--- /dev/null
+Base-2/theory.mma Base-2/preamble.ma Base-1/theory.ma
+Base-2/ext/tactics.mma Base-2/preamble.ma Base-1/ext/tactics.ma
+Base-2/ext/arith.mma Base-2/preamble.ma Base-1/ext/arith.ma
+Base-2/types/defs.mma Base-2/preamble.ma Base-1/types/defs.ma
+Base-2/types/props.mma Base-2/preamble.ma Base-1/types/props.ma
+Base-2/blt/defs.mma Base-2/preamble.ma Base-1/blt/defs.ma
+Base-2/blt/props.mma Base-2/preamble.ma Base-1/blt/props.ma
+Base-2/plist/defs.mma Base-2/preamble.ma Base-1/plist/defs.ma
+Base-2/plist/props.mma Base-2/preamble.ma Base-1/plist/props.ma
+++ /dev/null
-Base-2/theory.mma Base-2/preamble.ma Base-1/theory.ma
-Base-2/ext/tactics.mma Base-2/preamble.ma Base-1/ext/tactics.ma
-Base-2/ext/arith.mma Base-2/preamble.ma Base-1/ext/arith.ma
-Base-2/types/defs.mma Base-2/preamble.ma Base-1/types/defs.ma
-Base-2/types/props.mma Base-2/preamble.ma Base-1/types/props.ma
-Base-2/blt/defs.mma Base-2/preamble.ma Base-1/blt/defs.ma
-Base-2/blt/props.mma Base-2/preamble.ma Base-1/blt/props.ma
-Base-2/plist/defs.mma Base-2/preamble.ma Base-1/plist/defs.ma
-Base-2/plist/props.mma Base-2/preamble.ma Base-1/plist/props.ma
(* This file was automatically generated: do not edit *********************)
-include "preamble.ma".
+include "LambdaDelta-1/preamble.ma".
inductive A: Set \def
| ASort: nat \to (nat \to A)
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
inductive C: Set \def
| CSort: nat \to C
(* This file was automatically generated: do not edit *********************)
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
theorem clt_cong:
\forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t:
(* This file was automatically generated: do not edit *********************)
-include "preamble.ma".
+include "LambdaDelta-1/preamble.ma".
record G : Set \def {
next: (nat \to nat);
+++ /dev/null
-DIR=$(shell basename $$PWD)
-MATITAOPTIONS=-onepass
-
-$(DIR) all:
- ../../../matitac $(MATITAOPTIONS)
-$(DIR).opt opt all.opt:
- ../../../matitac.opt $(MATITAOPTIONS)
-clean:
- ../../../matitaclean
-clean.opt:
- ../../../matitaclean.opt
-depend:
- ../../../matitadep
-depend.opt:
- ../../../matitadep.opt
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
theorem terms_props__bind_dec:
\forall (b1: B).(\forall (b2: B).(or (eq B b1 b2) ((eq B b1 b2) \to (\forall
(* This file was automatically generated: do not edit *********************)
-include "preamble.ma".
+include "LambdaDelta-1/preamble.ma".
inductive B: Set \def
| Abbr: B
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
theorem not_abbr_abst:
not (eq B Abbr Abst)
(* This file was automatically generated: do not edit *********************)
-include "asucc/defs.ma".
+include "LambdaDelta-1/asucc/defs.ma".
definition aplus:
G \to (A \to (nat \to A))
(* This file was automatically generated: do not edit *********************)
-include "aplus/defs.ma".
+include "LambdaDelta-1/aplus/defs.ma".
-include "next_plus/props.ma".
+include "LambdaDelta-1/next_plus/props.ma".
theorem aplus_reg_r:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall
(* This file was automatically generated: do not edit *********************)
-include "A/defs.ma".
+include "LambdaDelta-1/A/defs.ma".
inductive aprem: nat \to (A \to (A \to Prop)) \def
| aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1))
(* This file was automatically generated: do not edit *********************)
-include "aprem/defs.ma".
+include "LambdaDelta-1/aprem/defs.ma".
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
theorem aprem_repl:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall
(* This file was automatically generated: do not edit *********************)
-include "arity/props.ma".
+include "LambdaDelta-1/arity/props.ma".
-include "arity/cimp.ma".
+include "LambdaDelta-1/arity/cimp.ma".
-include "aprem/props.ma".
+include "LambdaDelta-1/aprem/props.ma".
theorem arity_aprem:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
(* This file was automatically generated: do not edit *********************)
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
-include "cimp/props.ma".
+include "LambdaDelta-1/cimp/props.ma".
theorem arity_cimp_conf:
\forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
(* This file was automatically generated: do not edit *********************)
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
inductive arity (g: G): C \to (T \to (A \to Prop)) \def
| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O
(* This file was automatically generated: do not edit *********************)
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
-include "leq/asucc.ma".
+include "LambdaDelta-1/leq/asucc.ma".
-include "leq/fwd.ma".
+include "LambdaDelta-1/leq/fwd.ma".
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
theorem arity_gen_sort:
\forall (g: G).(\forall (c: C).(\forall (n: nat).(\forall (a: A).((arity g c
(* This file was automatically generated: do not edit *********************)
-include "arity/props.ma".
+include "LambdaDelta-1/arity/props.ma".
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
theorem arity_lift1:
\forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds:
(* This file was automatically generated: do not edit *********************)
-include "csuba/arity.ma".
+include "LambdaDelta-1/csuba/arity.ma".
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
-include "pr1/defs.ma".
+include "LambdaDelta-1/pr1/defs.ma".
-include "wcpr0/getl.ma".
+include "LambdaDelta-1/wcpr0/getl.ma".
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
-include "arity/subst0.ma".
+include "LambdaDelta-1/arity/subst0.ma".
theorem arity_sred_wcpr0_pr0:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (a: A).((arity g
(* This file was automatically generated: do not edit *********************)
-include "arity/fwd.ma".
+include "LambdaDelta-1/arity/fwd.ma".
theorem node_inh:
\forall (g: G).(\forall (n: nat).(\forall (k: nat).(ex_2 C T (\lambda (c:
(* This file was automatically generated: do not edit *********************)
-include "arity/props.ma".
+include "LambdaDelta-1/arity/props.ma".
-include "fsubst0/fwd.ma".
+include "LambdaDelta-1/fsubst0/fwd.ma".
-include "csubst0/getl.ma".
+include "LambdaDelta-1/csubst0/getl.ma".
-include "csubst0/props.ma".
+include "LambdaDelta-1/csubst0/props.ma".
-include "subst0/dec.ma".
+include "LambdaDelta-1/subst0/dec.ma".
-include "subst0/fwd.ma".
+include "LambdaDelta-1/subst0/fwd.ma".
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
theorem arity_gen_cvoid_subst0:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
(* This file was automatically generated: do not edit *********************)
-include "A/defs.ma".
+include "LambdaDelta-1/A/defs.ma".
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
definition asucc:
G \to (A \to A)
(* This file was automatically generated: do not edit *********************)
-include "asucc/defs.ma".
+include "LambdaDelta-1/asucc/defs.ma".
theorem asucc_gen_sort:
\forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A
(* This file was automatically generated: do not edit *********************)
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
definition cimp:
C \to (C \to Prop)
(* This file was automatically generated: do not edit *********************)
-include "cimp/defs.ma".
+include "LambdaDelta-1/cimp/defs.ma".
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
theorem cimp_flat_sx:
\forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v)
(* This file was automatically generated: do not edit *********************)
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
inductive clear: C \to (C \to Prop) \def
| clear_bind: \forall (b: B).(\forall (e: C).(\forall (u: T).(clear (CHead e
(* This file was automatically generated: do not edit *********************)
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
theorem drop_clear:
\forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to
(* This file was automatically generated: do not edit *********************)
-include "clear/defs.ma".
+include "LambdaDelta-1/clear/defs.ma".
theorem clear_gen_sort:
\forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P:
(* This file was automatically generated: do not edit *********************)
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
theorem clear_clear:
\forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
(* This file was automatically generated: do not edit *********************)
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
-include "s/defs.ma".
+include "LambdaDelta-1/s/defs.ma".
definition clen:
C \to nat
(* This file was automatically generated: do not edit *********************)
-include "clen/defs.ma".
+include "LambdaDelta-1/clen/defs.ma".
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
theorem getl_ctail_clen:
\forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n:
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
inductive cnt: T \to Prop \def
| cnt_sort: \forall (n: nat).(cnt (TSort n))
(* This file was automatically generated: do not edit *********************)
-include "cnt/defs.ma".
+include "LambdaDelta-1/cnt/defs.ma".
-include "lift/fwd.ma".
+include "LambdaDelta-1/lift/fwd.ma".
theorem cnt_lift:
\forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i
(* This file was automatically generated: do not edit *********************)
-include "csuba/getl.ma".
+include "LambdaDelta-1/csuba/getl.ma".
-include "csuba/props.ma".
+include "LambdaDelta-1/csuba/props.ma".
-include "arity/props.ma".
+include "LambdaDelta-1/arity/props.ma".
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
theorem csuba_arity:
\forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
(* This file was automatically generated: do not edit *********************)
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
theorem csuba_clear_conf:
\forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to
(* This file was automatically generated: do not edit *********************)
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
inductive csuba (g: G): C \to (C \to Prop) \def
| csuba_sort: \forall (n: nat).(csuba g (CSort n) (CSort n))
(* This file was automatically generated: do not edit *********************)
-include "csuba/fwd.ma".
+include "LambdaDelta-1/csuba/fwd.ma".
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
theorem csuba_drop_abbr:
\forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i
(* This file was automatically generated: do not edit *********************)
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
theorem csuba_gen_abbr:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g
(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g (CHead d1 (Bind Abbr) u) c)).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0
-(CHead d1 (Bind Abbr) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C
-c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind
-Abbr) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr)
-u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) |
-(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0)
-(CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k
-u0) (CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1)
-in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c0:
-C).((eq K k (Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to
-((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr)
-u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind
-Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead
-c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead
-d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq
-T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) t) c) \to
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr)
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2
-(Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c0:
-C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1
-c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2
-(Bind Abbr) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr)
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1
-(Bind Abbr) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5
-\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H3) in (False_ind ((eq C
-(CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g
-a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))])
-in (H0 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C c))))))).
+(csuba g (CHead d1 (Bind Abbr) u) c)).(insert_eq C (CHead d1 (Bind Abbr) u)
+(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
+C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda
+(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda
+(c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(let H2
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Abbr) u) H1) in (False_ind (ex2 C (\lambda (d2:
+C).(eq C (CSort n) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0)
+(CHead d1 (Bind Abbr) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u0) (CHead d1 (Bind Abbr) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H3)
+in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Abbr)
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2
+(Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C
+c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0
+c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr)
+u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2
+(refl_equal C (CHead c2 (Bind Abbr) u)) H10))) k H7) u0 H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1
+d2)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abbr) u))).(let H6 \def (eq_ind C
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1
+(Bind Abbr) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2
+(Bind Abbr) u0) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))
+H6)))))))))))) y c H0))) H))))).
theorem csuba_gen_void:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g
(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g (CHead d1 (Bind Void) u) c)).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0
-(CHead d1 (Bind Void) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C
-c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind
-Void) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void)
-u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) |
-(csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0)
-(CHead d1 (Bind Void) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k
-u0) (CHead d1 (Bind Void) u) H1) in ((let H4 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1)
-in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in (eq_ind C d1 (\lambda (c0:
-C).((eq K k (Bind Void)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to
-((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind
-Void))).(eq_ind K (Bind Void) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead
-c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead
-d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq
-T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Void) t) c) \to
-((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2
-(Bind Void) u) c)).(eq_ind C (CHead c2 (Bind Void) u) (\lambda (c0:
-C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Void) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1
-c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) u) (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2
-(Bind Void) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Void)
-H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1
-(Bind Void) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5
-\def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead d1 (Bind Void) u) H3) in (False_ind ((eq C
-(CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g
-a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))])
-in (H0 (refl_equal C (CHead d1 (Bind Void) u)) (refl_equal C c))))))).
+(csuba g (CHead d1 (Bind Void) u) c)).(insert_eq C (CHead d1 (Bind Void) u)
+(\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
+C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (\lambda
+(y: C).(\lambda (H0: (csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda
+(c1: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2:
+C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1
+d2))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1
+d2)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c1 k u0)
+(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H3)
+in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k t) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (eq_ind_r K (Bind Void)
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H9 \def (eq_ind C
+c1 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1
+d2))))) H2 d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0
+c2)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void)
+u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2
+(refl_equal C (CHead c2 (Bind Void) u)) H10))) k H7) u0 H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c2 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1
+d2)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow
+True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1
+(Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c2
+(Bind Abbr) u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))
+H6)))))))))))) y c H0))) H))))).
theorem csuba_gen_abst:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g
a))))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(let H0 \def (match H in csuba
-return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C
-c0 (CHead d1 (Bind Abst) u1)) \to ((eq C c1 c) \to (or (ex2 C (\lambda (d2:
+(H: (csuba g (CHead d1 (Bind Abst) u1) c)).(insert_eq C (CHead d1 (Bind Abst)
+u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(or (ex2 C (\lambda (d2:
C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0:
-(eq C (CSort n) (CHead d1 (Bind Abst) u1))).(\lambda (H1: (eq C (CSort n)
-c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H0) in (False_ind ((eq C
-(CSort n) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst)
+A).(arity g d2 u2 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g y
+c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind
+Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2 (Bind Abst)
u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2)))))
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abbr) u2)))))
(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))
-H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead
-c1 k u) (CHead d1 (Bind Abst) u1))).(\lambda (H2: (eq C (CHead c2 k u)
-c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
-(CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H4 \def (f_equal C K
-(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead d1
-(Bind Abst) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in
-C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in (eq_ind C
-d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to ((eq T u u1) \to ((eq C (CHead
-c2 k u) c) \to ((csuba g c0 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind
-Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1
-d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc
-g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst)
-(\lambda (k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1
-c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Abst) u1) H1) in (False_ind (or (ex2 C
+(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abst) u1))) (\lambda (d2:
+C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C (CSort n) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) (\lambda
-(H7: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Bind Abst)
-t) c) \to ((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2
-(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abst) u1) c)).(eq_ind C (CHead
-c2 (Bind Abst) u1) (\lambda (c0: C).((csuba g d1 c2) \to (or (ex2 C (\lambda
-(d2: C).(eq C c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0
+(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H2))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda
+(H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2:
+C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))
+(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2
(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity
g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
-A).(arity g d2 u2 a)))))))) (\lambda (H9: (csuba g d1 c2)).(or_introl (ex2 C
-(\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abbr)
-u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g
-a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
-a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead
-d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C
-(CHead c2 (Bind Abst) u1)) H9))) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k
-(Bind Abst) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst
-c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst)
-t) (CHead d1 (Bind Abst) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u)
-c)).((let H5 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0]))
-(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H3) in ((let H6 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 (Bind
-Abst) t) (CHead d1 (Bind Abst) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq
-T t u1) \to ((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g c0 c2) \to
-((arity g c0 t (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda
-(d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 a0)))))))))))) (\lambda (H7: (eq T t u1)).(eq_ind T u1
-(\lambda (t0: T).((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g d1 c2) \to
-((arity g d1 t0 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda
-(d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 a0))))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u)
-c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c0: C).((csuba g d1 c2) \to
-((arity g d1 u1 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda
-(d2: C).(eq C c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1
-d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c0
-(CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity
-g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 a0)))))))))) (\lambda (H9: (csuba g d1 c2)).(\lambda (H10:
-(arity g d1 u1 (asucc g a))).(\lambda (H11: (arity g c2 u a)).(or_intror (ex2
-C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abst) u1)))
-(\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda
+A).(arity g d2 u2 a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3:
+(eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(let H4 \def (f_equal C C
+(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0])) (CHead c1 k u) (CHead d1
+(Bind Abst) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
+\Rightarrow k0])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H3) in ((let H6
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u)
+(CHead d1 (Bind Abst) u1) H3) in (\lambda (H7: (eq K k (Bind Abst))).(\lambda
+(H8: (eq C c1 d1)).(eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2:
+C).(eq C (CHead c2 k t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g
+d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
+(CHead c2 k t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_:
+T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda
+(u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (eq_ind_r K (Bind Abst)
+(\lambda (k0: K).(or (ex2 C (\lambda (d2: C).(eq C (CHead c2 k0 u1) (CHead d2
+(Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 k0 u1) (CHead d2
+(Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
+d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+(asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2
+u2 a))))))) (let H9 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1
+(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) H2
+d1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1
+H8) in (or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst)
+u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2
+(Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))
+c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H10)))) k H7) u H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c1 (CHead d1 (Bind Abst) u1)) \to (or (ex2 C
+(\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba
+g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a:
+A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a: A).(arity g d2 u2 a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda
+(H3: (arity g c1 t (asucc g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u
+a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst)
+u1))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _) \Rightarrow c0]))
+(CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H5) in ((let H7 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind
+Abst) t) (CHead d1 (Bind Abst) u1) H5) in (\lambda (H8: (eq C c1 d1)).(let H9
+\def (eq_ind T t (\lambda (t0: T).(arity g c1 t0 (asucc g a))) H3 u1 H7) in
+(let H10 \def (eq_ind C c1 (\lambda (c0: C).(arity g c0 u1 (asucc g a))) H9
+d1 H8) in (let H11 \def (eq_ind C c1 (\lambda (c0: C).((eq C c0 (CHead d1
+(Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind
+Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2)))))
+(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0)))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0))))))))
+H2 d1 H8) in (let H12 \def (eq_ind C c1 (\lambda (c0: C).(csuba g c0 c2)) H1
+d1 H8) in (or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u)
+(CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr)
+u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
+(_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0:
+A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(a0: A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda
(u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr)
u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2))))
(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g
a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2
-a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
-A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_:
-C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))) c2 u a
-(refl_equal C (CHead c2 (Bind Abbr) u)) H9 H10 H11))))) c H8)) t (sym_eq T t
-u1 H7))) c1 (sym_eq C c1 d1 H6))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C
-(CHead d1 (Bind Abst) u1)) (refl_equal C c))))))).
+a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H12 H10 H4))))))))
+H6)))))))))))) y c H0))) H))))).
theorem csuba_gen_flat:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
C).(\lambda (_: T).(csuba g d1 d2)))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(let H0 \def (match H
-in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0
-c1)).((eq C c0 (CHead d1 (Flat f) u1)) \to ((eq C c1 c) \to (ex2_2 C T
+(f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(insert_eq C (CHead
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c0 c)) (\lambda (_: C).(ex2_2 C T
(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) with [(csuba_sort n)
-\Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u1))).(\lambda
-(H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e:
-C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Flat f) u1) H0) in
-(False_ind ((eq C (CSort n) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d1 d2))))) H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1:
-(eq C (CHead c1 k u) (CHead d1 (Flat f) u1))).(\lambda (H2: (eq C (CHead c2 k
-u) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow
-t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H4 \def (f_equal C K
-(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead d1
-(Flat f) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e in C
-return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _ _)
-\Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in (eq_ind C d1
-(\lambda (c0: C).((eq K k (Flat f)) \to ((eq T u u1) \to ((eq C (CHead c2 k
-u) c) \to ((csuba g c0 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d1 d2))))))))) (\lambda (H6: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda
-(k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) (\lambda (H7:
-(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Flat f) t) c) \to
-((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c
-(CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))))))) (\lambda (H8: (eq C (CHead c2 (Flat f) u1) c)).(eq_ind C (CHead c2
-(Flat f) u1) (\lambda (c0: C).((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d1 d2)))))) (\lambda (H9: (csuba g d1
-c2)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat
-f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
-d2))) c2 u1 (refl_equal C (CHead c2 (Flat f) u1)) H9)) c H8)) u (sym_eq T u
-u1 H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2
-H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C
-(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(\lambda (H4: (eq C (CHead
-c2 (Bind Abbr) u) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t)
-(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (CHead d1 (Flat f) u1) H3) in (False_ind ((eq C (CHead c2 (Bind
-Abbr) u) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity
-g c2 u a) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) H5))
-H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Flat f) u1)) (refl_equal C
-c)))))))).
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (\lambda (y: C).(\lambda (H0:
+(csuba g y c)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c0
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))) H2)))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u)
+(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k
+u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H3) in
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c1 d1)).(eq_ind_r T u1
+(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2
+k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c2 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d1 d2))))) (let H9 \def (eq_ind C c1
+(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d1 d2)))))) H2 d1 H8) in (let H10 \def (eq_ind C
+c1 (\lambda (c0: C).(csuba g c0 c2)) H1 d1 H8) in (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C
+(CHead c2 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1
+d2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C
+(CHead c1 (Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Bind Abbr) u) (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))
+H6)))))))))))) y c H0))) H)))))).
theorem csuba_gen_bind:
\forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
\def
\lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(let H0 \def
-(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba
-? c c0)).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
-(CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def
-(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to
-(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c c2)) (\lambda (_:
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e1 e2)))))) H2)) H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow
-(\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2:
-(eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e
-in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
-\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u)
-(CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 |
-(CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in
-(eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C
-(CHead c0 k u) c2) \to ((csuba g c c0) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
-(\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T
-u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C
-(CHead c0 (Bind b1) t) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead
-c0 (Bind b1) v1) (\lambda (c: C).((csuba g e1 c0) \to (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))
-(\lambda (H9: (csuba g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c:
-C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in
-(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
-(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0
-(Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1)
-H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c0 H0 t a
-H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead e1
-(Bind b1) v1))).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H5
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1
-(Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C B
-(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])]))
-(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H7 \def
+T).(csuba g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csuba g y
+c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
+v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (k: K).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind
-Abst) t) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: C).((eq B
-Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba
-g c c0) \to ((arity g c t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2)))))))))))) (\lambda (H8: (eq B Abst b1)).(eq_ind B Abst (\lambda (_:
-B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0)
-\to ((arity g e1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
+(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
+b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c3)) H1 e1 H8) in
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c1
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))).(\lambda (t:
+T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc g a))).(\lambda (u:
+T).(\lambda (_: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c1 (Bind Abst)
+t) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _
+_) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in
+((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in
+((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead
+c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abst
+b1)).(\lambda (H10: (eq C c1 e1)).(let H11 \def (eq_ind T t (\lambda (t0:
+T).(arity g c1 t0 (asucc g a))) H3 v1 H8) in (let H12 \def (eq_ind C c1
+(\lambda (c: C).(arity g c v1 (asucc g a))) H11 e1 H10) in (let H13 \def
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C
+T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind
b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1
-e2))))))))))) (\lambda (H9: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C
-(CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t0 (asucc
-g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H10:
-(eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u)
-(\lambda (c: C).((csuba g e1 c0) \to ((arity g e1 v1 (asucc g a)) \to ((arity
-g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
-T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H11: (csuba g e1
-c0)).(\lambda (_: (arity g e1 v1 (asucc g a))).(\lambda (_: (arity g c0 u
-a)).(let H14 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1)
-v1) c)) H (CHead c0 (Bind Abbr) u) H10) in (let H15 \def (eq_ind_r B b1
-(\lambda (b: B).(csuba g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u)))
-H14 Abst H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c0 u
-(refl_equal C (CHead c0 (Bind Abbr) u)) H11)))))) c2 H10)) t (sym_eq T t v1
-H9))) b1 H8)) c1 (sym_eq C c1 e1 H7))) H6)) H5)) H4 H0 H1 H2)))]) in (H0
-(refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
+e2))))))) H2 e1 H10) in (let H14 \def (eq_ind C c1 (\lambda (c: C).(csuba g c
+c3)) H1 e1 H10) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1
+(CHead e1 (Bind b) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) H13 Abst H9) in
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c3 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c3 u (refl_equal C (CHead c3
+(Bind Abbr) u)) H14))))))))) H7)) H6)))))))))))) y c2 H0))) H)))))).
theorem csuba_gen_abst_rev:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g c (CHead d1 (Bind Abst) u))).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c)
-\to ((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1:
-(eq C (CSort n) (CHead d1 (Bind Abst) u))).(eq_ind C (CSort n) (\lambda (c0:
-C).((eq C (CSort n) (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq
-C c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda
-(H2: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H3 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst)
-u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind
-Abst) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda
-(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u))).(eq_ind C (CHead c1 k
-u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Abst) u)) \to
-((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k
-u0) (CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C T (\lambda (e:
+(csuba g c (CHead d1 (Bind Abst) u))).(insert_eq C (CHead d1 (Bind Abst) u)
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
+C c (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda
+(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda
+(c1: C).((eq C c1 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c0 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) u))).(let H2
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Abst) u) H1) in (False_ind (ex2 C (\lambda (d2:
+C).(eq C (CSort n) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0)
+(CHead d1 (Bind Abst) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Abst) u) H3)
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c2 k u0) (CHead d1 (Bind Abst) u) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1
-(Bind Abst) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to
-((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1
-k u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-(\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0:
-K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead
-c1 k0 u0) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1))))))
-(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
-u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1
-d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) u) (CHead d2
-(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
-(Bind Abst) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Abst) H7)))
-c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t)
-c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst)
-u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2
-(Bind Abbr) u0) (CHead d1 (Bind Abst) u)) \to ((csuba g c1 c2) \to ((arity g
-c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0
-(CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda
-(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def
-(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead d1 (Bind Abst) u) H5) in (False_ind ((csuba g
-c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u)))
-(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0
-(refl_equal C c) (refl_equal C (CHead d1 (Bind Abst) u)))))))).
