))) (arity_appls_abbr g c d v i H0 vs (ASort n n0) H2) -(sn3_appls_abbr c d v i H0 vs H3)))) H1))))))))))) (\lambda (a0: A).(\lambda -(_: ((\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: -T).(\forall (c: C).((sc3 g a0 c (THeads (Flat Appl) vs (lift (S i) O v))) \to -((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a0 c (THeads (Flat Appl) vs -(TLRef i)))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: -TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: -C).((sc3 g a1 c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c -(CHead d (Bind Abbr) v)) \to (sc3 g a1 c (THeads (Flat Appl) vs (TLRef -i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda -(v: T).(\lambda (c: C).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs -(lift (S i) O v)) (AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 -d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat -Appl) w (lift1 is (THeads (Flat Appl) vs (lift (S i) O v)))))))))))).(\lambda -(H2: (getl i c (CHead d (Bind Abbr) v))).(let H3 \def H1 in (land_ind (arity -g c (THeads (Flat Appl) vs (lift (S i) O v)) (AHead a0 a1)) (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift -(S i) O v)))))))))) (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead -a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (TLRef i))))))))))) (\lambda (H4: (arity g c (THeads -(Flat Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) -\to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (lift -(S i) O v)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (TLRef i)) -(AHead a0 a1)) (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall -(is: PList).((drop1 is d0 c) \to (sc3 g a1 d0 (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (TLRef i)))))))))) (arity_appls_abbr g c d v i H2 vs -(AHead a0 a1) H4) (\lambda (d0: C).(\lambda (w: T).(\lambda (H6: (sc3 g a0 d0 -w)).(\lambda (is: PList).(\lambda (H7: (drop1 is d0 c)).(let H_x \def -(drop1_getl_trans is c d0 H7 Abbr d v i H2) in (let H8 \def H_x in (ex2_ind C -(\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is -i) d0 (CHead e2 (Bind Abbr) (lift1 (ptrans is i) v)))) (sc3 g a1 d0 (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (x: -C).(\lambda (_: (drop1 (ptrans is i) x d)).(\lambda (H10: (getl (trans is i) -d0 (CHead x (Bind Abbr) (lift1 (ptrans is i) v)))).(let H_y \def (H0 (TCons w -(lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is -(TLRef i))) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w t))) (eq_ind_r -T (TLRef (trans is i)) (\lambda (t: T).(sc3 g a1 d0 (THead (Flat Appl) w -(THeads (Flat Appl) (lifts1 is vs) t)))) (H_y (trans is i) x (lift1 (ptrans -is i) v) d0 (eq_ind T (lift1 is (lift (S i) O v)) (\lambda (t: T).(sc3 g a1 -d0 (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t)))) (eq_ind T -(lift1 is (THeads (Flat Appl) vs (lift (S i) O v))) (\lambda (t: T).(sc3 g a1 -d0 (THead (Flat Appl) w t))) (H5 d0 w H6 is H7) (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (lift (S i) O v))) (lifts1_flat Appl is (lift (S i) O v) -vs)) (lift (S (trans is i)) O (lift1 (ptrans is i) v)) (lift1_free is i v)) -H10) (lift1 is (TLRef i)) (lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs -(TLRef i))) (lifts1_flat Appl is (TLRef i) vs)))))) H8))))))))))) -H3))))))))))))) a)). -(* COMMENTS -Initial nodes: 1563 -END *) - -theorem sc3_cast: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall -(u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: -T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) -vs (THead (Flat Cast) u t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: -TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat -Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to -(sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: -T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | -(S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: -T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) -(sn3 c (THeads (Flat Appl) vs t)))).(nat_ind (\lambda (n1: nat).((sc3 g -(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow -(ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads -(Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land -(arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) -(sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) (\lambda (H1: -(sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: -(land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat -Appl) vs t)))).