X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fpr0%2Fprops.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fpr0%2Fprops.ma;h=7e0cd81903d5461ab62d4f2081a44eb4f645e15d;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma new file mode 100644 index 000000000..7e0cd8190 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/pr0/props.ma @@ -0,0 +1,1746 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +include "LambdaDelta-1/pr0/defs.ma". + +include "LambdaDelta-1/subst0/subst0.ma". + +theorem pr0_lift: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall +(d: nat).(pr0 (lift h d t1) (lift h d t2)))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t) +(lift h d t0)))))) (\lambda (t: T).(\lambda (h: nat).(\lambda (d: +nat).(pr0_refl (lift h d t))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda +(_: (pr0 u1 u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 +(lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda +(_: (pr0 t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 +(lift h d t3) (lift h d t4)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda +(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t3)) (\lambda (t: +T).(pr0 t (lift h d (THead k u2 t4)))) (eq_ind_r T (THead k (lift h d u2) +(lift h (s k d) t4)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k +d) t3)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d) +t3) (lift h (s k d) t4) (H3 h (s k d)) k) (lift h d (THead k u2 t4)) +(lift_head k u2 t4 h d)) (lift h d (THead k u1 t3)) (lift_head k u1 t3 h +d))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: +(pr0 v1 v2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h +d v1) (lift h d v2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 +t3 t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) +(lift h d t4)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead +(Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) (THead (Bind Abst) u +t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r +T (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s (Bind Abst) (s +(Flat Appl) d)) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) +(lift h d (THead (Bind Abbr) v2 t4)))) (eq_ind_r T (THead (Bind Abbr) (lift h +d v2) (lift h (s (Bind Abbr) d) t4)) (\lambda (t: T).(pr0 (THead (Flat Appl) +(lift h d v1) (THead (Bind Abst) (lift h (s (Flat Appl) d) u) (lift h (s +(Bind Abst) (s (Flat Appl) d)) t3))) t)) (pr0_beta (lift h (s (Flat Appl) d) +u) (lift h d v1) (lift h d v2) (H1 h d) (lift h (s (Bind Abst) (s (Flat Appl) +d)) t3) (lift h (s (Bind Abbr) d) t4) (H3 h (s (Bind Abbr) d))) (lift h d +(THead (Bind Abbr) v2 t4)) (lift_head (Bind Abbr) v2 t4 h d)) (lift h (s +(Flat Appl) d) (THead (Bind Abst) u t3)) (lift_head (Bind Abst) u t3 h (s +(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t3))) +(lift_head (Flat Appl) v1 (THead (Bind Abst) u t3) h d))))))))))))) (\lambda +(b: B).(\lambda (H0: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: +T).(\lambda (_: (pr0 v1 v2)).(\lambda (H2: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d v1) (lift h d v2)))))).(\lambda (u1: T).(\lambda (u2: +T).(\lambda (_: (pr0 u1 u2)).(\lambda (H4: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d u1) (lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (pr0 t3 t4)).(\lambda (H6: ((\forall (h: nat).(\forall (d: +nat).(pr0 (lift h d t3) (lift h d t4)))))).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Flat Appl) (lift h d v1) (lift h (s (Flat Appl) d) +(THead (Bind b) u1 t3))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead (Bind b) +(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t3)) +(\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t) (lift h d (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))))) (eq_ind_r T (THead +(Bind b) (lift h d u2) (lift h (s (Bind b) d) (THead (Flat Appl) (lift (S O) +O v2) t4))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead +(Bind b) (lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) +t3))) t)) (eq_ind_r T (THead (Flat Appl) (lift h (s (Bind b) d) (lift (S O) O +v2)) (lift h (s (Flat Appl) (s (Bind b) d)) t4)) (\lambda (t: T).(pr0 (THead +(Flat Appl) (lift h d v1) (THead (Bind b) (lift h (s (Flat Appl) d) u1) (lift +h (s (Bind b) (s (Flat Appl) d)) t3))) (THead (Bind b) (lift h d u2) t))) +(eq_ind nat (plus (S O) d) (\lambda (n: nat).(pr0 (THead (Flat Appl) (lift h +d v1) (THead (Bind b) (lift h d u1) (lift h n t3))) (THead (Bind b) (lift h d +u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t4))))) +(eq_ind_r T (lift (S O) O (lift h d v2)) (\lambda (t: T).(pr0 (THead (Flat +Appl) (lift h d v1) (THead (Bind b) (lift h d u1) (lift h (plus (S O) d) +t3))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O) +d) t4))))) (pr0_upsilon b H0 (lift h d v1) (lift h d v2) (H2 h d) (lift h d +u1) (lift h d u2) (H4 h d) (lift h (plus (S O) d) t3) (lift h (plus (S O) d) +t4) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_d +v2 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) (lift h (s (Bind b) +d) (THead (Flat Appl) (lift (S O) O v2) t4)) (lift_head (Flat Appl) (lift (S +O) O v2) t4 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t3)) +(lift_head (Bind b) u1 t3 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl) +v1 (THead (Bind b) u1 t3))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t3) +h d)))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 +u2)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d u1) +(lift h d u2)))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 +t4)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) +(lift h d t4)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t4 w)).(\lambda +(h: nat).(\lambda (d: nat).(eq_ind_r T (THead (Bind Abbr) (lift h d u1) (lift +h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) +u2 w)))) (eq_ind_r T (THead (Bind Abbr) (lift h d u2) (lift h (s (Bind Abbr) +d) w)) (\lambda (t: T).(pr0 (THead (Bind Abbr) (lift h d u1) (lift h (s (Bind +Abbr) d) t3)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S +d) t3) (lift h (S d) t4) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in +(eq_ind nat (minus (S d) (S O)) (\lambda (n: nat).(subst0 O (lift h n u2) +(lift h d' t4) (lift h d' w))) (subst0_lift_lt t4 w u2 O H4 (S d) (le_n_S O d +(le_O_n d)) h) d (eq_ind nat d (\lambda (n: nat).(eq nat n d)) (refl_equal +nat d) (minus d O) (minus_n_O d))))) (lift h d (THead (Bind Abbr) u2 w)) +(lift_head (Bind Abbr) u2 w h d)) (lift h d (THead (Bind Abbr) u1 t3)) +(lift_head (Bind Abbr) u1 t3 h d)))))))))))))) (\lambda (b: B).(\lambda (H0: +(not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0 t3 +t4)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t3) +(lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Bind b) (lift h d u) (lift h (s (Bind b) d) (lift (S +O) O t3))) (\lambda (t: T).(pr0 t (lift h d t4))) (eq_ind nat (plus (S O) d) +(\lambda (n: nat).(pr0 (THead (Bind b) (lift h d u) (lift h n (lift (S O) O +t3))) (lift h d t4))) (eq_ind_r T (lift (S O) O (lift h d t3)) (\lambda (t: +T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t4))) (pr0_zeta b H0 (lift +h d t3) (lift h d t4) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S +O) O t3)) (lift_d t3 h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S d))) +(lift h d (THead (Bind b) u (lift (S O) O t3))) (lift_head (Bind b) u (lift +(S O) O t3) h d))))))))))) (\lambda (t3: T).(\lambda (t4: T).(\lambda (_: +(pr0 t3 t4)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h +d t3) (lift h d t4)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: +nat).(eq_ind_r T (THead (Flat Cast) (lift h d u) (lift h (s (Flat Cast) d) +t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_tau (lift h (s (Flat Cast) +d) t3) (lift h d t4) (H1 h d) (lift h d u)) (lift h d (THead (Flat Cast) u +t3)) (lift_head (Flat Cast) u t3 h d))))))))) t1 t2 H))). + +theorem pr0_subst0_back: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: +T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t2))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 u1 t) \to (ex2 T +(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t4 t3))))))))) +(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 u1 +v)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: +T).(pr0 t (lift (S i0) O v))) (lift (S i0) O u1) (subst0_lref u1 i0) +(pr0_lift u1 v H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda +(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: +((\forall (u4: T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) +(\lambda (t: T).(pr0 t u3))))))).(\lambda (t: T).(\lambda (k: K).(\lambda +(u0: T).(\lambda (H2: (pr0 u0 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 +u1 t0)) (\lambda (t0: T).(pr0 t0 u3)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 +(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u3 t)))) (\lambda (x: +T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 x u3)).(ex_intro2 T +(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 +(THead k u3 t))) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp x u3 +H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda (v: +T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: (subst0 +(s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T +(\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t +t3))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind +T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t t3)) (ex2 +T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 t +(THead k u t3)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 +x)).