X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fsc3%2Fprops.ma;h=80da46d3cbac77ada65a26bcd7f3b06cb8811dbe;hb=57ae1762497a5f3ea75740e2908e04adb8642cc2;hp=67d97963b0f65f50b5f388bdfb2845652a2b90d1;hpb=57e01777627e779555dc091a58ec2f4f860ee51f;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma b/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma index 67d97963b..80da46d3c 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/sc3/props.ma @@ -36,258 +36,185 @@ include "basic_1/drop1/props.ma". include "basic_1/lift1/drop1.ma". -theorem sc3_arity_gen: +lemma 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).(let TMP_1 -\def (\lambda (a0: A).((sc3 g a0 c t) \to (arity g c t a0))) in (let TMP_8 -\def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (land (arity g c t -(ASort n n0)) (sn3 c t))).(let H0 \def H in (let TMP_2 \def (ASort n n0) in -(let TMP_3 \def (arity g c t TMP_2) in (let TMP_4 \def (sn3 c t) in (let -TMP_5 \def (ASort n n0) in (let TMP_6 \def (arity g c t TMP_5) in (let TMP_7 -\def (\lambda (H1: (arity g c t (ASort n n0))).(\lambda (_: (sn3 c t)).H1)) -in (land_ind TMP_3 TMP_4 TMP_6 TMP_7 H0))))))))))) in (let TMP_18 \def -(\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: + \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 -(let TMP_9 \def (AHead a0 a1) in (let TMP_10 \def (arity g c t TMP_9) in (let -TMP_14 \def (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (let TMP_11 \def (Flat Appl) in (let TMP_12 \def -(lift1 is t) in (let TMP_13 \def (THead TMP_11 w TMP_12) in (sc3 g a1 d -TMP_13))))))))) in (let TMP_15 \def (AHead a0 a1) in (let TMP_16 \def (arity -g c t TMP_15) in (let TMP_17 \def (\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)) in (land_ind TMP_10 TMP_14 TMP_16 TMP_17 -H2))))))))))))) in (A_ind TMP_1 TMP_8 TMP_18 a))))))). +(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: +lemma 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).(let TMP_1 \def (\lambda (a: A).(\forall (c: + \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))))))) in (let TMP_73 \def (\lambda (a2: A).(let TMP_2 \def (\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)))))))) in (let TMP_28 \def (\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 (let TMP_3 \def -(ASort n n0) in (let TMP_4 \def (arity g c t TMP_3) in (let TMP_5 \def (sn3 c -t) in (let TMP_6 \def (sc3 g a3 c t) in (let TMP_27 \def (\lambda (H3: (arity -g c t (ASort n n0))).(\lambda (H4: (sn3 c t)).(let TMP_7 \def (ASort n n0) in -(let H_y \def (arity_repl g c t TMP_7 H3 a3 H1) in (let H_x \def -(leq_gen_sort1 g n n0 a3 H1) in (let H5 \def H_x in (let TMP_12 \def (\lambda -(n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_8 \def (ASort n n0) -in (let TMP_9 \def (aplus g TMP_8 k) in (let TMP_10 \def (ASort h2 n2) in -(let TMP_11 \def (aplus g TMP_10 k) in (eq A TMP_9 TMP_11)))))))) in (let -TMP_14 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let -TMP_13 \def (ASort h2 n2) in (eq A a3 TMP_13))))) in (let TMP_15 \def (sc3 g -a3 c t) in (let TMP_26 \def (\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 TMP_16 \def (\lambda (e: -A).e) in (let TMP_17 \def (ASort x1 x0) in (let H8 \def (f_equal A A TMP_16 -a3 TMP_17 H7) in (let TMP_18 \def (\lambda (a: A).(arity g c t a)) in (let -TMP_19 \def (ASort x1 x0) in (let H9 \def (eq_ind A a3 TMP_18 H_y TMP_19 H8) -in (let TMP_20 \def (ASort x1 x0) in (let TMP_21 \def (\lambda (a: A).(sc3 g -a c t)) in (let TMP_22 \def (ASort x1 x0) in (let TMP_23 \def (arity g c t -TMP_22) in (let TMP_24 \def (sn3 c t) in (let TMP_25 \def (conj TMP_23 TMP_24 -H9 H4) in (eq_ind_r A TMP_20 TMP_21 TMP_25 a3 H8)))))))))))))))))) in -(ex2_3_ind nat nat nat TMP_12 TMP_14 TMP_15 TMP_26 H5))))))))))) in (land_ind -TMP_4 TMP_5 TMP_6 TMP_27 H2))))))))))))))) in (let TMP_72 \def (\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 +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 (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 +(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 (let TMP_29 \def (AHead a a0) in (let TMP_30 \def (arity g c t -TMP_29) in (let TMP_34 \def (\forall (d: C).(\forall (w: T).((sc3 g a d w) -\to (\forall (is: PList).((drop1 is d c) \to (let TMP_31 \def (Flat Appl) in -(let TMP_32 \def (lift1 is t) in (let TMP_33 \def (THead TMP_31 w TMP_32) in -(sc3 g a0 d TMP_33))))))))) in (let TMP_35 \def (sc3 g a3 c t) in (let TMP_71 -\def (\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 (let TMP_36 \def (\lambda -(a4: A).(\lambda (_: A).(leq g a a4))) in (let TMP_37 \def (\lambda (_: -A).(\lambda (a5: A).(leq g a0 a5))) in (let TMP_39 \def (\lambda (a4: -A).(\lambda (a5: A).(let TMP_38 \def (AHead a4 a5) in (eq A a3 TMP_38)))) in -(let TMP_40 \def (sc3 g a3 c t) in (let TMP_70 \def (\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 TMP_41 \def (\lambda (e: A).e) in (let -TMP_42 \def (AHead x0 x1) in (let H11 \def (f_equal A A TMP_41 a3 TMP_42 H10) -in (let TMP_43 \def (AHead x0 x1) in (let TMP_44 \def (\lambda (a4: A).(sc3 g -a4 c t)) in (let TMP_45 \def (AHead x0 x1) in (let TMP_46 \def (arity g c t -TMP_45) in (let TMP_50 \def (\forall (d: C).(\forall (w: T).((sc3 g x0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (let TMP_47 \def (Flat Appl) in -(let TMP_48 \def (lift1 is t) in (let TMP_49 \def (THead TMP_47 w TMP_48) in -(sc3 g x1 d TMP_49))))))))) in (let TMP_51 \def (AHead a a0) in (let TMP_52 -\def (AHead x0 x1) in (let TMP_53 \def (leq_head g a x0 H8 a0 x1 H9) in (let -TMP_54 \def (arity_repl g c t TMP_51 H5 TMP_52 TMP_53) in (let TMP_68 \def -(\lambda (d: C).(\lambda (w: T).(\lambda (H12: (sc3 g x0 d w)).(\lambda (is: -PList).(\lambda (H13: (drop1 is d c)).(let TMP_58 \def (\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)).(let -TMP_55 \def (AHead a a0) in (let TMP_56 \def (llt_head_dx a a0) in (let -TMP_57 \def (llt_trans a4 a0 TMP_55 H14 TMP_56) in (H1 a4 TMP_57 c0 t0 H15 a5 -H16))))))))))) in (let TMP_59 \def (Flat Appl) in (let TMP_60 \def (lift1 is -t) in (let TMP_61 \def (THead TMP_59 w TMP_60) in (let TMP_62 \def (AHead a -a0) in (let TMP_63 \def (llt_head_sx a a0) in (let TMP_64 \def (llt_repl g a -x0 H8 TMP_62 TMP_63) in (let TMP_65 \def (leq_sym g a x0 H8) in (let TMP_66 -\def (H1 x0 TMP_64 d w H12 a TMP_65) in (let TMP_67 \def (H6 d w TMP_66 is -H13) in (H0 TMP_58 d TMP_61 TMP_67 x1 H9)))))))))))))))) in (let TMP_69 \def -(conj TMP_46 TMP_50 TMP_54 TMP_68) in (eq_ind_r A TMP_43 TMP_44 TMP_69 a3 -H11)))))))))))))))))))) in (ex3_2_ind A A TMP_36 TMP_37 TMP_39 TMP_40 TMP_70 -H7)))))))))) in (land_ind TMP_30 TMP_34 TMP_35 TMP_71 H4))))))))))))))))) in -(A_ind TMP_2 TMP_28 TMP_72 a2))))) in (llt_wf_ind TMP_1 TMP_73 a1)))). +\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: +lemma 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).(let TMP_2 \def (\lambda (a0: A).(\forall (e: + \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 (let TMP_1 \def (lift h d t) in -(sc3 g a0 c TMP_1)))))))))) in (let TMP_21 \def (\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 (let TMP_3 \def (ASort n -n0) in (let TMP_4 \def (arity g e t TMP_3) in (let TMP_5 \def (sn3 e t) in -(let TMP_6 \def (lift h d t) in (let TMP_7 \def (ASort n n0) in (let TMP_8 -\def (arity g c TMP_6 TMP_7) in (let TMP_9 \def (lift h d t) in (let TMP_10 -\def (sn3 c TMP_9) in (let TMP_11 \def (land TMP_8 TMP_10) in (let TMP_20 -\def (\lambda (H2: (arity g e t (ASort n n0))).(\lambda (H3: (sn3 e t)).(let -TMP_12 \def (lift h d t) in (let TMP_13 \def (ASort n n0) in (let TMP_14 \def -(arity g c TMP_12 TMP_13) in (let TMP_15 \def (lift h d t) in (let TMP_16 -\def (sn3 c TMP_15) in (let TMP_17 \def (ASort n n0) in (let TMP_18 \def -(arity_lift g e t TMP_17 H2 c h d H0) in (let TMP_19 \def (sn3_lift e t H3 c -h d H0) in (conj TMP_14 TMP_16 TMP_18 TMP_19))))))))))) in (land_ind TMP_4 -TMP_5 TMP_11 TMP_20 H1))))))))))))))))))))) in (let TMP_60 \def (\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 (let TMP_22 \def (AHead a0 a1) in (let -TMP_23 \def (arity g e t TMP_22) in (let TMP_27 \def (\forall (d0: -C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 e) -\to (let TMP_24 \def (Flat Appl) in (let TMP_25 \def (lift1 is t) in (let -TMP_26 \def (THead TMP_24 w TMP_25) in (sc3 g a1 d0 TMP_26))))))))) in (let -TMP_28 \def (lift h d t) in (let TMP_29 \def (AHead a0 a1) in (let TMP_30 -\def (arity g c TMP_28 TMP_29) in (let TMP_35 \def (\forall (d0: C).(\forall -(w: T).