+in (\lambda (H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Abst)
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2
+(Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1
+c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst)
+u) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)) c1
+(refl_equal C (CHead c1 (Bind Abst) u)) H10))) k H7) u0 H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Abst) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Abst) u))).(let H6 \def (eq_ind C
+(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead
+d1 (Bind Abst) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1
+(Bind Abst) t) (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d1)))
+H6)))))))))))) c y H0))) H))))).
theorem csuba_gen_void_rev:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g c
Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H:
-(csuba g c (CHead d1 (Bind Void) u))).(let H0 \def (match H in csuba return
-(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C c0 c)
-\to ((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C c
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) with
-[(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1:
-(eq C (CSort n) (CHead d1 (Bind Void) u))).(eq_ind C (CSort n) (\lambda (c0:
-C).((eq C (CSort n) (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq
-C c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda
-(H2: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H3 \def (eq_ind C (CSort
-n) (\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void)
-u) H2) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind
-Void) u))) (\lambda (d2: C).(csuba g d2 d1))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) c)).(\lambda
-(H2: (eq C (CHead c2 k u0) (CHead d1 (Bind Void) u))).(eq_ind C (CHead c1 k
-u0) (\lambda (c0: C).((eq C (CHead c2 k u0) (CHead d1 (Bind Void) u)) \to
-((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda (H3: (eq C (CHead c2 k
-u0) (CHead d1 (Bind Void) u))).(let H4 \def (f_equal C T (\lambda (e:
+(csuba g c (CHead d1 (Bind Void) u))).(insert_eq C (CHead d1 (Bind Void) u)
+(\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2 C (\lambda (d2: C).(eq
+C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (\lambda
+(y: C).(\lambda (H0: (csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda
+(c1: C).((eq C c1 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda (d2: C).(eq C
+c0 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))) (\lambda
+(n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Void) u))).(let H2
+\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead d1 (Bind Void) u) H1) in (False_ind (ex2 C (\lambda (d2:
+C).(eq C (CSort n) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1))) H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1
+c2)).(\lambda (H2: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (k: K).(\lambda (u0: T).(\lambda (H3: (eq C (CHead c2 k u0)
+(CHead d1 (Bind Void) u))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C T (\lambda (e:
C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
(CHead _ _ t) \Rightarrow t])) (CHead c2 k u0) (CHead d1 (Bind Void) u) H3)
-in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
-(CHead c2 k u0) (CHead d1 (Bind Void) u) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u0) (CHead d1
-(Bind Void) u) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to
-((eq T u0 u) \to ((csuba g c1 c0) \to (ex2 C (\lambda (d2: C).(eq C (CHead c1
-k u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))
-(\lambda (H7: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0:
-K).((eq T u0 u) \to ((csuba g c1 d1) \to (ex2 C (\lambda (d2: C).(eq C (CHead
-c1 k0 u0) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1))))))
-(\lambda (H8: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((csuba g c1 d1) \to
-(ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) t) (CHead d2 (Bind Void)
-u))) (\lambda (d2: C).(csuba g d2 d1))))) (\lambda (H9: (csuba g c1
-d1)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void) u) (CHead d2
-(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
-(Bind Void) u)) H9)) u0 (sym_eq T u0 u H8))) k (sym_eq K k (Bind Void) H7)))
-c2 (sym_eq C c2 d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a
-H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t)
-c)).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void)
-u))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2
-(Bind Abbr) u0) (CHead d1 (Bind Void) u)) \to ((csuba g c1 c2) \to ((arity g
-c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c0
-(CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))))))) (\lambda
-(H5: (eq C (CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def
-(eq_ind C (CHead c2 (Bind Abbr) u0) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead d1 (Bind Void) u) H5) in (False_ind ((csuba g
-c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Void) u)))
-(\lambda (d2: C).(csuba g d2 d1)))))) H6))) c H3 H4 H0 H1 H2)))]) in (H0
-(refl_equal C c) (refl_equal C (CHead d1 (Bind Void) u)))))))).
+in (\lambda (H7: (eq K k (Bind Void))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
+T u (\lambda (t: T).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (eq_ind_r K (Bind Void)
+(\lambda (k0: K).(ex2 C (\lambda (d2: C).(eq C (CHead c1 k0 u) (CHead d2
+(Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))) (let H9 \def (eq_ind C
+c2 (\lambda (c0: C).((eq C c0 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1))))) H2 d1 H8) in (let H10 \def (eq_ind C c2 (\lambda (c0: C).(csuba g c1
+c0)) H1 d1 H8) in (ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Void)
+u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)) c1
+(refl_equal C (CHead c1 (Bind Void) u)) H10))) k H7) u0 H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_: (csuba g c1
+c2)).(\lambda (_: (((eq C c2 (CHead d1 (Bind Void) u)) \to (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2
+d1)))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u0: T).(\lambda (_: (arity g c2 u0 a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u0) (CHead d1 (Bind Void) u))).(let H6 \def (eq_ind C
+(CHead c2 (Bind Abbr) u0) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
+k in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B
+return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow
+False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead
+d1 (Bind Void) u) H5) in (False_ind (ex2 C (\lambda (d2: C).(eq C (CHead c1
+(Bind Abst) t) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d2 d1)))
+H6)))))))))))) c y H0))) H))))).
theorem csuba_gen_abbr_rev:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g c
a))))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(let H0 \def (match H in csuba
-return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0 c1)).((eq C
-c0 c) \to ((eq C c1 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2:
+(H: (csuba g c (CHead d1 (Bind Abbr) u1))).(insert_eq C (CHead d1 (Bind Abbr)
+u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(or (ex2 C (\lambda (d2:
C).(eq C c (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)))
(ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead
d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a: A).(arity g d1 u1 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda
-(H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr)
-u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort n) (CHead d1 (Bind
+(a: A).(arity g d1 u1 a))))))) (\lambda (y: C).(\lambda (H0: (csuba g c
+y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1 (CHead d1 (Bind
Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind Abbr)
u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))
-(\lambda (H2: (eq C (CSort n) (CHead d1 (Bind Abbr) u1))).(let H3 \def
-(eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Bind Abbr) u1) H2) in (False_ind (or (ex2 C (\lambda
-(d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g
-d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
-(CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H3))) c H0 H1))) | (csuba_head
-c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda
-(H2: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead c1 k
-u) (\lambda (c0: C).((eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1)) \to
-((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))))))
-(\lambda (H3: (eq C (CHead c2 k u) (CHead d1 (Bind Abbr) u1))).(let H4 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u)
-(CHead d1 (Bind Abbr) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3)
-in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0]))
-(CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in (eq_ind C d1 (\lambda (c0:
-C).((eq K k (Bind Abbr)) \to ((eq T u u1) \to ((csuba g c1 c0) \to (or (ex2 C
-(\lambda (d2: C).(eq C (CHead c1 k u) (CHead d2 (Bind Abbr) u1))) (\lambda
-(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C (CHead c1 k u) (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))))))))
-(\lambda (H7: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0:
-K).((eq T u u1) \to ((csuba g c1 d1) \to (or (ex2 C (\lambda (d2: C).(eq C
-(CHead c1 k0 u) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
-d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C
-(CHead c1 k0 u) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_:
-T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abbr)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Bind Abbr) u1) H1) in (False_ind (or (ex2 C
+(\lambda (d2: C).(eq C (CSort n) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
+C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
+(_: A).(eq C (CSort n) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))) H2)))) (\lambda
+(c1: C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C
+c2 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
+a))))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u)
+(CHead d1 (Bind Abbr) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u) (CHead d1 (Bind Abbr) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Bind Abbr) u1) H3)
+in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r
+T u1 (\lambda (t: T).(or (ex2 C (\lambda (d2: C).(eq C (CHead c1 k t) (CHead
+d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 k t) (CHead d2 (Bind
+Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2
+d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2
+(asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1
+u1 a))))))) (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex2 C (\lambda (d2:
+C).(eq C (CHead c1 k0 u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba
+g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq
+C (CHead c1 k0 u1) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda
+(_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2:
T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda
-(_: T).(\lambda (a: A).(arity g d1 u1 a))))))))) (\lambda (H8: (eq T u
-u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (or (ex2 C (\lambda
-(d2: C).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(_: T).(\lambda (a: A).(arity g d1 u1 a))))))) (let H9 \def (eq_ind C c2
+(\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1
+(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
+A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a:
+A).(arity g d2 u2 (asucc g a))))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(arity g d1 u1 a)))))))) H2 d1 H8) in (let H10 \def (eq_ind C c2
+(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (or_introl (ex2 C (\lambda
+(d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda
(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) t) (CHead d2 (Bind Abst)
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst)
+u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
+(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
+a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a)))))
+(ex_intro2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind
+Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1
+(Bind Abbr) u1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda (d2: C).(eq C c1 (CHead d2
+(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
+(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1 (CHead d2 (Bind Abst)
u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
(\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g
a))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1
-a)))))))) (\lambda (H9: (csuba g c1 d1)).(or_introl (ex2 C (\lambda (d2:
-C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1))) (\lambda (d2:
-C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda
-(_: A).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abst) u2))))) (\lambda
-(d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 (asucc g a))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 a))))) (ex_intro2 C
-(\lambda (d2: C).(eq C (CHead c1 (Bind Abbr) u1) (CHead d2 (Bind Abbr) u1)))
-(\lambda (d2: C).(csuba g d2 d1)) c1 (refl_equal C (CHead c1 (Bind Abbr) u1))
-H9))) u (sym_eq T u u1 H8))) k (sym_eq K k (Bind Abbr) H7))) c2 (sym_eq C c2
-d1 H6))) H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2)
-\Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4:
-(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(eq_ind C (CHead
-c1 (Bind Abst) t) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1
-(Bind Abbr) u1)) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to
-((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c0 (CHead d2 (Bind
-Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (_: A).(eq C c0 (CHead d2 (Bind Abst) u2)))))
-(\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0)))))
-(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0)))))))))))
-(\lambda (H5: (eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let
-H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead c2
-(Bind Abbr) u) (CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
+a))))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H3: (arity g c1 t (asucc
+g a))).(\lambda (u: T).(\lambda (H4: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u) (CHead d1 (Bind Abbr) u1))).(let H6 \def (f_equal C
+C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 (Bind Abbr) u)
-(CHead d1 (Bind Abbr) u1) H5) in (eq_ind C d1 (\lambda (c0: C).((eq T u u1)
-\to ((csuba g c1 c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a)
-\to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2
-(Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
+(CHead d1 (Bind Abbr) u1) H5) in ((let H7 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (CHead c2 (Bind Abbr) u) (CHead d1 (Bind
+Abbr) u1) H5) in (\lambda (H8: (eq C c2 d1)).(let H9 \def (eq_ind T u
+(\lambda (t0: T).(arity g c2 t0 a)) H4 u1 H7) in (let H10 \def (eq_ind C c2
+(\lambda (c0: C).(arity g c0 u1 a)) H9 d1 H8) in (let H11 \def (eq_ind C c2
+(\lambda (c0: C).((eq C c0 (CHead d1 (Bind Abbr) u1)) \to (or (ex2 C (\lambda
+(d2: C).(eq C c1 (CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2
+d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c1
(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0))))))))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1
-(\lambda (t0: T).((csuba g c1 d1) \to ((arity g c1 t (asucc g a)) \to ((arity
-g d1 t0 a) \to (or (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t)
-(CHead d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A
-(\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst)
-t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda
-(_: A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0)))))))))) (\lambda (H9: (csuba g c1 d1)).(\lambda
-(H10: (arity g c1 t (asucc g a))).(\lambda (H11: (arity g d1 u1
-a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead
-d2 (Bind Abbr) u1))) (\lambda (d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda
-(d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t)
-(CHead d2 (Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_:
-A).(csuba g d2 d1)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0:
-A).(arity g d2 u2 (asucc g a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda
-(a0: A).(arity g d1 u1 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda
-(u2: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
+(a0: A).(arity g d1 u1 a0)))))))) H2 d1 H8) in (let H12 \def (eq_ind C c2
+(\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (or_intror (ex2 C (\lambda
+(d2: C).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abbr) u1))) (\lambda
+(d2: C).(csuba g d2 d1))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
+T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst)
u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1))))
(\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g
a0))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1
-a0)))) c1 t a (refl_equal C (CHead c1 (Bind Abst) t)) H9 H10 H11))))) u
-(sym_eq T u u1 H8))) c2 (sym_eq C c2 d1 H7))) H6))) c H3 H4 H0 H1 H2)))]) in
-(H0 (refl_equal C c) (refl_equal C (CHead d1 (Bind Abbr) u1)))))))).
+a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_:
+A).(eq C (CHead c1 (Bind Abst) t) (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
+C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d1)))) (\lambda (d2:
+C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 (asucc g a0))))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 a0)))) c1 t a
+(refl_equal C (CHead c1 (Bind Abst) t)) H12 H3 H10)))))))) H6)))))))))))) c y
+H0))) H))))).
theorem csuba_gen_flat_rev:
\forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall
C).(\lambda (_: T).(csuba g d2 d1)))))))))
\def
\lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda
-(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(let H0 \def (match H
-in csuba return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csuba ? c0
-c1)).((eq C c0 c) \to ((eq C c1 (CHead d1 (Flat f) u1)) \to (ex2_2 C T
+(f: F).(\lambda (H: (csuba g c (CHead d1 (Flat f) u1))).(insert_eq C (CHead
+d1 (Flat f) u1) (\lambda (c0: C).(csuba g c c0)) (\lambda (_: C).(ex2_2 C T
(\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda
-(d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) with [(csuba_sort n)
-\Rightarrow (\lambda (H0: (eq C (CSort n) c)).(\lambda (H1: (eq C (CSort n)
-(CHead d1 (Flat f) u1))).(eq_ind C (CSort n) (\lambda (c0: C).((eq C (CSort
-n) (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba
-g d2 d1)))))) (\lambda (H2: (eq C (CSort n) (CHead d1 (Flat f) u1))).(let H3
-\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead d1 (Flat f) u1) H2) in (False_ind (ex2_2 C T (\lambda (d2:
-C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2)))) (\lambda (d2:
-C).(\lambda (_: T).(csuba g d2 d1)))) H3))) c H0 H1))) | (csuba_head c1 c2 H0
-k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) c)).(\lambda (H2: (eq C
-(CHead c2 k u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 k u) (\lambda
-(c0: C).((eq C (CHead c2 k u) (CHead d1 (Flat f) u1)) \to ((csuba g c1 c2)
-\to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H3:
-(eq C (CHead c2 k u) (CHead d1 (Flat f) u1))).(let H4 \def (f_equal C T
-(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat
-f) u1) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow
-k0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c2 | (CHead c0 _ _) \Rightarrow c0])) (CHead c2 k u) (CHead d1
-(Flat f) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq
-T u u1) \to ((csuba g c1 c0) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2:
-T).(eq C (CHead c1 k u) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1)))))))) (\lambda (H7: (eq K k (Flat f))).(eq_ind K
-(Flat f) (\lambda (k0: K).((eq T u u1) \to ((csuba g c1 d1) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 k0 u) (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) (\lambda (H8:
-(eq T u u1)).(eq_ind T u1 (\lambda (t: T).((csuba g c1 d1) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) t) (CHead d2 (Flat
-f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))))) (\lambda (H9:
-(csuba g c1 d1)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C
-(CHead c1 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda
-(_: T).(csuba g d2 d1))) c1 u1 (refl_equal C (CHead c1 (Flat f) u1)) H9)) u
-(sym_eq T u u1 H8))) k (sym_eq K k (Flat f) H7))) c2 (sym_eq C c2 d1 H6)))
-H5)) H4))) c H1 H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow
-(\lambda (H3: (eq C (CHead c1 (Bind Abst) t) c)).(\lambda (H4: (eq C (CHead
-c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(eq_ind C (CHead c1 (Bind Abst) t)
-(\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1)) \to
-((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to
-(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c0 (CHead d2 (Flat f)
-u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))))) (\lambda (H5:
-(eq C (CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind
-C (CHead c2 (Bind Abbr) u) (\lambda (e: C).(match e in C return (\lambda (_:
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (\lambda (y: C).(\lambda (H0:
+(csuba g c y)).(csuba_ind g (\lambda (c0: C).(\lambda (c1: C).((eq C c1
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c0 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead d1 (Flat f)
+u1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
+(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead d1 (Flat f) u1) H1) in (False_ind (ex2_2 C T
+(\lambda (d2: C).(\lambda (u2: T).(eq C (CSort n) (CHead d2 (Flat f) u2))))
+(\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1)))) H2)))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (H1: (csuba g c1 c2)).(\lambda (H2: (((eq C c2
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c2 k u)
+(CHead d1 (Flat f) u1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c2 | (CHead c0 _
+_) \Rightarrow c0])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in ((let H5
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c2 k
+u) (CHead d1 (Flat f) u1) H3) in ((let H6 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t) \Rightarrow t])) (CHead c2 k u) (CHead d1 (Flat f) u1) H3) in
+(\lambda (H7: (eq K k (Flat f))).(\lambda (H8: (eq C c2 d1)).(eq_ind_r T u1
+(\lambda (t: T).(ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1
+k t) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))) (eq_ind_r K (Flat f) (\lambda (k0: K).(ex2_2 C T (\lambda (d2:
+C).(\lambda (u2: T).(eq C (CHead c1 k0 u1) (CHead d2 (Flat f) u2)))) (\lambda
+(d2: C).(\lambda (_: T).(csuba g d2 d1))))) (let H9 \def (eq_ind C c2
+(\lambda (c0: C).((eq C c0 (CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2:
+C).(\lambda (_: T).(csuba g d2 d1)))))) H2 d1 H8) in (let H10 \def (eq_ind C
+c2 (\lambda (c0: C).(csuba g c1 c0)) H1 d1 H8) in (ex2_2_intro C T (\lambda
+(d2: C).(\lambda (u2: T).(eq C (CHead c1 (Flat f) u1) (CHead d2 (Flat f)
+u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))) c1 u1 (refl_equal C
+(CHead c1 (Flat f) u1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c2: C).(\lambda (_: (csuba g c1 c2)).(\lambda (_: (((eq C c2
+(CHead d1 (Flat f) u1)) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq
+C c1 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2
+d1))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g
+a))).(\lambda (u: T).(\lambda (_: (arity g c2 u a)).(\lambda (H5: (eq C
+(CHead c2 (Bind Abbr) u) (CHead d1 (Flat f) u1))).(let H6 \def (eq_ind C
+(CHead c2 (Bind Abbr) u) (\lambda (ee: C).(match ee in C return (\lambda (_:
C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match
k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat
-_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind ((csuba
-g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to (ex2_2 C T
-(\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2
-(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))))) H6)))
-c H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c) (refl_equal C (CHead d1 (Flat
-f) u1))))))))).
+_) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H5) in (False_ind (ex2_2 C
+T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c1 (Bind Abst) t) (CHead d2
+(Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d2 d1))))
+H6)))))))))))) c y H0))) H)))))).
theorem csuba_gen_bind_rev:
\forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))))
\def
\lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(let H0 \def
-(match H in csuba return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csuba
-? c c0)).((eq C c c2) \to ((eq C c0 (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n)
-c2)).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(eq_ind C (CSort
-n) (\lambda (c: C).((eq C (CSort n) (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1))))))) (\lambda (H2: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(let H3
-\def (eq_ind C (CSort n) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead e1 (Bind b1) v1) H2) in (False_ind (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))
-H3))) c2 H0 H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C
-(CHead c1 k u) c2)).(\lambda (H2: (eq C (CHead c0 k u) (CHead e1 (Bind b1)
-v1))).(eq_ind C (CHead c1 k u) (\lambda (c: C).((eq C (CHead c0 k u) (CHead
-e1 (Bind b1) v1)) \to ((csuba g c1 c0) \to (ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))))
-(\lambda (H3: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u)
+(v1: T).(\lambda (H: (csuba g c2 (CHead e1 (Bind b1) v1))).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csuba g c2 c)) (\lambda (_:
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csuba g e2 e1)))))) (\lambda (y: C).(\lambda (H0: (csuba g c2
+y)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c0 (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1)))))))) (\lambda (n: nat).(\lambda (H1: (eq
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
+v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e2 e1))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3 (CHead e1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (k: K).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c3 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u)
(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H3)
-in ((let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
-(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
-(CHead c0 k u) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c:
-C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((csuba g c1 c) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k u)
+(CHead _ k0 _) \Rightarrow k0])) (CHead c3 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c3 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
+b1))).(\lambda (H8: (eq C c3 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k t)
(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1))))))))) (\lambda (H7: (eq K k (Bind b1))).(eq_ind K (Bind
-b1) (\lambda (k0: K).((eq T u v1) \to ((csuba g c1 e1) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 u)
+T).(csuba g e2 e1)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 k0 v1)
(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csuba g e2 e1)))))))) (\lambda (H8: (eq T u v1)).(eq_ind T v1 (\lambda
-(t: T).((csuba g c1 e1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C (CHead c1 (Bind b1) t) (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))
-(\lambda (H9: (csuba g c1 e1)).(let H10 \def (eq_ind T u (\lambda (t: T).(eq
-C (CHead c1 k t) c2)) H1 v1 H8) in (let H11 \def (eq_ind K k (\lambda (k0:
-K).(eq C (CHead c1 k0 v1) c2)) H10 (Bind b1) H7) in (let H12 \def (eq_ind_r C
-c2 (\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind b1)
-v1) H11) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1
-(refl_equal C (CHead c1 (Bind b1) v1)) H9))))) u (sym_eq T u v1 H8))) k
-(sym_eq K k (Bind b1) H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4))) c2 H1 H2
-H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C
-(CHead c1 (Bind Abst) t) c2)).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u)
-(CHead e1 (Bind b1) v1))).(eq_ind C (CHead c1 (Bind Abst) t) (\lambda (c:
-C).((eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1)) \to ((csuba g c1
-c0) \to ((arity g c1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind
-b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1)))))))))) (\lambda (H5: (eq C (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1)
-v1))).(let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0]))
-(CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H7 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K return
-(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
-Abbr])])) (CHead c0 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in ((let H8
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0
-(Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (eq_ind C e1 (\lambda (c:
-C).((eq B Abbr b1) \to ((eq T u v1) \to ((csuba g c1 c) \to ((arity g c1 t
-(asucc g a)) \to ((arity g c u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda
-(e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2
-e1))))))))))) (\lambda (H9: (eq B Abbr b1)).(eq_ind B Abbr (\lambda (_:
-B).((eq T u v1) \to ((csuba g c1 e1) \to ((arity g c1 t (asucc g a)) \to
-((arity g e1 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda
-(_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))))) (\lambda
-(H10: (eq T u v1)).(eq_ind T v1 (\lambda (t0: T).((csuba g c1 e1) \to ((arity
-g c1 t (asucc g a)) \to ((arity g e1 t0 a) \to (ex2_3 B C T (\lambda (b2:
+T).(csuba g e2 e1)))))) (let H9 \def (eq_ind C c3 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H8) in
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csuba g e2 e1)))) b1 c1 v1 (refl_equal C (CHead c1
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (H1: (csuba g c1 c3)).(\lambda (H2: (((eq C c3
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1)))))))).(\lambda (t:
+T).(\lambda (a: A).(\lambda (_: (arity g c1 t (asucc g a))).(\lambda (u:
+T).(\lambda (H4: (arity g c3 u a)).(\lambda (H5: (eq C (CHead c3 (Bind Abbr)
+u) (CHead e1 (Bind b1) v1))).(let H6 \def (f_equal C C (\lambda (e: C).(match
+e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _
+_) \Rightarrow c])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in
+((let H7 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_:
+C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k
+in K return (\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _)
+\Rightarrow Abbr])])) (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in
+((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead
+c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1) H5) in (\lambda (H9: (eq B Abbr
+b1)).(\lambda (H10: (eq C c3 e1)).(let H11 \def (eq_ind T u (\lambda (t0:
+T).(arity g c3 t0 a)) H4 v1 H8) in (let H12 \def (eq_ind C c3 (\lambda (c:
+C).(arity g c v1 a)) H11 e1 H10) in (let H13 \def (eq_ind C c3 (\lambda (c:
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c1 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e2 e1))))))) H2 e1
+H10) in (let H14 \def (eq_ind C c3 (\lambda (c: C).(csuba g c1 c)) H1 e1 H10)
+in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b)
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq
+C c1 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda
+(_: T).(csuba g e2 e1))))))) H13 Abbr H9) in (ex2_3_intro B C T (\lambda (b2:
B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind Abst) t) (CHead e2
(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g
-e2 e1))))))))) (\lambda (H11: (csuba g c1 e1)).(\lambda (_: (arity g c1 t
-(asucc g a))).(\lambda (_: (arity g e1 v1 a)).(let H14 \def (eq_ind_r C c2
-(\lambda (c: C).(csuba g c (CHead e1 (Bind b1) v1))) H (CHead c1 (Bind Abst)
-t) H3) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead c1
-(Bind Abst) t) (CHead e1 (Bind b) v1))) H14 Abbr H9) in (ex2_3_intro B C T
-(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c1 (Bind
-Abst) t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csuba g e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind
-Abst) t)) H11)))))) u (sym_eq T u v1 H10))) b1 H9)) c0 (sym_eq C c0 e1 H8)))
-H7)) H6))) c2 H3 H4 H0 H1 H2)))]) in (H0 (refl_equal C c2) (refl_equal C
-(CHead e1 (Bind b1) v1))))))))).
+e2 e1)))) Abst c1 t (refl_equal C (CHead c1 (Bind Abst) t)) H14))))))))) H7))
+H6)))))))))))) c2 y H0))) H)))))).
(* This file was automatically generated: do not edit *********************)
-include "csuba/drop.ma".
+include "LambdaDelta-1/csuba/drop.ma".
-include "csuba/clear.ma".
+include "LambdaDelta-1/csuba/clear.ma".
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
theorem csuba_getl_abbr:
\forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
theorem csuba_refl:
\forall (g: G).(\forall (c: C).(csuba g c c))
(* This file was automatically generated: do not edit *********************)
-include "csubc/csuba.ma".
-
-include "arity/defs.ma".