(\lambda (H4: (pr0 x t3)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 +(THead k u t4) t)) (\lambda (t: T).(pr0 t (THead k u t3))) (THead k u x) +(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) x t3 H4 k))))) (H1 +u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda +(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: +T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: +T).(pr0 t u3))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: +T).((pr0 u4 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda +(t: T).(pr0 t t4))))))).(\lambda (u0: T).(\lambda (H4: (pr0 u0 v)).(ex2_ind T +(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t t4)) (ex2 T +(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t +(THead k u3 t4)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 +x)).(\lambda (H6: (pr0 x t4)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) +(\lambda (t: T).(pr0 t u3)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 +t3) t)) (\lambda (t: T).(pr0 t (THead k u3 t4)))) (\lambda (x0: T).(\lambda +(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 x0 u3)).(ex_intro2 T (\lambda +(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 t (THead k u3 +t4))) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp x0 u3 +H8 x t4 H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). + +theorem pr0_subst0_fwd: + \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst0 +i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: +T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))) +\def + \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda +(H: (subst0 i u2 t1 t2)).(subst0_ind (\lambda (n: nat).(\lambda (t: +T).(\lambda (t0: T).(\lambda (t3: T).(\forall (u1: T).((pr0 t u1) \to (ex2 T +(\lambda (t4: T).(subst0 n u1 t0 t4)) (\lambda (t4: T).(pr0 t3 t4))))))))) +(\lambda (v: T).(\lambda (i0: nat).(\lambda (u1: T).(\lambda (H0: (pr0 v +u1)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 (TLRef i0) t)) (\lambda (t: +T).(pr0 (lift (S i0) O v) t)) (lift (S i0) O u1) (subst0_lref u1 i0) +(pr0_lift v u1 H0 (S i0) O)))))) (\lambda (v: T).(\lambda (u3: T).(\lambda +(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: +((\forall (u4: T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) +(\lambda (t: T).(pr0 u3 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda +(u0: T).(\lambda (H2: (pr0 v u0)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u0 +u1 t0)) (\lambda (t0: T).(pr0 u3 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u0 +(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u3 t) t0))) (\lambda (x: +T).(\lambda (H3: (subst0 i0 u0 u1 x)).(\lambda (H4: (pr0 u3 x)).(ex_intro2 T +(\lambda (t0: T).(subst0 i0 u0 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 +(THead k u3 t) t0)) (THead k x t) (subst0_fst u0 x u1 i0 H3 t k) (pr0_comp u3 +x H4 t t (pr0_refl t) k))))) (H1 u0 H2)))))))))))) (\lambda (k: K).(\lambda +(v: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (i0: nat).(\lambda (_: +(subst0 (s k i0) v t4 t3)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to +(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 +t))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 v u1)).(ex2_ind +T (\lambda (t: T).(subst0 (s k i0) u1 t4 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 +T (\lambda (t: T).(subst0 i0 u1 (THead k u t4) t)) (\lambda (t: T).(pr0 +(THead k u t3) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t4 +x)).(\lambda (H4: (pr0 t3 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1 +(THead k u t4) t)) (\lambda (t: T).(pr0 (THead k u t3) t)) (THead k u x) +(subst0_snd k u1 x t4 i0 H3 u) (pr0_comp u u (pr0_refl u) t3 x H4 k))))) (H1 +u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u3: T).(\lambda +(i0: nat).(\lambda (_: (subst0 i0 v u1 u3)).(\lambda (H1: ((\forall (u4: +T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 i0 u4 u1 t)) (\lambda (t: +T).(pr0 u3 t))))))).(\lambda (k: K).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (_: (subst0 (s k i0) v t3 t4)).(\lambda (H3: ((\forall (u4: +T).((pr0 v u4) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u4 t3 t)) (\lambda +(t: T).(pr0 t4 t))))))).(\lambda (u0: T).(\lambda (H4: (pr0 v u0)).(ex2_ind T +(\lambda (t: T).(subst0 (s k i0) u0 t3 t)) (\lambda (t: T).(pr0 t4 t)) (ex2 T +(\lambda (t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead +k u3 t4) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u0 t3 +x)).(\lambda (H6: (pr0 t4 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u0 u1 t)) +(\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(subst0 i0 u0 (THead k u1 +t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) t))) (\lambda (x0: T).(\lambda +(H7: (subst0 i0 u0 u1 x0)).(\lambda (H8: (pr0 u3 x0)).(ex_intro2 T (\lambda +(t: T).(subst0 i0 u0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u3 t4) +t)) (THead k x0 x) (subst0_both u0 u1 x0 i0 H7 k t3 x H5) (pr0_comp u3 x0 H8 +t4 x H6 k))))) (H1 u0 H4))))) (H3 u0 H4))))))))))))))) i u2 t1 t2 H))))). + +theorem pr0_subst0: + \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall +(w1: T).(\forall (i: nat).((subst0 i v1 t1 w1) \to (\forall (v2: T).((pr0 v1 +v2) \to (or (pr0 w1 t2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t2 w2)))))))))))) +\def + \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(pr0_ind (\lambda +(t: T).(\lambda (t0: T).(\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 t w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +t0) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t0 +w2)))))))))))) (\lambda (t: T).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: +nat).(\lambda (H0: (subst0 i v1 t w1)).(\lambda (v2: T).(\lambda (H1: (pr0 v1 +v2)).(or_intror (pr0 w1 t) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t w2))) (ex2_sym T (subst0 i v2 t) (pr0 w1) (pr0_subst0_fwd +v1 t w1 i H0 v2 H1)))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: +(pr0 u1 u2)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 u1 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 u2 +w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 +t4)).(\lambda (H3: ((\forall (v1: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) \to (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))))))))))).(\lambda (k: K).(\lambda (v1: T).(\lambda (w1: T).(\lambda (i: +nat).(\lambda (H4: (subst0 i v1 (THead k u1 t3) w1)).(\lambda (v2: +T).(\lambda (H5: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda (u3: T).(eq T w1 +(THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3))) (ex2 T (\lambda (t5: +T).(eq T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) +(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) +(\lambda (u3: T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))) (or (pr0 w1 (THead k u2 t4)) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 (THead k +u2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u3: T).(eq T w1 (THead k u3 +t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq +T w1 (THead k u3 t3))) (\lambda (u3: T).(subst0 i v1 u1 u3)) (or (pr0 w1 +(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 +(THead k x t3))).(\lambda (H8: (subst0 i v1 u1 x)).(eq_ind_r T (THead k x t3) +(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t +w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x u2) +(ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) +(or (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead +k x t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda +(H9: (pr0 x u2)).(or_introl (pr0 (THead k x t3) (THead k u2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2))) (pr0_comp x u2 H9 t3 t4 H2 k))) (\lambda (H9: (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind +T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 +(THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x t3) +w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: +T).(\lambda (H10: (pr0 x x0)).(\lambda (H11: (subst0 i v2 u2 x0)).(or_intror +(pr0 (THead k x t3) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x +t3) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead k x t3) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2)) (THead k x0 t4) (pr0_comp x x0 H10 t3 t4 H2 k) +(subst0_fst v2 x0 u2 i H11 t4 k)))))) H9)) (H1 v1 x i H8 v2 H5)) w1 H7)))) +H6)) (\lambda (H6: (ex2 T (\lambda (t5: T).(eq T w1 (THead k u1 t5))) +(\lambda (t5: T).(subst0 (s k i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq +T w1 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i) v1 t3 t5)) (or (pr0 +w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda (H7: (eq T w1 +(THead k u1 x))).(\lambda (H8: (subst0 (s k i) v1 t3 x)).(eq_ind_r T (THead k +u1 x) (\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind +(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s k +i) v2 t4 w2))) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) +w2)))) (\lambda (H9: (pr0 x t4)).