((sc3 g a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (let -TMP_31 \def (Flat Appl) in (let TMP_32 \def (lift h d t) in (let TMP_33 \def -(lift1 is TMP_32) in (let TMP_34 \def (THead TMP_31 w TMP_33) in (sc3 g a1 d0 -TMP_34)))))))))) in (let TMP_36 \def (land TMP_30 TMP_35) in (let TMP_59 \def -(\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)))))))))).(let TMP_37 \def -(lift h d t) in (let TMP_38 \def (AHead a0 a1) in (let TMP_39 \def (arity g c -TMP_37 TMP_38) in (let TMP_44 \def (\forall (d0: C).(\forall (w: T).((sc3 g -a0 d0 w) \to (\forall (is: PList).((drop1 is d0 c) \to (let TMP_40 \def (Flat -Appl) in (let TMP_41 \def (lift h d t) in (let TMP_42 \def (lift1 is TMP_41) -in (let TMP_43 \def (THead TMP_40 w TMP_42) in (sc3 g a1 d0 TMP_43)))))))))) -in (let TMP_45 \def (AHead a0 a1) in (let TMP_46 \def (arity_lift g e t -TMP_45 H4 c h d H2) in (let TMP_58 \def (\lambda (d0: C).(\lambda (w: +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 TMP_47 \def (PConsTail is h d) in (let H_y \def (H5 d0 w H6 -TMP_47) in (let TMP_48 \def (PConsTail is h d) in (let TMP_49 \def (lift1 -TMP_48 t) in (let TMP_52 \def (\lambda (t0: T).(let TMP_50 \def (Flat Appl) -in (let TMP_51 \def (THead TMP_50 w t0) in (sc3 g a1 d0 TMP_51)))) in (let -TMP_53 \def (drop1_cons_tail c e h d H2 is d0 H7) in (let TMP_54 \def (H_y -TMP_53) in (let TMP_55 \def (lift h d t) in (let TMP_56 \def (lift1 is -TMP_55) in (let TMP_57 \def (lift1_cons_tail t h d is) in (eq_ind T TMP_49 -TMP_52 TMP_54 TMP_56 TMP_57)))))))))))))))) in (conj TMP_39 TMP_44 TMP_46 -TMP_58)))))))))) in (land_ind TMP_23 TMP_27 TMP_36 TMP_59 -H3)))))))))))))))))))))) in (A_ind TMP_2 TMP_21 TMP_60 a))))). +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: +lemma 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).(let -TMP_2 \def (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) -\to ((drop1 p c e) \to (let TMP_1 \def (lift1 p t) in (sc3 g a c TMP_1))))))) -in (let TMP_4 \def (\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 -(let TMP_3 \def (\lambda (c0: C).(sc3 g a c0 t)) in (eq_ind_r C e TMP_3 H c -H_y))))))) in (let TMP_13 \def (\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 (let -TMP_5 \def (\lambda (c2: C).(drop n n0 c c2)) in (let TMP_6 \def (\lambda -(c2: C).(drop1 p c2 e)) in (let TMP_7 \def (lift1 p t) in (let TMP_8 \def -(lift n n0 TMP_7) in (let TMP_9 \def (sc3 g a c TMP_8) in (let TMP_12 \def -(\lambda (x: C).(\lambda (H3: (drop n n0 c x)).(\lambda (H4: (drop1 p x -e)).(let TMP_10 \def (lift1 p t) in (let TMP_11 \def (H x t H0 H4) in -(sc3_lift g a x TMP_10 TMP_11 c n n0 H3)))))) in (ex2_ind C TMP_5 TMP_6 TMP_9 -TMP_12 H2))))))))))))))))) in (PList_ind TMP_2 TMP_4 TMP_13 hds))))))). + \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: +lemma 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).(let TMP_4 \def (\lambda (a0: A).(\forall -(vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: + \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 (let TMP_1 \def (Flat Appl) in (let TMP_2 \def -(TLRef i) in (let TMP_3 \def (THeads TMP_1 vs TMP_2) in (sc3 g a0 c -TMP_3)))))))))))) in (let TMP_39 \def (\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 -(let TMP_5 \def (Flat Appl) in (let TMP_6 \def (S i) in (let TMP_7 \def (lift -TMP_6 O v) in (let TMP_8 \def (THeads TMP_5 vs TMP_7) in (let TMP_9 \def -(ASort n n0) in (let TMP_10 \def (arity g c TMP_8 TMP_9) in (let TMP_11 \def -(Flat Appl) in (let TMP_12 \def (S i) in (let TMP_13 \def (lift TMP_12 O v) -in (let TMP_14 \def (THeads TMP_11 vs TMP_13) in (let TMP_15 \def (sn3 c -TMP_14) in (let TMP_16 \def (Flat Appl) in (let TMP_17 \def (TLRef i) in (let -TMP_18 \def (THeads TMP_16 vs TMP_17) in (let TMP_19 \def (ASort n n0) in -(let TMP_20 \def (arity g c TMP_18 TMP_19) in (let TMP_21 \def (Flat Appl) in -(let TMP_22 \def (TLRef i) in (let TMP_23 \def (THeads TMP_21 vs TMP_22) in -(let TMP_24 \def (sn3 c TMP_23) in (let TMP_25 \def (land TMP_20 TMP_24) in -(let TMP_38 \def (\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)))).(let TMP_26 \def (Flat Appl) in (let TMP_27 \def (TLRef i) in (let -TMP_28 \def (THeads TMP_26 vs TMP_27) in (let TMP_29 \def (ASort n n0) in -(let TMP_30 \def (arity g c TMP_28 TMP_29) in (let TMP_31 \def (Flat Appl) in -(let TMP_32 \def (TLRef i) in (let TMP_33 \def (THeads TMP_31 vs TMP_32) in -(let TMP_34 \def (sn3 c TMP_33) in (let TMP_35 \def (ASort n n0) in (let -TMP_36 \def (arity_appls_abbr g c d v i H0 vs TMP_35 H2) in (let TMP_37 \def -(sn3_appls_abbr c d v i H0 vs H3) in (conj TMP_30 TMP_34 TMP_36 -TMP_37))))))))))))))) in (land_ind TMP_10 TMP_15 TMP_25 TMP_38 -H1))))))))))))))))))))))))))))))))) in (let TMP_166 \def (\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: +(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 @@ -295,91 +222,42 @@ i)))))))))))).(\lambda (vs: TList).(\lambda (i: nat).(\lambda (d: C).(\lambda (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 (let TMP_40 \def -(Flat Appl) in (let TMP_41 \def (S i) in (let TMP_42 \def (lift TMP_41 O v) -in (let TMP_43 \def (THeads TMP_40 vs TMP_42) in (let TMP_44 \def (AHead a0 -a1) in (let TMP_45 \def (arity g c TMP_43 TMP_44) in (let TMP_53 \def -(\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 c) \to (let TMP_46 \def (Flat Appl) in (let TMP_47 \def -(Flat Appl) in (let TMP_48 \def (S i) in (let TMP_49 \def (lift TMP_48 O v) -in (let TMP_50 \def (THeads TMP_47 vs TMP_49) in (let TMP_51 \def (lift1 is -TMP_50) in (let TMP_52 \def (THead TMP_46 w TMP_51) in (sc3 g a1 d0 -TMP_52))))))))))))) in (let TMP_54 \def (Flat Appl) in (let TMP_55 \def -(TLRef i) in (let TMP_56 \def (THeads TMP_54 vs TMP_55) in (let TMP_57 \def -(AHead a0 a1) in (let TMP_58 \def (arity g c TMP_56 TMP_57) in (let TMP_65 -\def (\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 c) \to (let TMP_59 \def (Flat Appl) in (let TMP_60 \def -(Flat Appl) in (let TMP_61 \def (TLRef i) in (let TMP_62 \def (THeads TMP_60 -vs TMP_61) in (let TMP_63 \def (lift1 is TMP_62) in (let TMP_64 \def (THead -TMP_59 w TMP_63) in (sc3 g a1 d0 TMP_64)))))))))))) in (let TMP_66 \def (land -TMP_58 TMP_65) in (let TMP_165 \def (\lambda (H4: (arity g c (THeads (Flat -Appl) vs (lift (S i) O v)) (AHead a0 a1))).(\lambda (H5: ((\forall (d0: +(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)))))))))))).(let TMP_67 \def (Flat Appl) in (let TMP_68 \def (TLRef -i) in (let TMP_69 \def (THeads TMP_67 vs TMP_68) in (let TMP_70 \def (AHead -a0 a1) in (let TMP_71 \def (arity g c TMP_69 TMP_70) in (let TMP_78 \def -(\forall (d0: C).(\forall (w: T).((sc3 g a0 d0 w) \to (\forall (is: -PList).((drop1 is d0 c) \to (let TMP_72 \def (Flat Appl) in (let TMP_73 \def -(Flat Appl) in (let TMP_74 \def (TLRef i) in (let TMP_75 \def (THeads TMP_73 -vs TMP_74) in (let TMP_76 \def (lift1 is TMP_75) in (let TMP_77 \def (THead -TMP_72 w TMP_76) in (sc3 g a1 d0 TMP_77)))))))))))) in (let TMP_79 \def -(AHead a0 a1) in (let TMP_80 \def (arity_appls_abbr g c d v i H2 vs TMP_79 -H4) in (let TMP_164 \def (\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 (let -TMP_82 \def (\lambda (e2: C).(let TMP_81 \def (ptrans is i) in (drop1 TMP_81 -e2 d))) in (let TMP_88 \def (\lambda (e2: C).(let TMP_83 \def (trans is i) in -(let TMP_84 \def (Bind Abbr) in (let TMP_85 \def (ptrans is i) in (let TMP_86 -\def (lift1 TMP_85 v) in (let TMP_87 \def (CHead e2 TMP_84 TMP_86) in (getl -TMP_83 d0 TMP_87))))))) in (let TMP_89 \def (Flat Appl) in (let TMP_90 \def -(Flat Appl) in (let TMP_91 \def (TLRef i) in (let TMP_92 \def (THeads TMP_90 -vs TMP_91) in (let TMP_93 \def (lift1 is TMP_92) in (let TMP_94 \def (THead -TMP_89 w TMP_93) in (let TMP_95 \def (sc3 g a1 d0 TMP_94) in (let TMP_163 -\def (\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 -TMP_96 \def (lifts1 is vs) in (let TMP_97 \def (TCons w TMP_96) in (let H_y -\def (H0 TMP_97) in (let TMP_98 \def (Flat Appl) in (let TMP_99 \def (lifts1 -is vs) in (let TMP_100 \def (TLRef i) in (let TMP_101 \def (lift1 is TMP_100) -in (let TMP_102 \def (THeads TMP_98 TMP_99 TMP_101) in (let TMP_105 \def -(\lambda (t: T).(let TMP_103 \def (Flat Appl) in (let TMP_104 \def (THead -TMP_103 w t) in (sc3 g a1 d0 TMP_104)))) in (let TMP_106 \def (trans is i) in -(let TMP_107 \def (TLRef TMP_106) in (let TMP_113 \def (\lambda (t: T).(let -TMP_108 \def (Flat Appl) in (let TMP_109 \def (Flat Appl) in (let TMP_110 -\def (lifts1 is vs) in (let TMP_111 \def (THeads TMP_109 TMP_110 t) in (let -TMP_112 \def (THead TMP_108 w TMP_111) in (sc3 g a1 d0 TMP_112))))))) in (let -TMP_114 \def (trans is i) in (let TMP_115 \def (ptrans is i) in (let TMP_116 -\def (lift1 TMP_115 v) in (let TMP_117 \def (S i) in (let TMP_118 \def (lift -TMP_117 O v) in (let TMP_119 \def (lift1 is TMP_118) in (let TMP_125 \def -(\lambda (t: T).