+include "LambdaDelta-1/csubc/csuba.ma".
theorem csubc_arity_conf:
\forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to
(* This file was automatically generated: do not edit *********************)
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/fwd.ma".
theorem csubc_clear_conf:
\forall (g: G).(\forall (c1: C).(\forall (e1: C).((clear c1 e1) \to (\forall
e1)).(clear_ind (\lambda (c: C).(\lambda (c0: C).(\forall (c2: C).((csubc g c
c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c0
e2))))))) (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(\lambda (c2:
-C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H1 \def (match H0 in
-csubc return (\lambda (c: C).(\lambda (c0: C).(\lambda (_: (csubc ? c
-c0)).((eq C c (CHead e (Bind b) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda
-(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u)
-e2)))))))) with [(csubc_sort n) \Rightarrow (\lambda (H1: (eq C (CSort n)
-(CHead e (Bind b) u))).(\lambda (H2: (eq C (CSort n) c2)).((let H3 \def
-(eq_ind C (CSort n) (\lambda (e0: C).(match e0 in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead e (Bind b) u) H1) in (False_ind ((eq C (CSort n) c2) \to
-(ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e
-(Bind b) u) e2)))) H3)) H2))) | (csubc_head c0 c3 H1 k v) \Rightarrow
-(\lambda (H2: (eq C (CHead c0 k v) (CHead e (Bind b) u))).(\lambda (H3: (eq C
-(CHead c3 k v) c2)).((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 in
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t)
-\Rightarrow t])) (CHead c0 k v) (CHead e (Bind b) u) H2) in ((let H5 \def
-(f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k v)
-(CHead e (Bind b) u) H2) in ((let H6 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
-(CHead c _ _) \Rightarrow c])) (CHead c0 k v) (CHead e (Bind b) u) H2) in
-(eq_ind C e (\lambda (c: C).((eq K k (Bind b)) \to ((eq T v u) \to ((eq C
-(CHead c3 k v) c2) \to ((csubc g c c3) \to (ex2 C (\lambda (e2: C).(clear c2
-e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)))))))) (\lambda (H7:
-(eq K k (Bind b))).(eq_ind K (Bind b) (\lambda (k0: K).((eq T v u) \to ((eq C
-(CHead c3 k0 v) c2) \to ((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2
-e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))))))) (\lambda (H8:
-(eq T v u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Bind b) t) c2) \to
-((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
-C).(csubc g (CHead e (Bind b) u) e2)))))) (\lambda (H9: (eq C (CHead c3 (Bind
-b) u) c2)).(eq_ind C (CHead c3 (Bind b) u) (\lambda (c: C).((csubc g e c3)
-\to (ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e
-(Bind b) u) e2))))) (\lambda (H10: (csubc g e c3)).(ex_intro2 C (\lambda (e2:
-C).(clear (CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind
-b) u) e2)) (CHead c3 (Bind b) u) (clear_bind b c3 u) (csubc_head g e c3 H10
-(Bind b) u))) c2 H9)) v (sym_eq T v u H8))) k (sym_eq K k (Bind b) H7))) c0
-(sym_eq C c0 e H6))) H5)) H4)) H3 H1))) | (csubc_abst c0 c3 H1 v a H2 w H3)
-\Rightarrow (\lambda (H4: (eq C (CHead c0 (Bind Abst) v) (CHead e (Bind b)
-u))).(\lambda (H5: (eq C (CHead c3 (Bind Abbr) w) c2)).((let H6 \def (f_equal
-C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abst) v)
-(CHead e (Bind b) u) H4) in ((let H7 \def (f_equal C B (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abst
-| (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abst])])) (CHead c0 (Bind
-Abst) v) (CHead e (Bind b) u) H4) in ((let H8 \def (f_equal C C (\lambda (e0:
-C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
-(CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) v) (CHead e (Bind b) u)
-H4) in (eq_ind C e (\lambda (c: C).((eq B Abst b) \to ((eq T v u) \to ((eq C
-(CHead c3 (Bind Abbr) w) c2) \to ((csubc g c c3) \to ((sc3 g (asucc g a) c v)
-\to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
-C).(csubc g (CHead e (Bind b) u) e2)))))))))) (\lambda (H9: (eq B Abst
-b)).(eq_ind B Abst (\lambda (b0: B).((eq T v u) \to ((eq C (CHead c3 (Bind
-Abbr) w) c2) \to ((csubc g e c3) \to ((sc3 g (asucc g a) e v) \to ((sc3 g a
-c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g
-(CHead e (Bind b0) u) e2))))))))) (\lambda (H10: (eq T v u)).(eq_ind T u
-(\lambda (t: T).((eq C (CHead c3 (Bind Abbr) w) c2) \to ((csubc g e c3) \to
-((sc3 g (asucc g a) e t) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2:
-C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u)
-e2)))))))) (\lambda (H11: (eq C (CHead c3 (Bind Abbr) w) c2)).(eq_ind C
-(CHead c3 (Bind Abbr) w) (\lambda (c: C).((csubc g e c3) \to ((sc3 g (asucc g
-a) e u) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c e2))
-(\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2))))))) (\lambda (H12:
-(csubc g e c3)).(\lambda (H13: (sc3 g (asucc g a) e u)).(\lambda (H14: (sc3 g
-a c3 w)).(ex_intro2 C (\lambda (e2: C).(clear (CHead c3 (Bind Abbr) w) e2))
-(\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2)) (CHead c3 (Bind Abbr)
-w) (clear_bind Abbr c3 w) (csubc_abst g e c3 H12 u a H13 w H14))))) c2 H11))
-v (sym_eq T v u H10))) b H9)) c0 (sym_eq C c0 e H8))) H7)) H6)) H5 H1 H2
-H3)))]) in (H1 (refl_equal C (CHead e (Bind b) u)) (refl_equal C c2))))))))
-(\lambda (e: C).(\lambda (c: C).(\lambda (_: (clear e c)).(\lambda (H1:
-((\forall (c2: C).((csubc g e c2) \to (ex2 C (\lambda (e2: C).(clear c2 e2))
-(\lambda (e2: C).(csubc g c e2))))))).(\lambda (f: F).(\lambda (u:
-T).(\lambda (c2: C).(\lambda (H2: (csubc g (CHead e (Flat f) u) c2)).(let H3
-\def (match H2 in csubc return (\lambda (c0: C).(\lambda (c3: C).(\lambda (_:
-(csubc ? c0 c3)).((eq C c0 (CHead e (Flat f) u)) \to ((eq C c3 c2) \to (ex2 C
-(\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2)))))))) with
-[(csubc_sort n) \Rightarrow (\lambda (H3: (eq C (CSort n) (CHead e (Flat f)
-u))).(\lambda (H4: (eq C (CSort n) c2)).((let H5 \def (eq_ind C (CSort n)
-(\lambda (e0: C).(match e0 in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e (Flat f) u)
-H3) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(clear c2
-e2)) (\lambda (e2: C).(csubc g c e2)))) H5)) H4))) | (csubc_head c0 c3 H3 k
-v) \Rightarrow (\lambda (H4: (eq C (CHead c0 k v) (CHead e (Flat f)
-u))).(\lambda (H5: (eq C (CHead c3 k v) c2)).((let H6 \def (f_equal C T
-(\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c0 k v) (CHead e (Flat
-f) u) H4) in ((let H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C
-return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _)
-\Rightarrow k0])) (CHead c0 k v) (CHead e (Flat f) u) H4) in ((let H8 \def
-(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k v)
-(CHead e (Flat f) u) H4) in (eq_ind C e (\lambda (c4: C).((eq K k (Flat f))
-\to ((eq T v u) \to ((eq C (CHead c3 k v) c2) \to ((csubc g c4 c3) \to (ex2 C
-(\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))))))))
-(\lambda (H9: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T v
-u) \to ((eq C (CHead c3 k0 v) c2) \to ((csubc g e c3) \to (ex2 C (\lambda
-(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))))))) (\lambda (H10:
-(eq T v u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c3 (Flat f) t) c2) \to
-((csubc g e c3) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2:
-C).(csubc g c e2)))))) (\lambda (H11: (eq C (CHead c3 (Flat f) u)
-c2)).(eq_ind C (CHead c3 (Flat f) u) (\lambda (c4: C).((csubc g e c3) \to
-(ex2 C (\lambda (e2: C).(clear c4 e2)) (\lambda (e2: C).(csubc g c e2)))))
-(\lambda (H12: (csubc g e c3)).(let H_x \def (H1 c3 H12) in (let H13 \def H_x
-in (ex2_ind C (\lambda (e2: C).(clear c3 e2)) (\lambda (e2: C).(csubc g c
-e2)) (ex2 C (\lambda (e2: C).(clear (CHead c3 (Flat f) u) e2)) (\lambda (e2:
-C).(csubc g c e2))) (\lambda (x: C).(\lambda (H14: (clear c3 x)).(\lambda
-(H15: (csubc g c x)).(ex_intro2 C (\lambda (e2: C).(clear (CHead c3 (Flat f)
-u) e2)) (\lambda (e2: C).(csubc g c e2)) x (clear_flat c3 x H14 f u) H15))))
-H13)))) c2 H11)) v (sym_eq T v u H10))) k (sym_eq K k (Flat f) H9))) c0
-(sym_eq C c0 e H8))) H7)) H6)) H5 H3))) | (csubc_abst c0 c3 H3 v a H4 w H5)
-\Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind Abst) v) (CHead e (Flat f)
-u))).(\lambda (H7: (eq C (CHead c3 (Bind Abbr) w) c2)).((let H8 \def (eq_ind
-C (CHead c0 (Bind Abst) v) (\lambda (e0: C).(match e0 in C return (\lambda
-(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H6) in (False_ind ((eq
-C (CHead c3 (Bind Abbr) w) c2) \to ((csubc g c0 c3) \to ((sc3 g (asucc g a)
-c0 v) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda
-(e2: C).(csubc g c e2))))))) H8)) H7 H3 H4 H5)))]) in (H3 (refl_equal C
-(CHead e (Flat f) u)) (refl_equal C c2))))))))))) c1 e1 H)))).
+C).(\lambda (H0: (csubc g (CHead e (Bind b) u) c2)).(let H_x \def
+(csubc_gen_head_l g e c2 u (Bind b) H0) in (let H1 \def H_x in (or_ind (ex2 C
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
+(Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq
+C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda
+(_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3
+g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
+a c3 w))))) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g
+(CHead e (Bind b) u) e2))) (\lambda (H2: (ex2 C (\lambda (c3: C).(eq C c2
+(CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e c3)))).(ex2_ind C
+(\lambda (c3: C).(eq C c2 (CHead c3 (Bind b) u))) (\lambda (c3: C).(csubc g e
+c3)) (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g (CHead
+e (Bind b) u) e2))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind b)
+u))).(\lambda (H4: (csubc g e x)).(eq_ind_r C (CHead x (Bind b) u) (\lambda
+(c: C).(ex2 C (\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead
+e (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x (Bind b)
+u) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2)) (CHead x (Bind b)
+u) (clear_bind b x u) (csubc_head g e x H4 (Bind b) u)) c2 H3)))) H2))
+(\lambda (H2: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
+A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda
+(_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
+(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K (Bind b) (Bind Abst))))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(clear c2
+e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u) e2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H3: (eq K (Bind b) (Bind
+Abst))).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H5:
+(csubc g e x0)).(\lambda (H6: (sc3 g (asucc g x2) e u)).(\lambda (H7: (sc3 g
+x2 x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(ex2 C
+(\lambda (e2: C).(clear c e2)) (\lambda (e2: C).(csubc g (CHead e (Bind b) u)
+e2)))) (let H8 \def (f_equal K B (\lambda (e0: K).(match e0 in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b]))
+(Bind b) (Bind Abst) H3) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 C (\lambda
+(e2: C).(clear (CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g
+(CHead e (Bind b0) u) e2)))) (ex_intro2 C (\lambda (e2: C).(clear (CHead x0
+(Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g (CHead e (Bind Abst) u) e2))
+(CHead x0 (Bind Abbr) x1) (clear_bind Abbr x0 x1) (csubc_abst g e x0 H5 u x2
+H6 x1 H7)) b H8)) c2 H4))))))))) H2)) H1)))))))) (\lambda (e: C).(\lambda (c:
+C).(\lambda (_: (clear e c)).(\lambda (H1: ((\forall (c2: C).((csubc g e c2)
+\to (ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c
+e2))))))).(\lambda (f: F).(\lambda (u: T).(\lambda (c2: C).(\lambda (H2:
+(csubc g (CHead e (Flat f) u) c2)).(let H_x \def (csubc_gen_head_l g e c2 u
+(Flat f) H2) in (let H3 \def H_x in (or_ind (ex2 C (\lambda (c3: C).(eq C c2
+(CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e c3))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Flat f) (Bind
+Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3
+(Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g
+e c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e
+u)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))
+(ex2 C (\lambda (e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2)))
+(\lambda (H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 (Flat f) u)))
+(\lambda (c3: C).(csubc g e c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2
+(CHead c3 (Flat f) u))) (\lambda (c3: C).(csubc g e c3)) (ex2 C (\lambda (e2:
+C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x: C).(\lambda
+(H5: (eq C c2 (CHead x (Flat f) u))).(\lambda (H6: (csubc g e x)).(eq_ind_r C
+(CHead x (Flat f) u) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(clear c0 e2))
+(\lambda (e2: C).(csubc g c e2)))) (let H_x0 \def (H1 x H6) in (let H7 \def
+H_x0 in (ex2_ind C (\lambda (e2: C).(clear x e2)) (\lambda (e2: C).(csubc g c
+e2)) (ex2 C (\lambda (e2: C).(clear (CHead x (Flat f) u) e2)) (\lambda (e2:
+C).(csubc g c e2))) (\lambda (x0: C).(\lambda (H8: (clear x x0)).(\lambda
+(H9: (csubc g c x0)).(ex_intro2 C (\lambda (e2: C).(clear (CHead x (Flat f)
+u) e2)) (\lambda (e2: C).(csubc g c e2)) x0 (clear_flat x x0 H8 f u) H9))))
+H7))) c2 H5)))) H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Flat f) (Bind Abst))))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g e c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) e u)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda
+(e2: C).(clear c2 e2)) (\lambda (e2: C).(csubc g c e2))) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H5: (eq K (Flat f) (Bind
+Abst))).(\lambda (H6: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_:
+(csubc g e x0)).(\lambda (_: (sc3 g (asucc g x2) e u)).(\lambda (_: (sc3 g x2
+x0 x1)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C
+(\lambda (e2: C).(clear c0 e2)) (\lambda (e2: C).(csubc g c e2)))) (let H10
+\def (eq_ind K (Flat f) (\lambda (ee: K).(match ee in K return (\lambda (_:
+K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I
+(Bind Abst) H5) in (False_ind (ex2 C (\lambda (e2: C).(clear (CHead x0 (Bind
+Abbr) x1) e2)) (\lambda (e2: C).(csubc g c e2))) H10)) c2 H6))))))))) H4))
+H3))))))))))) c1 e1 H)))).
(* This file was automatically generated: do not edit *********************)
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/defs.ma".
-include "sc3/props.ma".
-
-include "csuba/defs.ma".
+include "LambdaDelta-1/sc3/props.ma".
theorem csubc_csuba:
\forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubc g c1 c2) \to (csuba
(* This file was automatically generated: do not edit *********************)
-include "sc3/defs.ma".
+include "LambdaDelta-1/sc3/defs.ma".
inductive csubc (g: G): C \to (C \to Prop) \def
| csubc_sort: \forall (n: nat).(csubc g (CSort n) (CSort n))
(* This file was automatically generated: do not edit *********************)
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/fwd.ma".
-include "sc3/props.ma".
+include "LambdaDelta-1/sc3/props.ma".
theorem csubc_drop_conf_O:
\forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (h: nat).((drop h
(((drop n O (CHead c k t) e1) \to (\forall (c2: C).((csubc g (CHead c k t)
c2) \to (ex2 C (\lambda (e2: C).(drop n O c2 e2)) (\lambda (e2: C).(csubc g
e1 e2)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e1)).(\lambda (c2:
-C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H3 \def (match H2 in csubc
-return (\lambda (c0: C).(\lambda (c3: C).(\lambda (_: (csubc ? c0 c3)).((eq C
-c0 (CHead c k t)) \to ((eq C c3 c2) \to (ex2 C (\lambda (e2: C).(drop (S n) O
-c2 e2)) (\lambda (e2: C).(csubc g e1 e2)))))))) with [(csubc_sort n0)
-\Rightarrow (\lambda (H3: (eq C (CSort n0) (CHead c k t))).(\lambda (H4: (eq
-C (CSort n0) c2)).((let H5 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e
-in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _
-_ _) \Rightarrow False])) I (CHead c k t) H3) in (False_ind ((eq C (CSort n0)
-c2) \to (ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc
-g e1 e2)))) H5)) H4))) | (csubc_head c0 c3 H3 k0 v) \Rightarrow (\lambda (H4:
-(eq C (CHead c0 k0 v) (CHead c k t))).(\lambda (H5: (eq C (CHead c3 k0 v)
-c2)).((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
-(_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0]))
-(CHead c0 k0 v) (CHead c k t) H4) in ((let H7 \def (f_equal C K (\lambda (e:
-C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 |
-(CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v) (CHead c k t) H4) in ((let
-H8 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0
-k0 v) (CHead c k t) H4) in (eq_ind C c (\lambda (c4: C).((eq K k0 k) \to ((eq
-T v t) \to ((eq C (CHead c3 k0 v) c2) \to ((csubc g c4 c3) \to (ex2 C
-(\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1
-e2)))))))) (\lambda (H9: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v
-t) \to ((eq C (CHead c3 k1 v) c2) \to ((csubc g c c3) \to (ex2 C (\lambda
-(e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda
-(H10: (eq T v t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 k t0) c2) \to
-((csubc g c c3) \to (ex2 C (\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda
-(e2: C).(csubc g e1 e2)))))) (\lambda (H11: (eq C (CHead c3 k t) c2)).(eq_ind
-C (CHead c3 k t) (\lambda (c4: C).((csubc g c c3) \to (ex2 C (\lambda (e2:
-C).(drop (S n) O c4 e2)) (\lambda (e2: C).(csubc g e1 e2))))) (\lambda (H12:
-(csubc g c c3)).(let H_x \def (H e1 (r k n) (drop_gen_drop k c e1 t n H1) c3
-H12) in (let H13 \def H_x in (ex2_ind C (\lambda (e2: C).(drop (r k n) O c3
-e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O
-(CHead c3 k t) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x:
-C).(\lambda (H14: (drop (r k n) O c3 x)).(\lambda (H15: (csubc g e1
-x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead c3 k t) e2)) (\lambda
-(e2: C).(csubc g e1 e2)) x (drop_drop k n c3 x H14 t) H15)))) H13)))) c2
-H11)) v (sym_eq T v t H10))) k0 (sym_eq K k0 k H9))) c0 (sym_eq C c0 c H8)))
-H7)) H6)) H5 H3))) | (csubc_abst c0 c3 H3 v a H4 w H5) \Rightarrow (\lambda
-(H6: (eq C (CHead c0 (Bind Abst) v) (CHead c k t))).(\lambda (H7: (eq C
-(CHead c3 (Bind Abbr) w) c2)).((let H8 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v |
-(CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind Abst) v) (CHead c k t) H6)
-in ((let H9 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
-(_: C).K) with [(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _)
-\Rightarrow k0])) (CHead c0 (Bind Abst) v) (CHead c k t) H6) in ((let H10
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0
-(Bind Abst) v) (CHead c k t) H6) in (eq_ind C c (\lambda (c4: C).((eq K (Bind
-Abst) k) \to ((eq T v t) \to ((eq C (CHead c3 (Bind Abbr) w) c2) \to ((csubc
-g c4 c3) \to ((sc3 g (asucc g a) c4 v) \to ((sc3 g a c3 w) \to (ex2 C
-(\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1
-e2)))))))))) (\lambda (H11: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst)
-(\lambda (_: K).((eq T v t) \to ((eq C (CHead c3 (Bind Abbr) w) c2) \to
-((csubc g c c3) \to ((sc3 g (asucc g a) c v) \to ((sc3 g a c3 w) \to (ex2 C
-(\lambda (e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1
-e2))))))))) (\lambda (H12: (eq T v t)).(eq_ind T t (\lambda (t0: T).((eq C
-(CHead c3 (Bind Abbr) w) c2) \to ((csubc g c c3) \to ((sc3 g (asucc g a) c
-t0) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2: C).(drop (S n) O c2 e2))
-(\lambda (e2: C).(csubc g e1 e2)))))))) (\lambda (H13: (eq C (CHead c3 (Bind
-Abbr) w) c2)).(eq_ind C (CHead c3 (Bind Abbr) w) (\lambda (c4: C).((csubc g c
-c3) \to ((sc3 g (asucc g a) c t) \to ((sc3 g a c3 w) \to (ex2 C (\lambda (e2:
-C).(drop (S n) O c4 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) (\lambda
-(H14: (csubc g c c3)).(\lambda (_: (sc3 g (asucc g a) c t)).(\lambda (_: (sc3
-g a c3 w)).(let H17 \def (eq_ind_r K k (\lambda (k0: K).(drop (r k0 n) O c
-e1)) (drop_gen_drop k c e1 t n H1) (Bind Abst) H11) in (let H18 \def
-(eq_ind_r K k (\lambda (k0: K).((drop n O (CHead c k0 t) e1) \to (\forall
-(c4: C).((csubc g (CHead c k0 t) c4) \to (ex2 C (\lambda (e2: C).(drop n O c4
-e2)) (\lambda (e2: C).(csubc g e1 e2))))))) H0 (Bind Abst) H11) in (let H_x
-\def (H e1 (r (Bind Abst) n) H17 c3 H14) in (let H19 \def H_x in (ex2_ind C
-(\lambda (e2: C).(drop (r (Bind Abst) n) O c3 e2)) (\lambda (e2: C).(csubc g
-e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O (CHead c3 (Bind Abbr) w) e2))
-(\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H20: (drop (r
-(Bind Abst) n) O c3 x)).(\lambda (H21: (csubc g e1 x)).(ex_intro2 C (\lambda
-(e2: C).(drop (S n) O (CHead c3 (Bind Abbr) w) e2)) (\lambda (e2: C).(csubc g
-e1 e2)) x (drop_drop (Bind Abbr) n c3 x H20 w) H21)))) H19)))))))) c2 H13)) v
-(sym_eq T v t H12))) k H11)) c0 (sym_eq C c0 c H10))) H9)) H8)) H7 H3 H4
-H5)))]) in (H3 (refl_equal C (CHead c k t)) (refl_equal C c2)))))))) h)))))))
-c1)).