(or_introl (pr0 (THead k u1 x) (THead k u2 +t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2))) (pr0_comp u1 u2 H0 x t4 H9 k))) +(\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s +k i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: +T).(subst0 (s k i) v2 t4 w2)) (or (pr0 (THead k u1 x) (THead k u2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x x0)).(\lambda +(H11: (subst0 (s k i) v2 t4 x0)).(or_intror (pr0 (THead k u1 x) (THead k u2 +t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k u1 x) w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 +(THead k u1 x) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) (THead +k u2 x0) (pr0_comp u1 u2 H0 x x0 H10 k) (subst0_snd k v2 x0 t4 i H11 u2)))))) +H9)) (H3 v1 x (s k i) H8 v2 H5)) w1 H7)))) H6)) (\lambda (H6: (ex3_2 T T +(\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s k i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda +(t5: T).(eq T w1 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 +i v1 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i) v1 t3 t5))) +(or (pr0 w1 (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda +(w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: +T).(\lambda (H7: (eq T w1 (THead k x0 x1))).(\lambda (H8: (subst0 i v1 u1 +x0)).(\lambda (H9: (subst0 (s k i) v1 t3 x1)).(eq_ind_r T (THead k x0 x1) +(\lambda (t: T).(or (pr0 t (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 t +w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))))) (or_ind (pr0 x1 +t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 +t4 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) +w2)))) (\lambda (H10: (pr0 x1 t4)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 +x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (H11: (pr0 x0 +u2)).(or_introl (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) +w2))) (pr0_comp x0 u2 H11 x1 t4 H10 k))) (\lambda (H11: (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k +x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: T).(\lambda +(H12: (pr0 x0 x)).(\lambda (H13: (subst0 i v2 u2 x)).(or_intror (pr0 (THead k +x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda +(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 +t4) w2)) (THead k x t4) (pr0_comp x0 x H12 x1 t4 H10 k) (subst0_fst v2 x u2 i +H13 t4 k)))))) H11)) (H1 v1 x0 i H8 v2 H5))) (\lambda (H10: (ex2 T (\lambda +(w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s k i) v2 t4 w2)) (or +(pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k +x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x: +T).(\lambda (H11: (pr0 x1 x)).(\lambda (H12: (subst0 (s k i) v2 t4 +x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 i v2 u2 w2))) (or (pr0 (THead k x0 x1) (THead k u2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 +(THead k u2 t4) w2)))) (\lambda (H13: (pr0 x0 u2)).(or_intror (pr0 (THead k +x0 x1) (THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda +(w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 +t4) w2)) (THead k u2 x) (pr0_comp x0 u2 H13 x1 x H11 k) (subst0_snd k v2 x t4 +i H12 u2)))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda +(w2: T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) +(\lambda (w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead k x0 x1) (THead k u2 +t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda (w2: +T).(subst0 i v2 (THead k u2 t4) w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 +x0 x2)).(\lambda (H15: (subst0 i v2 u2 x2)).(or_intror (pr0 (THead k x0 x1) +(THead k u2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead k x0 x1) w2)) (\lambda +(w2: T).(subst0 i v2 (THead k u2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 +(THead k x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead k u2 t4) w2)) +(THead k x2 x) (pr0_comp x0 x2 H14 x1 x H11 k) (subst0_both v2 u2 x2 i H15 k +t4 x H12)))))) H13)) (H1 v1 x0 i H8 v2 H5))))) H10)) (H3 v1 x1 (s k i) H9 v2 +H5)) w1 H7)))))) H6)) (subst0_gen_head k v1 u1 t3 w1 i H4))))))))))))))))) +(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H0: (pr0 v1 +v2)).(\lambda (H1: ((\forall (v3: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v3 v1 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 +v2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 +w2)))))))))))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 +t4)).(\lambda (H3: ((\forall (v3: T).(\forall (w1: T).(\forall (i: +nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 t4 +w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda (i: nat).(\lambda +(H4: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) +w1)).(\lambda (v3: T).(\lambda (H5: (pr0 v0 v3)).(or3_ind (ex2 T (\lambda +(u2: T).(eq T w1 (THead (Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda +(u2: T).(subst0 i v0 v1 u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat +Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind +Abst) u t3) t5))) (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 +(THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 v1 +u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead +(Bind Abst) u t3) t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2)))) (\lambda (H6: (ex2 T (\lambda (u2: T).(eq T w1 (THead +(Flat Appl) u2 (THead (Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 +u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Appl) u2 (THead +(Bind Abst) u t3)))) (\lambda (u2: T).(subst0 i v0 v1 u2)) (or (pr0 w1 (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda +(H7: (eq T w1 (THead (Flat Appl) x (THead (Bind Abst) u t3)))).(\lambda (H8: +(subst0 i v0 v1 x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind Abst) u +t3)) (\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda +(w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2))))) (or_ind (pr0 x v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: +T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind Abst) u +t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)))) (\lambda (H9: (pr0 x v2)).(or_introl (pr0 (THead +(Flat Appl) x (THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u x +v2 H9 t3 t4 H2))) (\lambda (H9: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda +(w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) +(\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x (THead +(Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: T).(subst0 +i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (pr0 x +x0)).(\lambda (H11: (subst0 i v3 v2 x0)).(or_intror (pr0 (THead (Flat Appl) x +(THead (Bind Abst) u t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x (THead (Bind Abst) u t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x0 t4) +(pr0_beta u x x0 H10 t3 t4 H2) (subst0_fst v3 x0 v2 i H11 t4 (Bind +Abbr))))))) H9)) (H1 v0 x i H8 v3 H5)) w1 H7)))) H6)) (\lambda (H6: (ex2 T +(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: +T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)))).(ex2_ind T +(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: +T).(subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) t5)) (or (pr0 w1 +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda +(H7: (eq T w1 (THead (Flat Appl) v1 x))).(\lambda (H8: (subst0 (s (Flat Appl) +i) v0 (THead (Bind Abst) u t3) x)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x +(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u +u2))) (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda +(t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T +(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 +t3 t5)))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) +(\lambda (H9: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 t3))) +(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda +(u2: T).(eq T x (THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat +Appl) i) v0 u u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead (Bind Abst) x0 +t3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(let H12 \def (eq_ind +T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) +x0 t3) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) +(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2))))) (or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 t3)) +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 t3 t4 H2)) w1 H12))))) H9)) (\lambda +(H9: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda +(t5: T).(eq T x (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind +Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (H10: (eq T x (THead +(Bind Abst) u x0))).(\lambda (H11: (subst0 (s (Bind Abst) (s (Flat Appl) i)) +v0 t3 x0)).(let H12 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat +Appl) v1 t))) H7 (THead (Bind Abst) u x0) H10) in (eq_ind_r T (THead (Flat +Appl) v1 (THead (Bind Abst) u x0)) (\lambda (t: T).(or (pr0 t (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 +t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead +(Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)))) (\lambda (H13: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta u v1 v2 H0 x0 t4 +H13))) (\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) +i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) u x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2)))) (\lambda (x1: T).