(let TMP_120 \def (Flat Appl) in (let TMP_121 \def (Flat -Appl) in (let TMP_122 \def (lifts1 is vs) in (let TMP_123 \def (THeads -TMP_121 TMP_122 t) in (let TMP_124 \def (THead TMP_120 w TMP_123) in (sc3 g -a1 d0 TMP_124))))))) in (let TMP_126 \def (Flat Appl) in (let TMP_127 \def (S -i) in (let TMP_128 \def (lift TMP_127 O v) in (let TMP_129 \def (THeads -TMP_126 vs TMP_128) in (let TMP_130 \def (lift1 is TMP_129) in (let TMP_133 -\def (\lambda (t: T).(let TMP_131 \def (Flat Appl) in (let TMP_132 \def -(THead TMP_131 w t) in (sc3 g a1 d0 TMP_132)))) in (let TMP_134 \def (H5 d0 w -H6 is H7) in (let TMP_135 \def (Flat Appl) in (let TMP_136 \def (lifts1 is -vs) in (let TMP_137 \def (S i) in (let TMP_138 \def (lift TMP_137 O v) in -(let TMP_139 \def (lift1 is TMP_138) in (let TMP_140 \def (THeads TMP_135 -TMP_136 TMP_139) in (let TMP_141 \def (S i) in (let TMP_142 \def (lift -TMP_141 O v) in (let TMP_143 \def (lifts1_flat Appl is TMP_142 vs) in (let -TMP_144 \def (eq_ind T TMP_130 TMP_133 TMP_134 TMP_140 TMP_143) in (let -TMP_145 \def (trans is i) in (let TMP_146 \def (S TMP_145) in (let TMP_147 -\def (ptrans is i) in (let TMP_148 \def (lift1 TMP_147 v) in (let TMP_149 -\def (lift TMP_146 O TMP_148) in (let TMP_150 \def (lift1_free is i v) in -(let TMP_151 \def (eq_ind T TMP_119 TMP_125 TMP_144 TMP_149 TMP_150) in (let -TMP_152 \def (H_y TMP_114 x TMP_116 d0 TMP_151 H10) in (let TMP_153 \def -(TLRef i) in (let TMP_154 \def (lift1 is TMP_153) in (let TMP_155 \def -(lift1_lref is i) in (let TMP_156 \def (eq_ind_r T TMP_107 TMP_113 TMP_152 -TMP_154 TMP_155) in (let TMP_157 \def (Flat Appl) in (let TMP_158 \def (TLRef -i) in (let TMP_159 \def (THeads TMP_157 vs TMP_158) in (let TMP_160 \def -(lift1 is TMP_159) in (let TMP_161 \def (TLRef i) in (let TMP_162 \def -(lifts1_flat Appl is TMP_161 vs) in (eq_ind_r T TMP_102 TMP_105 TMP_156 -TMP_160 TMP_162)))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in -(ex2_ind C TMP_82 TMP_88 TMP_95 TMP_163 H8)))))))))))))))))) in (conj TMP_71 -TMP_78 TMP_80 TMP_164)))))))))))) in (land_ind TMP_45 TMP_53 TMP_66 TMP_165 -H3)))))))))))))))))))))))))))) in (A_ind TMP_4 TMP_39 TMP_166 a))))). +(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 @@ -387,254 +265,136 @@ theorem sc3_cast: 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).(let TMP_5 \def (\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 (let TMP_1 \def (Flat Appl) in (let TMP_2 \def (Flat Cast) in (let -TMP_3 \def (THead TMP_2 u t) in (let TMP_4 \def (THeads TMP_1 vs TMP_3) in -(sc3 g a0 c TMP_4)))))))))))) in (let TMP_134 \def (\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)))).(let TMP_17 \def (\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 (let TMP_6 \def (Flat -Appl) in (let TMP_7 \def (Flat Cast) in (let TMP_8 \def (THead TMP_7 u t) in -(let TMP_9 \def (THeads TMP_6 vs TMP_8) in (let TMP_10 \def (ASort n1 n0) in -(let TMP_11 \def (arity g c TMP_9 TMP_10) in (let TMP_12 \def (Flat Appl) in -(let TMP_13 \def (Flat Cast) in (let TMP_14 \def (THead TMP_13 u t) in (let -TMP_15 \def (THeads TMP_12 vs TMP_14) in (let TMP_16 \def (sn3 c TMP_15) in -(land TMP_11 TMP_16))))))))))))))) in (let TMP_73 \def (\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 (let TMP_18 \def (Flat Appl) in (let TMP_19 \def -(THeads TMP_18 vs u) in (let TMP_20 \def (next g n0) in (let TMP_21 \def -(ASort O TMP_20) in (let TMP_22 \def (arity g c TMP_19 TMP_21) in (let TMP_23 -\def (Flat Appl) in (let TMP_24 \def (THeads TMP_23 vs u) in (let TMP_25 \def -(sn3 c TMP_24) in (let TMP_26 \def (Flat Appl) in (let TMP_27 \def (Flat -Cast) in (let TMP_28 \def (THead TMP_27 u t) in (let TMP_29 \def (THeads -TMP_26 vs TMP_28) in (let TMP_30 \def (ASort O n0) in (let TMP_31 \def (arity -g c TMP_29 TMP_30) in (let TMP_32 \def (Flat Appl) in (let TMP_33 \def (Flat -Cast) in (let TMP_34 \def (THead TMP_33 u t) in (let TMP_35 \def (THeads -TMP_32 vs TMP_34) in (let TMP_36 \def (sn3 c TMP_35) in (let TMP_37 \def -(land TMP_31 TMP_36) in (let TMP_72 \def (\lambda (H4: (arity g c (THeads + \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 (let TMP_38 \def (Flat Appl) in (let TMP_39 -\def (THeads TMP_38 vs t) in (let TMP_40 \def (ASort O n0) in (let TMP_41 -\def (arity g c TMP_39 TMP_40) in (let TMP_42 \def (Flat Appl) in (let TMP_43 -\def (THeads TMP_42 vs t) in (let TMP_44 \def (sn3 c TMP_43) in (let TMP_45 -\def (Flat Appl) in (let TMP_46 \def (Flat Cast) in (let TMP_47 \def (THead -TMP_46 u t) in (let TMP_48 \def (THeads TMP_45 vs TMP_47) in (let TMP_49 \def -(ASort O n0) in (let TMP_50 \def (arity g c TMP_48 TMP_49) in (let TMP_51 -\def (Flat Appl) in (let TMP_52 \def (Flat Cast) in (let TMP_53 \def (THead -TMP_52 u t) in (let TMP_54 \def (THeads TMP_51 vs TMP_53) in (let TMP_55 \def -(sn3 c TMP_54) in (let TMP_56 \def (land TMP_50 TMP_55) in (let TMP_71 \def -(\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort O n0))).(\lambda -(H8: (sn3 c (THeads (Flat Appl) vs t))).(let TMP_57 \def (Flat Appl) in (let -TMP_58 \def (Flat Cast) in (let TMP_59 \def (THead TMP_58 u t) in (let TMP_60 -\def (THeads TMP_57 vs TMP_59) in (let TMP_61 \def (ASort O n0) in (let -TMP_62 \def (arity g c TMP_60 TMP_61) in (let TMP_63 \def (Flat Appl) in (let -TMP_64 \def (Flat Cast) in (let TMP_65 \def (THead TMP_64 u t) in (let TMP_66 -\def (THeads TMP_63 vs TMP_65) in (let TMP_67 \def (sn3 c TMP_66) in (let -TMP_68 \def (ASort O n0) in (let TMP_69 \def (arity_appls_cast g c u t vs -TMP_68 H4 H7) in (let TMP_70 \def (sn3_appls_cast c vs u H5 t H8) in (conj -TMP_62 TMP_67 TMP_69 TMP_70))))))))))))))))) in (land_ind TMP_41 TMP_44 -TMP_56 TMP_71 H6)))))))))))))))))))))))) in (land_ind TMP_22 TMP_25 TMP_37 -TMP_72 H3))))))))))))))))))))))))) in (let TMP_133 \def (\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 (let TMP_74 \def -(Flat Appl) in (let TMP_75 \def (THeads TMP_74 vs u) in (let TMP_76 \def -(ASort n1 n0) in (let TMP_77 \def (arity g c TMP_75 TMP_76) in (let TMP_78 -\def (Flat Appl) in (let TMP_79 \def (THeads TMP_78 vs u) in (let TMP_80 \def -(sn3 c TMP_79) in (let TMP_81 \def (Flat Appl) in (let TMP_82 \def (Flat -Cast) in (let TMP_83 \def (THead TMP_82 u t) in (let TMP_84 \def (THeads -TMP_81 vs TMP_83) in (let TMP_85 \def (S n1) in (let TMP_86 \def (ASort -TMP_85 n0) in (let TMP_87 \def (arity g c TMP_84 TMP_86) in (let TMP_88 \def -(Flat Appl) in (let TMP_89 \def (Flat Cast) in (let TMP_90 \def (THead TMP_89 -u t) in (let TMP_91 \def (THeads TMP_88 vs TMP_90) in (let TMP_92 \def (sn3 c -TMP_91) in (let TMP_93 \def (land TMP_87 TMP_92) in (let TMP_132 \def -(\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 (let TMP_94 \def -(Flat Appl) in (let TMP_95 \def (THeads TMP_94 vs t) in (let TMP_96 \def (S -n1) in (let TMP_97 \def (ASort TMP_96 n0) in (let TMP_98 \def (arity g c -TMP_95 TMP_97) in (let TMP_99 \def (Flat Appl) in (let TMP_100 \def (THeads -TMP_99 vs t) in (let TMP_101 \def (sn3 c TMP_100) in (let TMP_102 \def (Flat -Appl) in (let TMP_103 \def (Flat Cast) in (let TMP_104 \def (THead TMP_103 u -t) in (let TMP_105 \def (THeads TMP_102 vs TMP_104) in (let TMP_106 \def (S -n1) in (let TMP_107 \def (ASort TMP_106 n0) in (let TMP_108 \def (arity g c -TMP_105 TMP_107) in (let TMP_109 \def (Flat Appl) in (let TMP_110 \def (Flat -Cast) in (let TMP_111 \def (THead TMP_110 u t) in (let TMP_112 \def (THeads -TMP_109 vs TMP_111) in (let TMP_113 \def (sn3 c TMP_112) in (let TMP_114 \def -(land TMP_108 TMP_113) in (let TMP_131 \def (\lambda (H7: (arity g c (THeads -(Flat Appl) vs t) (ASort (S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat -Appl) vs t))).