X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fsc3%2Fprops.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fsc3%2Fprops.ma;h=cd5ce1f7d42e880cd8a98e2cc4c35e211e57c277;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma new file mode 100644 index 000000000..cd5ce1f7d --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/sc3/props.ma @@ -0,0 +1,695 @@ +(**************************************************************************) +(* ___ *) +(* ||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/sc3/defs.ma". + +include "LambdaDelta-1/sn3/lift1.ma". + +include "LambdaDelta-1/nf2/lift1.ma". + +include "LambdaDelta-1/csuba/arity.ma". + +include "LambdaDelta-1/arity/lift1.ma". + +include "LambdaDelta-1/arity/aprem.ma". + +include "LambdaDelta-1/llt/props.ma". + +include "LambdaDelta-1/drop1/getl.ma". + +include "LambdaDelta-1/drop1/props.ma". + +include "LambdaDelta-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)))). + +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)). + +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)). + +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)))). + +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)). + +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)). + +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)). + +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))))))). + +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)))))))))). + +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))))). + +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))). +