(\lambda (H14: (pr0 x0 x1)).(\lambda +(H15: (subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 +(THead (Flat Appl) v1 (THead (Bind Abst) u x0)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 +T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u x0)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind +Abbr) v2 x1) (pr0_beta u v1 v2 H0 x0 x1 H14) (subst0_snd (Bind Abbr) v3 x1 t4 +i H15 v2)))))) H13)) (H3 v0 x0 (s (Bind Abst) (s (Flat Appl) i)) H11 v3 H5)) +w1 H12))))) H9)) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T x (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T x (THead (Bind Abst) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 +t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T x (THead (Bind Abst) +x0 x1))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x0)).(\lambda (H12: +(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x1)).(let H13 \def (eq_ind T +x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H7 (THead (Bind Abst) +x0 x1) H10) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) +(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda +(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead +(Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H14: +(pr0 x1 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2))) (pr0_beta x0 v1 v2 H0 x1 t4 H14))) (\lambda (H14: (ex2 T +(\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s +(Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) +(\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)) (or +(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 +t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 +x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) +(\lambda (x2: T).(\lambda (H15: (pr0 x1 x2)).(\lambda (H16: (subst0 (s (Bind +Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_intror (pr0 (THead (Flat Appl) v1 +(THead (Bind Abst) x0 x1)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) +(pr0_beta x0 v1 v2 H0 x1 x2 H15) (subst0_snd (Bind Abbr) v3 x2 t4 i H16 +v2)))))) H14)) (H3 v0 x1 (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 +H13))))))) H9)) (subst0_gen_head (Bind Abst) v0 u t3 x (s (Flat Appl) i) +H8))))) H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T +w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v0 +v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead +(Bind Abst) u t3) t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: +T).(eq T w1 (THead (Flat Appl) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i v0 v1 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat +Appl) i) v0 (THead (Bind Abst) u t3) t5))) (or (pr0 w1 (THead (Bind Abbr) v2 +t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda +(H7: (eq T w1 (THead (Flat Appl) x0 x1))).(\lambda (H8: (subst0 i v0 v1 +x0)).(\lambda (H9: (subst0 (s (Flat Appl) i) v0 (THead (Bind Abst) u t3) +x1)).(or3_ind (ex2 T (\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) +(\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u u2))) (ex2 T (\lambda (t5: +T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind +Abst) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u2: T).(\lambda +(t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))) (or (pr0 w1 (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H10: (ex2 T +(\lambda (u2: T).(eq T x1 (THead (Bind Abst) u2 t3))) (\lambda (u2: +T).(subst0 (s (Flat Appl) i) v0 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T x1 +(THead (Bind Abst) u2 t3))) (\lambda (u2: T).(subst0 (s (Flat Appl) i) v0 u +u2)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 w1 +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(x: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) x t3))).(\lambda (_: +(subst0 (s (Flat Appl) i) v0 u x)).(let H13 \def (eq_ind T x1 (\lambda (t: +T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) x t3) H11) in +(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (\lambda (t: +T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 +x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2)))) (\lambda (H14: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat Appl) x0 +(THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (pr0_beta x x0 v2 H14 t3 t4 +H2))) (\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda +(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind +Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead +(Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind Abbr) v2 t4) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 +x2)).(\lambda (H16: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) +x0 (THead (Bind Abst) x t3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x2 t4) +(pr0_beta x x0 x2 H15 t3 t4 H2) (subst0_fst v3 x2 v2 i H16 t4 (Bind +Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)) w1 H13))))) H10)) (\lambda (H10: +(ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda +(t5: T).(eq T x1 (THead (Bind Abst) u t5))) (\lambda (t5: T).(subst0 (s (Bind +Abst) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)))) (\lambda (x: T).(\lambda (H11: (eq T x1 (THead +(Bind Abst) u x))).(\lambda (H12: (subst0 (s (Bind Abst) (s (Flat Appl) i)) +v0 t3 x)).(let H13 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat +Appl) x0 t))) H7 (THead (Bind Abst) u x) H11) in (eq_ind_r T (THead (Flat +Appl) x0 (THead (Bind Abst) u x)) (\lambda (t: T).(or (pr0 t (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind Abbr) v2 t4) w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 +t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2)))) (\lambda (H14: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda +(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) (\lambda +(w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: (pr0 x0 +v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (pr0_beta u x0 v2 H15 x t4 H14))) (\lambda (H15: (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(x2: T).(\lambda (H16: (pr0 x0 x2)).(\lambda (H17: (subst0 i v3 v2 +x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)) (THead (Bind Abbr) x2 t4) (pr0_beta u x0 x2 H16 x t4 H14) +(subst0_fst v3 x2 v2 i H17 t4 (Bind Abbr))))))) H15)) (H1 v0 x0 i H8 v3 H5))) +(\lambda (H14: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 +(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 +t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind +Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x x2)).(\lambda (H16: (subst0 (s +(Bind Abst) (s (Flat Appl) i)) v3 t4 x2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead +(Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H17: +(pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) +(THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 +(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 +(THead (Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind +Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x2) (pr0_beta u x0 v2 H17 x x2 H15) +(subst0_snd (Bind Abbr) v3 x2 t4 i H16 v2)))) (\lambda (H17: (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(x3: T).(\lambda (H18: (pr0 x0 x3)).(\lambda (H19: (subst0 i v3 v2 +x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) u x)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) u x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)) (THead (Bind Abbr) x3 x2) (pr0_beta u x0 x3 H18 x x2 H15) +(subst0_both v3 v2 x3 i H19 (Bind Abbr) t4 x2 H16)))))) H17)) (H1 v0 x0 i H8 +v3 H5))))) H14)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) w1 +H13))))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5: +T).(eq T x1 (THead (Bind Abst) u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 (s (Flat Appl) i) v0 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 t5))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T x1 (THead (Bind Abst) u2 t5)))) +(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u u2))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 +t3 t5))) (or (pr0 w1 (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) +(\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T x1 (THead (Bind Abst) +x2 x3))).(\lambda (_: (subst0 (s (Flat Appl) i) v0 u x2)).(\lambda (H13: +(subst0 (s (Bind Abst) (s (Flat Appl) i)) v0 t3 x3)).(let H14 \def (eq_ind T +x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H7 (THead (Bind Abst) +x2 x3) H11) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) +(\lambda (t: T).(or (pr0 t (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) +w2))))) (or_ind (pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda +(w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead +(Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H15: +(pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) +(\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda (H16: (pr0 x0 +v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (pr0_beta x2 x0 v2 H16 x3 t4 H15))) (\lambda (H16: (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)))).(ex2_ind T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2)) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(x: T).(\lambda (H17: (pr0 x0 x)).