(let TMP_115 \def (Flat Appl) in (let TMP_116 \def (Flat Cast) -in (let TMP_117 \def (THead TMP_116 u t) in (let TMP_118 \def (THeads TMP_115 -vs TMP_117) in (let TMP_119 \def (S n1) in (let TMP_120 \def (ASort TMP_119 -n0) in (let TMP_121 \def (arity g c TMP_118 TMP_120) in (let TMP_122 \def -(Flat Appl) in (let TMP_123 \def (Flat Cast) in (let TMP_124 \def (THead -TMP_123 u t) in (let TMP_125 \def (THeads TMP_122 vs TMP_124) in (let TMP_126 -\def (sn3 c TMP_125) in (let TMP_127 \def (S n1) in (let TMP_128 \def (ASort -TMP_127 n0) in (let TMP_129 \def (arity_appls_cast g c u t vs TMP_128 H4 H7) -in (let TMP_130 \def (sn3_appls_cast c vs u H5 t H8) in (conj TMP_121 TMP_126 -TMP_129 TMP_130))))))))))))))))))) in (land_ind TMP_98 TMP_101 TMP_114 -TMP_131 H6)))))))))))))))))))))))))) in (land_ind TMP_77 TMP_80 TMP_93 -TMP_132 H3))))))))))))))))))))))))))) in (nat_ind TMP_17 TMP_73 TMP_133 n H -H0)))))))))))) in (let TMP_270 \def (\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 (let TMP_135 -\def (Flat Appl) in (let TMP_136 \def (THeads TMP_135 vs u) in (let TMP_137 -\def (asucc g a1) in (let TMP_138 \def (AHead a0 TMP_137) in (let TMP_139 -\def (arity g c TMP_136 TMP_138) in (let TMP_146 \def (\forall (d: +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 (let TMP_140 \def (asucc g a1) in (let TMP_141 \def (Flat Appl) in (let -TMP_142 \def (Flat Appl) in (let TMP_143 \def (THeads TMP_142 vs u) in (let -TMP_144 \def (lift1 is TMP_143) in (let TMP_145 \def (THead TMP_141 w -TMP_144) in (sc3 g TMP_140 d TMP_145)))))))))))) in (let TMP_147 \def (Flat -Appl) in (let TMP_148 \def (Flat Cast) in (let TMP_149 \def (THead TMP_148 u -t) in (let TMP_150 \def (THeads TMP_147 vs TMP_149) in (let TMP_151 \def -(AHead a0 a1) in (let TMP_152 \def (arity g c TMP_150 TMP_151) in (let -TMP_160 \def (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall -(is: PList).((drop1 is d c) \to (let TMP_153 \def (Flat Appl) in (let TMP_154 -\def (Flat Appl) in (let TMP_155 \def (Flat Cast) in (let TMP_156 \def (THead -TMP_155 u t) in (let TMP_157 \def (THeads TMP_154 vs TMP_156) in (let TMP_158 -\def (lift1 is TMP_157) in (let TMP_159 \def (THead TMP_153 w TMP_158) in -(sc3 g a1 d TMP_159))))))))))))) in (let TMP_161 \def (land TMP_152 TMP_160) -in (let TMP_269 \def (\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 (let TMP_162 \def (Flat Appl) in (let TMP_163 -\def (THeads TMP_162 vs t) in (let TMP_164 \def (AHead a0 a1) in (let TMP_165 -\def (arity g c TMP_163 TMP_164) in (let TMP_171 \def (\forall (d: +\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 (let TMP_166 \def (Flat Appl) in (let TMP_167 \def (Flat Appl) in (let -TMP_168 \def (THeads TMP_167 vs t) in (let TMP_169 \def (lift1 is TMP_168) in -(let TMP_170 \def (THead TMP_166 w TMP_169) in (sc3 g a1 d TMP_170))))))))))) -in (let TMP_172 \def (Flat Appl) in (let TMP_173 \def (Flat Cast) in (let -TMP_174 \def (THead TMP_173 u t) in (let TMP_175 \def (THeads TMP_172 vs -TMP_174) in (let TMP_176 \def (AHead a0 a1) in (let TMP_177 \def (arity g c -TMP_175 TMP_176) in (let TMP_185 \def (\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (let TMP_178 \def (Flat -Appl) in (let TMP_179 \def (Flat Appl) in (let TMP_180 \def (Flat Cast) in -(let TMP_181 \def (THead TMP_180 u t) in (let TMP_182 \def (THeads TMP_179 vs -TMP_181) in (let TMP_183 \def (lift1 is TMP_182) in (let TMP_184 \def (THead -TMP_178 w TMP_183) in (sc3 g a1 d TMP_184))))))))))))) in (let TMP_186 \def -(land TMP_177 TMP_185) in (let TMP_268 \def (\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))))))))))).(let -TMP_187 \def (Flat Appl) in (let TMP_188 \def (Flat Cast) in (let TMP_189 -\def (THead TMP_188 u t) in (let TMP_190 \def (THeads TMP_187 vs TMP_189) in -(let TMP_191 \def (AHead a0 a1) in (let TMP_192 \def (arity g c TMP_190 -TMP_191) in (let TMP_200 \def (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) -\to (\forall (is: PList).((drop1 is d c) \to (let TMP_193 \def (Flat Appl) in -(let TMP_194 \def (Flat Appl) in (let TMP_195 \def (Flat Cast) in (let -TMP_196 \def (THead TMP_195 u t) in (let TMP_197 \def (THeads TMP_194 vs -TMP_196) in (let TMP_198 \def (lift1 is TMP_197) in (let TMP_199 \def (THead -TMP_193 w TMP_198) in (sc3 g a1 d TMP_199))))))))))))) in (let TMP_201 \def -(AHead a0 a1) in (let TMP_202 \def (arity_appls_cast g c u t vs TMP_201 H4 -H7) in (let TMP_267 \def (\lambda (d: C).(\lambda (w: T).(\lambda (H9: (sc3 g -a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let TMP_203 -\def (lifts1 is vs) in (let TMP_204 \def (TCons w TMP_203) in (let H_y \def -(H0 TMP_204) in (let TMP_205 \def (Flat Appl) in (let TMP_206 \def (lifts1 is -vs) in (let TMP_207 \def (Flat Cast) in (let TMP_208 \def (THead TMP_207 u t) -in (let TMP_209 \def (lift1 is TMP_208) in (let TMP_210 \def (THeads TMP_205 -TMP_206 TMP_209) in (let TMP_213 \def (\lambda (t0: T).(let TMP_211 \def -(Flat Appl) in (let TMP_212 \def (THead TMP_211 w t0) in (sc3 g a1 d -TMP_212)))) in (let TMP_214 \def (Flat Cast) in (let TMP_215 \def (lift1 is -u) in (let TMP_216 \def (lift1 is t) in (let TMP_217 \def (THead TMP_214 -TMP_215 TMP_216) in (let TMP_223 \def (\lambda (t0: T).(let TMP_218 \def -(Flat Appl) in (let TMP_219 \def (Flat Appl) in (let TMP_220 \def (lifts1 is -vs) in (let TMP_221 \def (THeads TMP_219 TMP_220 t0) in (let TMP_222 \def -(THead TMP_218 w TMP_221) in (sc3 g a1 d TMP_222))))))) in (let TMP_224 \def -(lift1 is u) in (let TMP_225 \def (Flat Appl) in (let TMP_226 \def (THeads -TMP_225 vs u) in (let TMP_227 \def (lift1 is TMP_226) in (let TMP_231 \def -(\lambda (t0: T).(let TMP_228 \def (asucc g a1) in (let TMP_229 \def (Flat -Appl) in (let TMP_230 \def (THead TMP_229 w t0) in (sc3 g TMP_228 d -TMP_230))))) in (let TMP_232 \def (H5 d w H9 is H10) in (let TMP_233 \def -(Flat Appl) in (let TMP_234 \def (lifts1 is vs) in (let TMP_235 \def (lift1 -is u) in (let TMP_236 \def (THeads TMP_233 TMP_234 TMP_235) in (let TMP_237 -\def (lifts1_flat Appl is u vs) in (let TMP_238 \def (eq_ind T TMP_227 -TMP_231 TMP_232 TMP_236 TMP_237) in (let TMP_239 \def (lift1 is t) in (let -TMP_240 \def (Flat Appl) in (let TMP_241 \def (THeads TMP_240 vs t) in (let -TMP_242 \def (lift1 is TMP_241) in (let TMP_245 \def (\lambda (t0: T).(let -TMP_243 \def (Flat Appl) in (let TMP_244 \def (THead TMP_243 w t0) in (sc3 g -a1 d TMP_244)))) in (let TMP_246 \def (H8 d w H9 is H10) in (let TMP_247 \def -(Flat Appl) in (let TMP_248 \def (lifts1 is vs) in (let TMP_249 \def (lift1 -is t) in (let TMP_250 \def (THeads TMP_247 TMP_248 TMP_249) in (let TMP_251 -\def (lifts1_flat Appl is t vs) in (let TMP_252 \def (eq_ind T TMP_242 -TMP_245 TMP_246 TMP_250 TMP_251) in (let TMP_253 \def (H_y d TMP_224 TMP_238 -TMP_239 TMP_252) in (let TMP_254 \def (Flat Cast) in (let TMP_255 \def (THead -TMP_254 u t) in (let TMP_256 \def (lift1 is TMP_255) in (let TMP_257 \def -(lift1_flat Cast is u t) in (let TMP_258 \def (eq_ind_r T TMP_217 TMP_223 -TMP_253 TMP_256 TMP_257) in (let TMP_259 \def (Flat Appl) in (let TMP_260 -\def (Flat Cast) in (let TMP_261 \def (THead TMP_260 u t) in (let TMP_262 -\def (THeads TMP_259 vs TMP_261) in (let TMP_263 \def (lift1 is TMP_262) in -(let TMP_264 \def (Flat Cast) in (let TMP_265 \def (THead TMP_264 u t) in -(let TMP_266 \def (lifts1_flat Appl is TMP_265 vs) in (eq_ind_r T TMP_210 -TMP_213 TMP_258 TMP_263 -TMP_266))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (conj -TMP_192 TMP_200 TMP_202 TMP_267))))))))))))) in (land_ind TMP_165 TMP_171 -TMP_186 TMP_268 H6)))))))))))))))))) in (land_ind TMP_139 TMP_146 TMP_161 -TMP_269 H3))))))))))))))))))))))))))) in (A_ind TMP_5 TMP_134 TMP_270 a))))). +\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: +fact 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).(let TMP_5 \def (\lambda (a0: A).(let TMP_1 -\def (\forall (c: C).(\forall (t: T).((sc3 g a0 c t) \to (sn3 c t)))) in (let -TMP_4 \def (\forall (vs: TList).(\forall (i: nat).(let TMP_2 \def (Flat Appl) -in (let TMP_3 \def (TLRef i) in (let t \def (THeads TMP_2 vs TMP_3) in + \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)))))))))) in (land TMP_1 TMP_4)))) in (let TMP_34 \def (\lambda -(n: nat).(\lambda (n0: nat).(let TMP_6 \def (\forall (c: C).(\forall (t: -T).((land (arity g c t (ASort n n0)) (sn3 c t)) \to (sn3 c t)))) in (let -TMP_16 \def (\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 (let TMP_7 \def (Flat Appl) in (let TMP_8 \def (TLRef i) in -(let TMP_9 \def (THeads TMP_7 vs TMP_8) in (let TMP_10 \def (ASort n n0) in -(let TMP_11 \def (arity g c TMP_9 TMP_10) in (let TMP_12 \def (Flat Appl) in -(let TMP_13 \def (TLRef i) in (let TMP_14 \def (THeads TMP_12 vs TMP_13) in -(let TMP_15 \def (sn3 c TMP_14) in (land TMP_11 TMP_15)))))))))))))))) in -(let TMP_22 \def (\lambda (c: C).(\lambda (t: T).(\lambda (H: (land (arity g -c t (ASort n n0)) (sn3 c t))).(let H0 \def H in (let TMP_17 \def (ASort n n0) -in (let TMP_18 \def (arity g c t TMP_17) in (let TMP_19 \def (sn3 c t) in -(let TMP_20 \def (sn3 c t) in (let TMP_21 \def (\lambda (_: (arity g c t -(ASort n n0))).(\lambda (H2: (sn3 c t)).H2)) in (land_ind TMP_18 TMP_19 -TMP_20 TMP_21 H0)))))))))) in (let TMP_33 \def (\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)).