+C).(\lambda (H2: (csubc g (CHead c k t) c2)).(let H_x \def (csubc_gen_head_l
+g c c2 t k H2) in (let H3 \def H_x in (or_ind (ex2 C (\lambda (c3: C).(eq C
+c2 (CHead c3 k t))) (\lambda (c3: C).(csubc g c c3))) (ex5_3 C T A (\lambda
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))) (ex2 C (\lambda
+(e2: C).(drop (S n) O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda
+(H4: (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k t))) (\lambda (c3:
+C).(csubc g c c3)))).(ex2_ind C (\lambda (c3: C).(eq C c2 (CHead c3 k t)))
+(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (e2: C).(drop (S n) O c2
+e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x: C).(\lambda (H5: (eq C
+c2 (CHead x k t))).(\lambda (H6: (csubc g c x)).(eq_ind_r C (CHead x k t)
+(\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop (S n) O c0 e2)) (\lambda (e2:
+C).(csubc g e1 e2)))) (let H_x0 \def (H e1 (r k n) (drop_gen_drop k c e1 t n
+H1) x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (e2: C).(drop (r k n) O
+x e2)) (\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n)
+O (CHead x k t) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0:
+C).(\lambda (H8: (drop (r k n) O x x0)).(\lambda (H9: (csubc g e1
+x0)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x k t) e2)) (\lambda
+(e2: C).(csubc g e1 e2)) x0 (drop_drop k n x x0 H8 t) H9)))) H7))) c2 H5))))
+H4)) (\lambda (H4: (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
+A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a: A).(sc3 g (asucc g a) c t)))) (\lambda (c3: C).(\lambda (w: T).(\lambda
+(a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w:
+T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c c3)))) (\lambda (_: C).(\lambda
+(_: T).(\lambda (a: A).(sc3 g (asucc g a) c t)))) (\lambda (c3: C).(\lambda
+(w: T).(\lambda (a: A).(sc3 g a c3 w)))) (ex2 C (\lambda (e2: C).(drop (S n)
+O c2 e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (x2: A).(\lambda (H5: (eq K k (Bind Abst))).(\lambda (H6: (eq C
+c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (csubc g c x0)).(\lambda (_:
+(sc3 g (asucc g x2) c t)).(\lambda (_: (sc3 g x2 x0 x1)).(eq_ind_r C (CHead
+x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop (S n) O c0
+e2)) (\lambda (e2: C).(csubc g e1 e2)))) (let H10 \def (eq_ind K k (\lambda
+(k0: K).(drop (r k0 n) O c e1)) (drop_gen_drop k c e1 t n H1) (Bind Abst) H5)
+in (let H11 \def (eq_ind K k (\lambda (k0: K).((drop n O (CHead c k0 t) e1)
+\to (\forall (c3: C).((csubc g (CHead c k0 t) c3) \to (ex2 C (\lambda (e2:
+C).(drop n O c3 e2)) (\lambda (e2: C).(csubc g e1 e2))))))) H0 (Bind Abst)
+H5) in (let H_x0 \def (H e1 (r (Bind Abst) n) H10 x0 H7) in (let H12 \def
+H_x0 in (ex2_ind C (\lambda (e2: C).(drop (r (Bind Abst) n) O x0 e2))
+(\lambda (e2: C).(csubc g e1 e2)) (ex2 C (\lambda (e2: C).(drop (S n) O
+(CHead x0 (Bind Abbr) x1) e2)) (\lambda (e2: C).(csubc g e1 e2))) (\lambda
+(x: C).(\lambda (H13: (drop (r (Bind Abst) n) O x0 x)).(\lambda (H14: (csubc
+g e1 x)).(ex_intro2 C (\lambda (e2: C).(drop (S n) O (CHead x0 (Bind Abbr)
+x1) e2)) (\lambda (e2: C).(csubc g e1 e2)) x (drop_drop (Bind Abbr) n x0 x
+H13 x1) H14)))) H12))))) c2 H6))))))))) H4)) H3)))))))) h))))))) c1)).
theorem drop_csubc_trans:
\forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall
x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1:
C).(csubc g (CHead c k t0) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r
T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n)
-c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H9 \def (match
-H6 in csubc return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0
-c1)).((eq C c0 (CHead x0 k x1)) \to ((eq C c1 e1) \to (ex2 C (\lambda (c3:
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n)
-x1)) c3)))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H9: (eq C (CSort
-n0) (CHead x0 k x1))).(\lambda (H10: (eq C (CSort n0) e1)).((let H11 \def
-(eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead x0 k x1) H9) in (False_ind ((eq C (CSort n0) e1) \to (ex2 C
-(\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k
-(lift h (r k n) x1)) c1)))) H11)) H10))) | (csubc_head c1 c0 H9 k0 v)
-\Rightarrow (\lambda (H10: (eq C (CHead c1 k0 v) (CHead x0 k x1))).(\lambda
-(H11: (eq C (CHead c0 k0 v) e1)).((let H12 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v |
-(CHead _ _ t0) \Rightarrow t0])) (CHead c1 k0 v) (CHead x0 k x1) H10) in
-((let H13 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_:
-C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
-(CHead c1 k0 v) (CHead x0 k x1) H10) in ((let H14 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1
-| (CHead c3 _ _) \Rightarrow c3])) (CHead c1 k0 v) (CHead x0 k x1) H10) in
-(eq_ind C x0 (\lambda (c3: C).((eq K k0 k) \to ((eq T v x1) \to ((eq C (CHead
-c0 k0 v) e1) \to ((csubc g c3 c0) \to (ex2 C (\lambda (c4: C).(drop h (S n)
-c4 e1)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) x1)) c4))))))))
-(\lambda (H15: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to
-((eq C (CHead c0 k1 v) e1) \to ((csubc g x0 c0) \to (ex2 C (\lambda (c3:
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n)
-x1)) c3))))))) (\lambda (H16: (eq T v x1)).(eq_ind T x1 (\lambda (t0: T).((eq
-C (CHead c0 k t0) e1) \to ((csubc g x0 c0) \to (ex2 C (\lambda (c3: C).(drop
-h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1))
-c3)))))) (\lambda (H17: (eq C (CHead c0 k x1) e1)).(eq_ind C (CHead c0 k x1)
-(\lambda (c3: C).((csubc g x0 c0) \to (ex2 C (\lambda (c4: C).(drop h (S n)
-c4 c3)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n) x1)) c4)))))
-(\lambda (H18: (csubc g x0 c0)).(let H_x \def (H x0 (r k n) h H5 c0 H18) in
-(let H19 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c0))
-(\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3
-(CHead c0 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1))
-c3))) (\lambda (x: C).(\lambda (H20: (drop h (r k n) x c0)).(\lambda (H21:
-(csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c0 k
-x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)) (CHead x
-k (lift h (r k n) x1)) (drop_skip k h n x c0 H20 x1) (csubc_head g c x H21 k
-(lift h (r k n) x1)))))) H19)))) e1 H17)) v (sym_eq T v x1 H16))) k0 (sym_eq
-K k0 k H15))) c1 (sym_eq C c1 x0 H14))) H13)) H12)) H11 H9))) | (csubc_abst
-c1 c0 H9 v a H10 w H11) \Rightarrow (\lambda (H12: (eq C (CHead c1 (Bind
-Abst) v) (CHead x0 k x1))).(\lambda (H13: (eq C (CHead c0 (Bind Abbr) w)
-e1)).((let H14 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow
-t0])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H12) in ((let H15 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _) \Rightarrow k0])) (CHead
-c1 (Bind Abst) v) (CHead x0 k x1) H12) in ((let H16 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c1 | (CHead c3 _ _) \Rightarrow c3])) (CHead c1 (Bind Abst) v)
-(CHead x0 k x1) H12) in (eq_ind C x0 (\lambda (c3: C).((eq K (Bind Abst) k)
-\to ((eq T v x1) \to ((eq C (CHead c0 (Bind Abbr) w) e1) \to ((csubc g c3 c0)
-\to ((sc3 g (asucc g a) c3 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c4:
-C).(drop h (S n) c4 e1)) (\lambda (c4: C).(csubc g (CHead c k (lift h (r k n)
-x1)) c4)))))))))) (\lambda (H17: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst)
-(\lambda (k0: K).((eq T v x1) \to ((eq C (CHead c0 (Bind Abbr) w) e1) \to
-((csubc g x0 c0) \to ((sc3 g (asucc g a) x0 v) \to ((sc3 g a c0 w) \to (ex2 C
-(\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k0
-(lift h (r k0 n) x1)) c3))))))))) (\lambda (H18: (eq T v x1)).(eq_ind T x1
-(\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) w) e1) \to ((csubc g x0 c0) \to
-((sc3 g (asucc g a) x0 t0) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c3:
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift
-h (r (Bind Abst) n) x1)) c3)))))))) (\lambda (H19: (eq C (CHead c0 (Bind
-Abbr) w) e1)).(eq_ind C (CHead c0 (Bind Abbr) w) (\lambda (c3: C).((csubc g
-x0 c0) \to ((sc3 g (asucc g a) x0 x1) \to ((sc3 g a c0 w) \to (ex2 C (\lambda
-(c4: C).(drop h (S n) c4 c3)) (\lambda (c4: C).(csubc g (CHead c (Bind Abst)
-(lift h (r (Bind Abst) n) x1)) c4))))))) (\lambda (H20: (csubc g x0
-c0)).(\lambda (H21: (sc3 g (asucc g a) x0 x1)).(\lambda (H22: (sc3 g a c0
-w)).(let H23 \def (eq_ind_r K k (\lambda (k0: K).(\forall (h0: nat).((drop h0
-n (CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3:
-C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c3: C).(drop h0 n c3
-e3)) (\lambda (c3: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c3))))))))
-H8 (Bind Abst) H17) in (let H24 \def (eq_ind_r K k (\lambda (k0: K).(drop h
-(r k0 n) c x0)) H5 (Bind Abst) H17) in (let H_x \def (H x0 (r (Bind Abst) n)
-h H24 c0 H20) in (let H25 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r
-(Bind Abst) n) c3 c0)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3:
-C).(drop h (S n) c3 (CHead c0 (Bind Abbr) w))) (\lambda (c3: C).(csubc g
-(CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3))) (\lambda (x:
-C).(\lambda (H26: (drop h (r (Bind Abst) n) x c0)).(\lambda (H27: (csubc g c
-x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c0 (Bind Abbr) w)))
-(\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1))
-c3)) (CHead x (Bind Abbr) (lift h n w)) (drop_skip_bind h n x c0 H26 Abbr w)
-(csubc_abst g c x H27 (lift h (r (Bind Abst) n) x1) a (sc3_lift g (asucc g a)
-x0 x1 H21 c h (r (Bind Abst) n) H24) (lift h n w) (sc3_lift g a c0 w H22 x h
-n H26)))))) H25)))))))) e1 H19)) v (sym_eq T v x1 H18))) k H17)) c1 (sym_eq C
-c1 x0 H16))) H15)) H14)) H13 H9 H10 H11)))]) in (H9 (refl_equal C (CHead x0 k
-x1)) (refl_equal C e1))) t H4))))))))) (drop_gen_skip_l c e2 t h n k
+c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H_x \def
+(csubc_gen_head_l g x0 e1 x1 k H6) in (let H9 \def H_x in (or_ind (ex2 C
+(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1
+(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g x0 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
+(asucc g a) x0 x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
+a c3 w))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
+C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (H10: (ex2 C
+(\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3: C).(csubc g x0
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C e1 (CHead c3 k x1))) (\lambda (c3:
+C).(csubc g x0 c3)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda
+(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x:
+C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x0
+x)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop
+h (S n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1))
+c1)))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in
+(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g
+c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
+(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2:
+C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g c
+x2)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
+(c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)) (CHead x2 k (lift h (r
+k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g c x2 H15 k (lift h (r k
+n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0 c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0 x1)))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))).(ex5_3_ind C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C e1 (CHead c3 (Bind
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x0
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) x0
+x1)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))
+(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g
+(CHead c k (lift h (r k n) x1)) c1))) (\lambda (x2: C).(\lambda (x3:
+T).(\lambda (x4: A).(\lambda (H11: (eq K k (Bind Abst))).(\lambda (H12: (eq C
+e1 (CHead x2 (Bind Abbr) x3))).(\lambda (H13: (csubc g x0 x2)).(\lambda (H14:
+(sc3 g (asucc g x4) x0 x1)).(\lambda (H15: (sc3 g x4 x2 x3)).(eq_ind_r C
+(CHead x2 (Bind Abbr) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S
+n) c1 c0)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1))))
+(let H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n
+(CHead c k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3:
+C).((csubc g (CHead x0 k0 x1) e3) \to (ex2 C (\lambda (c1: C).(drop h0 n c1
+e3)) (\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1))))))))
+H8 (Bind Abst) H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r
+k0 n) c x0)) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0:
+K).(ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3)))
+(\lambda (c1: C).(csubc g (CHead c k0 (lift h (r k0 n) x1)) c1)))) (let H_x0
+\def (H x0 (r (Bind Abst) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind
+C (\lambda (c1: C).(drop h (r (Bind Abst) n) c1 x2)) (\lambda (c1: C).(csubc
+g c c1)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abbr) x3)))
+(\lambda (c1: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1))
+c1))) (\lambda (x: C).(\lambda (H19: (drop h (r (Bind Abst) n) x
+x2)).(\lambda (H20: (csubc g c x)).(ex_intro2 C (\lambda (c1: C).(drop h (S
+n) c1 (CHead x2 (Bind Abbr) x3))) (\lambda (c1: C).(csubc g (CHead c (Bind
+Abst) (lift h (r (Bind Abst) n) x1)) c1)) (CHead x (Bind Abbr) (lift h n x3))
+(drop_skip_bind h n x x2 H19 Abbr x3) (csubc_abst g c x H20 (lift h (r (Bind
+Abst) n) x1) x4 (sc3_lift g (asucc g x4) x0 x1 H14 c h (r (Bind Abst) n) H17)
+(lift h n x3) (sc3_lift g x4 x2 x3 H15 x h n H19)))))) H18))) k H11))) e1
+H12))))))))) H10)) H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k
H1)))))))) d))))))) c2)).
theorem csubc_drop_conf_rev:
k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda (c1: C).(csubc
g c1 (CHead c k t0))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h
(r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1))
-(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H9 \def (match H6 in
-csubc return (\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (csubc ? c0
-c1)).((eq C c0 e1) \to ((eq C c1 (CHead x0 k x1)) \to (ex2 C (\lambda (c3:
-C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k
-n) x1)))))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H9: (eq C (CSort
-n0) e1)).(\lambda (H10: (eq C (CSort n0) (CHead x0 k x1))).(eq_ind C (CSort
-n0) (\lambda (c0: C).((eq C (CSort n0) (CHead x0 k x1)) \to (ex2 C (\lambda
-(c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h
-(r k n) x1))))))) (\lambda (H11: (eq C (CSort n0) (CHead x0 k x1))).(let H12
-\def (eq_ind C (CSort n0) (\lambda (e: C).(match e in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead x0 k x1) H11) in (False_ind (ex2 C (\lambda (c1: C).(drop h
-(S n) c1 (CSort n0))) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n)
-x1))))) H12))) e1 H9 H10))) | (csubc_head c1 c0 H9 k0 v) \Rightarrow (\lambda
-(H10: (eq C (CHead c1 k0 v) e1)).(\lambda (H11: (eq C (CHead c0 k0 v) (CHead
-x0 k x1))).(eq_ind C (CHead c1 k0 v) (\lambda (c3: C).((eq C (CHead c0 k0 v)
-(CHead x0 k x1)) \to ((csubc g c1 c0) \to (ex2 C (\lambda (c4: C).(drop h (S
-n) c4 c3)) (\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) x1))))))))
-(\lambda (H12: (eq C (CHead c0 k0 v) (CHead x0 k x1))).(let H13 \def (f_equal
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k0 v) (CHead x0 k
-x1) H12) in ((let H14 \def (f_equal C K (\lambda (e: C).(match e in C return
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _)
-\Rightarrow k1])) (CHead c0 k0 v) (CHead x0 k x1) H12) in ((let H15 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c0 | (CHead c3 _ _) \Rightarrow c3])) (CHead c0 k0 v)
-(CHead x0 k x1) H12) in (eq_ind C x0 (\lambda (c3: C).((eq K k0 k) \to ((eq T
-v x1) \to ((csubc g c1 c3) \to (ex2 C (\lambda (c4: C).(drop h (S n) c4
-(CHead c1 k0 v))) (\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n)
-x1))))))))) (\lambda (H16: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v
-x1) \to ((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead
-c1 k1 v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))
-(\lambda (H17: (eq T v x1)).(eq_ind T x1 (\lambda (t0: T).((csubc g c1 x0)
-\to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k t0))) (\lambda (c3:
-C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))) (\lambda (H18: (csubc g
-c1 x0)).(let H19 \def (eq_ind T v (\lambda (t0: T).(eq C (CHead c1 k0 t0)
-e1)) H10 x1 H17) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(eq C (CHead
-c1 k1 x1) e1)) H19 k H16) in (let H_x \def (H x0 (r k n) h H5 c1 H18) in (let
-H21 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c1)) (\lambda
-(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k
-x1))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))
-(\lambda (x: C).(\lambda (H22: (drop h (r k n) x c1)).(\lambda (H23: (csubc g
-x c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1)))
-(\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))) (CHead x k
-(lift h (r k n) x1)) (drop_skip k h n x c1 H22 x1) (csubc_head g x c H23 k
-(lift h (r k n) x1)))))) H21)))))) v (sym_eq T v x1 H17))) k0 (sym_eq K k0 k
-H16))) c0 (sym_eq C c0 x0 H15))) H14)) H13))) e1 H10 H11 H9))) | (csubc_abst
-c1 c0 H9 v a H10 w H11) \Rightarrow (\lambda (H12: (eq C (CHead c1 (Bind
-Abst) v) e1)).(\lambda (H13: (eq C (CHead c0 (Bind Abbr) w) (CHead x0 k
-x1))).(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (c3: C).((eq C (CHead c0
-(Bind Abbr) w) (CHead x0 k x1)) \to ((csubc g c1 c0) \to ((sc3 g (asucc g a)
-c1 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c4: C).(drop h (S n) c4 c3))
-(\lambda (c4: C).(csubc g c4 (CHead c k (lift h (r k n) x1)))))))))) (\lambda
-(H14: (eq C (CHead c0 (Bind Abbr) w) (CHead x0 k x1))).(let H15 \def (f_equal
-C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
-\Rightarrow w | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind Abbr) w)
-(CHead x0 k x1) H14) in ((let H16 \def (f_equal C K (\lambda (e: C).(match e
-in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow (Bind Abbr) |
-(CHead _ k0 _) \Rightarrow k0])) (CHead c0 (Bind Abbr) w) (CHead x0 k x1)
-H14) in ((let H17 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c3 _ _)
-\Rightarrow c3])) (CHead c0 (Bind Abbr) w) (CHead x0 k x1) H14) in (eq_ind C
-x0 (\lambda (c3: C).((eq K (Bind Abbr) k) \to ((eq T w x1) \to ((csubc g c1
-c3) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c3 w) \to (ex2 C (\lambda
-(c4: C).(drop h (S n) c4 (CHead c1 (Bind Abst) v))) (\lambda (c4: C).(csubc g
-c4 (CHead c k (lift h (r k n) x1))))))))))) (\lambda (H18: (eq K (Bind Abbr)
-k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T w x1) \to ((csubc g c1 x0)
-\to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 w) \to (ex2 C (\lambda (c3:
-C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3
-(CHead c k0 (lift h (r k0 n) x1)))))))))) (\lambda (H19: (eq T w x1)).(eq_ind
-T x1 (\lambda (t0: T).((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to
-((sc3 g a x0 t0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind
-Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind
-Abbr) n) x1))))))))) (\lambda (H20: (csubc g c1 x0)).(\lambda (H21: (sc3 g
-(asucc g a) c1 v)).(\lambda (H22: (sc3 g a x0 x1)).(let H23 \def (eq_ind_r K
-k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c k0 (lift h (r k0
-n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3 (CHead x0 k0 x1))
-\to (ex2 C (\lambda (c3: C).(drop h0 n c3 e3)) (\lambda (c3: C).(csubc g c3
-(CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr) H18) in (let H24
-\def (eq_ind_r K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5 (Bind Abbr)
-H18) in (let H_x \def (H x0 (r (Bind Abbr) n) h H24 c1 H20) in (let H25 \def
-H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abbr) n) c3 c1)) (\lambda
-(c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1
-(Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r
-(Bind Abbr) n) x1))))) (\lambda (x: C).(\lambda (H26: (drop h (r (Bind Abbr)
-n) x c1)).(\lambda (H27: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h
-(S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c
-(Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n
-v)) (drop_skip_bind h n x c1 H26 Abst v) (csubc_abst g x c H27 (lift h n v) a
-(sc3_lift g (asucc g a) c1 v H21 x h n H26) (lift h (r (Bind Abbr) n) x1)
-(sc3_lift g a x0 x1 H22 c h (r (Bind Abbr) n) H24)))))) H25)))))))) w (sym_eq
-T w x1 H19))) k H18)) c0 (sym_eq C c0 x0 H17))) H16)) H15))) e1 H12 H13 H9
-H10 H11)))]) in (H9 (refl_equal C e1) (refl_equal C (CHead x0 k x1)))) t
-H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+(\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H_x \def
+(csubc_gen_head_r g x0 e1 x1 k H6) in (let H9 \def H_x in (or_ind (ex2 C
+(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1
+x0))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abbr))))) (\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1
+(CHead c1 (Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c1 x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g
+(asucc g a) c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a
+x0 x1))))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1:
+C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (H10: (ex2 C
+(\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1: C).(csubc g c1
+x0)))).(ex2_ind C (\lambda (c1: C).(eq C e1 (CHead c1 k x1))) (\lambda (c1:
+C).(csubc g c1 x0)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x:
+C).(\lambda (H11: (eq C e1 (CHead x k x1))).(\lambda (H12: (csubc g x
+x0)).(eq_ind_r C (CHead x k x1) (\lambda (c0: C).(ex2 C (\lambda (c1:
+C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k
+n) x1)))))) (let H_x0 \def (H x0 (r k n) h H5 x H12) in (let H13 \def H_x0 in
+(ex2_ind C (\lambda (c1: C).(drop h (r k n) c1 x)) (\lambda (c1: C).(csubc g
+c1 c)) (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) (\lambda (x2:
+C).(\lambda (H14: (drop h (r k n) x2 x)).(\lambda (H15: (csubc g x2
+c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x k x1))) (\lambda
+(c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))) (CHead x2 k (lift h (r
+k n) x1)) (drop_skip k h n x2 x H14 x1) (csubc_head g x2 c H15 k (lift h (r k
+n) x1)))))) H13))) e1 H11)))) H10)) (\lambda (H10: (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1:
+C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind Abst) v)))))
+(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 x0)))) (\lambda
+(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))))).(ex5_3_ind C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C e1 (CHead c1 (Bind
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+x0)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a x0 x1)))) (ex2
+C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead
+c k (lift h (r k n) x1))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4:
+A).(\lambda (H11: (eq K k (Bind Abbr))).(\lambda (H12: (eq C e1 (CHead x2
+(Bind Abst) x3))).(\lambda (H13: (csubc g x2 x0)).(\lambda (H14: (sc3 g
+(asucc g x4) x2 x3)).(\lambda (H15: (sc3 g x4 x0 x1)).(eq_ind_r C (CHead x2
+(Bind Abst) x3) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop h (S n) c1
+c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))) (let
+H16 \def (eq_ind K k (\lambda (k0: K).(\forall (h0: nat).((drop h0 n (CHead c
+k0 (lift h (r k0 n) x1)) (CHead x0 k0 x1)) \to (\forall (e3: C).((csubc g e3
+(CHead x0 k0 x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e3)) (\lambda
+(c1: C).(csubc g c1 (CHead c k0 (lift h (r k0 n) x1)))))))))) H8 (Bind Abbr)
+H11) in (let H17 \def (eq_ind K k (\lambda (k0: K).(drop h (r k0 n) c x0)) H5
+(Bind Abbr) H11) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda
+(c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1: C).(csubc
+g c1 (CHead c k0 (lift h (r k0 n) x1)))))) (let H_x0 \def (H x0 (r (Bind
+Abbr) n) h H17 x2 H13) in (let H18 \def H_x0 in (ex2_ind C (\lambda (c1:
+C).(drop h (r (Bind Abbr) n) c1 x2)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C
+(\lambda (c1: C).(drop h (S n) c1 (CHead x2 (Bind Abst) x3))) (\lambda (c1:
+C).(csubc g c1 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))))
+(\lambda (x: C).(\lambda (H19: (drop h (r (Bind Abbr) n) x x2)).(\lambda
+(H20: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop h (S n) c1 (CHead x2
+(Bind Abst) x3))) (\lambda (c1: C).(csubc g c1 (CHead c (Bind Abbr) (lift h
+(r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n x3)) (drop_skip_bind
+h n x x2 H19 Abst x3) (csubc_abst g x c H20 (lift h n x3) x4 (sc3_lift g
+(asucc g x4) x2 x3 H14 x h n H19) (lift h (r (Bind Abbr) n) x1) (sc3_lift g
+x4 x0 x1 H15 c h (r (Bind Abbr) n) H17)))))) H18))) k H11))) e1 H12)))))))))
+H10)) H9))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d)))))))
+c2)).
(* This file was automatically generated: do not edit *********************)
-include "csubc/drop.ma".
+include "LambdaDelta-1/csubc/drop.ma".
theorem drop1_csubc_trans:
\forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2:
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/csubc/defs.ma".
+
+theorem csubc_gen_sort_l:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g (CSort n) x) \to
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g
+(CSort n) x)).(insert_eq C (CSort n) (\lambda (c: C).(csubc g c x)) (\lambda
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c))))
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def
+(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with
+[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0)
+(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort
+n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C
+c2 c1)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c1 k v)
+(CSort n))).(let H4 \def (eq_ind C (CHead c1 k v) (\lambda (ee: C).(match ee
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
+_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c2 k v)
+(CHead c1 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
+(csubc g c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2
+c1)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1
+v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead
+c1 (Bind Abst) v) (CSort n))).(let H6 \def (eq_ind C (CHead c1 (Bind Abst) v)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H5) in
+(False_ind (eq C (CHead c2 (Bind Abbr) w) (CHead c1 (Bind Abst) v))
+H6)))))))))))) y x H0))) H)))).