(let H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs -u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads -(Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat -Appl) vs u))).(let H6 \def H2 in (land_ind (arity g c (THeads (Flat Appl) vs -t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat -Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat -Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs -t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort -O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) -(arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t -H8)))) H6)))) H3)))) (\lambda (n1: nat).(\lambda (_: (((sc3 g (match n1 with -[O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c -(THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) -(ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Cast) u t)))))))).(\lambda (H1: (sc3 g (ASort n1 -n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads -(Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let -H3 \def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) -(sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) -(ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def -H2 in (land_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 -c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead -(Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead -(Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort -(S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g -c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c -(THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs -(ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))))) n -H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall -(c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to -(\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c -(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: -A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 -g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c -(THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead -(Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: -T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc -g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 -is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land -(arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall -(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 -d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 -\def H1 in (land_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g -a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 -is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 -g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u -t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 -(asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d -w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead -(Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 -in (land_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) -(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity -g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) -(AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is -(THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c -u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: -(sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y -\def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d -(THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 -is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) -(lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat -Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w -t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) -(lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat -Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w -H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl -is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) -(lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl -is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)). -(* COMMENTS -Initial nodes: 2625 -END *) - -theorem sc3_props__sc3_sn3_abst: - \forall (g: G).(\forall (a: A).(land (\forall (c: C).