(let TMP_23 \def (Flat Appl) in (let TMP_24 \def (TLRef i) in -(let TMP_25 \def (THeads TMP_23 vs TMP_24) in (let TMP_26 \def (ASort n n0) -in (let TMP_27 \def (arity g c TMP_25 TMP_26) in (let TMP_28 \def (Flat Appl) -in (let TMP_29 \def (TLRef i) in (let TMP_30 \def (THeads TMP_28 vs TMP_29) -in (let TMP_31 \def (sn3 c TMP_30) in (let TMP_32 \def (sn3_appls_lref c i H0 -vs H1) in (conj TMP_27 TMP_31 H TMP_32))))))))))))))))) in (conj TMP_6 TMP_16 -TMP_22 TMP_33))))))) in (let TMP_254 \def (\lambda (a0: A).(\lambda (H: (land +(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 @@ -642,210 +402,121 @@ vs (TLRef i)) a0) \to ((nf2 c (TLRef i)) \to ((sns3 c vs) \to (sc3 g a0 c (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))))))))))).(let TMP_35 \def -(\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)))) -in (let TMP_48 \def (\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 (let TMP_36 \def (Flat Appl) in (let TMP_37 -\def (TLRef i) in (let TMP_38 \def (THeads TMP_36 vs TMP_37) in (let TMP_39 -\def (AHead a0 a1) in (let TMP_40 \def (arity g c TMP_38 TMP_39) in (let -TMP_47 \def (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (let TMP_41 \def (Flat Appl) in (let TMP_42 \def -(Flat Appl) in (let TMP_43 \def (TLRef i) in (let TMP_44 \def (THeads TMP_42 -vs TMP_43) in (let TMP_45 \def (lift1 is TMP_44) in (let TMP_46 \def (THead -TMP_41 w TMP_45) in (sc3 g a1 d TMP_46)))))))))))) in (land TMP_40 -TMP_47))))))))))))) in (let TMP_130 \def (\lambda (c: C).(\lambda (t: +(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 (let TMP_49 -\def (\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 t0)))) -in (let TMP_53 \def (\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 (let TMP_50 \def (Flat Appl) in (let TMP_51 \def (TLRef -i) in (let TMP_52 \def (THeads TMP_50 vs TMP_51) in (sc3 g a0 c0 -TMP_52)))))))))) in (let TMP_54 \def (sn3 c t) in (let TMP_129 \def (\lambda -(_: ((\forall (c0: C).(\forall (t0: T).((sc3 g a0 c0 t0) \to (sn3 c0 +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 (let TMP_55 \def (\forall (c0: C).(\forall -(t0: T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))) in (let TMP_59 \def (\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 (let -TMP_56 \def (Flat Appl) in (let TMP_57 \def (TLRef i) in (let TMP_58 \def -(THeads TMP_56 vs TMP_57) in (sc3 g a1 c0 TMP_58)))))))))) in (let TMP_60 -\def (sn3 c t) in (let TMP_128 \def (\lambda (H6: ((\forall (c0: C).(\forall -(t0: T).((sc3 g a1 c0 t0) \to (sn3 c0 t0)))))).(\lambda (_: ((\forall (vs: +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 (let TMP_61 -\def (AHead a0 a1) in (let TMP_62 \def (arity g c t TMP_61) in (let TMP_66 -\def (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: -PList).((drop1 is d c) \to (let TMP_63 \def (Flat Appl) in (let TMP_64 \def -(lift1 is t) in (let TMP_65 \def (THead TMP_63 w TMP_64) in (sc3 g a1 d -TMP_65))))))))) in (let TMP_67 \def (sn3 c t) in (let TMP_127 \def (\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 TMP_68 \def (AHead a0 a1) -in (let H_y \def (arity_aprem g c t TMP_68 H9 O a0) in (let TMP_69 \def -(aprem_zero a0 a1) in (let H11 \def (H_y TMP_69) in (let TMP_70 \def (\lambda -(d: C).(\lambda (_: T).(\lambda (j: nat).(drop j O d c)))) in (let TMP_72 -\def (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(let TMP_71 \def -(asucc g a0) in (arity g d u TMP_71))))) in (let TMP_73 \def (sn3 c t) in -(let TMP_126 \def (\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 TMP_74 \def (Bind Abst) in (let TMP_75 \def (CHead x0 TMP_74 x1) -in (let TMP_76 \def (TLRef O) in (let TMP_77 \def (Bind Abst) in (let TMP_78 -\def (CHead x0 TMP_77 x1) in (let TMP_79 \def (Bind Abst) in (let TMP_80 \def -(CHead x0 TMP_79 x1) in (let TMP_81 \def (getl_refl Abst x0 x1) in (let -TMP_82 \def (arity_abst g TMP_80 x0 x1 O TMP_81 a0 H13) in (let TMP_83 \def -(Bind Abst) in (let TMP_84 \def (CHead x0 TMP_83 x1) in (let TMP_85 \def -(getl_refl Abst x0 x1) in (let TMP_86 \def (nf2_lref_abst TMP_84 x0 x1 O -TMP_85) in (let TMP_87 \def (H4 TNil O TMP_78 TMP_82 TMP_86 I) in (let TMP_88 -\def (S x2) in (let TMP_89 \def (PCons TMP_88 O PNil) in (let H_y0 \def (H10 -TMP_75 TMP_76 TMP_87 TMP_89) in (let TMP_90 \def (Bind Abst) in (let TMP_91 -\def (CHead x0 TMP_90 x1) in (let TMP_92 \def (Flat Appl) in (let TMP_93 \def -(TLRef O) in (let TMP_94 \def (S x2) in (let TMP_95 \def (lift TMP_94 O t) in -(let TMP_96 \def (THead TMP_92 TMP_93 TMP_95) in (let TMP_97 \def (Bind Abst) -in (let TMP_98 \def (CHead x0 TMP_97 x1) in (let TMP_99 \def (S x2) in (let -TMP_100 \def (Bind Abst) in (let TMP_101 \def (drop_drop TMP_100 x2 x0 c H12 -x1) in (let TMP_102 \def (drop1_nil c) in (let TMP_103 \def (drop1_cons -TMP_98 c TMP_99 O TMP_101 c PNil TMP_102) in (let TMP_104 \def (H_y0 TMP_103) -in (let H_y1 \def (H6 TMP_91 TMP_96 TMP_104) in (let TMP_105 \def (Bind Abst) -in (let TMP_106 \def (CHead x0 TMP_105 x1) in (let TMP_107 \def (TLRef O) in -(let TMP_108 \def (S x2) in (let TMP_109 \def (lift TMP_108 O t) in (let H_x -\def (sn3_gen_flat Appl TMP_106 TMP_107 TMP_109 H_y1) in (let H14 \def H_x in -(let TMP_110 \def (Bind Abst) in (let TMP_111 \def (CHead x0 TMP_110 x1) in -(let TMP_112 \def (TLRef O) in (let TMP_113 \def (sn3 TMP_111 TMP_112) in -(let TMP_114 \def (Bind Abst) in (let TMP_115 \def (CHead x0 TMP_114 x1) in -(let TMP_116 \def (S x2) in (let TMP_117 \def (lift TMP_116 O t) in (let -TMP_118 \def (sn3 TMP_115 TMP_117) in (let TMP_119 \def (sn3 c t) in (let -TMP_125 \def (\lambda (_: (sn3 (CHead x0 (Bind Abst) x1) (TLRef O))).(\lambda -(H16: (sn3 (CHead x0 (Bind Abst) x1) (lift (S x2) O t))).(let TMP_120 \def -(Bind Abst) in (let TMP_121 \def (CHead x0 TMP_120 x1) in (let TMP_122 \def -(S x2) in (let TMP_123 \def (Bind Abst) in (let TMP_124 \def (drop_drop -TMP_123 x2 x0 c H12 x1) in (sn3_gen_lift TMP_121 t TMP_122 O H16 c -TMP_124)))))))) in (land_ind TMP_113 TMP_118 TMP_119 TMP_125 -H14))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (ex2_3_ind C -T nat TMP_70 TMP_72 TMP_73 TMP_126 H11))))))))))) in (land_ind TMP_62 TMP_66 -TMP_67 TMP_127 H8))))))))) in (land_ind TMP_55 TMP_59 TMP_60 TMP_128 -H5)))))))) in (land_ind TMP_49 TMP_53 TMP_54 TMP_129 H2))))))))) in (let -TMP_253 \def (\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)).(let TMP_131 \def (Flat -Appl) in (let TMP_132 \def (TLRef i) in (let TMP_133 \def (THeads TMP_131 vs -TMP_132) in (let TMP_134 \def (AHead a0 a1) in (let TMP_135 \def (arity g c -TMP_133 TMP_134) in (let TMP_142 \def (\forall (d: C).(\forall (w: T).((sc3 g -a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (let TMP_136 \def (Flat -Appl) in (let TMP_137 \def (Flat Appl) in (let TMP_138 \def (TLRef i) in (let -TMP_139 \def (THeads TMP_137 vs TMP_138) in (let TMP_140 \def (lift1 is -TMP_139) in (let TMP_141 \def (THead TMP_136 w TMP_140) in (sc3 g a1 d -TMP_141)))))))))))) in (let TMP_252 \def (\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 (let TMP_143 \def (\forall (c0: C).(\forall (t: -T).((sc3 g a0 c0 t) \to (sn3 c0 t)))) in (let TMP_147 \def (\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 (let -TMP_144 \def (Flat Appl) in (let TMP_145 \def (TLRef i0) in (let TMP_146 \def -(THeads TMP_144 vs0 TMP_145) in (sc3 g a0 c0 TMP_146)))))))))) in (let -TMP_148 \def (Flat Appl) in (let TMP_149 \def (Flat Appl) in (let TMP_150 -\def (TLRef i) in (let TMP_151 \def (THeads TMP_149 vs TMP_150) in (let -TMP_152 \def (lift1 is TMP_151) in (let TMP_153 \def (THead TMP_148 w -TMP_152) in (let TMP_154 \def (sc3 g a1 d TMP_153) in (let TMP_251 \def -(\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 (let TMP_155 \def (\forall (c0: C).(\forall -(t: T).((sc3 g a1 c0 t) \to (sn3 c0 t)))) in (let TMP_159 \def (\forall (vs0: +(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)) a1) \to ((nf2 c0 (TLRef i0)) \to ((sns3 c0 vs0) \to (let -TMP_156 \def (Flat Appl) in (let TMP_157 \def (TLRef i0) in (let TMP_158 \def -(THeads TMP_156 vs0 TMP_157) in (sc3 g a1 c0 TMP_158)))))))))) in (let -TMP_160 \def (Flat Appl) in (let TMP_161 \def (Flat Appl) in (let TMP_162 -\def (TLRef i) in (let TMP_163 \def (THeads TMP_161 vs TMP_162) in (let -TMP_164 \def (lift1 is TMP_163) in (let TMP_165 \def (THead TMP_160 w -TMP_164) in (let TMP_166 \def (sc3 g a1 d TMP_165) in (let TMP_250 \def -(\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 TMP_167 \def (lifts1 is vs) in (let TMP_168 \def -(TCons w TMP_167) in (let H_y \def (H11 TMP_168) in (let TMP_169 \def (Flat -Appl) in (let TMP_170 \def (lifts1 is vs) in (let TMP_171 \def (TLRef i) in -(let TMP_172 \def (lift1 is TMP_171) in (let TMP_173 \def (THeads TMP_169 -TMP_170 TMP_172) in (let TMP_176 \def (\lambda (t: T).