+
+theorem csubc_gen_head_l:
+ \forall (g: G).(\forall (c1: C).(\forall (x: C).(\forall (v: T).(\forall (k:
+K).((csubc g (CHead c1 k v) x) \to (or (ex2 C (\lambda (c2: C).(eq C x (CHead
+c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c2:
+C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind Abbr) w)))))
+(\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
+(c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))))))))
+\def
+ \lambda (g: G).(\lambda (c1: C).(\lambda (x: C).(\lambda (v: T).(\lambda (k:
+K).(\lambda (H: (csubc g (CHead c1 k v) x)).(insert_eq C (CHead c1 k v)
+(\lambda (c: C).(csubc g c x)) (\lambda (_: C).(or (ex2 C (\lambda (c2:
+C).(eq C x (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C x (CHead c2 (Bind
+Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))))
+(\lambda (y: C).(\lambda (H0: (csubc g y x)).(csubc_ind g (\lambda (c:
+C).(\lambda (c0: C).((eq C c (CHead c1 k v)) \to (or (ex2 C (\lambda (c2:
+C).(eq C c0 (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C c0 (CHead c2 (Bind
+Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w)))))))))
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c1 k v))).(let H2 \def
+(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead c1 k v) H1) in (False_ind (or (ex2 C (\lambda (c2: C).(eq C
+(CSort n) (CHead c2 k v))) (\lambda (c2: C).(csubc g c1 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c2: C).(\lambda (w: T).(\lambda (_: A).(eq C (CSort n) (CHead c2
+(Bind Abbr) w))))) (\lambda (c2: C).(\lambda (_: T).(\lambda (_: A).(csubc g
+c1 c2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a)
+c1 v)))) (\lambda (c2: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c2 w))))))
+H2)))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1: (csubc g c0
+c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or (ex2 C (\lambda (c3:
+C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3
+w))))))))).(\lambda (k0: K).(\lambda (v0: T).(\lambda (H3: (eq C (CHead c0 k0
+v0) (CHead c1 k v))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
+\Rightarrow c])) (CHead c0 k0 v0) (CHead c1 k v) H3) in ((let H5 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v0)
+(CHead c1 k v) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v0 | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 k0 v0) (CHead c1 k v) H3) in (\lambda (H7: (eq K
+k0 k)).(\lambda (H8: (eq C c0 c1)).(eq_ind_r T v (\lambda (t: T).(or (ex2 C
+(\lambda (c3: C).(eq C (CHead c2 k0 t) (CHead c3 k v))) (\lambda (c3:
+C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C (CHead c2 k0 t) (CHead c3 (Bind Abbr) w))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w))))))) (eq_ind_r K k
+(\lambda (k1: K).(or (ex2 C (\lambda (c3: C).(eq C (CHead c2 k1 v) (CHead c3
+k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k1 v) (CHead c3 (Bind
+Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))))
+(let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k v)) \to (or
+(ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g
+c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2
+(CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g
+(asucc g a) c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g
+a c3 w)))))))) H2 c1 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(csubc
+g c c2)) H1 c1 H8) in (or_introl (ex2 C (\lambda (c3: C).(eq C (CHead c2 k v)
+(CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (_: A).(eq C (CHead c2 k v) (CHead c3 (Bind Abbr)
+w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v))))
+(\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3 w)))))
+(ex_intro2 C (\lambda (c3: C).(eq C (CHead c2 k v) (CHead c3 k v))) (\lambda
+(c3: C).(csubc g c1 c3)) c2 (refl_equal C (CHead c2 k v)) H10)))) k0 H7) v0
+H6)))) H5)) H4))))))))) (\lambda (c0: C).(\lambda (c2: C).(\lambda (H1:
+(csubc g c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k v)) \to (or (ex2 C
+(\lambda (c3: C).(eq C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3)))
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind
+Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C c2 (CHead c3
+(Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g
+c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g (asucc g a)
+c1 v)))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a: A).(sc3 g a c3
+w))))))))).(\lambda (v0: T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a)
+c0 v0)).(\lambda (w: T).(\lambda (H4: (sc3 g a c2 w)).(\lambda (H5: (eq C
+(CHead c0 (Bind Abst) v0) (CHead c1 k v))).(let H6 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0
+| (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5)
+in ((let H7 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow (Bind Abst) | (CHead _ k0 _)
+\Rightarrow k0])) (CHead c0 (Bind Abst) v0) (CHead c1 k v) H5) in ((let H8
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow v0 | (CHead _ _ t) \Rightarrow t])) (CHead c0
+(Bind Abst) v0) (CHead c1 k v) H5) in (\lambda (H9: (eq K (Bind Abst)
+k)).(\lambda (H10: (eq C c0 c1)).(let H11 \def (eq_ind T v0 (\lambda (t:
+T).(sc3 g (asucc g a) c0 t)) H3 v H8) in (let H12 \def (eq_ind C c0 (\lambda
+(c: C).(sc3 g (asucc g a) c v)) H11 c1 H10) in (let H13 \def (eq_ind C c0
+(\lambda (c: C).((eq C c (CHead c1 k v)) \to (or (ex2 C (\lambda (c3: C).(eq
+C c2 (CHead c3 k v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abst)))))
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind
+Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0)
+c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3
+w0)))))))) H2 c1 H10) in (let H14 \def (eq_ind C c0 (\lambda (c: C).(csubc g
+c c2)) H1 c1 H10) in (let H15 \def (eq_ind_r K k (\lambda (k0: K).((eq C c1
+(CHead c1 k0 v)) \to (or (ex2 C (\lambda (c3: C).(eq C c2 (CHead c3 k0 v)))
+(\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0:
+T).(\lambda (_: A).(eq C c2 (CHead c3 (Bind Abbr) w0))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda
+(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))))))) H13 (Bind
+Abst) H9) in (eq_ind K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (c3:
+C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 k0 v))) (\lambda (c3: C).(csubc g
+c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
+k0 (Bind Abst))))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (_: A).(eq C
+(CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr) w0))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v)))) (\lambda
+(c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0))))))) (or_intror
+(ex2 C (\lambda (c3: C).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abst)
+v))) (\lambda (c3: C).(csubc g c1 c3))) (ex5_3 C T A (\lambda (_: C).(\lambda
+(_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3:
+C).(\lambda (w0: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3
+(Bind Abbr) w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g
+c1 c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g
+a0) c1 v)))) (\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3
+w0))))) (ex5_3_intro C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_:
+A).(eq K (Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w0:
+T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) w) (CHead c3 (Bind Abbr)
+w0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c3))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g (asucc g a0) c1 v))))
+(\lambda (c3: C).(\lambda (w0: T).(\lambda (a0: A).(sc3 g a0 c3 w0)))) c2 w a
+(refl_equal K (Bind Abst)) (refl_equal C (CHead c2 (Bind Abbr) w)) H14 H12
+H4)) k H9))))))))) H7)) H6)))))))))))) y x H0))) H)))))).
+
+theorem csubc_gen_sort_r:
+ \forall (g: G).(\forall (x: C).(\forall (n: nat).((csubc g x (CSort n)) \to
+(eq C x (CSort n)))))
+\def
+ \lambda (g: G).(\lambda (x: C).(\lambda (n: nat).(\lambda (H: (csubc g x
+(CSort n))).(insert_eq C (CSort n) (\lambda (c: C).(csubc g x c)) (\lambda
+(c: C).(eq C x c)) (\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g
+(\lambda (c: C).(\lambda (c0: C).((eq C c0 (CSort n)) \to (eq C c c0))))
+(\lambda (n0: nat).(\lambda (H1: (eq C (CSort n0) (CSort n))).(let H2 \def
+(f_equal C nat (\lambda (e: C).(match e in C return (\lambda (_: C).nat) with
+[(CSort n1) \Rightarrow n1 | (CHead _ _ _) \Rightarrow n0])) (CSort n0)
+(CSort n) H1) in (eq_ind_r nat n (\lambda (n1: nat).(eq C (CSort n1) (CSort
+n1))) (refl_equal C (CSort n)) n0 H2)))) (\lambda (c1: C).(\lambda (c2:
+C).(\lambda (_: (csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C
+c1 c2)))).(\lambda (k: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c2 k v)
+(CSort n))).(let H4 \def (eq_ind C (CHead c2 k v) (\lambda (ee: C).(match ee
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead
+_ _ _) \Rightarrow True])) I (CSort n) H3) in (False_ind (eq C (CHead c1 k v)
+(CHead c2 k v)) H4))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (_:
+(csubc g c1 c2)).(\lambda (_: (((eq C c2 (CSort n)) \to (eq C c1
+c2)))).(\lambda (v: T).(\lambda (a: A).(\lambda (_: (sc3 g (asucc g a) c1
+v)).(\lambda (w: T).(\lambda (_: (sc3 g a c2 w)).(\lambda (H5: (eq C (CHead
+c2 (Bind Abbr) w) (CSort n))).(let H6 \def (eq_ind C (CHead c2 (Bind Abbr) w)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H5) in
+(False_ind (eq C (CHead c1 (Bind Abst) v) (CHead c2 (Bind Abbr) w))
+H6)))))))))))) x y H0))) H)))).
+
+theorem csubc_gen_head_r:
+ \forall (g: G).(\forall (c2: C).(\forall (x: C).(\forall (w: T).(\forall (k:
+K).((csubc g x (CHead c2 k w)) \to (or (ex2 C (\lambda (c1: C).(eq C x (CHead
+c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c1:
+C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind Abst) v)))))
+(\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1 c2)))) (\lambda
+(c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1 v)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))))))))
+\def
+ \lambda (g: G).(\lambda (c2: C).(\lambda (x: C).(\lambda (w: T).(\lambda (k:
+K).(\lambda (H: (csubc g x (CHead c2 k w))).(insert_eq C (CHead c2 k w)
+(\lambda (c: C).(csubc g x c)) (\lambda (_: C).(or (ex2 C (\lambda (c1:
+C).(eq C x (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C x (CHead c1 (Bind
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))))
+(\lambda (y: C).(\lambda (H0: (csubc g x y)).(csubc_ind g (\lambda (c:
+C).(\lambda (c0: C).((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c1:
+C).(eq C c (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C c (CHead c1 (Bind
+Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g c1
+c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c1
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w)))))))))
+(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead c2 k w))).(let H2 \def
+(eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
+C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
+False])) I (CHead c2 k w) H1) in (False_ind (or (ex2 C (\lambda (c1: C).(eq C
+(CSort n) (CHead c1 k w))) (\lambda (c1: C).(csubc g c1 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c1: C).(\lambda (v: T).(\lambda (_: A).(eq C (CSort n) (CHead c1
+(Bind Abst) v))))) (\lambda (c1: C).(\lambda (_: T).(\lambda (_: A).(csubc g
+c1 c2)))) (\lambda (c1: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a)
+c1 v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))
+H2)))) (\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1
+c0)).(\lambda (H2: (((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c3:
+C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A
+(\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr)))))
+(\lambda (c3: C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind
+Abst) v))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3
+c2)))) (\lambda (c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3
+v)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2
+w))))))))).(\lambda (k0: K).(\lambda (v: T).(\lambda (H3: (eq C (CHead c0 k0
+v) (CHead c2 k w))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
+\Rightarrow c])) (CHead c0 k0 v) (CHead c2 k w) H3) in ((let H5 \def (f_equal
+C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 v) (CHead c2 k
+w) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return
+(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow
+t])) (CHead c0 k0 v) (CHead c2 k w) H3) in (\lambda (H7: (eq K k0
+k)).(\lambda (H8: (eq C c0 c2)).(eq_ind_r T w (\lambda (t: T).(or (ex2 C
+(\lambda (c3: C).(eq C (CHead c1 k0 t) (CHead c3 k w))) (\lambda (c3:
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_:
+A).(eq C (CHead c1 k0 t) (CHead c3 (Bind Abst) v0))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3:
+C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_:
+C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))) (eq_ind_r K k
+(\lambda (k1: K).(or (ex2 C (\lambda (c3: C).(eq C (CHead c1 k1 w) (CHead c3
+k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3:
+C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 k1 w) (CHead c3 (Bind
+Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3
+c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a: A).(sc3 g (asucc g a)
+c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2
+w))))))) (let H9 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w))
+\to (or (ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3:
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_:
+A).(eq C c1 (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0:
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(sc3 g a c2 w)))))))) H2 c2 H8) in (let H10 \def (eq_ind C
+c0 (\lambda (c: C).(csubc g c1 c)) H1 c2 H8) in (or_introl (ex2 C (\lambda
+(c3: C).(eq C (CHead c1 k w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3
+c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C
+(CHead c1 k w) (CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0:
+T).(\lambda (a: A).(sc3 g (asucc g a) c3 v0)))) (\lambda (_: C).(\lambda (_:
+T).(\lambda (a: A).(sc3 g a c2 w))))) (ex_intro2 C (\lambda (c3: C).(eq C
+(CHead c1 k w) (CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2)) c1
+(refl_equal C (CHead c1 k w)) H10)))) k0 H7) v H6)))) H5)) H4)))))))))
+(\lambda (c1: C).(\lambda (c0: C).(\lambda (H1: (csubc g c1 c0)).(\lambda
+(H2: (((eq C c0 (CHead c2 k w)) \to (or (ex2 C (\lambda (c3: C).(eq C c1
+(CHead c3 k w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K k (Bind Abbr))))) (\lambda (c3:
+C).(\lambda (v: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda
+(c3: C).(\lambda (v: T).(\lambda (a: A).(sc3 g (asucc g a) c3 v)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a: A).(sc3 g a c2 w))))))))).(\lambda (v:
+T).(\lambda (a: A).(\lambda (H3: (sc3 g (asucc g a) c1 v)).(\lambda (w0:
+T).(\lambda (H4: (sc3 g a c0 w0)).(\lambda (H5: (eq C (CHead c0 (Bind Abbr)
+w0) (CHead c2 k w))).(let H6 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
+\Rightarrow c])) (CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H7
+\def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
+with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k0 _) \Rightarrow k0]))
+(CHead c0 (Bind Abbr) w0) (CHead c2 k w) H5) in ((let H8 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow w0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abbr) w0)
+(CHead c2 k w) H5) in (\lambda (H9: (eq K (Bind Abbr) k)).(\lambda (H10: (eq
+C c0 c2)).(let H11 \def (eq_ind T w0 (\lambda (t: T).(sc3 g a c0 t)) H4 w H8)
+in (let H12 \def (eq_ind C c0 (\lambda (c: C).(sc3 g a c w)) H11 c2 H10) in
+(let H13 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c2 k w)) \to (or
+(ex2 C (\lambda (c3: C).(eq C c1 (CHead c3 k w))) (\lambda (c3: C).(csubc g
+c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K k
+(Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_: A).(eq C c1
+(CHead c3 (Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3
+g (asucc g a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0:
+A).(sc3 g a0 c2 w)))))))) H2 c2 H10) in (let H14 \def (eq_ind C c0 (\lambda
+(c: C).(csubc g c1 c)) H1 c2 H10) in (let H15 \def (eq_ind_r K k (\lambda
+(k0: K).((eq C c2 (CHead c2 k0 w)) \to (or (ex2 C (\lambda (c3: C).(eq C c1
+(CHead c3 k0 w))) (\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda
+(_: C).(\lambda (_: T).(\lambda (_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3:
+C).(\lambda (v0: T).(\lambda (_: A).(eq C c1 (CHead c3 (Bind Abst) v0)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda
+(c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0))))
+(\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))))))) H13
+(Bind Abbr) H9) in (eq_ind K (Bind Abbr) (\lambda (k0: K).(or (ex2 C (\lambda
+(c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 k0 w))) (\lambda (c3:
+C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda
+(_: A).(eq K k0 (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda
+(_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) v0))))) (\lambda
+(c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3:
+C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w))))))) (or_intror (ex2
+C (\lambda (c3: C).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abbr) w)))
+(\lambda (c3: C).(csubc g c3 c2))) (ex5_3 C T A (\lambda (_: C).(\lambda (_:
+T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abbr))))) (\lambda (c3:
+C).(\lambda (v0: T).(\lambda (_: A).(eq C (CHead c1 (Bind Abst) v) (CHead c3
+(Bind Abst) v0))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g
+c3 c2)))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g
+a0) c3 v0)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2
+w))))) (ex5_3_intro C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq
+K (Bind Abbr) (Bind Abbr))))) (\lambda (c3: C).(\lambda (v0: T).(\lambda (_:
+A).(eq C (CHead c1 (Bind Abst) v) (CHead c3 (Bind Abst) v0))))) (\lambda (c3:
+C).(\lambda (_: T).(\lambda (_: A).(csubc g c3 c2)))) (\lambda (c3:
+C).(\lambda (v0: T).(\lambda (a0: A).(sc3 g (asucc g a0) c3 v0)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a0: A).(sc3 g a0 c2 w)))) c1 v a (refl_equal
+K (Bind Abbr)) (refl_equal C (CHead c1 (Bind Abst) v)) H14 H3 H12)) k
+H9))))))))) H7)) H6)))))))))))) x y H0))) H)))))).
+
(* This file was automatically generated: do not edit *********************)
-include "csubc/drop.ma".
+include "LambdaDelta-1/csubc/drop.ma".
-include "csubc/clear.ma".
+include "LambdaDelta-1/csubc/clear.ma".
theorem csubc_getl_conf:
\forall (g: G).(\forall (c1: C).(\forall (e1: C).(\forall (i: nat).((getl i
(* This file was automatically generated: do not edit *********************)
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/defs.ma".
-include "sc3/props.ma".
+include "LambdaDelta-1/sc3/props.ma".
theorem csubc_refl:
\forall (g: G).(\forall (c: C).(csubc g c c))
(* This file was automatically generated: do not edit *********************)
-include "csubst0/fwd.ma".
+include "LambdaDelta-1/csubst0/fwd.ma".
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
theorem csubst0_clear_O:
\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to
(* This file was automatically generated: do not edit *********************)
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
inductive csubst0: nat \to (T \to (C \to (C \to Prop))) \def
| csubst0_snd: \forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1:
(* This file was automatically generated: do not edit *********************)
-include "csubst0/fwd.ma".
+include "LambdaDelta-1/csubst0/fwd.ma".
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
-include "s/props.ma".
+include "LambdaDelta-1/s/props.ma".
theorem csubst0_drop_gt:
\forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csubst0/defs.ma".
+include "LambdaDelta-1/csubst0/defs.ma".
theorem csubst0_gen_sort:
\forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0
(* This file was automatically generated: do not edit *********************)
-include "csubst0/clear.ma".
+include "LambdaDelta-1/csubst0/clear.ma".
-include "csubst0/drop.ma".
+include "LambdaDelta-1/csubst0/drop.ma".
-include "getl/fwd.ma".
+include "LambdaDelta-1/getl/fwd.ma".
theorem csubst0_getl_ge:
\forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csubst0/defs.ma".
+include "LambdaDelta-1/csubst0/defs.ma".
theorem csubst0_snd_bind:
\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csubst0/defs.ma".
+include "LambdaDelta-1/csubst0/defs.ma".
inductive csubst1 (i: nat) (v: T) (c1: C): C \to Prop \def
| csubst1_refl: csubst1 i v c1 c1
(* This file was automatically generated: do not edit *********************)
-include "csubst1/defs.ma".
+include "LambdaDelta-1/csubst1/defs.ma".
-include "csubst0/fwd.ma".
+include "LambdaDelta-1/csubst0/fwd.ma".
-include "subst1/props.ma".
+include "LambdaDelta-1/subst1/props.ma".
theorem csubst1_gen_head:
\forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csubst1/props.ma".
+include "LambdaDelta-1/csubst1/props.ma".
-include "csubst0/getl.ma".
+include "LambdaDelta-1/csubst0/getl.ma".
-include "csubst0/props.ma".
+include "LambdaDelta-1/csubst0/props.ma".
-include "subst1/props.ma".
+include "LambdaDelta-1/subst1/props.ma".
-include "drop/props.ma".
+include "LambdaDelta-1/drop/props.ma".
theorem csubst1_getl_ge:
\forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csubst1/defs.ma".
+include "LambdaDelta-1/csubst1/defs.ma".
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
theorem csubst1_head:
\forall (k: K).(\forall (i: nat).(\forall (v: T).(\forall (u1: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csubt/defs.ma".
+include "LambdaDelta-1/csubt/defs.ma".
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
theorem csubt_clear_conf:
\forall (g: G).(\forall (c1: C).(\forall (c2: C).((csubt g c1 c2) \to
(* This file was automatically generated: do not edit *********************)
-include "ty3/defs.ma".
+include "LambdaDelta-1/ty3/defs.ma".
inductive csubt (g: G): C \to (C \to Prop) \def
| csubt_sort: \forall (n: nat).(csubt g (CSort n) (CSort n))
(* This file was automatically generated: do not edit *********************)
-include "csubt/defs.ma".
+include "LambdaDelta-1/csubt/defs.ma".
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
theorem csubt_drop_flat:
\forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csubt/defs.ma".
-
-theorem csubt_inv_coq:
- \forall (g: G).(\forall (c1: C).(\forall (c2: C).(\forall (P: ((G \to (C \to
-(C \to Prop))))).((((csubt g c1 c2) \to (\forall (n: nat).((eq C (CSort n)
-c1) \to ((eq C (CSort n) c2) \to (P g c1 c2)))))) \to ((((csubt g c1 c2) \to
-(\forall (c0: C).(\forall (c3: C).(\forall (k: K).(\forall (u: T).((eq C
-(CHead c0 k u) c1) \to ((eq C (CHead c3 k u) c2) \to ((csubt g c0 c3) \to (P
-g c1 c2)))))))))) \to ((((csubt g c1 c2) \to (\forall (c0: C).(\forall (c3:
-C).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).((eq C (CHead c0 (Bind
-Void) u1) c1) \to ((eq C (CHead c3 (Bind b) u2) c2) \to ((csubt g c0 c3) \to
-((not (eq B b Void)) \to (P g c1 c2)))))))))))) \to ((((csubt g c1 c2) \to
-(\forall (c0: C).(\forall (c3: C).(\forall (u: T).(\forall (t: T).((eq C
-(CHead c0 (Bind Abst) t) c1) \to ((eq C (CHead c3 (Bind Abbr) u) c2) \to
-((csubt g c0 c3) \to ((ty3 g c3 u t) \to (P g c1 c2))))))))))) \to ((csubt g
-c1 c2) \to (P g c1 c2)))))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (P: ((G \to (C \to
-(C \to Prop))))).(\lambda (H: (((csubt g c1 c2) \to (\forall (n: nat).((eq C
-(CSort n) c1) \to ((eq C (CSort n) c2) \to (P g c1 c2))))))).(\lambda (H0:
-(((csubt g c1 c2) \to (\forall (c0: C).(\forall (c3: C).(\forall (k:
-K).(\forall (u: T).((eq C (CHead c0 k u) c1) \to ((eq C (CHead c3 k u) c2)
-\to ((csubt g c0 c3) \to (P g c1 c2))))))))))).(\lambda (H1: (((csubt g c1
-c2) \to (\forall (c0: C).(\forall (c3: C).(\forall (b: B).(\forall (u1:
-T).(\forall (u2: T).((eq C (CHead c0 (Bind Void) u1) c1) \to ((eq C (CHead c3
-(Bind b) u2) c2) \to ((csubt g c0 c3) \to ((not (eq B b Void)) \to (P g c1
-c2))))))))))))).(\lambda (H2: (((csubt g c1 c2) \to (\forall (c0: C).(\forall
-(c3: C).(\forall (u: T).(\forall (t: T).((eq C (CHead c0 (Bind Abst) t) c1)
-\to ((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csubt g c0 c3) \to ((ty3 g c3 u
-t) \to (P g c1 c2)))))))))))).(\lambda (H3: (csubt g c1 c2)).(let H4 \def
-(match H3 in csubt return (\lambda (c: C).(\lambda (c0: C).(\lambda (_:
-(csubt ? c c0)).((eq C c c1) \to ((eq C c0 c2) \to (P g c1 c2)))))) with
-[(csubt_sort n) \Rightarrow (\lambda (H4: (eq C (CSort n) c1)).(\lambda (H5:
-(eq C (CSort n) c2)).(H H3 n H4 H5))) | (csubt_head c0 c3 H4 k u) \Rightarrow
-(\lambda (H5: (eq C (CHead c0 k u) c1)).(\lambda (H6: (eq C (CHead c3 k u)
-c2)).(H0 H3 c0 c3 k u H5 H6 H4))) | (csubt_void c0 c3 H4 b H5 u1 u2)
-\Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind Void) u1) c1)).(\lambda (H7:
-(eq C (CHead c3 (Bind b) u2) c2)).(H1 H3 c0 c3 b u1 u2 H6 H7 H4 H5))) |
-(csubt_abst c0 c3 H4 u t H5) \Rightarrow (\lambda (H6: (eq C (CHead c0 (Bind
-Abst) t) c1)).(\lambda (H7: (eq C (CHead c3 (Bind Abbr) u) c2)).(H2 H3 c0 c3
-u t H6 H7 H4 H5)))]) in (H4 (refl_equal C c1) (refl_equal C c2))))))))))).
+include "LambdaDelta-1/csubt/defs.ma".
theorem csubt_gen_abbr:
\forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csubt g
(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))))
\def
\lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda
-(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(csubt_inv_coq g (CHead e1 (Bind
-Abbr) v) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0: C).(ex2 C (\lambda
-(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g0 e1
-e2)))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda (n:
-nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind Abbr) v))).(\lambda (H2:
-(eq C (CSort n) c2)).(let H3 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g
-(CHead e1 (Bind Abbr) v) c)) H0 (CSort n) H2) in (let H4 \def (eq_ind_r C c2
-(\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H (CSort n) H2) in
-(eq_ind C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e2: C).(eq C c (CHead e2
-(Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H5 \def (eq_ind C
-(CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with
-[(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1
-(Bind Abbr) v) H1) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CSort n)
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)) c2
-H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda (c0:
-C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u: T).(\lambda (H1: (eq C
-(CHead c0 k u) (CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c3 k u)
-c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def (eq_ind_r C c2 (\lambda (c:
-C).(csubt g (CHead e1 (Bind Abbr) v) c)) H0 (CHead c3 k u) H2) in (let H5
-\def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H
-(CHead c3 k u) H2) in (eq_ind C (CHead c3 k u) (\lambda (c: C).(ex2 C
-(\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g
-e1 e2)))) (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
-c])) (CHead c0 k u) (CHead e1 (Bind Abbr) v) H1) in ((let H7 \def (f_equal C
-K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1
-(Bind Abbr) v) H1) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in
-C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
-\Rightarrow t])) (CHead c0 k u) (CHead e1 (Bind Abbr) v) H1) in (\lambda (H9:
-(eq K k (Bind Abbr))).(\lambda (H10: (eq C c0 e1)).(let H11 \def (eq_ind T u
-(\lambda (t: T).(csubt g (CHead e1 (Bind Abbr) v) (CHead c3 k t))) H5 v H8)
-in (let H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind Abbr)
-v) (CHead c3 k t))) H4 v H8) in (eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda
-(e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v))) (\lambda (e2:
-C).(csubt g e1 e2)))) (let H13 \def (eq_ind K k (\lambda (k0: K).(csubt g
-(CHead e1 (Bind Abbr) v) (CHead c3 k0 v))) H11 (Bind Abbr) H9) in (let H14
-\def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind Abbr) v) (CHead c3
-k0 v))) H12 (Bind Abbr) H9) in (eq_ind_r K (Bind Abbr) (\lambda (k0: K).(ex2
-C (\lambda (e2: C).(eq C (CHead c3 k0 v) (CHead e2 (Bind Abbr) v))) (\lambda
-(e2: C).(csubt g e1 e2)))) (let H15 \def (eq_ind C c0 (\lambda (c: C).(csubt
-g c c3)) H3 e1 H10) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind
-Abbr) v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3
-(refl_equal C (CHead c3 (Bind Abbr) v)) H15)) k H9))) u H8)))))) H7)) H6)) c2
-H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda
-(c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1 (Bind Abbr)
-v))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (_: (csubt g c0
-c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C c2 (\lambda
-(c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H0 (CHead c3 (Bind b) u2) H3) in
-(let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v)
-c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind b) u2) (\lambda
-(c: C).(ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda
-(e2: C).(csubt g e1 e2)))) (let H7 \def (eq_ind C (CHead c0 (Bind Void) u1)
-(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+(H: (csubt g (CHead e1 (Bind Abbr) v) c2)).(insert_eq C (CHead e1 (Bind Abbr)
+v) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(ex2 C (\lambda (e2:
+C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))))
+(\lambda (y: C).(\lambda (H0: (csubt g y c2)).(csubt_ind g (\lambda (c:
+C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
+(e2: C).(eq C c0 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
+e2)))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind
+Abbr) v))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C
+return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
+\Rightarrow False])) I (CHead e1 (Bind Abbr) v) H1) in (False_ind (ex2 C
+(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csubt g e1 e2))) H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2:
+C).(csubt g e1 e2)))))).(\lambda (k: K).(\lambda (u: T).(\lambda (H3: (eq C
+(CHead c1 k u) (CHead e1 (Bind Abbr) v))).(let H4 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c1
+| (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H3)
+in ((let H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0]))
+(CHead c1 k u) (CHead e1 (Bind Abbr) v) H3) in ((let H6 \def (f_equal C T
+(\lambda (e: C).(match e in C return (\lambda (_: C).T) with [(CSort _)
+\Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind
+Abbr) v) H3) in (\lambda (H7: (eq K k (Bind Abbr))).(\lambda (H8: (eq C c1
+e1)).(eq_ind_r T v (\lambda (t: T).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k
+t) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (eq_ind_r K
+(Bind Abbr) (\lambda (k0: K).(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v)
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))) (let H9 \def
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abbr) v)) \to (ex2 C
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt
+g e1 e2))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g
+c c3)) H1 e1 H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr)
+v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)) c3
+(refl_equal C (CHead c3 (Bind Abbr) v)) H10))) k H7) u H6)))) H5))
+H4))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1
+c3)).(\lambda (_: (((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda
+(e2: C).(eq C c3 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1
+e2)))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1
+(Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda
+(ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr)
-v) H2) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2)
-(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H7)) c2
-H3))))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abbr) v) c2)).(\lambda
-(c0: C).(\lambda (c3: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (eq C
-(CHead c0 (Bind Abst) t) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C
-(CHead c3 (Bind Abbr) u) c2)).(\lambda (_: (csubt g c0 c3)).(\lambda (_: (ty3
-g c3 u t)).(let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1
-(Bind Abbr) v) c)) H0 (CHead c3 (Bind Abbr) u) H3) in (let H6 \def (eq_ind_r
-C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abbr) v) c)) H (CHead c3 (Bind
-Abbr) u) H3) in (eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: C).(ex2 C
-(\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g
-e1 e2)))) (let H7 \def (eq_ind C (CHead c0 (Bind Abst) t) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
-with [(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with
-[Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) |
-(Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind
-(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr)
-v))) (\lambda (e2: C).(csubt g e1 e2))) H7)) c2 H3)))))))))))) H))))).