(\lambda (H18: (subst0 i v3 v2 +x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)) (THead (Bind Abbr) x t4) (pr0_beta x2 x0 x H17 x3 t4 H15) +(subst0_fst v3 x v2 i H18 t4 (Bind Abbr))))))) H16)) (H1 v0 x0 i H8 v3 H5))) +(\lambda (H15: (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 +(s (Bind Abst) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: +T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s (Bind Abst) (s (Flat Appl) i)) v3 +t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)))) (\lambda (x: T).(\lambda (H16: (pr0 x3 x)).(\lambda (H17: (subst0 +(s (Bind Abst) (s (Flat Appl) i)) v3 t4 x)).(or_ind (pr0 x0 v2) (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead (Bind Abbr) v2 t4)) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 t4) w2)))) (\lambda +(H18: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) +x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) v2 x) (pr0_beta x2 x0 v2 H18 x3 x +H16) (subst0_snd (Bind Abbr) v3 x t4 i H17 v2)))) (\lambda (H18: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) x2 x3)) (THead +(Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind Abbr) v2 +t4) w2)))) (\lambda (x4: T).(\lambda (H19: (pr0 x0 x4)).(\lambda (H20: +(subst0 i v3 v2 x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind Abst) +x2 x3)) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind Abst) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind Abbr) v2 t4) w2)) (THead (Bind Abbr) x4 x) (pr0_beta x2 x0 x4 H19 x3 x +H16) (subst0_both v3 v2 x4 i H20 (Bind Abbr) t4 x H17)))))) H18)) (H1 v0 x0 i +H8 v3 H5))))) H15)) (H3 v0 x3 (s (Bind Abst) (s (Flat Appl) i)) H13 v3 H5)) +w1 H14))))))) H10)) (subst0_gen_head (Bind Abst) v0 u t3 x1 (s (Flat Appl) i) +H9))))))) H6)) (subst0_gen_head (Flat Appl) v0 v1 (THead (Bind Abst) u t3) w1 +i H4))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b +Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H1: (pr0 v1 v2)).(\lambda +(H2: ((\forall (v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 v1 +w1) \to (\forall (v4: T).((pr0 v3 v4) \to (or (pr0 w1 v2) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v4 v2 w2)))))))))))).(\lambda +(u1: T).(\lambda (u2: T).(\lambda (H3: (pr0 u1 u2)).(\lambda (H4: ((\forall +(v3: T).(\forall (w1: T).(\forall (i: nat).((subst0 i v3 u1 w1) \to (\forall +(v4: T).((pr0 v3 v4) \to (or (pr0 w1 u2) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) +(\lambda (w2: T).(subst0 i v4 u2 w2)))))))))))).(\lambda (t3: T).(\lambda +(t4: T).(\lambda (H5: (pr0 t3 t4)).(\lambda (H6: ((\forall (v3: T).(\forall +(w1: T).(\forall (i: nat).((subst0 i v3 t3 w1) \to (\forall (v4: T).((pr0 v3 +v4) \to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v4 t4 w2)))))))))))).(\lambda (v0: T).(\lambda (w1: T).(\lambda +(i: nat).(\lambda (H7: (subst0 i v0 (THead (Flat Appl) v1 (THead (Bind b) u1 +t3)) w1)).(\lambda (v3: T).(\lambda (H8: (pr0 v0 v3)).(or3_ind (ex2 T +(\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) +(\lambda (u3: T).(subst0 i v0 v1 u3))) (ex2 T (\lambda (t5: T).(eq T w1 +(THead (Flat Appl) v1 t5))) (\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 +(THead (Bind b) u1 t3) t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq +T w1 (THead (Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i +v0 v1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 +(THead (Bind b) u1 t3) t5)))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda +(w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4)) w2)))) (\lambda (H9: (ex2 T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) +u3 (THead (Bind b) u1 t3)))) (\lambda (u3: T).(subst0 i v0 v1 u3)))).(ex2_ind +T (\lambda (u3: T).(eq T w1 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)))) +(\lambda (u3: T).(subst0 i v0 v1 u3)) (or (pr0 w1 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq T w1 (THead (Flat +Appl) x (THead (Bind b) u1 t3)))).(\lambda (H11: (subst0 i v0 v1 +x)).(eq_ind_r T (THead (Flat Appl) x (THead (Bind b) u1 t3)) (\lambda (t: +T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) +(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x +v2) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2))) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (H12: (pr0 x v2)).(or_introl (pr0 (THead (Flat Appl) x (THead (Bind +b) u1 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 +T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2))) (pr0_upsilon b H0 x v2 H12 u1 u2 H3 t3 t4 H5))) (\lambda +(H12: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x0: T).(\lambda (H13: (pr0 x x0)).(\lambda (H14: (subst0 i v3 v2 +x0)).(or_intror (pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x (THead (Bind b) u1 t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x (THead (Bind b) +u1 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O x0) t4)) (pr0_upsilon b H0 x x0 H13 u1 u2 H3 t3 t4 H5) (subst0_snd +(Bind b) v3 (THead (Flat Appl) (lift (S O) O x0) t4) (THead (Flat Appl) (lift +(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x0) (lift (S O) O v2) (s (Bind +b) i) (subst0_lift_ge_s v2 x0 v3 i H14 O (le_O_n i) b) t4 (Flat Appl)) +u2)))))) H12)) (H2 v0 x i H11 v3 H8)) w1 H10)))) H9)) (\lambda (H9: (ex2 T +(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: +T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)))).(ex2_ind T +(\lambda (t5: T).(eq T w1 (THead (Flat Appl) v1 t5))) (\lambda (t5: +T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) t5)) (or (pr0 w1 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H10: (eq +T w1 (THead (Flat Appl) v1 x))).(\lambda (H11: (subst0 (s (Flat Appl) i) v0 +(THead (Bind b) u1 t3) x)).(or3_ind (ex2 T (\lambda (u3: T).(eq T x (THead +(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (ex2 +T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 +(s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T (\lambda (u3: +T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or +(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H12: (ex2 T +(\lambda (u3: T).(eq T x (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s +(Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x (THead (Bind +b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or (pr0 w1 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda +(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq +T x (THead (Bind b) x0 t3))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 +x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 +t))) H10 (THead (Bind b) x0 t3) H13) in (eq_ind_r T (THead (Flat Appl) v1 +(THead (Bind b) x0 t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2))))) (or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 +w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or (pr0 (THead +(Flat Appl) v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H16: (pr0 x0 +u2)).(or_introl (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H16 t3 t4 H5))) (\lambda (H16: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 +u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 +(s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) +x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda +(H18: (subst0 (s (Flat Appl) i) v3 u2 x1)).(or_intror (pr0 (THead (Flat Appl) +v1 (THead (Bind b) x0 t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) +O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind +b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead +(Flat Appl) v1 (THead (Bind b) x0 t3)) w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead +(Bind b) x1 (THead (Flat Appl) (lift (S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 +H1 x0 x1 H17 t3 t4 H5) (subst0_fst v3 x1 u2 i H18 (THead (Flat Appl) (lift (S +O) O v2) t4) (Bind b))))))) H16)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8)) w1 +H15))))) H12)) (\lambda (H12: (ex2 T (\lambda (t5: T).(eq T x (THead (Bind b) +u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 +t5)))).(ex2_ind T (\lambda (t5: T).(eq T x (THead (Bind b) u1 t5))) (\lambda +(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) (or (pr0 w1 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x0: T).(\lambda (H13: (eq T x +(THead (Bind b) u1 x0))).(\lambda (H14: (subst0 (s (Bind b) (s (Flat Appl) +i)) v0 t3 x0)).(let H15 \def (eq_ind T x (\lambda (t: T).(eq T w1 (THead +(Flat Appl) v1 t))) H10 (THead (Bind b) u1 x0) H13) in (eq_ind_r T (THead +(Flat Appl) v1 (THead (Bind b) u1 x0)) (\lambda (t: T).(or (pr0 t (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 +w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (H16: (pr0 x0 t4)).(or_introl (pr0 (THead (Flat Appl) v1 (THead +(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 +x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 x0 t4 H16))) +(\lambda (H16: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 +(s (Bind b) (s (Flat Appl) i)) v3 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 +x0 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 w2)) +(or (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 x0)) (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x1: T).(\lambda (H17: (pr0 x0 x1)).(\lambda (H18: (subst0 (s (Bind +b) (s (Flat Appl) i)) v3 t4 x1)).