(let TMP_174 \def (Flat -Appl) in (let TMP_175 \def (THead TMP_174 w t) in (sc3 g a1 d TMP_175)))) in -(let TMP_177 \def (trans is i) in (let TMP_178 \def (TLRef TMP_177) in (let -TMP_184 \def (\lambda (t: T).(let TMP_179 \def (Flat Appl) in (let TMP_180 -\def (Flat Appl) in (let TMP_181 \def (lifts1 is vs) in (let TMP_182 \def -(THeads TMP_180 TMP_181 t) in (let TMP_183 \def (THead TMP_179 w TMP_182) in -(sc3 g a1 d TMP_183))))))) in (let TMP_185 \def (trans is i) in (let TMP_186 -\def (TLRef i) in (let TMP_187 \def (lift1 is TMP_186) in (let TMP_193 \def -(\lambda (t: T).(let TMP_188 \def (Flat Appl) in (let TMP_189 \def (Flat -Appl) in (let TMP_190 \def (lifts1 is vs) in (let TMP_191 \def (THeads -TMP_189 TMP_190 t) in (let TMP_192 \def (THead TMP_188 w TMP_191) in (arity g -d TMP_192 a1))))))) in (let TMP_194 \def (Flat Appl) in (let TMP_195 \def -(TLRef i) in (let TMP_196 \def (THeads TMP_194 vs TMP_195) in (let TMP_197 -\def (lift1 is TMP_196) in (let TMP_200 \def (\lambda (t: T).(let TMP_198 -\def (Flat Appl) in (let TMP_199 \def (THead TMP_198 w t) in (arity g d -TMP_199 a1)))) in (let TMP_201 \def (sc3_arity_gen g d w a0 H4) in (let -TMP_202 \def (Flat Appl) in (let TMP_203 \def (TLRef i) in (let TMP_204 \def -(THeads TMP_202 vs TMP_203) in (let TMP_205 \def (lift1 is TMP_204) in (let -TMP_206 \def (AHead a0 a1) in (let TMP_207 \def (Flat Appl) in (let TMP_208 -\def (TLRef i) in (let TMP_209 \def (THeads TMP_207 vs TMP_208) in (let -TMP_210 \def (arity_lift1 g TMP_206 c is d TMP_209 H5 H1) in (let TMP_211 -\def (arity_appl g d w a0 TMP_201 TMP_205 a1 TMP_210) in (let TMP_212 \def -(Flat Appl) in (let TMP_213 \def (lifts1 is vs) in (let TMP_214 \def (TLRef -i) in (let TMP_215 \def (lift1 is TMP_214) in (let TMP_216 \def (THeads -TMP_212 TMP_213 TMP_215) in (let TMP_217 \def (TLRef i) in (let TMP_218 \def -(lifts1_flat Appl is TMP_217 vs) in (let TMP_219 \def (eq_ind T TMP_197 -TMP_200 TMP_211 TMP_216 TMP_218) in (let TMP_220 \def (trans is i) in (let -TMP_221 \def (TLRef TMP_220) in (let TMP_222 \def (lift1_lref is i) in (let -TMP_223 \def (eq_ind T TMP_187 TMP_193 TMP_219 TMP_221 TMP_222) in (let -TMP_224 \def (TLRef i) in (let TMP_225 \def (lift1 is TMP_224) in (let -TMP_226 \def (\lambda (t: T).(nf2 d t)) in (let TMP_227 \def (TLRef i) in -(let TMP_228 \def (nf2_lift1 c is d TMP_227 H5 H2) in (let TMP_229 \def -(trans is i) in (let TMP_230 \def (TLRef TMP_229) in (let TMP_231 \def -(lift1_lref is i) in (let TMP_232 \def (eq_ind T TMP_225 TMP_226 TMP_228 -TMP_230 TMP_231) in (let TMP_233 \def (sn3 d w) in (let TMP_234 \def (lifts1 -is vs) in (let TMP_235 \def (sns3 d TMP_234) in (let TMP_236 \def (H7 d w H4) -in (let TMP_237 \def (sns3_lifts1 c is d H5 vs H3) in (let TMP_238 \def (conj -TMP_233 TMP_235 TMP_236 TMP_237) in (let TMP_239 \def (H_y TMP_185 d TMP_223 -TMP_232 TMP_238) in (let TMP_240 \def (TLRef i) in (let TMP_241 \def (lift1 -is TMP_240) in (let TMP_242 \def (lift1_lref is i) in (let TMP_243 \def -(eq_ind_r T TMP_178 TMP_184 TMP_239 TMP_241 TMP_242) in (let TMP_244 \def -(Flat Appl) in (let TMP_245 \def (TLRef i) in (let TMP_246 \def (THeads -TMP_244 vs TMP_245) in (let TMP_247 \def (lift1 is TMP_246) in (let TMP_248 -\def (TLRef i) in (let TMP_249 \def (lifts1_flat Appl is TMP_248 vs) in -(eq_ind_r T TMP_173 TMP_176 TMP_243 TMP_247 -TMP_249))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -)) in (land_ind TMP_155 TMP_159 TMP_166 TMP_250 H9)))))))))))))) in (land_ind -TMP_143 TMP_147 TMP_154 TMP_251 H6))))))))))))))))) in (conj TMP_135 TMP_142 -H1 TMP_252)))))))))))))) in (conj TMP_35 TMP_48 TMP_130 TMP_253))))))))) in -(A_ind TMP_5 TMP_34 TMP_254 a))))). +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: +lemma 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 (let TMP_1 \def (\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to -(sn3 c0 t0)))) in (let TMP_5 \def (\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 (let TMP_2 \def (Flat Appl) in (let -TMP_3 \def (TLRef i) in (let TMP_4 \def (THeads TMP_2 vs TMP_3) in (sc3 g a -c0 TMP_4)))))))))) in (let TMP_6 \def (sn3 c t) in (let TMP_7 \def (\lambda -(H1: ((\forall (c0: C).(\forall (t0: T).((sc3 g a c0 t0) \to (sn3 c0 +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))) in (land_ind TMP_1 TMP_5 TMP_6 TMP_7 H0))))))))))). +i))))))))))).(H1 c t H))) H0))))))). -theorem sc3_abst: +lemma 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)))))))))) @@ -853,20 +524,16 @@ i)) \to ((sns3 c vs) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))) \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 (let TMP_1 \def (\forall -(c0: C).(\forall (t: T).((sc3 g a c0 t) \to (sn3 c0 t)))) in (let TMP_5 \def -(\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 (let TMP_2 \def (Flat Appl) in (let TMP_3 \def (TLRef i0) in (let -TMP_4 \def (THeads TMP_2 vs0 TMP_3) in (sc3 g a c0 TMP_4)))))))))) in (let -TMP_6 \def (Flat Appl) in (let TMP_7 \def (TLRef i) in (let TMP_8 \def -(THeads TMP_6 vs TMP_7) in (let TMP_9 \def (sc3 g a c TMP_8) in (let TMP_10 -\def (\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))) in (land_ind TMP_1 TMP_5 TMP_9 TMP_10 -H2))))))))))))))))). +(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: @@ -876,154 +543,73 @@ T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (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).(let TMP_5 \def (\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 (let TMP_1 \def (Flat Appl) in (let TMP_2 \def (Bind b) in (let TMP_3 -\def (THead TMP_2 v t) in (let TMP_4 \def (THeads TMP_1 vs TMP_3) in (sc3 g a -c TMP_4)))))))))))) in (let TMP_50 \def (\lambda (n: nat).(\lambda (n0: +(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 (let TMP_6 \def (Bind b) in (let TMP_7 \def (CHead c TMP_6 v) in (let -TMP_8 \def (Flat Appl) in (let TMP_9 \def (S O) in (let TMP_10 \def (lifts -TMP_9 O vs) in (let TMP_11 \def (THeads TMP_8 TMP_10 t) in (let TMP_12 \def -(ASort n n0) in (let TMP_13 \def (arity g TMP_7 TMP_11 TMP_12) in (let TMP_14 -\def (Bind b) in (let TMP_15 \def (CHead c TMP_14 v) in (let TMP_16 \def -(Flat Appl) in (let TMP_17 \def (S O) in (let TMP_18 \def (lifts TMP_17 O vs) -in (let TMP_19 \def (THeads TMP_16 TMP_18 t) in (let TMP_20 \def (sn3 TMP_15 -TMP_19) in (let TMP_21 \def (Flat Appl) in (let TMP_22 \def (Bind b) in (let -TMP_23 \def (THead TMP_22 v t) in (let TMP_24 \def (THeads TMP_21 vs TMP_23) -in (let TMP_25 \def (ASort n n0) in (let TMP_26 \def (arity g c TMP_24 -TMP_25) in (let TMP_27 \def (Flat Appl) in (let TMP_28 \def (Bind b) in (let -TMP_29 \def (THead TMP_28 v t) in (let TMP_30 \def (THeads TMP_27 vs TMP_29) -in (let TMP_31 \def (sn3 c TMP_30) in (let TMP_32 \def (land TMP_26 TMP_31) -in (let TMP_49 \def (\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))).(let TMP_33 \def (Flat -Appl) in (let TMP_34 \def (Bind b) in (let TMP_35 \def (THead TMP_34 v t) in -(let TMP_36 \def (THeads TMP_33 vs TMP_35) in (let TMP_37 \def (ASort n n0) -in (let TMP_38 \def (arity g c TMP_36 TMP_37) in (let TMP_39 \def (Flat Appl) -in (let TMP_40 \def (Bind b) in (let TMP_41 \def (THead TMP_40 v t) in (let -TMP_42 \def (THeads TMP_39 vs TMP_41) in (let TMP_43 \def (sn3 c TMP_42) in -(let TMP_44 \def (sc3_arity_gen g c v a1 H1) in (let TMP_45 \def (ASort n n0) -in (let TMP_46 \def (arity_appls_bind g b H c v a1 TMP_44 t vs TMP_45 H3) in -(let TMP_47 \def (sc3_sn3 g a1 c v H1) in (let TMP_48 \def (sn3_appls_bind b -H c v TMP_47 vs t H4) in (conj TMP_38 TMP_43 TMP_46 TMP_48))))))))))))))))))) -in (land_ind TMP_13 TMP_20 TMP_32 TMP_49 -H2)))))))))))))))))))))))))))))))))))))) in (let TMP_206 \def (\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) +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 (let TMP_51 -\def (Bind b) in (let TMP_52 \def (CHead c TMP_51 v) in (let TMP_53 \def -(Flat Appl) in (let TMP_54 \def (S O) in (let TMP_55 \def (lifts TMP_54 O vs) -in (let TMP_56 \def (THeads TMP_53 TMP_55 t) in (let TMP_57 \def (AHead a a0) -in (let TMP_58 \def (arity g TMP_52 TMP_56 TMP_57) in (let TMP_66 \def +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 (CHead c (Bind b) v)) \to (let TMP_59 \def (Flat Appl) in -(let TMP_60 \def (Flat Appl) in (let TMP_61 \def (S O) in (let TMP_62 \def -(lifts TMP_61 O vs) in (let TMP_63 \def (THeads TMP_60 TMP_62 t) in (let -TMP_64 \def (lift1 is TMP_63) in (let TMP_65 \def (THead TMP_59 w TMP_64) in -(sc3 g a0 d TMP_65))))))))))))) in (let TMP_67 \def (Flat Appl) in (let -TMP_68 \def (Bind b) in (let TMP_69 \def (THead TMP_68 v t) in (let TMP_70 -\def (THeads TMP_67 vs TMP_69) in (let TMP_71 \def (AHead a a0) in (let -TMP_72 \def (arity g c TMP_70 TMP_71) in (let TMP_80 \def (\forall (d: +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 (let TMP_73 \def (Flat Appl) in (let TMP_74 \def (Flat Appl) in (let -TMP_75 \def (Bind b) in (let TMP_76 \def (THead TMP_75 v t) in (let TMP_77 -\def (THeads TMP_74 vs TMP_76) in (let TMP_78 \def (lift1 is TMP_77) in (let -TMP_79 \def (THead TMP_73 w TMP_78) in (sc3 g a0 d TMP_79))))))))))))) in -(let TMP_81 \def (land TMP_72 TMP_80) in (let TMP_205 \def (\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))))))))))).