+v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2)
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2))) H5)))))))))))
+(\lambda (c1: C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_:
+(((eq C c1 (CHead e1 (Bind Abbr) v)) \to (ex2 C (\lambda (e2: C).(eq C c3
+(CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))))).(\lambda (u:
+T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (H4: (eq C (CHead c1
+(Bind Abst) t) (CHead e1 (Bind Abbr) v))).(let H5 \def (eq_ind C (CHead c1
+(Bind Abst) t) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind
+Abbr) v) H4) in (False_ind (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind
+Abbr) u) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csubt g e1 e2)))
+H5)))))))))) y c2 H0))) H))))).
theorem csubt_gen_abst:
\forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csubt g
e2 v2 v1)))))))))
\def
\lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda
-(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(csubt_inv_coq g (CHead e1 (Bind
-Abst) v1) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0: C).(or (ex2 C
-(\lambda (e2: C).(eq C c0 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
-g0 e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c0 (CHead e2
-(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g0 e1 e2)))
-(\lambda (e2: C).(\lambda (v2: T).(ty3 g0 e2 v2 v1)))))))) (\lambda (H0:
-(csubt g (CHead e1 (Bind Abst) v1) c2)).(\lambda (n: nat).(\lambda (H1: (eq C
-(CSort n) (CHead e1 (Bind Abst) v1))).(\lambda (H2: (eq C (CSort n) c2)).(let
-H3 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c))
-H0 (CSort n) H2) in (let H4 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g
-(CHead e1 (Bind Abst) v1) c)) H (CSort n) H2) in (eq_ind C (CSort n) (\lambda
-(c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1)))
-(\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2:
-T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))
-(let H5 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead e1 (Bind Abst) v1) H1) in (False_ind (or (ex2 C
-(\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda (e2:
-C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
+(H: (csubt g (CHead e1 (Bind Abst) v1) c2)).(insert_eq C (CHead e1 (Bind
+Abst) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_: C).(or (ex2 C (\lambda
+(e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1
+e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind
+Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (\lambda (y: C).(\lambda (H0:
+(csubt g y c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead
+e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c0 (CHead e2 (Bind
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C c0 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
+e2 v2 v1)))))))) (\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1
+(Bind Abst) v1))).(let H2 \def (eq_ind C (CSort n) (\lambda (ee: C).(match ee
+in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _
+_ _) \Rightarrow False])) I (CHead e1 (Bind Abst) v1) H1) in (False_ind (or
+(ex2 C (\lambda (e2: C).(eq C (CSort n) (CHead e2 (Bind Abst) v1))) (\lambda
+(e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
(CSort n) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))
-H5)) c2 H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abst) v1)
-c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u:
-T).(\lambda (H1: (eq C (CHead c0 k u) (CHead e1 (Bind Abst) v1))).(\lambda
-(H2: (eq C (CHead c3 k u) c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0
-(CHead c3 k u) H2) in (let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g
-(CHead e1 (Bind Abst) v1) c)) H (CHead c3 k u) H2) in (eq_ind C (CHead c3 k
-u) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst)
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda
-(v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))
-(let H6 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
-C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead
-c0 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H7 \def (f_equal C K (\lambda
-(e: C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k
-| (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u) (CHead e1 (Bind Abst) v1)
-H1) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow
-t])) (CHead c0 k u) (CHead e1 (Bind Abst) v1) H1) in (\lambda (H9: (eq K k
-(Bind Abst))).(\lambda (H10: (eq C c0 e1)).(let H11 \def (eq_ind T u (\lambda
-(t: T).(csubt g (CHead e1 (Bind Abst) v1) (CHead c3 k t))) H5 v1 H8) in (let
-H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind Abst) v1)
-(CHead c3 k t))) H4 v1 H8) in (eq_ind_r T v1 (\lambda (t: T).(or (ex2 C
-(\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2 (Bind Abst) v1))) (\lambda
-(e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c3 k t) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))
-(let H13 \def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind Abst) v1)
-(CHead c3 k0 v1))) H11 (Bind Abst) H9) in (let H14 \def (eq_ind K k (\lambda
-(k0: K).(csubt g (CHead e1 (Bind Abst) v1) (CHead c3 k0 v1))) H12 (Bind Abst)
-H9) in (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex2 C (\lambda (e2:
-C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
-g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0
-v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
-e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) (let H15 \def
-(eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H3 e1 H10) in (or_introl (ex2 C
-(\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1)))
+H2)))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1
+c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C
+(\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
+g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))).(\lambda (k:
+K).(\lambda (u: T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind Abst)
+v1))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
+(CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in ((let H5 \def (f_equal C K
+(\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1
+(Bind Abst) v1) H3) in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t)
+\Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H3) in (\lambda
+(H7: (eq K k (Bind Abst))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1
+(\lambda (t: T).(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 k t) (CHead e2
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c3 k t) (CHead e2 (Bind Abbr) v2))))
+(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda
+(v2: T).(ty3 g e2 v2 v1)))))) (eq_ind_r K (Bind Abst) (\lambda (k0: K).(or
+(ex2 C (\lambda (e2: C).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abst) v1)))
(\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2:
-T).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+T).(eq C (CHead c3 k0 v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1)))) (ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1)
-(CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal
-C (CHead c3 (Bind Abst) v1)) H15))) k H9))) u H8)))))) H7)) H6)) c2
-H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind Abst) v1) c2)).(\lambda
-(c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1 (Bind Abst)
-v1))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (_: (csubt g
-c0 c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C c2
-(\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0 (CHead c3 (Bind b)
-u2) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1
-(Bind Abst) v1) c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind
-b) u2) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind
+e2 v2 v1)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1
+(Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind
Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1)))))) (let H7 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (ee:
-C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).Prop)
-with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_: B).Prop) with
-[Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) |
-(Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H2) in (False_ind
-(or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind
-Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind Abbr) v2))))
-(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda
-(v2: T).(ty3 g e2 v2 v1))))) H7)) c2 H3))))))))))))) (\lambda (H0: (csubt g
-(CHead e1 (Bind Abst) v1) c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (u:
-T).(\lambda (t: T).(\lambda (H2: (eq C (CHead c0 (Bind Abst) t) (CHead e1
-(Bind Abst) v1))).(\lambda (H3: (eq C (CHead c3 (Bind Abbr) u) c2)).(\lambda
-(H1: (csubt g c0 c3)).(\lambda (H4: (ty3 g c3 u t)).(let H5 \def (eq_ind_r C
-c2 (\lambda (c: C).(csubt g (CHead e1 (Bind Abst) v1) c)) H0 (CHead c3 (Bind
-Abbr) u) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead
-e1 (Bind Abst) v1) c)) H (CHead c3 (Bind Abbr) u) H3) in (eq_ind C (CHead c3
-(Bind Abbr) u) (\lambda (c: C).(or (ex2 C (\lambda (e2: C).(eq C c (CHead e2
+e2 v2 v1))))))) H2 e1 H8) in (let H10 \def (eq_ind C c1 (\lambda (c:
+C).(csubt g c c3)) H1 e1 H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C
+(CHead c3 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt
+g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind
+Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))
+(ex_intro2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abst) v1) (CHead e2 (Bind
+Abst) v1))) (\lambda (e2: C).(csubt g e1 e2)) c3 (refl_equal C (CHead c3
+(Bind Abst) v1)) H10)))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (_: (csubt g c1 c3)).(\lambda (_: (((eq C c1
+(CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead e2
(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1)))))) (let H7 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
-c])) (CHead c0 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in ((let H8 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind
-Abst) t) (CHead e1 (Bind Abst) v1) H2) in (\lambda (H9: (eq C c0 e1)).(let
-H10 \def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H4 v1 H8) in (let H11
-\def (eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H1 e1 H9) in (or_intror
-(ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abst)
-v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda
-(e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2:
-T).(ty3 g e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq
-C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
+e2 v2 v1)))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda
+(u1: T).(\lambda (u2: T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead
+e1 (Bind Abst) v1))).(let H5 \def (eq_ind C (CHead c1 (Bind Void) u1)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow False | (CHead _ k _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind b0) \Rightarrow (match b0 in B return (\lambda (_:
+B).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void
+\Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst)
+v1) H4) in (False_ind (or (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind b)
+u2) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T
+(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2
+(Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))
+(\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))) H5))))))))))) (\lambda
+(c1: C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C
+c1 (CHead e1 (Bind Abst) v1)) \to (or (ex2 C (\lambda (e2: C).(eq C c3 (CHead
+e2 (Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda
+(e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2:
C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g
-e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H11 H10)))))) H7))
-c2 H3)))))))))))) H))))).
+e2 v2 v1)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c3 u
+t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst)
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
+(CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H4) in ((let H6 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1 (Bind
+Abst) t) (CHead e1 (Bind Abst) v1) H4) in (\lambda (H7: (eq C c1 e1)).(let H8
+\def (eq_ind T t (\lambda (t0: T).(ty3 g c3 u t0)) H3 v1 H6) in (let H9 \def
+(eq_ind C c1 (\lambda (c: C).((eq C c (CHead e1 (Bind Abst) v1)) \to (or (ex2
+C (\lambda (e2: C).(eq C c3 (CHead e2 (Bind Abst) v1))) (\lambda (e2:
+C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c3
+(CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1
+e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) H2 e1 H7) in
+(let H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H7) in
+(or_intror (ex2 C (\lambda (e2: C).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
+(Bind Abst) v1))) (\lambda (e2: C).(csubt g e1 e2))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr)
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2:
+C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Bind Abbr)
+v2)))) (\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))) (\lambda (e2:
+C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c3 u (refl_equal C (CHead c3 (Bind
+Abbr) u)) H10 H8))))))) H5)))))))))) y c2 H0))) H))))).
theorem csubt_gen_bind:
\forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall
(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))))))
\def
\lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda
-(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(csubt_inv_coq g
-(CHead e1 (Bind b1) v1) c2 (\lambda (g0: G).(\lambda (_: C).(\lambda (c0:
-C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c0
+(v1: T).(\lambda (H: (csubt g (CHead e1 (Bind b1) v1) c2)).(insert_eq C
+(CHead e1 (Bind b1) v1) (\lambda (c: C).(csubt g c c2)) (\lambda (_:
+C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2
(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csubt g0 e1 e2)))))))) (\lambda (H0: (csubt g (CHead e1 (Bind b1) v1)
-c2)).(\lambda (n: nat).(\lambda (H1: (eq C (CSort n) (CHead e1 (Bind b1)
-v1))).(\lambda (H2: (eq C (CSort n) c2)).(let H3 \def (eq_ind_r C c2 (\lambda
-(c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CSort n) H2) in (let H4 \def
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H (CSort
-n) H2) in (eq_ind C (CSort n) (\lambda (c: C).(ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))) (let H5
-\def (eq_ind C (CSort n) (\lambda (ee: C).(match ee in C return (\lambda (_:
-C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow
-False])) I (CHead e1 (Bind b1) v1) H1) in (False_ind (ex2_3 B C T (\lambda
-(b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CSort n) (CHead e2 (Bind b2)
-v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))
-H5)) c2 H2))))))) (\lambda (H0: (csubt g (CHead e1 (Bind b1) v1)
-c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (k: K).(\lambda (u:
-T).(\lambda (H1: (eq C (CHead c0 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2:
-(eq C (CHead c3 k u) c2)).(\lambda (H3: (csubt g c0 c3)).(let H4 \def
-(eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CHead
-c3 k u) H2) in (let H5 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1
-(Bind b1) v1) c)) H (CHead c3 k u) H2) in (eq_ind C (CHead c3 k u) (\lambda
-(c: C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
-c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
-T).(csubt g e1 e2)))))) (let H6 \def (f_equal C C (\lambda (e: C).(match e in
-C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
-\Rightarrow c])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H1) in ((let H7 \def
-(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
-[(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k u)
-(CHead e1 (Bind b1) v1) H1) in ((let H8 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
-(CHead _ _ t) \Rightarrow t])) (CHead c0 k u) (CHead e1 (Bind b1) v1) H1) in
-(\lambda (H9: (eq K k (Bind b1))).(\lambda (H10: (eq C c0 e1)).(let H11 \def
-(eq_ind T u (\lambda (t: T).(csubt g (CHead e1 (Bind b1) v1) (CHead c3 k t)))
-H5 v1 H8) in (let H12 \def (eq_ind T u (\lambda (t: T).(csubt g (CHead e1
-(Bind b1) v1) (CHead c3 k t))) H4 v1 H8) in (eq_ind_r T v1 (\lambda (t:
-T).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c3 k t) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2)))))) (let H13 \def (eq_ind K k (\lambda
-(k0: K).(csubt g (CHead e1 (Bind b1) v1) (CHead c3 k0 v1))) H11 (Bind b1) H9)
-in (let H14 \def (eq_ind K k (\lambda (k0: K).(csubt g (CHead e1 (Bind b1)
-v1) (CHead c3 k0 v1))) H12 (Bind b1) H9) in (eq_ind_r K (Bind b1) (\lambda
-(k0: K).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
-(CHead c3 k0 v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
-C).(\lambda (_: T).(csubt g e1 e2)))))) (let H15 \def (eq_ind C c0 (\lambda
-(c: C).(csubt g c c3)) H3 e1 H10) in (ex2_3_intro B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b1) v1) (CHead e2
-(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
-e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3 (Bind b1) v1)) H15)) k H9))) u
-H8)))))) H7)) H6)) c2 H2))))))))))) (\lambda (H0: (csubt g (CHead e1 (Bind
-b1) v1) c2)).(\lambda (c0: C).(\lambda (c3: C).(\lambda (b: B).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (H2: (eq C (CHead c0 (Bind Void) u1) (CHead e1
-(Bind b1) v1))).(\lambda (H3: (eq C (CHead c3 (Bind b) u2) c2)).(\lambda (H1:
-(csubt g c0 c3)).(\lambda (_: (not (eq B b Void))).(let H5 \def (eq_ind_r C
-c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0 (CHead c3 (Bind b)
-u2) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1
-(Bind b1) v1) c)) H (CHead c3 (Bind b) u2) H3) in (eq_ind C (CHead c3 (Bind
-b) u2) (\lambda (c: C).(ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
-C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))) (let H7 \def
+T).(csubt g e1 e2)))))) (\lambda (y: C).(\lambda (H0: (csubt g y
+c2)).(csubt_ind g (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind
+b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c0 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2)))))))) (\lambda (n: nat).(\lambda (H1: (eq
+C (CSort n) (CHead e1 (Bind b1) v1))).(let H2 \def (eq_ind C (CSort n)
+(\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop) with [(CSort _)
+\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1)
+v1) H1) in (False_ind (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C (CSort n) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csubt g e1 e2))))) H2)))) (\lambda (c1: C).(\lambda
+(c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1
+(Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
+(v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (k: K).(\lambda (u:
+T).(\lambda (H3: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(let H4 \def
(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind
-Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H8 \def (f_equal C B (\lambda
-(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Void | (CHead _ k _) \Rightarrow (match k in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Void])])) (CHead c0 (Bind
-Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H9 \def (f_equal C T (\lambda
-(e: C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u1
-| (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Void) u1) (CHead e1 (Bind
-b1) v1) H2) in (\lambda (H10: (eq B Void b1)).(\lambda (H11: (eq C c0
-e1)).(let H12 \def (eq_ind C c0 (\lambda (c: C).(csubt g c c3)) H1 e1 H11) in
-(let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csubt g (CHead e1 (Bind b0)
-v1) (CHead c3 (Bind b) u2))) H6 Void H10) in (let H14 \def (eq_ind_r B b1
-(\lambda (b0: B).(csubt g (CHead e1 (Bind b0) v1) (CHead c3 (Bind b) u2))) H5
-Void H10) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda
-(v2: T).(eq C (CHead c3 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_:
-B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))) b c3 u2 (refl_equal C
-(CHead c3 (Bind b) u2)) H12))))))) H8)) H7)) c2 H3))))))))))))) (\lambda (H0:
-(csubt g (CHead e1 (Bind b1) v1) c2)).(\lambda (c0: C).(\lambda (c3:
-C).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (eq C (CHead c0 (Bind Abst)
-t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c3 (Bind Abbr) u)
-c2)).(\lambda (H1: (csubt g c0 c3)).(\lambda (H4: (ty3 g c3 u t)).(let H5
-\def (eq_ind_r C c2 (\lambda (c: C).(csubt g (CHead e1 (Bind b1) v1) c)) H0
-(CHead c3 (Bind Abbr) u) H3) in (let H6 \def (eq_ind_r C c2 (\lambda (c:
-C).(csubt g (CHead e1 (Bind b1) v1) c)) H (CHead c3 (Bind Abbr) u) H3) in
-(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: C).(ex2_3 B C T (\lambda (b2:
-B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2)))))
-(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))) (let H7
-\def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C)
-with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0
-(Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H8 \def (f_equal C B
-(\lambda (e: C).(match e in C return (\lambda (_: C).B) with [(CSort _)
-\Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return (\lambda
-(_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])]))
-(CHead c0 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H9 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 (Bind
-Abst) t) (CHead e1 (Bind b1) v1) H2) in (\lambda (H10: (eq B Abst
-b1)).(\lambda (H11: (eq C c0 e1)).(let H12 \def (eq_ind T t (\lambda (t0:
-T).(ty3 g c3 u t0)) H4 v1 H9) in (let H13 \def (eq_ind C c0 (\lambda (c:
-C).(csubt g c c3)) H1 e1 H11) in (let H14 \def (eq_ind_r B b1 (\lambda (b:
-B).(csubt g (CHead e1 (Bind b) v1) (CHead c3 (Bind Abbr) u))) H6 Abst H10) in
-(let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csubt g (CHead e1 (Bind b) v1)
-(CHead c3 (Bind Abbr) u))) H5 Abst H10) in (ex2_3_intro B C T (\lambda (b2:
+[(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u)
+(CHead e1 (Bind b1) v1) H3) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k0 _) \Rightarrow k0])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H3)
+in ((let H6 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda
+(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t]))
+(CHead c1 k u) (CHead e1 (Bind b1) v1) H3) in (\lambda (H7: (eq K k (Bind
+b1))).(\lambda (H8: (eq C c1 e1)).(eq_ind_r T v1 (\lambda (t: T).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k t)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2)))))) (eq_ind_r K (Bind b1) (\lambda (k0: K).(ex2_3 B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 k0 v1)
+(CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2)))))) (let H9 \def (eq_ind C c1 (\lambda (c: C).((eq C c
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1 H8) in (let
+H10 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H8) in
+(ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C
+(CHead c3 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda
+(e2: C).(\lambda (_: T).(csubt g e1 e2)))) b1 c3 v1 (refl_equal C (CHead c3
+(Bind b1) v1)) H10))) k H7) u H6)))) H5)) H4))))))))) (\lambda (c1:
+C).(\lambda (c3: C).(\lambda (H1: (csubt g c1 c3)).(\lambda (H2: (((eq C c1
+(CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2:
+C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_:
+B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2)))))))).(\lambda (b:
+B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2:
+T).(\lambda (H4: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1)
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
+(CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in ((let H7
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1
+(Bind Void) u1) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Void
+b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind C c1 (\lambda (c:
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1
+H9) in (let H11 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9)
+in (let H12 \def (eq_ind_r B b1 (\lambda (b0: B).((eq C e1 (CHead e1 (Bind
+b0) v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2:
+T).(eq C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2:
+C).(\lambda (_: T).(csubt g e1 e2))))))) H10 Void H8) in (ex2_3_intro B C T
+(\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind b)
+u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_:
+T).(csubt g e1 e2)))) b c3 u2 (refl_equal C (CHead c3 (Bind b) u2))
+H11))))))) H6)) H5))))))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H1:
+(csubt g c1 c3)).(\lambda (H2: (((eq C c1 (CHead e1 (Bind b1) v1)) \to (ex2_3
+B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2
+(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
+e1 e2)))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H3: (ty3 g c3 u
+t)).(\lambda (H4: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1)
+v1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c]))
+(CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H4) in ((let H6 \def
+(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
+[(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow
+Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H4) in ((let H7
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c1
+(Bind Abst) t) (CHead e1 (Bind b1) v1) H4) in (\lambda (H8: (eq B Abst
+b1)).(\lambda (H9: (eq C c1 e1)).(let H10 \def (eq_ind T t (\lambda (t0:
+T).(ty3 g c3 u t0)) H3 v1 H7) in (let H11 \def (eq_ind C c1 (\lambda (c:
+C).((eq C c (CHead e1 (Bind b1) v1)) \to (ex2_3 B C T (\lambda (b2:
+B).(\lambda (e2: C).(\lambda (v2: T).(eq C c3 (CHead e2 (Bind b2) v2)))))
+(\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g e1 e2))))))) H2 e1
+H9) in (let H12 \def (eq_ind C c1 (\lambda (c: C).(csubt g c c3)) H1 e1 H9)
+in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).((eq C e1 (CHead e1 (Bind b)
+v1)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq
+C c3 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda
+(_: T).(csubt g e1 e2))))))) H11 Abst H8) in (ex2_3_intro B C T (\lambda (b2:
B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c3 (Bind Abbr) u) (CHead e2
(Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csubt g
-e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H13)))))))) H8))
-H7)) c2 H3)))))))))))) H)))))).
+e1 e2)))) Abbr c3 u (refl_equal C (CHead c3 (Bind Abbr) u)) H12)))))))) H6))
+H5)))))))))) y c2 H0))) H)))))).
(* This file was automatically generated: do not edit *********************)
-include "csubt/fwd.ma".
+include "LambdaDelta-1/csubt/fwd.ma".
-include "csubt/clear.ma".
+include "LambdaDelta-1/csubt/clear.ma".
-include "csubt/drop.ma".
+include "LambdaDelta-1/csubt/drop.ma".
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
theorem csubt_getl_abbr:
\forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "csubt/getl.ma".
+include "LambdaDelta-1/csubt/getl.ma".
-include "pc3/left.ma".
+include "LambdaDelta-1/pc3/left.ma".
theorem csubt_pr2:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
(* This file was automatically generated: do not edit *********************)
-include "csubt/defs.ma".
+include "LambdaDelta-1/csubt/defs.ma".
theorem csubt_refl:
\forall (g: G).(\forall (c: C).(csubt g c c))
(* This file was automatically generated: do not edit *********************)
-include "csubt/pc3.ma".
+include "LambdaDelta-1/csubt/pc3.ma".
-include "csubt/props.ma".
+include "LambdaDelta-1/csubt/props.ma".
theorem csubt_ty3:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
(* This file was automatically generated: do not edit *********************)
-include "tlt/defs.ma".
+include "LambdaDelta-1/tlt/defs.ma".
-include "iso/defs.ma".
+include "LambdaDelta-1/iso/defs.ma".
-include "clen/defs.ma".
+include "LambdaDelta-1/clen/defs.ma".
-include "flt/defs.ma".
+include "LambdaDelta-1/flt/defs.ma".
-include "cnt/defs.ma".
+include "LambdaDelta-1/cnt/defs.ma".
-include "cimp/defs.ma".
+include "LambdaDelta-1/cimp/defs.ma".
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
-include "csubst1/defs.ma".
+include "LambdaDelta-1/csubst1/defs.ma".
-include "fsubst0/defs.ma".
+include "LambdaDelta-1/fsubst0/defs.ma".
-include "next_plus/defs.ma".
+include "LambdaDelta-1/next_plus/defs.ma".
-include "tau1/defs.ma".
+include "LambdaDelta-1/tau1/defs.ma".
-include "llt/defs.ma".
+include "LambdaDelta-1/llt/defs.ma".
-include "aprem/defs.ma".
+include "LambdaDelta-1/aprem/defs.ma".
-include "ex0/defs.ma".
+include "LambdaDelta-1/ex0/defs.ma".
-include "wcpr0/defs.ma".
+include "LambdaDelta-1/wcpr0/defs.ma".
-include "csuba/defs.ma".
+include "LambdaDelta-1/csuba/defs.ma".
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
-include "ex2/defs.ma".
+include "LambdaDelta-1/ex2/defs.ma".