(or_intror (pr0 (THead (Flat Appl) v1 (THead +(Bind b) u1 x0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 +x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) v1 (THead (Bind b) u1 x0)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) x1)) (pr0_upsilon b H0 v1 v2 H1 u1 u2 H3 +x0 x1 H17) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O v2) x1) +(THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) v3 x1 t4 +(s (Bind b) i) H18 (lift (S O) O v2)) u2)))))) H16)) (H6 v0 x0 (s (Bind b) (s +(Flat Appl) i)) H14 v3 H8)) w1 H15))))) H12)) (\lambda (H12: (ex3_2 T T +(\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind b) u3 t5)))) (\lambda +(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x (THead (Bind +b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 +u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) +v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda +(x0: T).(\lambda (x1: T).(\lambda (H13: (eq T x (THead (Bind b) x0 +x1))).(\lambda (H14: (subst0 (s (Flat Appl) i) v0 u1 x0)).(\lambda (H15: +(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x1)).(let H16 \def (eq_ind T x +(\lambda (t: T).(eq T w1 (THead (Flat Appl) v1 t))) H10 (THead (Bind b) x0 +x1) H13) in (eq_ind_r T (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) +(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) +O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind +(pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s +(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) v1 (THead +(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 +x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (H17: (pr0 x1 t4)).(or_ind (pr0 x0 u2) +(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) +i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (H18: (pr0 x0 u2)).(or_introl (pr0 (THead (Flat Appl) v1 +(THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) +x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 v1 v2 H1 x0 u2 H18 x1 t4 +H17))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 (s (Flat Appl) i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 +w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead +(Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 +(THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda +(H19: (pr0 x0 x2)).(\lambda (H20: (subst0 (s (Flat Appl) i) v3 u2 +x2)).(or_intror (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind +b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift +(S O) O v2) t4)) (pr0_upsilon b H0 v1 v2 H1 x0 x2 H19 x1 t4 H17) (subst0_fst +v3 x2 u2 i H20 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))))) H18)) +(H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: +T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: T).(subst0 (s +(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) v1 (THead +(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 +x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x1 +x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind +(pr0 x0 u2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s +(Flat Appl) i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 +x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 u2)).(or_intror (pr0 (THead (Flat +Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead +(Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 v1 v2 +H1 x0 u2 H20 x1 x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S +O) O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat +Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: +(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) +i) v3 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 (s (Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) v1 (THead +(Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 +x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (x3: T).(\lambda (H21: (pr0 x0 +x3)).(\lambda (H22: (subst0 (s (Flat Appl) i) v3 u2 x3)).(or_intror (pr0 +(THead (Flat Appl) v1 (THead (Bind b) x0 x1)) (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) +v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) v1 (THead (Bind b) x0 x1)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)) (THead (Bind b) x3 (THead (Flat Appl) (lift (S O) O v2) x2)) +(pr0_upsilon b H0 v1 v2 H1 x0 x3 H21 x1 x2 H18) (subst0_both v3 u2 x3 i H22 +(Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat Appl) (lift (S +O) O v2) x2) (subst0_snd (Flat Appl) v3 x2 t4 (s (Bind b) i) H19 (lift (S O) +O v2)))))))) H20)) (H4 v0 x0 (s (Flat Appl) i) H14 v3 H8))))) H17)) (H6 v0 x1 +(s (Bind b) (s (Flat Appl) i)) H15 v3 H8)) w1 H16))))))) H12)) +(subst0_gen_head (Bind b) v0 u1 t3 x (s (Flat Appl) i) H11))))) H9)) (\lambda +(H9: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead (Flat Appl) +u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) u1 t3) +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T w1 (THead +(Flat Appl) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v0 v1 u3))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat Appl) i) v0 (THead (Bind b) +u1 t3) t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda +(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0 +x1))).(\lambda (H11: (subst0 i v0 v1 x0)).(\lambda (H12: (subst0 (s (Flat +Appl) i) v0 (THead (Bind b) u1 t3) x1)).(or3_ind (ex2 T (\lambda (u3: T).(eq +T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 +u1 u3))) (ex2 T (\lambda (t5: T).(eq T x1 (THead (Bind b) u1 t5))) (\lambda +(t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5))) (ex3_2 T T +(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda +(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)))) (or +(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H13: (ex2 T +(\lambda (u3: T).(eq T x1 (THead (Bind b) u3 t3))) (\lambda (u3: T).(subst0 +(s (Flat Appl) i) v0 u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T x1 (THead +(Bind b) u3 t3))) (\lambda (u3: T).(subst0 (s (Flat Appl) i) v0 u1 u3)) (or +(pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda +(H14: (eq T x1 (THead (Bind b) x t3))).(\lambda (H15: (subst0 (s (Flat Appl) +i) v0 u1 x)).(let H16 \def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Flat +Appl) x0 t))) H10 (THead (Bind b) x t3) H14) in (eq_ind_r T (THead (Flat +Appl) x0 (THead (Bind b) x t3)) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x u2) (ex2 T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 w2))) (or +(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat +Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H17: +(pr0 x u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda +(w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) +x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 v2)).(or_introl (pr0 (THead (Flat +Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2))) (pr0_upsilon b H0 x0 v2 H18 x u2 H17 +t3 t4 H5))) (\lambda (H18: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 i v3 v2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda +(w2: T).(subst0 i v3 v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x +t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda +(H20: (subst0 i v3 v2 x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind +b) x t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 +T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 +(THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 x u2 H17 t3 t4 +H5) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead +(Flat Appl) (lift (S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S +O) O v2) (s (Bind b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 +(Flat Appl)) u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s +(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x +t3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x x2)).(\lambda +(H19: (subst0 (s (Flat Appl) i) v3 u2 x2)).(or_ind (pr0 x0 v2) (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 +(THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) +x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 +v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift +(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H20 x x2 H18 t3 t4 H5) (subst0_fst +v3 x2 u2 i H19 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda +(H20: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 +x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) x t3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x2 (THead (Flat Appl) (lift +(S O) O x3) t4)) (pr0_upsilon b H0 x0 x3 H21 x x2 H18 t3 t4 H5) (subst0_both +v3 u2 x2 i H19 (Bind b) (THead (Flat Appl) (lift (S O) O v2) t4) (THead (Flat +Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O +v2) (s (Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) t4 (Flat +Appl)))))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H4 v0 x (s (Flat Appl) +i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T +x1 (THead (Bind b) u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat +Appl) i)) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T x1 (THead (Bind b) +u1 t5))) (\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 t5)) +(or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) +(ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (x: +T).(\lambda (H14: (eq T x1 (THead (Bind b) u1 x))).(\lambda (H15: (subst0 (s +(Bind b) (s (Flat Appl) i)) v0 t3 x)).(let H16 \def (eq_ind T x1 (\lambda (t: +T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) u1 x) H14) in +(eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (\lambda (t: T).