(let TMP_82 \def (Flat Appl) in (let TMP_83 \def (Bind b) in -(let TMP_84 \def (THead TMP_83 v t) in (let TMP_85 \def (THeads TMP_82 vs -TMP_84) in (let TMP_86 \def (AHead a a0) in (let TMP_87 \def (arity g c -TMP_85 TMP_86) in (let TMP_95 \def (\forall (d: C).(\forall (w: T).((sc3 g a -d w) \to (\forall (is: PList).((drop1 is d c) \to (let TMP_88 \def (Flat -Appl) in (let TMP_89 \def (Flat Appl) in (let TMP_90 \def (Bind b) in (let -TMP_91 \def (THead TMP_90 v t) in (let TMP_92 \def (THeads TMP_89 vs TMP_91) -in (let TMP_93 \def (lift1 is TMP_92) in (let TMP_94 \def (THead TMP_88 w -TMP_93) in (sc3 g a0 d TMP_94))))))))))))) in (let TMP_96 \def (sc3_arity_gen -g c v a1 H3) in (let TMP_97 \def (AHead a a0) in (let TMP_98 \def -(arity_appls_bind g b H c v a1 TMP_96 t vs TMP_97 H5) in (let TMP_204 \def -(\lambda (d: C).(\lambda (w: T).(\lambda (H7: (sc3 g a d w)).(\lambda (is: -PList).(\lambda (H8: (drop1 is d c)).(let TMP_99 \def (lifts1 is vs) in (let -TMP_100 \def (TCons w TMP_99) in (let H_y \def (H1 TMP_100) in (let TMP_101 -\def (Flat Appl) in (let TMP_102 \def (lifts1 is vs) in (let TMP_103 \def -(Bind b) in (let TMP_104 \def (THead TMP_103 v t) in (let TMP_105 \def (lift1 -is TMP_104) in (let TMP_106 \def (THeads TMP_101 TMP_102 TMP_105) in (let -TMP_109 \def (\lambda (t0: T).(let TMP_107 \def (Flat Appl) in (let TMP_108 -\def (THead TMP_107 w t0) in (sc3 g a0 d TMP_108)))) in (let TMP_110 \def -(Bind b) in (let TMP_111 \def (lift1 is v) in (let TMP_112 \def (Ss is) in -(let TMP_113 \def (lift1 TMP_112 t) in (let TMP_114 \def (THead TMP_110 -TMP_111 TMP_113) in (let TMP_120 \def (\lambda (t0: T).(let TMP_115 \def -(Flat Appl) in (let TMP_116 \def (Flat Appl) in (let TMP_117 \def (lifts1 is -vs) in (let TMP_118 \def (THeads TMP_116 TMP_117 t0) in (let TMP_119 \def -(THead TMP_115 w TMP_118) in (sc3 g a0 d TMP_119))))))) in (let TMP_121 \def -(lift1 is v) in (let TMP_122 \def (Ss is) in (let TMP_123 \def (lift1 TMP_122 -t) in (let TMP_124 \def (Ss is) in (let TMP_125 \def (S O) in (let TMP_126 -\def (lifts TMP_125 O vs) in (let TMP_127 \def (lifts1 TMP_124 TMP_126) in -(let TMP_139 \def (\lambda (t0: TList).(let TMP_128 \def (Bind b) in (let -TMP_129 \def (lift1 is v) in (let TMP_130 \def (CHead d TMP_128 TMP_129) in -(let TMP_131 \def (Flat Appl) in (let TMP_132 \def (S O) in (let TMP_133 \def -(lift TMP_132 O w) in (let TMP_134 \def (Flat Appl) in (let TMP_135 \def (Ss -is) in (let TMP_136 \def (lift1 TMP_135 t) in (let TMP_137 \def (THeads -TMP_134 t0 TMP_136) in (let TMP_138 \def (THead TMP_131 TMP_133 TMP_137) in -(sc3 g a0 TMP_130 TMP_138))))))))))))) in (let TMP_140 \def (Ss is) in (let -TMP_141 \def (Flat Appl) in (let TMP_142 \def (S O) in (let TMP_143 \def -(lifts TMP_142 O vs) in (let TMP_144 \def (THeads TMP_141 TMP_143 t) in (let -TMP_145 \def (lift1 TMP_140 TMP_144) in (let TMP_153 \def (\lambda (t0: -T).(let TMP_146 \def (Bind b) in (let TMP_147 \def (lift1 is v) in (let -TMP_148 \def (CHead d TMP_146 TMP_147) in (let TMP_149 \def (Flat Appl) in -(let TMP_150 \def (S O) in (let TMP_151 \def (lift TMP_150 O w) in (let -TMP_152 \def (THead TMP_149 TMP_151 t0) in (sc3 g a0 TMP_148 TMP_152))))))))) -in (let TMP_154 \def (Bind b) in (let TMP_155 \def (lift1 is v) in (let -TMP_156 \def (CHead d TMP_154 TMP_155) in (let TMP_157 \def (S O) in (let -TMP_158 \def (lift TMP_157 O w) in (let TMP_159 \def (Bind b) in (let TMP_160 -\def (lift1 is v) in (let TMP_161 \def (CHead d TMP_159 TMP_160) in (let -TMP_162 \def (S O) in (let TMP_163 \def (Bind b) in (let TMP_164 \def -(drop_refl d) in (let TMP_165 \def (lift1 is v) in (let TMP_166 \def -(drop_drop TMP_163 O d d TMP_164 TMP_165) in (let TMP_167 \def (sc3_lift g a -d w H7 TMP_161 TMP_162 O TMP_166) in (let TMP_168 \def (Ss is) in (let -TMP_169 \def (drop1_skip_bind b c is d v H8) in (let TMP_170 \def (H6 TMP_156 -TMP_158 TMP_167 TMP_168 TMP_169) in (let TMP_171 \def (Flat Appl) in (let -TMP_172 \def (Ss is) in (let TMP_173 \def (S O) in (let TMP_174 \def (lifts -TMP_173 O vs) in (let TMP_175 \def (lifts1 TMP_172 TMP_174) in (let TMP_176 -\def (Ss is) in (let TMP_177 \def (lift1 TMP_176 t) in (let TMP_178 \def -(THeads TMP_171 TMP_175 TMP_177) in (let TMP_179 \def (Ss is) in (let TMP_180 -\def (S O) in (let TMP_181 \def (lifts TMP_180 O vs) in (let TMP_182 \def -(lifts1_flat Appl TMP_179 t TMP_181) in (let TMP_183 \def (eq_ind T TMP_145 -TMP_153 TMP_170 TMP_178 TMP_182) in (let TMP_184 \def (S O) in (let TMP_185 -\def (lifts1 is vs) in (let TMP_186 \def (lifts TMP_184 O TMP_185) in (let -TMP_187 \def (lifts1_xhg is vs) in (let TMP_188 \def (eq_ind TList TMP_127 -TMP_139 TMP_183 TMP_186 TMP_187) in (let TMP_189 \def (sc3_lift1 g c a1 is d -v H3 H8) in (let TMP_190 \def (H_y d TMP_121 TMP_123 TMP_188 TMP_189) in (let -TMP_191 \def (Bind b) in (let TMP_192 \def (THead TMP_191 v t) in (let -TMP_193 \def (lift1 is TMP_192) in (let TMP_194 \def (lift1_bind b is v t) in -(let TMP_195 \def (eq_ind_r T TMP_114 TMP_120 TMP_190 TMP_193 TMP_194) in -(let TMP_196 \def (Flat Appl) in (let TMP_197 \def (Bind b) in (let TMP_198 -\def (THead TMP_197 v t) in (let TMP_199 \def (THeads TMP_196 vs TMP_198) in -(let TMP_200 \def (lift1 is TMP_199) in (let TMP_201 \def (Bind b) in (let -TMP_202 \def (THead TMP_201 v t) in (let TMP_203 \def (lifts1_flat Appl is -TMP_202 vs) in (eq_ind_r T TMP_106 TMP_109 TMP_195 TMP_200 -TMP_203))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -)))))))))))))))) in (conj TMP_87 TMP_95 TMP_98 TMP_204)))))))))))))) in -(land_ind TMP_58 TMP_66 TMP_81 TMP_205 H4)))))))))))))))))))))))))))))) in -(A_ind TMP_5 TMP_50 TMP_206 a2)))))))). +\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: @@ -1032,167 +618,80 @@ TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 c (THeads 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).(let TMP_7 \def (\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 (let TMP_1 \def -(Flat Appl) in (let TMP_2 \def (Flat Appl) in (let TMP_3 \def (Bind Abst) in -(let TMP_4 \def (THead TMP_3 w t) in (let TMP_5 \def (THead TMP_2 v TMP_4) in -(let TMP_6 \def (THeads TMP_1 vs TMP_5) in (sc3 g a c TMP_6)))))))))))))))) -in (let TMP_59 \def (\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 (let TMP_8 \def (Flat Appl) in (let TMP_9 \def (Bind Abbr) in (let TMP_10 -\def (THead TMP_9 v t) in (let TMP_11 \def (THeads TMP_8 vs TMP_10) in (let -TMP_12 \def (ASort n n0) in (let TMP_13 \def (arity g c TMP_11 TMP_12) in -(let TMP_14 \def (Flat Appl) in (let TMP_15 \def (Bind Abbr) in (let TMP_16 -\def (THead TMP_15 v t) in (let TMP_17 \def (THeads TMP_14 vs TMP_16) in (let -TMP_18 \def (sn3 c TMP_17) in (let TMP_19 \def (Flat Appl) in (let TMP_20 -\def (Flat Appl) in (let TMP_21 \def (Bind Abst) in (let TMP_22 \def (THead -TMP_21 w t) in (let TMP_23 \def (THead TMP_20 v TMP_22) in (let TMP_24 \def -(THeads TMP_19 vs TMP_23) in (let TMP_25 \def (ASort n n0) in (let TMP_26 -\def (arity g c TMP_24 TMP_25) in (let TMP_27 \def (Flat Appl) in (let TMP_28 -\def (Flat Appl) in (let TMP_29 \def (Bind Abst) in (let TMP_30 \def (THead -TMP_29 w t) in (let TMP_31 \def (THead TMP_28 v TMP_30) in (let TMP_32 \def -(THeads TMP_27 vs TMP_31) in (let TMP_33 \def (sn3 c TMP_32) in (let TMP_34 -\def (land TMP_26 TMP_33) in (let TMP_58 \def (\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)))).