-include "csubc/defs.ma".
+include "LambdaDelta-1/csubc/defs.ma".
-include "pc1/defs.ma".
+include "LambdaDelta-1/pc1/defs.ma".
-include "ex1/defs.ma".
+include "LambdaDelta-1/ex1/defs.ma".
-include "csubt/defs.ma".
+include "LambdaDelta-1/csubt/defs.ma".
+++ /dev/null
-theory.ma ex0/props.ma ex1/props.ma ex2/props.ma pr3/wcpr0.ma subst0/tlt.ma tau1/cnt.ma ty3/dec.ma ty3/nf2.ma ty3/tau0.ma wcpr0/fwd.ma
-spare.ma theory.ma
-preamble.ma ../Base-1/theory.ma
-definitions.ma aprem/defs.ma cimp/defs.ma clen/defs.ma cnt/defs.ma csuba/defs.ma csubc/defs.ma csubst1/defs.ma csubt/defs.ma ex0/defs.ma ex1/defs.ma ex2/defs.ma flt/defs.ma fsubst0/defs.ma iso/defs.ma llt/defs.ma next_plus/defs.ma nf2/defs.ma pc1/defs.ma subst1/defs.ma tau1/defs.ma tlt/defs.ma wcpr0/defs.ma
-ex0/defs.ma A/defs.ma G/defs.ma
-ex0/props.ma aplus/props.ma ex0/defs.ma leq/defs.ma
-ex2/defs.ma C/defs.ma
-ex2/props.ma arity/fwd.ma ex2/defs.ma nf2/defs.ma pr2/fwd.ma
-tlist/defs.ma T/defs.ma
-tlist/props.ma tlist/defs.ma
-csubt/defs.ma ty3/defs.ma
-csubt/fwd.ma csubt/defs.ma
-csubt/props.ma csubt/defs.ma
-csubt/clear.ma clear/fwd.ma csubt/defs.ma
-csubt/drop.ma csubt/defs.ma drop/fwd.ma
-csubt/getl.ma csubt/clear.ma csubt/drop.ma csubt/fwd.ma getl/clear.ma
-csubt/pc3.ma csubt/getl.ma pc3/left.ma
-csubt/ty3.ma csubt/pc3.ma csubt/props.ma
-csubc/defs.ma sc3/defs.ma
-csubc/props.ma csubc/defs.ma sc3/props.ma
-csubc/csuba.ma csuba/defs.ma csubc/defs.ma sc3/props.ma
-csubc/drop.ma csubc/defs.ma sc3/props.ma
-csubc/drop1.ma csubc/drop.ma
-csubc/clear.ma csubc/defs.ma
-csubc/getl.ma csubc/clear.ma csubc/drop.ma
-csubc/arity.ma arity/defs.ma csubc/csuba.ma
-ty3/fsubst0.ma csubst0/props.ma getl/getl.ma pc3/fsubst0.ma ty3/props.ma
-ty3/defs.ma G/defs.ma pc3/defs.ma
-ty3/fwd.ma pc3/props.ma ty3/defs.ma
-ty3/props.ma pc3/fwd.ma ty3/fwd.ma
-ty3/subst1.ma csubst1/fwd.ma csubst1/getl.ma getl/getl.ma pc3/fwd.ma pc3/subst1.ma ty3/props.ma
-ty3/pr3_props.ma ty3/pr3.ma
-ty3/pr3.ma csubt/ty3.ma pc1/props.ma pc3/pc1.ma pc3/wcpr0.ma ty3/fsubst0.ma ty3/subst1.ma
-ty3/tau0.ma tau0/defs.ma ty3/pr3_props.ma
-ty3/arity.ma arity/pr3.ma asucc/fwd.ma ty3/pr3_props.ma
-ty3/dec.ma getl/dec.ma getl/flt.ma pc3/dec.ma ty3/pr3_props.ma
-ty3/arity_props.ma sc3/arity.ma ty3/arity.ma ty3/fwd.ma
-ty3/nf2.ma nf2/arity.ma pc3/nf2.ma ty3/arity.ma
-pc3/defs.ma pr3/defs.ma
-pc3/props.ma pc3/defs.ma pr3/pr3.ma
-pc3/fsubst0.ma csubst0/getl.ma fsubst0/defs.ma pc3/left.ma
-pc3/pc1.ma pc1/defs.ma pc3/defs.ma pr3/pr1.ma
-pc3/wcpr0.ma pc3/props.ma wcpr0/getl.ma
-pc3/left.ma pc3/props.ma
-pc3/fwd.ma pc3/props.ma pr3/fwd.ma
-pc3/nf2.ma nf2/pr3.ma pc3/defs.ma
-pc3/subst1.ma pc3/props.ma pr3/subst1.ma
-pc3/dec.ma nf2/fwd.ma ty3/arity_props.ma ty3/pr3.ma
-pc1/defs.ma pr1/defs.ma
-pc1/props.ma pc1/defs.ma pr1/pr1.ma
-ex1/defs.ma C/defs.ma
-ex1/props.ma arity/defs.ma ex1/defs.ma leq/props.ma nf2/pr3.ma nf2/props.ma pc3/fwd.ma ty3/fwd.ma
-sc3/defs.ma arity/defs.ma drop1/defs.ma sn3/defs.ma
-sc3/props.ma arity/aprem.ma arity/lift1.ma csuba/arity.ma drop1/getl.ma drop1/props.ma lift1/props.ma llt/props.ma nf2/lift1.ma sc3/defs.ma sn3/lift1.ma
-sc3/arity.ma csubc/arity.ma csubc/drop1.ma csubc/getl.ma csubc/props.ma
-sn3/defs.ma pr3/defs.ma
-sn3/nf2.ma nf2/dec.ma nf2/pr3.ma sn3/defs.ma
-sn3/props.ma iso/props.ma nf2/iso.ma pr3/iso.ma sn3/fwd.ma sn3/nf2.ma
-sn3/lift1.ma drop1/defs.ma lift1/fwd.ma sn3/props.ma
-sn3/fwd.ma pr3/props.ma sn3/defs.ma
-nf2/defs.ma pr2/defs.ma
-nf2/fwd.ma T/props.ma nf2/defs.ma pr2/clen.ma subst0/dec.ma
-nf2/props.ma nf2/defs.ma pr2/fwd.ma
-nf2/arity.ma arity/subst0.ma nf2/fwd.ma
-nf2/pr3.ma nf2/defs.ma pr3/pr3.ma
-nf2/lift1.ma drop1/defs.ma nf2/props.ma
-nf2/iso.ma iso/props.ma nf2/pr3.ma pr3/fwd.ma
-nf2/dec.ma C/props.ma nf2/defs.ma pr0/dec.ma pr2/clen.ma pr2/fwd.ma
-csuba/defs.ma arity/defs.ma
-csuba/fwd.ma csuba/defs.ma
-csuba/props.ma csuba/defs.ma
-csuba/arity.ma T/props.ma arity/props.ma csuba/getl.ma csuba/props.ma
-csuba/clear.ma clear/fwd.ma csuba/defs.ma
-csuba/drop.ma csuba/fwd.ma drop/fwd.ma
-csuba/getl.ma csuba/clear.ma csuba/drop.ma getl/clear.ma
-wcpr0/getl.ma getl/props.ma wcpr0/defs.ma
-wcpr0/fwd.ma wcpr0/defs.ma
-wcpr0/defs.ma C/defs.ma pr0/defs.ma
-pr3/defs.ma pr2/defs.ma
-pr3/subst1.ma pr2/subst1.ma pr3/defs.ma
-pr3/props.ma pr1/props.ma pr2/props.ma pr3/pr1.ma
-pr3/pr1.ma pr1/defs.ma pr3/defs.ma
-pr3/wcpr0.ma pr3/props.ma wcpr0/getl.ma
-pr3/fwd.ma pr2/fwd.ma pr3/props.ma
-pr3/pr3.ma pr2/pr2.ma pr3/props.ma
-pr3/iso.ma iso/props.ma pr3/fwd.ma tlist/props.ma
-pr2/defs.ma getl/defs.ma pr0/defs.ma
-pr2/props.ma getl/clear.ma getl/drop.ma pr0/props.ma pr2/defs.ma
-pr2/fwd.ma getl/clear.ma getl/drop.ma pr0/fwd.ma pr2/defs.ma
-pr2/subst1.ma csubst1/fwd.ma csubst1/getl.ma getl/drop.ma pr0/fwd.ma pr0/subst1.ma pr2/defs.ma subst1/subst1.ma
-pr2/clen.ma clen/getl.ma pr2/props.ma
-pr2/pr2.ma getl/props.ma pr0/pr0.ma pr2/defs.ma
-pr1/defs.ma pr0/defs.ma
-pr1/props.ma T/props.ma pr0/subst1.ma pr1/defs.ma subst1/props.ma
-pr1/pr1.ma pr0/pr0.ma pr1/props.ma
-pr0/subst1.ma pr0/props.ma subst1/defs.ma
-pr0/defs.ma subst0/defs.ma
-pr0/fwd.ma pr0/props.ma
-pr0/props.ma pr0/defs.ma subst0/subst0.ma
-pr0/pr0.ma lift/tlt.ma pr0/fwd.ma
-pr0/dec.ma T/dec.ma T/props.ma pr0/fwd.ma subst0/dec.ma
-tau1/defs.ma tau0/defs.ma
-tau1/props.ma tau0/props.ma tau1/defs.ma
-tau1/cnt.ma cnt/props.ma tau1/props.ma
-arity/defs.ma getl/defs.ma leq/defs.ma
-arity/fwd.ma arity/defs.ma getl/drop.ma leq/asucc.ma leq/fwd.ma
-arity/subst0.ma arity/props.ma csubst0/getl.ma csubst0/props.ma fsubst0/fwd.ma getl/getl.ma subst0/dec.ma subst0/fwd.ma
-arity/props.ma arity/fwd.ma
-arity/pr3.ma arity/subst0.ma csuba/arity.ma pr0/fwd.ma pr1/defs.ma pr3/defs.ma wcpr0/getl.ma
-arity/lift1.ma arity/props.ma drop1/defs.ma
-arity/cimp.ma arity/defs.ma cimp/props.ma
-arity/aprem.ma aprem/props.ma arity/cimp.ma arity/props.ma
-aprem/defs.ma A/defs.ma
-aprem/props.ma aprem/defs.ma leq/defs.ma
-llt/defs.ma A/defs.ma
-llt/props.ma leq/defs.ma llt/defs.ma
-leq/defs.ma aplus/defs.ma
-leq/asucc.ma aplus/props.ma leq/props.ma
-leq/props.ma aplus/props.ma leq/defs.ma
-leq/fwd.ma leq/defs.ma
-aplus/defs.ma asucc/defs.ma
-aplus/props.ma aplus/defs.ma next_plus/props.ma
-asucc/defs.ma A/defs.ma G/defs.ma
-asucc/fwd.ma asucc/defs.ma
-A/defs.ma preamble.ma
-fsubst0/defs.ma csubst0/defs.ma subst0/defs.ma
-fsubst0/fwd.ma fsubst0/defs.ma
-csubst1/defs.ma csubst0/defs.ma
-csubst1/props.ma csubst1/defs.ma subst1/defs.ma
-csubst1/fwd.ma csubst0/fwd.ma csubst1/defs.ma subst1/props.ma
-csubst1/getl.ma csubst0/getl.ma csubst0/props.ma csubst1/props.ma drop/props.ma subst1/props.ma
-csubst0/defs.ma C/defs.ma subst0/defs.ma
-csubst0/fwd.ma csubst0/defs.ma
-csubst0/props.ma csubst0/defs.ma
-csubst0/drop.ma csubst0/fwd.ma drop/fwd.ma s/props.ma
-csubst0/clear.ma clear/fwd.ma csubst0/fwd.ma
-csubst0/getl.ma csubst0/clear.ma csubst0/drop.ma getl/fwd.ma
-subst0/defs.ma lift/defs.ma
-subst0/fwd.ma lift/props.ma subst0/defs.ma
-subst0/subst0.ma subst0/props.ma
-subst0/props.ma lift/props.ma subst0/fwd.ma
-subst0/tlt.ma lift/props.ma lift/tlt.ma subst0/defs.ma
-subst0/dec.ma lift/props.ma subst0/defs.ma
-tau0/defs.ma G/defs.ma getl/defs.ma
-tau0/props.ma getl/drop.ma tau0/defs.ma
-subst1/subst1.ma subst0/subst0.ma subst1/fwd.ma
-subst1/defs.ma subst0/defs.ma
-subst1/fwd.ma subst0/props.ma subst1/defs.ma
-subst1/props.ma subst0/props.ma subst1/defs.ma
-cimp/defs.ma getl/defs.ma
-cimp/props.ma cimp/defs.ma getl/getl.ma
-getl/defs.ma clear/defs.ma drop/defs.ma
-getl/props.ma clear/props.ma drop/props.ma getl/fwd.ma
-getl/fwd.ma clear/fwd.ma drop/fwd.ma getl/defs.ma
-getl/drop.ma clear/drop.ma getl/props.ma r/props.ma
-getl/clear.ma clear/drop.ma getl/props.ma
-getl/getl.ma getl/clear.ma getl/drop.ma
-getl/dec.ma getl/props.ma
-getl/flt.ma clear/props.ma flt/props.ma getl/fwd.ma
-clear/defs.ma C/defs.ma
-clear/fwd.ma clear/defs.ma
-clear/props.ma clear/fwd.ma
-clear/drop.ma clear/fwd.ma drop/fwd.ma
-drop1/defs.ma drop/defs.ma lift1/defs.ma
-drop1/props.ma drop/props.ma drop1/defs.ma getl/defs.ma
-drop1/getl.ma drop1/defs.ma getl/drop.ma
-drop/defs.ma C/defs.ma lift/defs.ma r/defs.ma
-drop/fwd.ma drop/defs.ma
-drop/props.ma drop/fwd.ma lift/props.ma r/props.ma
-lift1/defs.ma lift/defs.ma
-lift1/fwd.ma lift/fwd.ma lift1/defs.ma
-lift1/props.ma drop1/defs.ma lift/props.ma lift1/defs.ma
-cnt/defs.ma T/defs.ma
-cnt/props.ma cnt/defs.ma lift/fwd.ma
-lift/defs.ma T/defs.ma s/defs.ma tlist/defs.ma
-lift/fwd.ma lift/defs.ma
-lift/props.ma lift/fwd.ma s/props.ma tlist/defs.ma
-lift/tlt.ma lift/fwd.ma tlt/props.ma
-iso/props.ma iso/fwd.ma
-iso/fwd.ma iso/defs.ma tlist/defs.ma
-iso/defs.ma T/defs.ma
-G/defs.ma preamble.ma
-flt/props.ma C/props.ma flt/defs.ma
-flt/defs.ma C/defs.ma
-tlt/props.ma tlt/defs.ma
-tlt/defs.ma T/defs.ma
-T/props.ma T/defs.ma
-T/dec.ma T/defs.ma
-T/defs.ma preamble.ma
-s/props.ma s/defs.ma
-s/defs.ma T/defs.ma
-C/props.ma C/defs.ma T/props.ma
-C/defs.ma T/defs.ma
-r/props.ma r/defs.ma s/defs.ma
-r/defs.ma T/defs.ma
-next_plus/props.ma next_plus/defs.ma
-next_plus/defs.ma G/defs.ma
-clen/getl.ma clen/defs.ma getl/props.ma
-clen/defs.ma C/defs.ma s/defs.ma
-../Base-1/theory.ma
(* This file was automatically generated: do not edit *********************)
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
-include "r/defs.ma".
+include "LambdaDelta-1/r/defs.ma".
inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def
| drop_refl: \forall (c: C).(drop O O c c)
(* This file was automatically generated: do not edit *********************)
-include "drop/defs.ma".
+include "LambdaDelta-1/drop/defs.ma".
theorem drop_gen_sort:
\forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop
(* This file was automatically generated: do not edit *********************)
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
-include "lift/props.ma".
+include "LambdaDelta-1/lift/props.ma".
-include "r/props.ma".
+include "LambdaDelta-1/r/props.ma".
theorem drop_skip_bind:
\forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
(* This file was automatically generated: do not edit *********************)
-include "drop/defs.ma".
+include "LambdaDelta-1/drop/defs.ma".
-include "lift1/defs.ma".
+include "LambdaDelta-1/lift1/defs.ma".
inductive drop1: PList \to (C \to (C \to Prop)) \def
| drop1_nil: \forall (c: C).(drop1 PNil c c)
(* This file was automatically generated: do not edit *********************)
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
theorem drop1_getl_trans:
\forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
(* This file was automatically generated: do not edit *********************)
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
-include "drop/props.ma".
+include "LambdaDelta-1/drop/props.ma".
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
theorem drop1_skip_bind:
\forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c:
(* This file was automatically generated: do not edit *********************)
-include "A/defs.ma".
+include "LambdaDelta-1/A/defs.ma".
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
definition gz:
G
(* This file was automatically generated: do not edit *********************)
-include "ex0/defs.ma".
+include "LambdaDelta-1/ex0/defs.ma".
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
-include "aplus/props.ma".
+include "LambdaDelta-1/aplus/props.ma".
theorem aplus_gz_le:
\forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A
(* This file was automatically generated: do not edit *********************)
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
definition ex1_c:
C
(* This file was automatically generated: do not edit *********************)
-include "ex1/defs.ma".
+include "LambdaDelta-1/ex1/defs.ma".
-include "ty3/fwd.ma".
+include "LambdaDelta-1/ty3/fwd.ma".
-include "pc3/fwd.ma".
+include "LambdaDelta-1/pc3/fwd.ma".
-include "nf2/pr3.ma".
+include "LambdaDelta-1/nf2/pr3.ma".
-include "nf2/props.ma".
+include "LambdaDelta-1/nf2/props.ma".
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
-include "leq/props.ma".
+include "LambdaDelta-1/leq/props.ma".
theorem ex1__leq_sort_SS:
\forall (g: G).(\forall (k: nat).(\forall (n: nat).(leq g (ASort k n) (asucc
(* This file was automatically generated: do not edit *********************)
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
definition ex2_c:
C
(* This file was automatically generated: do not edit *********************)
-include "ex2/defs.ma".
+include "LambdaDelta-1/ex2/defs.ma".
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
-include "pr2/fwd.ma".
+include "LambdaDelta-1/pr2/fwd.ma".
-include "arity/fwd.ma".
+include "LambdaDelta-1/arity/fwd.ma".
theorem ex2_nf2:
nf2 ex2_c ex2_t
(* This file was automatically generated: do not edit *********************)
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
definition fweight:
C \to (T \to nat)
(* This file was automatically generated: do not edit *********************)
-include "flt/defs.ma".
+include "LambdaDelta-1/flt/defs.ma".
-include "C/props.ma".
+include "LambdaDelta-1/C/props.ma".
theorem flt_thead_sx:
\forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t: T).(flt c u c
(* This file was automatically generated: do not edit *********************)
-include "csubst0/defs.ma".
-
-include "subst0/defs.ma".
+include "LambdaDelta-1/csubst0/defs.ma".
inductive fsubst0 (i: nat) (v: T) (c1: C) (t1: T): C \to (T \to Prop) \def
| fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1
(* This file was automatically generated: do not edit *********************)
-include "fsubst0/defs.ma".
+include "LambdaDelta-1/fsubst0/defs.ma".
theorem fsubst0_gen_base:
\forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
-include "clear/drop.ma".
+include "LambdaDelta-1/clear/drop.ma".
theorem clear_getl_trans:
\forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to
(* This file was automatically generated: do not edit *********************)
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
theorem getl_dec:
\forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda
(* This file was automatically generated: do not edit *********************)
-include "drop/defs.ma".
+include "LambdaDelta-1/drop/defs.ma".
-include "clear/defs.ma".
+include "LambdaDelta-1/clear/defs.ma".
inductive getl (h: nat) (c1: C) (c2: C): Prop \def
| getl_intro: \forall (e: C).((drop h O c1 e) \to ((clear e c2) \to (getl h
(* This file was automatically generated: do not edit *********************)
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
-include "clear/drop.ma".
-
-include "r/props.ma".
+include "LambdaDelta-1/clear/drop.ma".
theorem getl_drop:
\forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
(* This file was automatically generated: do not edit *********************)
-include "getl/fwd.ma".
+include "LambdaDelta-1/getl/fwd.ma".
-include "clear/props.ma".
+include "LambdaDelta-1/clear/props.ma".
-include "flt/props.ma".
+include "LambdaDelta-1/flt/props.ma".
theorem getl_flt:
\forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i:
(* This file was automatically generated: do not edit *********************)
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
-include "drop/fwd.ma".
+include "LambdaDelta-1/drop/fwd.ma".
-include "clear/fwd.ma".
+include "LambdaDelta-1/clear/fwd.ma".
theorem getl_gen_all:
\forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2
(* This file was automatically generated: do not edit *********************)
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
theorem getl_conf_le:
\forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall
(* This file was automatically generated: do not edit *********************)
-include "getl/fwd.ma".
+include "LambdaDelta-1/getl/fwd.ma".
-include "drop/props.ma".
+include "LambdaDelta-1/drop/props.ma".
-include "clear/props.ma".
+include "LambdaDelta-1/clear/props.ma".
theorem getl_refl:
\forall (b: B).(\forall (c: C).(\forall (u: T).(getl O (CHead c (Bind b) u)
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
inductive iso: T \to (T \to Prop) \def
| iso_sort: \forall (n1: nat).(\forall (n2: nat).(iso (TSort n1) (TSort n2)))
(* This file was automatically generated: do not edit *********************)
-include "iso/defs.ma".
+include "LambdaDelta-1/iso/defs.ma".
-include "tlist/defs.ma".
+include "LambdaDelta-1/tlist/defs.ma".
theorem iso_flats_lref_bind_false:
\forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "iso/fwd.ma".
+include "LambdaDelta-1/iso/fwd.ma".
theorem iso_refl:
\forall (t: T).(iso t t)
(* This file was automatically generated: do not edit *********************)
-include "leq/props.ma".
-
-include "aplus/props.ma".
+include "LambdaDelta-1/leq/props.ma".
theorem asucc_repl:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g
(* This file was automatically generated: do not edit *********************)
-include "aplus/defs.ma".
+include "LambdaDelta-1/aplus/defs.ma".
inductive leq (g: G): A \to (A \to Prop) \def
| leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall
(* This file was automatically generated: do not edit *********************)
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
theorem leq_gen_sort:
\forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq
(* This file was automatically generated: do not edit *********************)
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
-include "aplus/props.ma".
+include "LambdaDelta-1/aplus/props.ma".
theorem ahead_inj_snd:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/tlist/defs.ma".
-include "tlist/defs.ma".
-
-include "s/defs.ma".
+include "LambdaDelta-1/s/defs.ma".
definition lref_map:
((nat \to nat)) \to (nat \to (T \to T))
(* This file was automatically generated: do not edit *********************)
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
theorem lift_sort:
\forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort
(* This file was automatically generated: do not edit *********************)
-include "tlist/defs.ma".
+include "LambdaDelta-1/lift/fwd.ma".
-include "lift/fwd.ma".
-
-include "s/props.ma".
+include "LambdaDelta-1/s/props.ma".
theorem thead_x_lift_y_y:
\forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall
(* This file was automatically generated: do not edit *********************)
-include "lift/fwd.ma".
+include "LambdaDelta-1/lift/fwd.ma".
-include "tlt/props.ma".
+include "LambdaDelta-1/tlt/props.ma".
theorem lift_weight_map:
\forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
(* This file was automatically generated: do not edit *********************)
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
definition trans:
PList \to (nat \to nat)
(* This file was automatically generated: do not edit *********************)
-include "lift1/defs.ma".
+include "LambdaDelta-1/lift1/defs.ma".
-include "lift/fwd.ma".
+include "LambdaDelta-1/lift/fwd.ma".
theorem lift1_sort:
\forall (n: nat).(\forall (is: PList).(eq T (lift1 is (TSort n)) (TSort n)))
(* This file was automatically generated: do not edit *********************)
-include "lift1/defs.ma".
+include "LambdaDelta-1/lift/props.ma".
-include "lift/props.ma".
-
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
theorem lift1_lift1:
\forall (is1: PList).(\forall (is2: PList).(\forall (t: T).(eq T (lift1 is1
(* This file was automatically generated: do not edit *********************)
-include "A/defs.ma".
+include "LambdaDelta-1/A/defs.ma".
definition lweight:
A \to nat
(* This file was automatically generated: do not edit *********************)
-include "llt/defs.ma".
+include "LambdaDelta-1/llt/defs.ma".
-include "leq/defs.ma".
+include "LambdaDelta-1/leq/defs.ma".
theorem lweight_repl:
\forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat
(* This file was automatically generated: do not edit *********************)
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
definition next_plus:
G \to (nat \to (nat \to nat))
(* This file was automatically generated: do not edit *********************)
-include "next_plus/defs.ma".
+include "LambdaDelta-1/next_plus/defs.ma".
theorem next_plus_assoc:
\forall (g: G).(\forall (n: nat).(\forall (h1: nat).(\forall (h2: nat).(eq
(* This file was automatically generated: do not edit *********************)
-include "nf2/fwd.ma".
+include "LambdaDelta-1/nf2/fwd.ma".
-include "arity/subst0.ma".
+include "LambdaDelta-1/arity/subst0.ma".
theorem arity_nf2_inv_all:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
(* This file was automatically generated: do not edit *********************)
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
-include "pr2/clen.ma".
+include "LambdaDelta-1/pr2/clen.ma".
-include "pr2/fwd.ma".
+include "LambdaDelta-1/pr2/fwd.ma".
-include "pr0/dec.ma".
+include "LambdaDelta-1/pr0/dec.ma".
-include "C/props.ma".
+include "LambdaDelta-1/C/props.ma".
theorem nf2_dec:
\forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq
(* This file was automatically generated: do not edit *********************)
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
definition nf2:
C \to (T \to Prop)
(* This file was automatically generated: do not edit *********************)
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
-include "pr2/clen.ma".