(\forall (t: T).((sc3 g -a c t) \to (sn3 c t)))) (\forall (vs: TList).(\forall (i: nat).(let t \def -(THeads (Flat Appl) vs (TLRef i)) in (\forall (c: C).((arity g c t a) \to -((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a c t)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(land (\forall (c: -C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall (vs: -TList).(\forall (i: nat).(let t \def (THeads (Flat Appl) vs (TLRef i)) in -(\forall (c: C).((arity g c t a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to -(sc3 g a0 c t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(conj (\forall -(c: C).(\forall (t: T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 -c t)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c -(THeads (Flat Appl) vs (TLRef i)) (ASort n n0)) \to ((nf2 c (TLRef i)) \to -((sns3 c vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n -n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i)))))))))) (\lambda (c: -C).(\lambda (t: T).(\lambda (H: (land (arity g c t (ASort n n0)) (sn3 c -t))).(let H0 \def H in (land_ind (arity g c t (ASort n n0)) (sn3 c t) (sn3 c -t) (\lambda (_: (arity g c t (ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) -H0))))) (\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H: -(arity g c (THeads (Flat Appl) vs (TLRef i)) (ASort n n0))).(\lambda (H0: -(nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(conj (arity g c (THeads (Flat -Appl) vs (TLRef i)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (TLRef i))) H -(sn3_appls_lref c i H0 vs H1))))))))))) (\lambda (a0: A).(\lambda (H: (land -(\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) (\forall -(vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads (Flat Appl) -vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c -(THeads (Flat Appl) vs (TLRef i))))))))))).(\lambda (a1: A).(\lambda (H0: -(land (\forall (c: C).(\forall (t: T).((sc3 g a1 c t) \to (sn3 c t)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads -(Flat Appl) vs (TLRef i)) a1) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to -(sc3 g a1 c (THeads (Flat Appl) vs (TLRef i))))))))))).(conj (\forall (c: -C).(\forall (t: T).((land (arity g c t (AHead a0 a1)) (\forall (d: -C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a1 d (THead (Flat Appl) w (lift1 is t))))))))) \to (sn3 c t)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c: C).((arity g c (THeads -(Flat Appl) vs (TLRef i)) (AHead a0 a1)) \to ((nf2 c (TLRef i)) \to ((sns3 c -vs) \to (land (arity g c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1)) -(\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (TLRef i))))))))))))))))) (\lambda (c: C).(\lambda (t: -T).(\lambda (H1: (land (arity g c t (AHead a0 a1)) (\forall (d: C).(\forall -(w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 -d (THead (Flat Appl) w (lift1 is t)))))))))).(let H2 \def H in (land_ind -(\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) -(\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads -(Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to -(sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) (\lambda (_: -((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 -t0)))))).(\lambda (H4: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a0) \to ((nf2 c0 (TLRef i)) -\to ((sns3 c0 vs) \to (sc3 g a0 c0 (THeads (Flat Appl) vs (TLRef -i))))))))))).(let H5 \def H0 in (land_ind (\forall (c0: C).(\forall (t0: -T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) (\forall (vs: TList).(\forall (i: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a1) \to -((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 (THeads (Flat Appl) vs -(TLRef i))))))))) (sn3 c t) (\lambda (H6: ((\forall (c0: C).(\forall (t0: -T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: -TList).(\forall (i: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs -(TLRef i)) a1) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 vs) \to (sc3 g a1 c0 -(THeads (Flat Appl) vs (TLRef i))))))))))).(let H8 \def H1 in (land_ind -(arity g c t (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is t)))))))) (sn3 c t) (\lambda (H9: (arity g c t (AHead a0 -a1))).(\lambda (H10: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to -(\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is t)))))))))).