(or +(pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind (pr0 x t4) (ex2 T +(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat +Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda +(w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (H17: (pr0 x t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat +Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x0 +v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (pr0_upsilon b H0 x0 v2 H18 u1 u2 H3 x t4 H17))) (\lambda (H18: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x2: T).(\lambda (H19: (pr0 x0 x2)).(\lambda (H20: (subst0 i v3 v2 +x2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O x2) t4)) (pr0_upsilon b H0 x0 x2 H19 u1 u2 H3 x t4 H17) (subst0_snd +(Bind b) v3 (THead (Flat Appl) (lift (S O) O x2) t4) (THead (Flat Appl) (lift +(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x2) (lift (S O) O v2) (s (Bind +b) i) (subst0_lift_ge_s v2 x2 v3 i H20 O (le_O_n i) b) t4 (Flat Appl)) +u2)))))) H18)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H17: (ex2 T (\lambda (w2: +T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s +(Bind b) (s (Flat Appl) i)) v3 t4 w2)) (or (pr0 (THead (Flat Appl) x0 (THead +(Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) +(ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) +w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (x2: T).(\lambda (H18: (pr0 x +x2)).(\lambda (H19: (subst0 (s (Bind b) (s (Flat Appl) i)) v3 t4 x2)).(or_ind +(pr0 x0 v2) (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i +v3 v2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (H20: (pr0 x0 v2)).(or_intror (pr0 (THead (Flat Appl) x0 +(THead (Bind b) u1 x)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O +v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) +u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead +(Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) x2)) (pr0_upsilon b H0 x0 v2 +H20 u1 u2 H3 x x2 H18) (subst0_snd (Bind b) v3 (THead (Flat Appl) (lift (S O) +O v2) x2) (THead (Flat Appl) (lift (S O) O v2) t4) i (subst0_snd (Flat Appl) +v3 x2 t4 (s (Bind b) i) H19 (lift (S O) O v2)) u2)))) (\lambda (H20: (ex2 T +(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead (Bind b) +u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x3: T).(\lambda (H21: (pr0 x0 x3)).(\lambda (H22: (subst0 i v3 v2 +x3)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) u1 x)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) u1 x)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O x3) x2)) (pr0_upsilon b H0 x0 x3 H21 u1 u2 H3 x x2 H18) (subst0_snd +(Bind b) v3 (THead (Flat Appl) (lift (S O) O x3) x2) (THead (Flat Appl) (lift +(S O) O v2) t4) i (subst0_both v3 (lift (S O) O v2) (lift (S O) O x3) (s +(Bind b) i) (subst0_lift_ge_s v2 x3 v3 i H22 O (le_O_n i) b) (Flat Appl) t4 +x2 H19) u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))))) H17)) (H6 v0 x (s (Bind b) +(s (Flat Appl) i)) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex3_2 T T +(\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead (Bind b) u3 t5)))) (\lambda +(u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) v0 u1 u3))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T x1 (THead +(Bind b) u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s (Flat Appl) i) +v0 u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Bind b) (s (Flat +Appl) i)) v0 t3 t5))) (or (pr0 w1 (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H14: (eq T x1 (THead (Bind +b) x2 x3))).(\lambda (H15: (subst0 (s (Flat Appl) i) v0 u1 x2)).(\lambda +(H16: (subst0 (s (Bind b) (s (Flat Appl) i)) v0 t3 x3)).(let H17 \def (eq_ind +T x1 (\lambda (t: T).(eq T w1 (THead (Flat Appl) x0 t))) H10 (THead (Bind b) +x2 x3) H14) in (eq_ind_r T (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) +(\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) +O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v3 +(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2))))) (or_ind +(pr0 x3 t4) (ex2 T (\lambda (w2: T).(pr0 x3 w2)) (\lambda (w2: T).(subst0 (s +(Bind b) (s (Flat Appl) i)) v3 t4 w2))) (or (pr0 (THead (Flat Appl) x0 (THead +(Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) +t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 +x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4)) w2)))) (\lambda (H18: (pr0 x3 t4)).(or_ind (pr0 x2 u2) +(ex2 T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) +i) v3 u2 w2))) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2)))) (\lambda (H19: (pr0 x2 u2)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead (Flat +Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead +(Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead +(Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H20: (pr0 x0 +v2)).(or_introl (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (pr0_upsilon b H0 x0 v2 H20 x2 u2 H19 x3 t4 H18))) (\lambda (H20: (ex2 +T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x: T).(\lambda (H21: (pr0 x0 x)).(\lambda (H22: (subst0 i v3 v2 +x)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) u2 (THead (Flat Appl) (lift +(S O) O x) t4)) (pr0_upsilon b H0 x0 x H21 x2 u2 H19 x3 t4 H18) (subst0_snd +(Bind b) v3 (THead (Flat Appl) (lift (S O) O x) t4) (THead (Flat Appl) (lift +(S O) O v2) t4) i (subst0_fst v3 (lift (S O) O x) (lift (S O) O v2) (s (Bind +b) i) (subst0_lift_ge_s v2 x v3 i H22 O (le_O_n i) b) t4 (Flat Appl)) +u2)))))) H20)) (H2 v0 x0 i H11 v3 H8))) (\lambda (H19: (ex2 T (\lambda (w2: +T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s (Flat Appl) i) v3 u2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x2 w2)) (\lambda (w2: T).(subst0 (s +(Flat Appl) i) v3 u2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 +x3)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T +(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) +(\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S +O) O v2) t4)) w2)))) (\lambda (x: T).(\lambda (H20: (pr0 x2 x)).(\lambda +(H21: (subst0 (s (Flat Appl) i) v3 u2 x)).(or_ind (pr0 x0 v2) (ex2 T (\lambda +(w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 w2))) (or (pr0 (THead +(Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind b) u2 (THead (Flat Appl) +(lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 +(THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 +(THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) (\lambda (H22: (pr0 x0 +v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat +Appl) (lift (S O) O v2) t4)) w2)) (THead (Bind b) x (THead (Flat Appl) (lift +(S O) O v2) t4)) (pr0_upsilon b H0 x0 v2 H22 x2 x H20 x3 t4 H18) (subst0_fst +v3 x u2 i H21 (THead (Flat Appl) (lift (S O) O v2) t4) (Bind b))))) (\lambda +(H22: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 +v2 w2)) (or (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead (Bind +b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 +(THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i +v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))) +(\lambda (x4: T).(\lambda (H23: (pr0 x0 x4)).(\lambda (H24: (subst0 i v3 v2 +x4)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) (THead +(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: +T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x2 x3)) w2)) (\lambda (w2: +T).(subst0 i v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) +w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Appl) x0 (THead (Bind +b) x2 x3)) w2)) (\lambda (w2: T).(subst0 i v3 (THead 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(w2: +T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (H12: +(pr0 x0 u2)).(or_introl (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 +w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: +T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (pr0_delta x0 u2 H12 x1 t4 H11 +w H4))) (\lambda (H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: +T).(subst0 i v2 u2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda +(w2: T).(subst0 i v2 u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x: +T).(\lambda (H13: (pr0 x0 x)).(\lambda (H14: (subst0 i v2 u2 x)).(ex2_ind T +(\lambda (t: T).(subst0 O x t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 +w t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 +O x t4 x2)).(\lambda (H16: (subst0 (S (plus i O)) v2 w x2)).(let H17 \def +(f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in +(let H18 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w +x2)) H16 (S i) H17) in (or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x x2) (pr0_delta x0 x +H13 x1 t4 H11 x2 H15) (subst0_both v2 u2 x i H14 (Bind Abbr) w x2 H18)))))))) +(subst0_subst0_back t4 w u2 O H4 x v2 i H14))))) H12)) (H1 v1 x0 i H9 v2 +H6))) (\lambda (H11: (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda (w2: +T).(subst0 (s (Bind Abbr) i) v2 t4 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x1 +w2)) (\lambda (w2: T).(subst0 (s (Bind Abbr) i) v2 t4 w2)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x: T).(\lambda (H12: (pr0 x1 x)).(\lambda (H13: +(subst0 (s (Bind Abbr) i) v2 t4 x)).