X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLevel-1%2FLambdaDelta%2Fsc3%2Fprops.ma;h=c1d3787b8cb2ac628abeac0e939a43c3cc5f293e;hb=6bb836528dd25934608f400779ef3edc8bc9bf92;hp=86a674344b9b4650a25d88c476c66be5126ec3ba;hpb=6d2ca42f41faba2789b2bd61792e8b43aca40713;p=helm.git diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma index 86a674344..c1d3787b8 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props.ma @@ -14,7 +14,7 @@ (* This file was automatically generated: do not edit *********************) -set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/sc3/props". +set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sc3/props". include "sc3/defs.ma". @@ -22,14 +22,20 @@ include "sn3/lift1.ma". include "nf2/lift1.ma". +include "csuba/arity.ma". + include "arity/lift1.ma". include "arity/aprem.ma". include "llt/props.ma". +include "drop1/getl.ma". + include "drop1/props.ma". +include "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))))) @@ -223,12 +229,77 @@ c2 (lift1 p t) (H c2 t H0 H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)))). -axiom sc3_abbr: +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 (and_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 (and_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 @@ -244,54 +315,58 @@ Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to 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)))).((match n in nat return (\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))))))) with [O -\Rightarrow (\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 (and_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 -(and_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)))) | (S n1) \Rightarrow -(\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 (and_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 (and_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))))]) 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: +(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 (and_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 (and_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 (and_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 (and_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 @@ -338,22 +413,6 @@ 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)). -axiom 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))))))))))))) -. - -axiom 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)))))))))))))) -. - 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 @@ -432,56 +491,56 @@ 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 H14 \def (sn3_gen_flat Appl (CHead x0 (Bind Abst) -x1) (TLRef O) (lift (S x2) O t) H_y1) in (and_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 (and_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 (and_ind (\forall (c0: -C).(\forall (t: T).((sc3 g a1 c0 t) \to (sn3 c0 t)))) (\forall (vs0: +(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 (and_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 (and_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)) a1) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a1 +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 (_: ((\forall (c0: -C).(\forall (t: T).((sc3 g a1 c0 t) \to (sn3 c0 t)))))).(\lambda (H11: +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)) 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)). +(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 (and_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 @@ -518,3 +577,163 @@ t)))))).(\lambda (H4: ((\forall (vs0: TList).(\forall (i0: nat).(let t \def ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (sc3 g a c0 t)))))))))).(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 (and_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 (and_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 (and_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 (and_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))). +