(let H_y \def (arity_aprem g c t (AHead a0 a1) H9 O a0) -in (let H11 \def (H_y (aprem_zero a0 a1)) in (ex2_3_ind C T nat (\lambda (d: -C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) (\lambda (d: -C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a0))))) (sn3 c t) -(\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H12: (drop x2 -O x0 c)).(\lambda (H13: (arity g x0 x1 (asucc g a0))).(let H_y0 \def (H10 -(CHead x0 (Bind Abst) x1) (TLRef O) (H4 TNil O (CHead x0 (Bind Abst) x1) -(arity_abst g (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1) a0 -H13) (nf2_lref_abst (CHead x0 (Bind Abst) x1) x0 x1 O (getl_refl Abst x0 x1)) -I) (PCons (S x2) O PNil)) in (let H_y1 \def (H6 (CHead x0 (Bind Abst) x1) -(THead (Flat Appl) (TLRef O) (lift (S x2) O t)) (H_y0 (drop1_cons (CHead x0 -(Bind Abst) x1) c (S x2) O (drop_drop (Bind Abst) x2 x0 c H12 x1) c PNil -(drop1_nil c)))) in (let H_x \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) -x1) (TLRef O) (lift (S x2) O t) H_y1) in (let H14 \def H_x in (land_ind (sn3 -(CHead x0 (Bind Abst) x1) (TLRef O)) (sn3 (CHead x0 (Bind Abst) x1) (lift (S -x2) O t)) (sn3 c t) (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef -O))).(\lambda (H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O -t))).(sn3_gen_lift (CHead x0 (Bind Abst) x1) t (S x2) O H16 c (drop_drop -(Bind Abst) x2 x0 c H12 x1)))) H14)))))))))) H11))))) H8)))) H5)))) H2))))) -(\lambda (vs: TList).(\lambda (i: nat).(\lambda (c: C).(\lambda (H1: (arity g -c (THeads (Flat Appl) vs (TLRef i)) (AHead a0 a1))).(\lambda (H2: (nf2 c -(TLRef i))).(\lambda (H3: (sns3 c vs)).(conj (arity g c (THeads (Flat Appl) -vs (TLRef i)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w -(lift1 is (THeads (Flat Appl) vs (TLRef i)))))))))) H1 (\lambda (d: -C).(\lambda (w: T).(\lambda (H4: (sc3 g a0 d w)).(\lambda (is: -PList).(\lambda (H5: (drop1 is d c)).(let H6 \def H in (land_ind (\forall -(c0: C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) (\forall (vs0: -TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) -vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a0 -c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a1 d (THead (Flat Appl) -w (lift1 is (THeads (Flat Appl) vs (TLRef i))))) (\lambda (H7: ((\forall (c0: -C).(\forall (t: T).((sc3 g a0 c0 t) \to (sn3 c0 t)))))).(\lambda (_: -((\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 -(THeads (Flat Appl) vs0 (TLRef i0)) a0) \to ((nf2 c0 (TLRef i0)) \to ((sns3 -c0 vs0) \to (sc3 g a0 c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))))).(let H9 -\def H0 in (land_ind (\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) \to -(sn3 c0 t)))) (\forall (vs0: TList).(\forall (i0: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) \to ((nf2 c0 (TLRef -i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat Appl) vs0 (TLRef -i0))))))))) (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs -(TLRef i))))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a1 c0 t) -\to (sn3 c0 t)))))).(\lambda (H11: ((\forall (vs0: TList).(\forall (i0: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a1) -\to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 c0 (THeads (Flat -Appl) vs0 (TLRef i0))))))))))).(let H_y \def (H11 (TCons w (lifts1 is vs))) -in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) -(\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w t))) (eq_ind_r T (TLRef -(trans is i)) (\lambda (t: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat -Appl) (lifts1 is vs) t)))) (H_y (trans is i) d (eq_ind T (lift1 is (TLRef i)) -(\lambda (t: T).(arity g d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 -is vs) t)) a1)) (eq_ind T (lift1 is (THeads (Flat Appl) vs (TLRef i))) -(\lambda (t: T).(arity g d (THead (Flat Appl) w t) a1)) (arity_appl g d w a0 -(sc3_arity_gen g d w a0 H4) (lift1 is (THeads (Flat Appl) vs (TLRef i))) a1 -(arity_lift1 g (AHead a0 a1) c is d (THeads (Flat Appl) vs (TLRef i)) H5 H1)) -(THeads (Flat Appl) (lifts1 is vs) (lift1 is (TLRef i))) (lifts1_flat Appl is -(TLRef i) vs)) (TLRef (trans is i)) (lift1_lref is i)) (eq_ind T (lift1 is -(TLRef i)) (\lambda (t: T).