(let TMP_35 \def -(Flat Appl) in (let TMP_36 \def (Flat Appl) in (let TMP_37 \def (Bind Abst) -in (let TMP_38 \def (THead TMP_37 w t) in (let TMP_39 \def (THead TMP_36 v -TMP_38) in (let TMP_40 \def (THeads TMP_35 vs TMP_39) in (let TMP_41 \def -(ASort n n0) in (let TMP_42 \def (arity g c TMP_40 TMP_41) in (let TMP_43 -\def (Flat Appl) in (let TMP_44 \def (Flat Appl) in (let TMP_45 \def (Bind -Abst) in (let TMP_46 \def (THead TMP_45 w t) in (let TMP_47 \def (THead -TMP_44 v TMP_46) in (let TMP_48 \def (THeads TMP_43 vs TMP_47) in (let TMP_49 -\def (sn3 c TMP_48) in (let TMP_50 \def (sc3_arity_gen g c v a1 H0) in (let -TMP_51 \def (asucc g a1) in (let TMP_52 \def (sc3_arity_gen g c w TMP_51 H1) -in (let TMP_53 \def (ASort n n0) in (let TMP_54 \def (arity_appls_appl g c v -a1 TMP_50 w TMP_52 t vs TMP_53 H3) in (let TMP_55 \def (asucc g a1) in (let -TMP_56 \def (sc3_sn3 g TMP_55 c w H1) in (let TMP_57 \def (sn3_appls_beta c v -t vs H4 w TMP_56) in (conj TMP_42 TMP_49 TMP_54 -TMP_57)))))))))))))))))))))))))) in (land_ind TMP_13 TMP_18 TMP_34 TMP_58 -H2)))))))))))))))))))))))))))))))))))))))) in (let TMP_226 \def (\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 + \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 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 (let TMP_60 -\def (Flat Appl) in (let TMP_61 \def (Bind Abbr) in (let TMP_62 \def (THead -TMP_61 v t) in (let TMP_63 \def (THeads TMP_60 vs TMP_62) in (let TMP_64 \def -(AHead a a0) in (let TMP_65 \def (arity g c TMP_63 TMP_64) in (let TMP_73 -\def (\forall (d: C).(\forall (w0: T).((sc3 g a d w0) \to (\forall (is: -PList).((drop1 is d c) \to (let TMP_66 \def (Flat Appl) in (let TMP_67 \def -(Flat Appl) in (let TMP_68 \def (Bind Abbr) in (let TMP_69 \def (THead TMP_68 -v t) in (let TMP_70 \def (THeads TMP_67 vs TMP_69) in (let TMP_71 \def (lift1 -is TMP_70) in (let TMP_72 \def (THead TMP_66 w0 TMP_71) in (sc3 g a0 d -TMP_72))))))))))))) in (let TMP_74 \def (Flat Appl) in (let TMP_75 \def (Flat -Appl) in (let TMP_76 \def (Bind Abst) in (let TMP_77 \def (THead TMP_76 w t) -in (let TMP_78 \def (THead TMP_75 v TMP_77) in (let TMP_79 \def (THeads -TMP_74 vs TMP_78) in (let TMP_80 \def (AHead a a0) in (let TMP_81 \def (arity -g c TMP_79 TMP_80) in (let TMP_91 \def (\forall (d: C).(\forall (w0: T).((sc3 -g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (let TMP_82 \def (Flat -Appl) in (let TMP_83 \def (Flat Appl) in (let TMP_84 \def (Flat Appl) in (let -TMP_85 \def (Bind Abst) in (let TMP_86 \def (THead TMP_85 w t) in (let TMP_87 -\def (THead TMP_84 v TMP_86) in (let TMP_88 \def (THeads TMP_83 vs TMP_87) in -(let TMP_89 \def (lift1 is TMP_88) in (let TMP_90 \def (THead TMP_82 w0 -TMP_89) in (sc3 g a0 d TMP_90))))))))))))))) in (let TMP_92 \def (land TMP_81 -TMP_91) in (let TMP_225 \def (\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)))))))))))).(let TMP_93 \def (Flat Appl) in (let TMP_94 \def -(Flat Appl) in (let TMP_95 \def (Bind Abst) in (let TMP_96 \def (THead TMP_95 -w t) in (let TMP_97 \def (THead TMP_94 v TMP_96) in (let TMP_98 \def (THeads -TMP_93 vs TMP_97) in (let TMP_99 \def (AHead a a0) in (let TMP_100 \def -(arity g c TMP_98 TMP_99) in (let TMP_110 \def (\forall (d: C).(\forall (w0: -T).((sc3 g a d w0) \to (\forall (is: PList).((drop1 is d c) \to (let TMP_101 -\def (Flat Appl) in (let TMP_102 \def (Flat Appl) in (let TMP_103 \def (Flat -Appl) in (let TMP_104 \def (Bind Abst) in (let TMP_105 \def (THead TMP_104 w -t) in (let TMP_106 \def (THead TMP_103 v TMP_105) in (let TMP_107 \def -(THeads TMP_102 vs TMP_106) in (let TMP_108 \def (lift1 is TMP_107) in (let -TMP_109 \def (THead TMP_101 w0 TMP_108) in (sc3 g a0 d TMP_109))))))))))))))) -in (let TMP_111 \def (sc3_arity_gen g c v a1 H2) in (let TMP_112 \def (asucc -g a1) in (let TMP_113 \def (sc3_arity_gen g c w TMP_112 H3) in (let TMP_114 -\def (AHead a a0) in (let TMP_115 \def (arity_appls_appl g c v a1 TMP_111 w -TMP_113 t vs TMP_114 H5) in (let TMP_224 \def (\lambda (d: C).(\lambda (w0: -T).(\lambda (H7: (sc3 g a d w0)).(\lambda (is: PList).(\lambda (H8: (drop1 is -d c)).(let TMP_116 \def (Flat Appl) in (let TMP_117 \def (lifts1 is vs) in -(let TMP_118 \def (Flat Appl) in (let TMP_119 \def (Bind Abst) in (let -TMP_120 \def (THead TMP_119 w t) in (let TMP_121 \def (THead TMP_118 v -TMP_120) in (let TMP_122 \def (lift1 is TMP_121) in (let TMP_123 \def (THeads -TMP_116 TMP_117 TMP_122) in (let TMP_126 \def (\lambda (t0: T).(let TMP_124 -\def (Flat Appl) in (let TMP_125 \def (THead TMP_124 w0 t0) in (sc3 g a0 d -TMP_125)))) in (let TMP_127 \def (Flat Appl) in (let TMP_128 \def (lift1 is -v) in (let TMP_129 \def (Bind Abst) in (let TMP_130 \def (THead TMP_129 w t) -in (let TMP_131 \def (lift1 is TMP_130) in (let TMP_132 \def (THead TMP_127 -TMP_128 TMP_131) in (let TMP_138 \def (\lambda (t0: T).(let TMP_133 \def -(Flat Appl) in (let TMP_134 \def (Flat Appl) in (let TMP_135 \def (lifts1 is -vs) in (let TMP_136 \def (THeads TMP_134 TMP_135 t0) in (let TMP_137 \def -(THead TMP_133 w0 TMP_136) in (sc3 g a0 d TMP_137))))))) in (let TMP_139 \def -(Bind Abst) in (let TMP_140 \def (lift1 is w) in (let TMP_141 \def (Ss is) in -(let TMP_142 \def (lift1 TMP_141 t) in (let TMP_143 \def (THead TMP_139 -TMP_140 TMP_142) in (let TMP_152 \def (\lambda (t0: T).(let TMP_144 \def -(Flat Appl) in (let TMP_145 \def (Flat Appl) in (let TMP_146 \def (lifts1 is -vs) in (let TMP_147 \def (Flat Appl) in (let TMP_148 \def (lift1 is v) in -(let TMP_149 \def (THead TMP_147 TMP_148 t0) in (let TMP_150 \def (THeads -TMP_145 TMP_146 TMP_149) in (let TMP_151 \def (THead TMP_144 w0 TMP_150) in -(sc3 g a0 d TMP_151)))))))))) in (let TMP_153 \def (lifts1 is vs) in (let -TMP_154 \def (TCons w0 TMP_153) in (let H_y \def (H0 TMP_154) in (let TMP_155 -\def (lift1 is v) in (let TMP_156 \def (Ss is) in (let TMP_157 \def (lift1 -TMP_156 t) in (let TMP_158 \def (Bind Abbr) in (let TMP_159 \def (THead -TMP_158 v t) in (let TMP_160 \def (lift1 is TMP_159) in (let TMP_166 \def -(\lambda (t0: T).(let TMP_161 \def (Flat Appl) in (let TMP_162 \def (Flat -Appl) in (let TMP_163 \def (lifts1 is vs) in (let TMP_164 \def (THeads -TMP_162 TMP_163 t0) in (let TMP_165 \def (THead TMP_161 w0 TMP_164) in (sc3 g -a0 d TMP_165))))))) in (let TMP_167 \def (Flat Appl) in (let TMP_168 \def -(Bind Abbr) in (let TMP_169 \def (THead TMP_168 v t) in (let TMP_170 \def -(THeads TMP_167 vs TMP_169) in (let TMP_171 \def (lift1 is TMP_170) in (let -TMP_174 \def (\lambda (t0: T).(let TMP_172 \def (Flat Appl) in (let TMP_173 -\def (THead TMP_172 w0 t0) in (sc3 g a0 d TMP_173)))) in (let TMP_175 \def -(H6 d w0 H7 is H8) in (let TMP_176 \def (Flat Appl) in (let TMP_177 \def -(lifts1 is vs) in (let TMP_178 \def (Bind Abbr) in (let TMP_179 \def (THead -TMP_178 v t) in (let TMP_180 \def (lift1 is TMP_179) in (let TMP_181 \def -(THeads TMP_176 TMP_177 TMP_180) in (let TMP_182 \def (Bind Abbr) in (let -TMP_183 \def (THead TMP_182 v t) in (let TMP_184 \def (lifts1_flat Appl is -TMP_183 vs) in (let TMP_185 \def (eq_ind T TMP_171 TMP_174 TMP_175 TMP_181 -TMP_184) in (let TMP_186 \def (Bind Abbr) in (let TMP_187 \def (lift1 is v) -in (let TMP_188 \def (Ss is) in (let TMP_189 \def (lift1 TMP_188 t) in (let -TMP_190 \def (THead TMP_186 TMP_187 TMP_189) in (let TMP_191 \def (lift1_bind -Abbr is v t) in (let TMP_192 \def (eq_ind T TMP_160 TMP_166 TMP_185 TMP_190 -TMP_191) in (let TMP_193 \def (sc3_lift1 g c a1 is d v H2 H8) in (let TMP_194 -\def (lift1 is w) in (let TMP_195 \def (asucc g a1) in (let TMP_196 \def -(sc3_lift1 g c TMP_195 is d w H3 H8) in (let TMP_197 \def (H_y d TMP_155 -TMP_157 TMP_192 TMP_193 TMP_194 TMP_196) in (let TMP_198 \def (Bind Abst) in -(let TMP_199 \def (THead TMP_198 w t) in (let TMP_200 \def (lift1 is TMP_199) -in (let TMP_201 \def (lift1_bind Abst is w t) in (let TMP_202 \def (eq_ind_r -T TMP_143 TMP_152 TMP_197 TMP_200 TMP_201) in (let TMP_203 \def (Flat Appl) -in (let TMP_204 \def (Bind Abst) in (let TMP_205 \def (THead TMP_204 w t) in -(let TMP_206 \def (THead TMP_203 v TMP_205) in (let TMP_207 \def (lift1 is -TMP_206) in (let TMP_208 \def (Bind Abst) in (let TMP_209 \def (THead TMP_208 -w t) in (let TMP_210 \def (lift1_flat Appl is v TMP_209) in (let TMP_211 \def -(eq_ind_r T TMP_132 TMP_138 TMP_202 TMP_207 TMP_210) in (let TMP_212 \def -(Flat Appl) in (let TMP_213 \def (Flat Appl) in (let TMP_214 \def (Bind Abst) -in (let TMP_215 \def (THead TMP_214 w t) in (let TMP_216 \def (THead TMP_213 -v TMP_215) in (let TMP_217 \def (THeads TMP_212 vs TMP_216) in (let TMP_218 -\def (lift1 is TMP_217) in (let TMP_219 \def (Flat Appl) in (let TMP_220 \def -(Bind Abst) in (let TMP_221 \def (THead TMP_220 w t) in (let TMP_222 \def -(THead TMP_219 v TMP_221) in (let TMP_223 \def (lifts1_flat Appl is TMP_222 -vs) in (eq_ind_r T TMP_123 TMP_126 TMP_211 TMP_218 -TMP_223))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -)))))))))))))))))))))) in (conj TMP_100 TMP_110 TMP_115 -TMP_224)))))))))))))))))) in (land_ind TMP_65 TMP_73 TMP_92 TMP_225 -H4)))))))))))))))))))))))))))))))) in (A_ind TMP_7 TMP_59 TMP_226 a2)))))). +(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))).