(or_ind (pr0 x0 u2) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 w2))) (or (pr0 (THead (Bind +Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead +(Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 +w) w2)))) (\lambda (H14: (pr0 x0 u2)).(ex2_ind T (\lambda (t: T).(subst0 O u2 +x t)) (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 w t)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (subst0 O u2 x +x2)).(\lambda (H16: (subst0 (s (Bind Abbr) i) v2 w x2)).(or_intror (pr0 +(THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: +T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead +(Bind Abbr) u2 w) w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) +x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)) +(THead (Bind Abbr) u2 x2) (pr0_delta x0 u2 H14 x1 x H12 x2 H15) (subst0_snd +(Bind Abbr) v2 x2 w i H16 u2)))))) (subst0_confluence_neq t4 x v2 (s (Bind +Abbr) i) H13 w u2 O H4 (sym_not_eq nat O (S i) (O_S i))))) (\lambda (H14: +(ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 u2 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 +u2 w2)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda (x2: T).(\lambda (H15: (pr0 x0 +x2)).(\lambda (H16: (subst0 i v2 u2 x2)).(ex2_ind T (\lambda (t: T).(subst0 O +x2 t4 t)) (\lambda (t: T).(subst0 (S (plus i O)) v2 w t)) (or (pr0 (THead +(Bind Abbr) x0 x1) (THead (Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 +(THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind +Abbr) u2 w) w2)))) (\lambda (x3: T).(\lambda (H17: (subst0 O x2 t4 +x3)).(\lambda (H18: (subst0 (S (plus i O)) v2 w x3)).(let H19 \def (f_equal +nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H20 +\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n v2 w x3)) H18 (S +i) H19) in (ex2_ind T (\lambda (t: T).(subst0 (s (Bind Abbr) i) v2 x3 t)) +(\lambda (t: T).(subst0 O x2 x t)) (or (pr0 (THead (Bind Abbr) x0 x1) (THead +(Bind Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) +w2)) (\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2)))) (\lambda +(x4: T).(\lambda (H21: (subst0 (s (Bind Abbr) i) v2 x3 x4)).(\lambda (H22: +(subst0 O x2 x x4)).(or_intror (pr0 (THead (Bind Abbr) x0 x1) (THead (Bind +Abbr) u2 w)) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) +(\lambda (w2: T).(subst0 i v2 (THead (Bind Abbr) u2 w) w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind Abbr) x0 x1) w2)) (\lambda (w2: T).(subst0 +i v2 (THead (Bind Abbr) u2 w) w2)) (THead (Bind Abbr) x2 x4) (pr0_delta x0 x2 +H15 x1 x H12 x4 H22) (subst0_both v2 u2 x2 i H16 (Bind Abbr) w x4 +(subst0_trans x3 w v2 (s (Bind Abbr) i) H20 x4 H21))))))) +(subst0_confluence_neq t4 x3 x2 O H17 x v2 (s (Bind Abbr) i) H13 (O_S +i)))))))) (subst0_subst0_back t4 w u2 O H4 x2 v2 i H16))))) H14)) (H1 v1 x0 i +H9 v2 H6))))) H11)) (H3 v1 x1 (s (Bind Abbr) i) H10 v2 H6)) w1 H8)))))) H7)) +(subst0_gen_head (Bind Abbr) v1 u1 t3 w1 i H5)))))))))))))))))) (\lambda (b: +B).(\lambda (H0: (not (eq B b Abst))).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H1: (pr0 t3 t4)).(\lambda (H2: ((\forall (v1: T).(\forall (w1: +T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) +\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda +(w1: T).(\lambda (i: nat).(\lambda (H3: (subst0 i v1 (THead (Bind b) u (lift +(S O) O t3)) w1)).(\lambda (v2: T).(\lambda (H4: (pr0 v1 v2)).(or3_ind (ex2 T +(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda +(u2: T).(subst0 i v1 u u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) +u t5))) (\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 +t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: +T).(\lambda (t5: T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))) (or +(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i +v2 t4 w2)))) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T w1 (THead (Bind b) +u2 (lift (S O) O t3)))) (\lambda (u2: T).(subst0 i v1 u u2)))).(ex2_ind T +(\lambda (u2: T).(eq T w1 (THead (Bind b) u2 (lift (S O) O t3)))) (\lambda +(u2: T).(subst0 i v1 u u2)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 +w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: +(eq T w1 (THead (Bind b) x (lift (S O) O t3)))).(\lambda (_: (subst0 i v1 u +x)).(eq_ind_r T (THead (Bind b) x (lift (S O) O t3)) (\lambda (t: T).(or (pr0 +t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2))))) (or_introl (pr0 (THead (Bind b) x (lift (S O) O t3)) t4) (ex2 T +(\lambda (w2: T).(pr0 (THead (Bind b) x (lift (S O) O t3)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 t3 t4 H1 x)) w1 H6)))) H5)) (\lambda +(H5: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5)))).(ex2_ind T (\lambda +(t5: T).(eq T w1 (THead (Bind b) u t5))) (\lambda (t5: T).(subst0 (s (Bind b) +i) v1 (lift (S O) O t3) t5)) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 +w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x: T).(\lambda (H6: +(eq T w1 (THead (Bind b) u x))).(\lambda (H7: (subst0 (s (Bind b) i) v1 (lift +(S O) O t3) x)).(ex2_ind T (\lambda (t5: T).(eq T x (lift (S O) O t5))) +(\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or (pr0 w1 +t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift (S O) O x0))).(\lambda +(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(let H10 \def (eq_ind T +x (\lambda (t: T).(eq T w1 (THead (Bind b) u t))) H6 (lift (S O) O x0) H8) in +(eq_ind_r T (THead (Bind b) u (lift (S O) O x0)) (\lambda (t: T).(or (pr0 t +t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2))))) (let H11 \def (eq_ind_r nat (minus i O) (\lambda (n: nat).(subst0 n +v1 t3 x0)) H9 i (minus_n_O i)) in (or_ind (pr0 x0 t4) (ex2 T (\lambda (w2: +T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or (pr0 (THead (Bind +b) u (lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u +(lift (S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda +(H12: (pr0 x0 t4)).(or_introl (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) +(ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) +(\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x0 t4 H12 u))) (\lambda +(H12: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4 +w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 +t4 w2)) (or (pr0 (THead (Bind b) u (lift (S O) O x0)) t4) (ex2 T (\lambda +(w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))) (\lambda (x1: T).(\lambda (H13: (pr0 x0 +x1)).(\lambda (H14: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u +(lift (S O) O x0)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) u (lift +(S O) O x0)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T +(\lambda (w2: T).(pr0 (THead (Bind b) u (lift (S O) O x0)) w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)) x1 (pr0_zeta b H0 x0 x1 H13 u) H14))))) H12)) (H2 v1 +x0 i H11 v2 H4))) w1 H10))))) (subst0_gen_lift_ge v1 t3 x (s (Bind b) i) (S +O) O H7 (le_n_S O i (le_O_n i))))))) H5)) (\lambda (H5: (ex3_2 T T (\lambda +(u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda (u2: +T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))))).(ex3_2_ind T T +(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Bind b) u2 t5)))) (\lambda +(u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s (Bind b) i) v1 (lift (S O) O t3) t5))) (or (pr0 w1 t4) (ex2 T +(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) +(\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T w1 (THead (Bind b) x0 +x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H8: (subst0 (s (Bind b) i) +v1 (lift (S O) O t3) x1)).(ex2_ind T (\lambda (t5: T).(eq T x1 (lift (S O) O +t5))) (\lambda (t5: T).(subst0 (minus (s (Bind b) i) (S O)) v1 t3 t5)) (or +(pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i +v2 t4 w2)))) (\lambda (x: T).(\lambda (H9: (eq T x1 (lift (S O) O +x))).(\lambda (H10: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x)).(let H11 +\def (eq_ind T x1 (\lambda (t: T).(eq T w1 (THead (Bind b) x0 t))) H6 (lift +(S O) O x) H9) in (eq_ind_r T (THead (Bind b) x0 (lift (S O) O x)) (\lambda +(t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2))))) (let H12 \def (eq_ind_r nat (minus i O) (\lambda +(n: nat).(subst0 n v1 t3 x)) H10 i (minus_n_O i)) in (or_ind (pr0 x t4) (ex2 +T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (or +(pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda (w2: 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+T).(subst0 i v2 t4 w2)) x2 (pr0_zeta b H0 x x2 H14 x0) H15))))) H13)) (H2 v1 +x i H12 v2 H4))) w1 H11))))) (subst0_gen_lift_ge v1 t3 x1 (s (Bind b) i) (S +O) O H8 (le_n_S O i (le_O_n i))))))))) H5)) (subst0_gen_head (Bind b) v1 u +(lift (S O) O t3) w1 i H3))))))))))))))) (\lambda (t3: T).(\lambda (t4: +T).(\lambda (H0: (pr0 t3 t4)).(\lambda (H1: ((\forall (v1: T).(\forall (w1: +T).(\forall (i: nat).((subst0 i v1 t3 w1) \to (\forall (v2: T).((pr0 v1 v2) +\to (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: +T).(subst0 i v2 t4 w2)))))))))))).(\lambda (u: T).(\lambda (v1: T).(\lambda +(w1: T).(\lambda (i: nat).(\lambda (H2: (subst0 i v1 (THead (Flat Cast) u t3) +w1)).(\lambda (v2: T).(\lambda (H3: (pr0 v1 v2)).(or3_ind (ex2 T (\lambda +(u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u +u2))) (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda +(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))) (ex3_2 T T (\lambda (u2: +T).(\lambda 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