(nf2 d t)) (nf2_lift1 c is d (TLRef i) H5 H2) -(TLRef (trans is i)) (lift1_lref is i)) (conj (sn3 d w) (sns3 d (lifts1 is -vs)) (H7 d w H4) (sns3_lifts1 c is d H5 vs H3))) (lift1 is (TLRef i)) -(lift1_lref is i)) (lift1 is (THeads (Flat Appl) vs (TLRef i))) (lifts1_flat -Appl is (TLRef i) vs))))) H9)))) H6))))))))))))))))))) a)). -(* COMMENTS -Initial nodes: 2737 -END *) - -theorem sc3_sn3: - \forall (g: G).(\forall (a: A).(\forall (c: C).(\forall (t: T).((sc3 g a c -t) \to (sn3 c t))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (c: C).(\lambda (t: T).(\lambda (H: -(sc3 g a c t)).(let H_x \def (sc3_props__sc3_sn3_abst g a) in (let H0 \def -H_x in (land_ind (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 -c0 t0)))) (\forall (vs: TList).(\forall (i: nat).(\forall (c0: C).((arity g -c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) \to ((sns3 c0 -vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef i))))))))) (sn3 c t) -(\lambda (H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 -t0)))))).(\lambda (_: ((\forall (vs: TList).(\forall (i: nat).(\forall (c0: -C).((arity g c0 (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c0 (TLRef i)) -\to ((sns3 c0 vs) \to (sc3 g a c0 (THeads (Flat Appl) vs (TLRef -i))))))))))).(H1 c t H))) H0))))))). -(* COMMENTS -Initial nodes: 203 -END *) - -theorem sc3_abst: - \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall -(i: nat).((arity g c (THeads (Flat Appl) vs (TLRef i)) a) \to ((nf2 c (TLRef -i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))) -\def - \lambda (g: G).(\lambda (a: A).(\lambda (vs: TList).(\lambda (c: C).(\lambda -(i: nat).(\lambda (H: (arity g c (THeads (Flat Appl) vs (TLRef i)) -a)).(\lambda (H0: (nf2 c (TLRef i))).(\lambda (H1: (sns3 c vs)).(let H_x \def -(sc3_props__sc3_sn3_abst g a) in (let H2 \def H_x in (land_ind (\forall (c0: -C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) (\forall (vs0: -TList).(\forall (i0: nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) -vs0 (TLRef i0)) a) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a -c0 (THeads (Flat Appl) vs0 (TLRef i0))))))))) (sc3 g a c (THeads (Flat Appl) -vs (TLRef i))) (\lambda (_: ((\forall (c0: C).(\forall (t: T).((sc3 g a c0 t) -\to (sn3 c0 t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: -nat).(\forall (c0: C).((arity g c0 (THeads (Flat Appl) vs0 (TLRef i0)) a) \to -((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 (THeads (Flat Appl) -vs0 (TLRef i0))))))))))).(H4 vs i c H H0 H1))) H2)))))))))). -(* COMMENTS -Initial nodes: 249 -END *) - -theorem sc3_bind: - \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: -A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: -T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts -(S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs -(THead (Bind b) v t))))))))))))) -\def - \lambda (g: G).(\lambda (b: B).(\lambda (H: (not (eq B b Abst))).(\lambda -(a1: A).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (vs: TList).(\forall -(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads -(Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads -(Flat Appl) vs (THead (Bind b) v t)))))))))) (\lambda (n: nat).(\lambda (n0: -nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: -T).(\lambda (H0: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat -Appl) (lifts (S O) O vs) t)))).(\lambda (H1: (sc3 g a1 c v)).(let H2 \def H0 -in (land_ind (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O -vs) t) (ASort n n0)) (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S -O) O vs) t)) (land (arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) -(ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t)))) (\lambda -(H3: (arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) -(ASort n n0))).(\lambda (H4: (sn3 (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Bind -b) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Bind b) v t))) -(arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 H1) t vs (ASort n n0) -H3) (sn3_appls_bind b H c v (sc3_sn3 g a1 c v H1) vs t H4)))) H2)))))))))) -(\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall -(v: T).(\forall (t: T).((sc3 g a (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a c (THeads (Flat Appl) -vs (THead (Bind b) v t))))))))))).(\lambda (a0: A).(\lambda (H1: ((\forall -(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 (CHead -c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) -\to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Bind b) v -t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: T).(\lambda -(t: T).(\lambda (H2: (land (arity g (CHead c (Bind b) v) (THeads (Flat Appl) -(lifts (S O) O vs) t) (AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g -a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) -t))))))))))).(\lambda (H3: (sc3 g a1 c v)).(let H4 \def H2 in (land_ind -(arity g (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) -(AHead a a0)) (\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall -(is: PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat -Appl) w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))) (land -(arity g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) -(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (THead (Bind b) v t))))))))))) (\lambda (H5: (arity g (CHead c -(Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t) (AHead a a0))).(\lambda -(H6: ((\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d (CHead c (Bind b) v)) \to (sc3 g a0 d (THead (Flat Appl) -w (lift1 is (THeads (Flat Appl) (lifts (S O) O vs) t))))))))))).(conj (arity -g c (THeads (Flat Appl) vs (THead (Bind b) v t)) (AHead a a0)) (\forall (d: -C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: PList).((drop1 is d c) -\to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead -(Bind b) v t)))))))))) (arity_appls_bind g b H c v a1 (sc3_arity_gen g c v a1 -H3) t vs (AHead a a0) H5) (\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 -g a d w)).(\lambda (is: PList).(\lambda (H8: (drop1 is d c)).(let H_y \def -(H1 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is -vs) (lift1 is (THead (Bind b) v t))) (\lambda (t0: T).(sc3 g a0 d (THead -(Flat Appl) w t0))) (eq_ind_r T (THead (Bind b) (lift1 is v) (lift1 (Ss is) -t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w (THeads (Flat Appl) -(lifts1 is vs) t0)))) (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind TList -(lifts1 (Ss is) (lifts (S O) O vs)) (\lambda (t0: TList).(sc3 g a0 (CHead d -(Bind b) (lift1 is v)) (THead (Flat Appl) (lift (S O) O w) (THeads (Flat -Appl) t0 (lift1 (Ss is) t))))) (eq_ind T (lift1 (Ss is) (THeads (Flat Appl) -(lifts (S O) O vs) t)) (\lambda (t0: T).(sc3 g a0 (CHead d (Bind b) (lift1 is -v)) (THead (Flat Appl) (lift (S O) O w) t0))) (H6 (CHead d (Bind b) (lift1 is -v)) (lift (S O) O w) (sc3_lift g a d w H7 (CHead d (Bind b) (lift1 is v)) (S -O) O (drop_drop (Bind b) O d d (drop_refl d) (lift1 is v))) (Ss is) -(drop1_skip_bind b c is d v H8)) (THeads (Flat Appl) (lifts1 (Ss is) (lifts -(S O) O vs)) (lift1 (Ss is) t)) (lifts1_flat Appl (Ss is) t (lifts (S O) O -vs))) (lifts (S O) O (lifts1 is vs)) (lifts1_xhg is vs)) (sc3_lift1 g c a1 is -d v H3 H8)) (lift1 is (THead (Bind b) v t)) (lift1_bind b is v t)) (lift1 is -(THeads (Flat Appl) vs (THead (Bind b) v t))) (lifts1_flat Appl is (THead -(Bind b) v t) vs))))))))))) H4)))))))))))) a2))))). -(* COMMENTS -Initial nodes: 1797 -END *) - -theorem sc3_appl: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (vs: -TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads -(Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: -T).((sc3 g (asucc g a1) c w) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead -(Flat Appl) v (THead (Bind Abst) w t)))))))))))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(A_ind (\lambda (a: -A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 -g a c (THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) -\to (\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a c (THeads (Flat -Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))))))))))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (v: -T).(\lambda (t: T).(\lambda (H: (land (arity g c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead -(Bind Abbr) v t))))).(\lambda (H0: (sc3 g a1 c v)).(\lambda (w: T).(\lambda -(H1: (sc3 g (asucc g a1) c w)).(let H2 \def H in (land_ind (arity g c (THeads -(Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n n0)) (sn3 c (THeads (Flat -Appl) vs (THead (Bind Abbr) v t))) (land (arity g c (THeads (Flat Appl) vs -(THead (Flat Appl) v (THead (Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads -(Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))))) (\lambda (H3: -(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (ASort n -n0))).(\lambda (H4: (sn3 c (THeads (Flat Appl) vs (THead (Bind Abbr) v -t)))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v (THead -(Bind Abst) w t))) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat -Appl) v (THead (Bind Abst) w t)))) (arity_appls_appl g c v a1 (sc3_arity_gen -g c v a1 H0) w (sc3_arity_gen g c w (asucc g a1) H1) t vs (ASort n n0) H3) -(sn3_appls_beta c v t vs H4 w (sc3_sn3 g (asucc g a1) c w H1))))) -H2)))))))))))) (\lambda (a: A).(\lambda (_: ((\forall (vs: TList).(\forall -(c: C).(\forall (v: T).(\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs -(THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to (\forall (w: T).((sc3 g -(asucc g a1) c w) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t)))))))))))))).(\lambda (a0: A).(\lambda (H0: ((\forall -(vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a0 c -(THeads (Flat Appl) vs (THead (Bind Abbr) v t))) \to ((sc3 g a1 c v) \to -(\forall (w: T).((sc3 g (asucc g a1) c w) \to (sc3 g a0 c (THeads (Flat Appl) -vs (THead (Flat Appl) v (THead (Bind Abst) w t)))))))))))))).(\lambda (vs: -TList).(\lambda (c: C).(\lambda (v: T).(\lambda (t: T).(\lambda (H1: (land -(arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) (AHead a a0)) -(\forall (d: C).(\forall (w: T).((sc3 g a d w) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w (lift1 is (THeads -(Flat Appl) vs (THead (Bind Abbr) v t)))))))))))).(\lambda (H2: (sc3 g a1 c -v)).(\lambda (w: T).(\lambda (H3: (sc3 g (asucc g a1) c w)).(let H4 \def H1 -in (land_ind (arity g c (THeads (Flat Appl) vs (THead (Bind Abbr) v t)) -(AHead a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall -(is: PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is -(THeads (Flat Appl) vs (THead (Bind Abbr) v t)))))))))) (land (arity g c -(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w t))) (AHead -a a0)) (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: -PList).((drop1 is d c) \to (sc3 g a0 d (THead (Flat Appl) w0 (lift1 is -(THeads (Flat Appl) vs (THead (Flat Appl) v (THead (Bind Abst) w -t)))))))))))) (\lambda (H5: (arity g c (THeads (Flat Appl) vs (THead (Bind -Abbr) v t)) (AHead a a0))).(\lambda (H6: ((\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Bind Abbr) v -t)))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t))) (AHead a a0)) (\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a0 d -(THead (Flat Appl) w0 (lift1 is (THeads (Flat Appl) vs (THead (Flat Appl) v -(THead (Bind Abst) w t))))))))))) (arity_appls_appl g c v a1 (sc3_arity_gen g -c v a1 H2) w (sc3_arity_gen g c w (asucc g a1) H3) t vs (AHead a a0) H5) -(\lambda (d: C).(\lambda (w0: T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: -PList).(\lambda (H8: (drop1 is d c)).(eq_ind_r T (THeads (Flat Appl) (lifts1 -is vs) (lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t)))) (\lambda -(t0: T).(sc3 g a0 d (THead (Flat Appl) w0 t0))) (eq_ind_r T (THead (Flat -Appl) (lift1 is v) (lift1 is (THead (Bind Abst) w t))) (\lambda (t0: T).(sc3 -g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) t0)))) -(eq_ind_r T (THead (Bind Abst) (lift1 is w) (lift1 (Ss is) t)) (\lambda (t0: -T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads (Flat Appl) (lifts1 is vs) -(THead (Flat Appl) (lift1 is v) t0))))) (let H_y \def (H0 (TCons w0 (lifts1 -is vs))) in (H_y d (lift1 is v) (lift1 (Ss is) t) (eq_ind T (lift1 is (THead -(Bind Abbr) v t)) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 (THeads -(Flat Appl) (lifts1 is vs) t0)))) (eq_ind T (lift1 is (THeads (Flat Appl) vs -(THead (Bind Abbr) v t))) (\lambda (t0: T).(sc3 g a0 d (THead (Flat Appl) w0 -t0))) (H6 d w0 H7 is H8) (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead -(Bind Abbr) v t))) (lifts1_flat Appl is (THead (Bind Abbr) v t) vs)) (THead -(Bind Abbr) (lift1 is v) (lift1 (Ss is) t)) (lift1_bind Abbr is v t)) -(sc3_lift1 g c a1 is d v H2 H8) (lift1 is w) (sc3_lift1 g c (asucc g a1) is d -w H3 H8))) (lift1 is (THead (Bind Abst) w t)) (lift1_bind Abst is w t)) -(lift1 is (THead (Flat Appl) v (THead (Bind Abst) w t))) (lift1_flat Appl is -v (THead (Bind Abst) w t))) (lift1 is (THeads (Flat Appl) vs (THead (Flat -Appl) v (THead (Bind Abst) w t)))) (lifts1_flat Appl is (THead (Flat Appl) v -(THead (Bind Abst) w t)) vs)))))))))) H4)))))))))))))) a2))). -(* COMMENTS -Initial nodes: 1901 -END *) -