+include "LambdaDelta-1/pr2/clen.ma".
-include "subst0/dec.ma".
+include "LambdaDelta-1/subst0/dec.ma".
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
theorem nf2_gen_lref:
\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
(* This file was automatically generated: do not edit *********************)
-include "nf2/pr3.ma".
+include "LambdaDelta-1/nf2/pr3.ma".
-include "pr3/fwd.ma".
+include "LambdaDelta-1/pr3/fwd.ma".
-include "iso/props.ma".
+include "LambdaDelta-1/iso/props.ma".
theorem nf2_iso_appls_lref:
\forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs:
(* This file was automatically generated: do not edit *********************)
-include "nf2/props.ma".
+include "LambdaDelta-1/nf2/props.ma".
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
theorem nf2_lift1:
\forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1
(* This file was automatically generated: do not edit *********************)
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
-include "pr3/pr3.ma".
+include "LambdaDelta-1/pr3/pr3.ma".
theorem nf2_pr3_unfold:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c
(* This file was automatically generated: do not edit *********************)
-include "nf2/defs.ma".
+include "LambdaDelta-1/nf2/defs.ma".
-include "pr2/fwd.ma".
+include "LambdaDelta-1/pr2/fwd.ma".
theorem nf2_sort:
\forall (c: C).(\forall (n: nat).(nf2 c (TSort n)))
(* This file was automatically generated: do not edit *********************)
-include "pr1/defs.ma".
+include "LambdaDelta-1/pr1/defs.ma".
definition pc1:
T \to (T \to Prop)
(* This file was automatically generated: do not edit *********************)
-include "pc1/defs.ma".
+include "LambdaDelta-1/pc1/defs.ma".
-include "pr1/pr1.ma".
+include "LambdaDelta-1/pr1/pr1.ma".
theorem pc1_pr0_r:
\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1)
(pr1_pr0 t2 t1 H)))).
+theorem pc1_refl:
+ \forall (t: T).(pc1 t t)
+\def
+ \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0:
+T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
+
theorem pc1_pr0_u:
\forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2
t3) \to (pc1 t1 t3)))))
(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2)
H3)))) H1)))))).
-theorem pc1_refl:
- \forall (t: T).(pc1 t t)
-\def
- \lambda (t: T).(ex_intro2 T (\lambda (t0: T).(pr1 t t0)) (\lambda (t0:
-T).(pr1 t t0)) t (pr1_refl t) (pr1_refl t)).
-
theorem pc1_s:
\forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
\def
(* This file was automatically generated: do not edit *********************)
-include "ty3/arity_props.ma".
+include "LambdaDelta-1/ty3/arity_props.ma".
-include "ty3/pr3.ma".
-
-include "nf2/fwd.ma".
+include "LambdaDelta-1/nf2/fwd.ma".
theorem pc3_dec:
\forall (g: G).(\forall (c: C).(\forall (u1: T).(\forall (t1: T).((ty3 g c
(* This file was automatically generated: do not edit *********************)
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
definition pc3:
C \to (T \to (T \to Prop))
(* This file was automatically generated: do not edit *********************)
-include "pc3/left.ma".
+include "LambdaDelta-1/pc3/left.ma".
-include "fsubst0/defs.ma".
+include "LambdaDelta-1/fsubst0/defs.ma".
-include "csubst0/getl.ma".
+include "LambdaDelta-1/csubst0/getl.ma".
theorem pc3_pr2_fsubst0:
\forall (c1: C).(\forall (t1: T).(\forall (t: T).((pr2 c1 t1 t) \to (\forall
(* This file was automatically generated: do not edit *********************)
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
-include "pr3/fwd.ma".
+include "LambdaDelta-1/pr3/fwd.ma".
theorem pc3_gen_sort:
\forall (c: C).(\forall (m: nat).(\forall (n: nat).((pc3 c (TSort m) (TSort
(* This file was automatically generated: do not edit *********************)
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
theorem pc3_ind_left__pc3_left_pr3:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to
(* This file was automatically generated: do not edit *********************)
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
-include "nf2/pr3.ma".
+include "LambdaDelta-1/nf2/pr3.ma".
theorem pc3_nf2:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to ((nf2 c
(* This file was automatically generated: do not edit *********************)
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
-include "pc1/defs.ma".
+include "LambdaDelta-1/pc1/defs.ma".
-include "pr3/pr1.ma".
+include "LambdaDelta-1/pr3/pr1.ma".
theorem pc3_pc1:
\forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (c: C).(pc3 c t1
(* This file was automatically generated: do not edit *********************)
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
-include "pr3/pr3.ma".
+include "LambdaDelta-1/pr3/pr3.ma".
theorem clear_pc3_trans:
\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pc3 c2 t1 t2) \to
t0)).(\lambda (t2: T).(\lambda (H0: (pr3 c t2 t0)).(ex_intro2 T (\lambda (t:
T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t2 t)) t0 H H0)))))).
-theorem pc3_pr2_u:
- \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall
-(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
-\def
- \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1
-t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
-(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
-t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
-x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t))
-x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
-
theorem pc3_refl:
\forall (c: C).(\forall (t: T).(pc3 c t t))
\def
t2) t)) (THead k u x) (pr3_head_12 c u u (pr3_refl c u) k t1 x H1)
(pr3_head_12 c u u (pr3_refl c u) k t2 x H2))))) H0))))))).
+theorem pc3_pr2_u:
+ \forall (c: C).(\forall (t2: T).(\forall (t1: T).((pr2 c t1 t2) \to (\forall
+(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
+\def
+ \lambda (c: C).(\lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr2 c t1
+t2)).(\lambda (t3: T).(\lambda (H0: (pc3 c t2 t3)).(let H1 \def H0 in
+(ex2_ind T (\lambda (t: T).(pr3 c t2 t)) (\lambda (t: T).(pr3 c t3 t)) (pc3 c
+t1 t3) (\lambda (x: T).(\lambda (H2: (pr3 c t2 x)).(\lambda (H3: (pr3 c t3
+x)).(ex_intro2 T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: T).(pr3 c t3 t))
+x (pr3_sing c t2 t1 H x H2) H3)))) H1))))))).
+
theorem pc3_t:
\forall (t2: T).(\forall (c: C).(\forall (t1: T).((pc3 c t1 t2) \to (\forall
(t3: T).((pc3 c t2 t3) \to (pc3 c t1 t3))))))
\def
\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c:
C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
-(CHead c k u2) t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0
-(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pc3 (CHead c k u1) t1
-t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0
-(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3
-t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2)
-\to ((pr0 t0 t3) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pc3
-(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda
-(t: T).((pr0 t1 t) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1
-t2)).(pc3_pr2_r (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3
-(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2)
-H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda
-(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq
-T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t
-t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i
-u t3 t) \to (pc3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2)
-(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pc3
-(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda
-(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
-((subst0 i u t3 t4) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i
-(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
-(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2)
-(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pc3 (CHead c k u1) t1
-t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr)
-u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k0: K).((clear
-(CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t1 t2)))
-(\lambda (b: B).(\lambda (H14: (clear (CHead c (Bind b) u2) (CHead d (Bind
-Abbr) u))).(let H15 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
-c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c
-(CHead d (Bind Abbr) u) u2 H14)) in ((let H16 \def (f_equal C B (\lambda (e:
-C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr |
-(CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with
-[(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind
-Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2
-H14)) in ((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow
-t4])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c
-(CHead d (Bind Abbr) u) u2 H14)) in (\lambda (H18: (eq B Abbr b)).(\lambda
-(_: (eq C d c)).(let H20 \def (eq_ind T u (\lambda (t4: T).(subst0 O t4 t3
-t2)) H13 u2 H17) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0)
-u1) t1 t2)) (ex2_ind T (\lambda (t4: T).(subst0 O u1 t3 t4)) (\lambda (t4:
-T).(pr0 t2 t4)) (pc3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda
-(H21: (subst0 O u1 t3 x)).(\lambda (H22: (pr0 t2 x)).(pc3_pr3_t (CHead c
-(Bind Abbr) u1) t1 x (pr3_pr2 (CHead c (Bind Abbr) u1) t1 x (pr2_delta (CHead
-c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H21)) t2 (pr3_pr2
-(CHead c (Bind Abbr) u1) t2 x (pr2_free (CHead c (Bind Abbr) u1) t2 x
-H22)))))) (pr0_subst0_fwd u2 t3 t2 O H20 u1 H)) b H18))))) H16)) H15))))
-(\lambda (f: F).(\lambda (H14: (clear (CHead c (Flat f) u2) (CHead d (Bind
-Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t1 t2 (pc3_pr2_r (CHead d
-(Bind Abbr) u) t1 t2 (pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr
-d u) t1 t3 H10 t2 H13)) (CHead c (Flat f) u1) (clear_flat c (CHead d (Bind
-Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H14) f u1)))) k
-(getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H12)))) (\lambda (i0:
-nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to
-((subst0 i0 u t3 t2) \to (pc3 (CHead c k u1) t1 t2))))).(\lambda (H12: (getl
-(S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 (S i0)
-u t3 t2)).(K_ind (\lambda (k0: K).((((getl i0 (CHead c k0 u2) (CHead d (Bind
-Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k0 u1) t1 t2)))) \to
-((getl (r k0 i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t1
-t2)))) (\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d
-(Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Bind b) u1) t1
-t2))))).(\lambda (H14: (getl (r (Bind b) i0) c (CHead d (Bind Abbr)
-u))).(pc3_pr2_r (CHead c (Bind b) u1) t1 t2 (pr2_delta (CHead c (Bind b) u1)
-d u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) H14 u1) t1 t3 H10
-t2 H13))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2)
-(CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Flat f)
-u1) t1 t2))))).(\lambda (H14: (getl (r (Flat f) i0) c (CHead d (Bind Abbr)
-u))).(pc3_pr2_r (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d
-u (r (Flat f) i0) H14 t1 t3 H10 t2 H13) f u1))))) k IHi (getl_gen_S k c
-(CHead d (Bind Abbr) u) u2 i0 H12)))))) i H9 H11)))) t (sym_eq T t t2 H8)))
-t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H1 H2
-H3))))]) in (H1 (refl_equal C (CHead c k u2)) (refl_equal T t1) (refl_equal T
-t2)))))))))).
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0
+t1 t2)) (\lambda (_: C).(pc3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda
+(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq C c0 (CHead c k u2)) \to (pc3 (CHead c k u1) t t0))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3:
+(eq C c0 (CHead c k u2))).(let H4 \def (f_equal C C (\lambda (e: C).e) c0
+(CHead c k u2) H3) in (pc3_pr2_r (CHead c k u1) t3 t4 (pr2_free (CHead c k
+u1) t3 t4 H2)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H4: (subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k
+u2))).(let H6 \def (f_equal C C (\lambda (e: C).e) c0 (CHead c k u2) H5) in
+(let H7 \def (eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr)
+u))) H2 (CHead c k u2) H6) in (nat_ind (\lambda (n: nat).((getl n (CHead c k
+u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pc3 (CHead c k u1)
+t3 t)))) (\lambda (H8: (getl O (CHead c k u2) (CHead d (Bind Abbr)
+u))).(\lambda (H9: (subst0 O u t4 t)).(K_ind (\lambda (k0: K).((clear (CHead
+c k0 u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t))) (\lambda
+(b: B).(\lambda (H10: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr)
+u))).(let H11 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1]))
+(CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d
+(Bind Abbr) u) u2 H10)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match
+e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _
+k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H10)) in
+((let H13 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
+C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) (CHead
+d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind
+Abbr) u) u2 H10)) in (\lambda (H14: (eq B Abbr b)).(\lambda (_: (eq C d
+c)).(let H16 \def (eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H9 u2 H13)
+in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t3 t))
+(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t t0))
+(pc3 (CHead c (Bind Abbr) u1) t3 t) (\lambda (x: T).(\lambda (H17: (subst0 O
+u1 t4 x)).(\lambda (H18: (pr0 t x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t3 x
+(pr3_pr2 (CHead c (Bind Abbr) u1) t3 x (pr2_delta (CHead c (Bind Abbr) u1) c
+u1 O (getl_refl Abbr c u1) t3 t4 H3 x H17)) t (pr3_pr2 (CHead c (Bind Abbr)
+u1) t x (pr2_free (CHead c (Bind Abbr) u1) t x H18)))))) (pr0_subst0_fwd u2
+t4 t O H16 u1 H)) b H14))))) H12)) H11)))) (\lambda (f: F).(\lambda (H10:
+(clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans
+(CHead d (Bind Abbr) u) t3 t (pc3_pr2_r (CHead d (Bind Abbr) u) t3 t
+(pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t3 t4 H3 t H9))
+(CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f
+c (CHead d (Bind Abbr) u) u2 H10) f u1)))) k (getl_gen_O (CHead c k u2)
+(CHead d (Bind Abbr) u) H8)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0
+(CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t) \to (pc3
+(CHead c k u1) t3 t))))).(\lambda (H8: (getl (S i0) (CHead c k u2) (CHead d
+(Bind Abbr) u))).(\lambda (H9: (subst0 (S i0) u t4 t)).(K_ind (\lambda (k0:
+K).((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4
+t) \to (pc3 (CHead c k0 u1) t3 t)))) \to ((getl (r k0 i0) c (CHead d (Bind
+Abbr) u)) \to (pc3 (CHead c k0 u1) t3 t)))) (\lambda (b: B).(\lambda (_:
+(((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u
+t4 t) \to (pc3 (CHead c (Bind b) u1) t3 t))))).(\lambda (H10: (getl (r (Bind
+b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t3 t
+(pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d
+(Bind Abbr) u) H10 u1) t3 t4 H3 t H9))))) (\lambda (f: F).(\lambda (_:
+(((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u
+t4 t) \to (pc3 (CHead c (Flat f) u1) t3 t))))).(\lambda (H10: (getl (r (Flat
+f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t3 t
+(pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0) H10 t3 t4 H3 t H9) f
+u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H8)))))) i H7
+H4)))))))))))))) y t1 t2 H1))) H0)))))))).
theorem pc3_pr2_pr2_t:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall
(CHead c k u1) t1 t2))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2
-u1)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u2)
-\to ((eq T t0 u1) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2
-(CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2)))))))))))) with
-[(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2:
-(eq T t1 u2)).(\lambda (H3: (eq T t2 u1)).(eq_ind C c (\lambda (_: C).((eq T
-t1 u2) \to ((eq T t2 u1) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4:
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3
-t4))))))))) (\lambda (H4: (eq T t1 u2)).(eq_ind T u2 (\lambda (t: T).((eq T
-t2 u1) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
-K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))) (\lambda
-(H5: (eq T t2 u1)).(eq_ind T u1 (\lambda (t: T).((pr0 u2 t) \to (\forall (t3:
-T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3
-(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u2 u1)).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H7: (pr2 (CHead c k u2) t0
-t3)).(pc3_pr0_pr2_t u1 u2 H6 c t0 t3 k H7)))))) t2 (sym_eq T t2 u1 H5))) t1
-(sym_eq T t1 u2 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u
-i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq
-T t1 u2)).(\lambda (H5: (eq T t u1)).(eq_ind C c (\lambda (c1: C).((eq T t1
-u2) \to ((eq T t u1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1
-t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
-K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))))))
-(\lambda (H6: (eq T t1 u2)).(eq_ind T u2 (\lambda (t0: T).((eq T t u1) \to
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t)
-\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3
-t4) \to (pc3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t
-u1)).(eq_ind T u1 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to
-((pr0 u2 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4:
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3
-t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
-(pr0 u2 t2)).(\lambda (H10: (subst0 i u t2 u1)).(\lambda (t0: T).(\lambda
-(t3: T).(\lambda (k: K).(\lambda (H11: (pr2 (CHead c k u2) t0 t3)).(let H12
-\def (match H11 in pr2 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5:
-T).(\lambda (_: (pr2 c1 t4 t5)).((eq C c1 (CHead c k u2)) \to ((eq T t4 t0)
-\to ((eq T t5 t3) \to (pc3 (CHead c k u1) t0 t3)))))))) with [(pr2_free c1 t4
-t5 H12) \Rightarrow (\lambda (H13: (eq C c1 (CHead c k u2))).(\lambda (H14:
-(eq T t4 t0)).(\lambda (H15: (eq T t5 t3)).(eq_ind C (CHead c k u2) (\lambda
-(_: C).((eq T t4 t0) \to ((eq T t5 t3) \to ((pr0 t4 t5) \to (pc3 (CHead c k
-u1) t0 t3))))) (\lambda (H16: (eq T t4 t0)).(eq_ind T t0 (\lambda (t6:
-T).((eq T t5 t3) \to ((pr0 t6 t5) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda
-(H17: (eq T t5 t3)).(eq_ind T t3 (\lambda (t6: T).((pr0 t0 t6) \to (pc3
-(CHead c k u1) t0 t3))) (\lambda (H18: (pr0 t0 t3)).(pc3_pr2_r (CHead c k u1)
-t0 t3 (pr2_free (CHead c k u1) t0 t3 H18))) t5 (sym_eq T t5 t3 H17))) t4
-(sym_eq T t4 t0 H16))) c1 (sym_eq C c1 (CHead c k u2) H13) H14 H15 H12)))) |
-(pr2_delta c1 d0 u0 i0 H12 t4 t5 H13 t6 H14) \Rightarrow (\lambda (H15: (eq C
-c1 (CHead c k u2))).(\lambda (H16: (eq T t4 t0)).(\lambda (H17: (eq T t6
-t3)).(eq_ind C (CHead c k u2) (\lambda (c2: C).((eq T t4 t0) \to ((eq T t6
-t3) \to ((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t4 t5) \to ((subst0
-i0 u0 t5 t6) \to (pc3 (CHead c k u1) t0 t3))))))) (\lambda (H18: (eq T t4
-t0)).(eq_ind T t0 (\lambda (t7: T).((eq T t6 t3) \to ((getl i0 (CHead c k u2)
-(CHead d0 (Bind Abbr) u0)) \to ((pr0 t7 t5) \to ((subst0 i0 u0 t5 t6) \to
-(pc3 (CHead c k u1) t0 t3)))))) (\lambda (H19: (eq T t6 t3)).(eq_ind T t3
-(\lambda (t7: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to
-((pr0 t0 t5) \to ((subst0 i0 u0 t5 t7) \to (pc3 (CHead c k u1) t0 t3)))))
-(\lambda (H20: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda
-(H21: (pr0 t0 t5)).(\lambda (H22: (subst0 i0 u0 t5 t3)).(nat_ind (\lambda (n:
-nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
-t3) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H23: (getl O (CHead c k u2)
-(CHead d0 (Bind Abbr) u0))).(\lambda (H24: (subst0 O u0 t5 t3)).(K_ind
-(\lambda (k0: K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pc3
-(CHead c k0 u1) t0 t3))) (\lambda (b: B).(\lambda (H25: (clear (CHead c (Bind
-b) u2) (CHead d0 (Bind Abbr) u0))).(let H26 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 |
-(CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b)
-u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H27 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+u1)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t) t1 t2) \to (pc3
+(CHead c0 k t0) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k:
+K).(\lambda (H1: (pr2 (CHead c0 k t1) t0 t3)).(pc3_pr0_pr2_t t2 t1 H0 c0 t0
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda
+(H3: (pr2 (CHead c0 k t1) t0 t3)).(insert_eq C (CHead c0 k t1) (\lambda (c1:
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pc3 (CHead c0 k t) t0 t3)) (\lambda (y:
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4:
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t1)) \to (pc3 (CHead c0 k t) t4
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t1))).(pc3_pr2_r (CHead c0 k t) t4
+t5 (pr2_free (CHead c0 k t) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C
+c1 (CHead c0 k t1))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t1) H8) in (nat_ind (\lambda (n:
+nat).((getl n (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
+t6) \to (pc3 (CHead c0 k t) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t1)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
+(\lambda (k0: K).((clear (CHead c0 k0 t1) (CHead d0 (Bind Abbr) u0)) \to (pc3
+(CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
+(Bind b) t1) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0
+| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind
+b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H14
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2)
-(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H28 \def
+\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t1)
+(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t1 H12)) in ((let H15 \def
(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
-Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0)
-u2 H25)) in (\lambda (H29: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H31
-\def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t3)) H24 u2 H28) in
-(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t0 t3)) (ex2_ind
-T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t3 t7)) (pc3
-(CHead c (Bind Abbr) u1) t0 t3) (\lambda (x: T).(\lambda (H32: (subst0 O t2
-t5 x)).(\lambda (H33: (pr0 t3 x)).(ex2_ind T (\lambda (t7: T).(subst0 O u1 t5
-t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c (Bind
-Abbr) u1) t0 t3) (\lambda (x0: T).(\lambda (H34: (subst0 O u1 t5
-x0)).(\lambda (H35: (subst0 (S (plus i O)) u x x0)).(let H36 \def (f_equal
-nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H37
-\def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H35 (S
-i) H36) in (pc3_pr2_u (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c
-(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t0 t5 H21 x0 H34) t3 (pc3_pr2_x
-(CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i)
-(getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) t3 x H33 x0
-H37)))))))) (subst0_subst0_back t5 x t2 O H32 u1 u i H10))))) (pr0_subst0_fwd
-u2 t5 t3 O H31 t2 H9)) b H29))))) H27)) H26)))) (\lambda (f: F).(\lambda
-(H25: (clear (CHead c (Flat f) u2) (CHead d0 (Bind Abbr)
-u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t0 t3 (pc3_pr2_r (CHead d0
-(Bind Abbr) u0) t0 t3 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl
-Abbr d0 u0) t0 t5 H21 t3 H24)) (CHead c (Flat f) u1) (clear_flat c (CHead d0
-(Bind Abbr) u0) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H25) f
-u1)))) k (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H23))))
-(\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0 (Bind
-Abbr) u0)) \to ((subst0 i1 u0 t5 t3) \to (pc3 (CHead c k u1) t0
-t3))))).(\lambda (H23: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr)
-u0))).(\lambda (H24: (subst0 (S i1) u0 t5 t3)).(K_ind (\lambda (k0: K).((getl
-(r k0 i1) c (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k0 u1) t0 t3)))
-(\lambda (b: B).(\lambda (H25: (getl (r (Bind b) i1) c (CHead d0 (Bind Abbr)
-u0))).(pc3_pr2_r (CHead c (Bind b) u1) t0 t3 (pr2_delta (CHead c (Bind b) u1)
-d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind Abbr) u0) H25 u1) t0 t5
-H21 t3 H24)))) (\lambda (f: F).(\lambda (H25: (getl (r (Flat f) i1) c (CHead
-d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c (Flat f) u1) t0 t3 (pr2_cflat c t0
-t3 (pr2_delta c d0 u0 (r (Flat f) i1) H25 t0 t5 H21 t3 H24) f u1)))) k
-(getl_gen_S k c (CHead d0 (Bind Abbr) u0) u2 i1 H23)))))) i0 H20 H22)))) t6
-(sym_eq T t6 t3 H19))) t4 (sym_eq T t4 t0 H18))) c1 (sym_eq C c1 (CHead c k
-u2) H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C (CHead c k u2))
-(refl_equal T t0) (refl_equal T t3)))))))))) t (sym_eq T t u1 H7))) t1
-(sym_eq T t1 u2 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
-(refl_equal C c) (refl_equal T u2) (refl_equal T u1)))))).
+Abbr) u0) (CHead c0 (Bind b) t1) (clear_gen_bind b c0 (CHead d0 (Bind Abbr)
+u0) t1 H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let
+H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t1 H15) in
+(eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c0 (Bind b0) t) t4 t6)) (ex2_ind
+T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(pr0 t6 t7)) (pc3
+(CHead c0 (Bind Abbr) t) t4 t6) (\lambda (x: T).(\lambda (H19: (subst0 O t2
+t5 x)).(\lambda (H20: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(subst0 O t t5
+t7)) (\lambda (t7: T).(subst0 (S (plus i O)) u x t7)) (pc3 (CHead c0 (Bind
+Abbr) t) t4 t6) (\lambda (x0: T).(\lambda (H21: (subst0 O t t5 x0)).(\lambda
+(H22: (subst0 (S (plus i O)) u x x0)).(let H23 \def (f_equal nat nat S (plus
+i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat
+(S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H22 (S i) H23) in
+(pc3_pr2_u (CHead c0 (Bind Abbr) t) x0 t4 (pr2_delta (CHead c0 (Bind Abbr) t)
+c0 t O (getl_refl Abbr c0 t) t4 t5 H6 x0 H21) t6 (pc3_pr2_x (CHead c0 (Bind
+Abbr) t) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t) d u (S i) (getl_head (Bind
+Abbr) i c0 (CHead d (Bind Abbr) u) H0 t) t6 x H20 x0 H24))))))))
+(subst0_subst0_back t5 x t2 O H19 t u i H2))))) (pr0_subst0_fwd t1 t5 t6 O
+H18 t2 H1)) b H16))))) H14)) H13)))) (\lambda (f: F).(\lambda (H12: (clear
+(CHead c0 (Flat f) t1) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0
+(Bind Abbr) u0) t4 t6 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t4 t6 (pr2_delta
+(CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t4 t5 H6 t6 H11))
+(CHead c0 (Flat f) t) (clear_flat c0 (CHead d0 (Bind Abbr) u0)
+(clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t1 H12) f t)))) k (getl_gen_O
+(CHead c0 k t1) (CHead d0 (Bind Abbr) u0) H10)))) (\lambda (i1: nat).(\lambda
+(_: (((getl i1 (CHead c0 k t1) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0
+t5 t6) \to (pc3 (CHead c0 k t) t4 t6))))).(\lambda (H10: (getl (S i1) (CHead
+c0 k t1) (CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 (S i1) u0 t5
+t6)).(K_ind (\lambda (k0: K).((getl (r k0 i1) c0 (CHead d0 (Bind Abbr) u0))
+\to (pc3 (CHead c0 k0 t) t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (r
+(Bind b) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c0 (Bind b) t)
+t4 t6 (pr2_delta (CHead c0 (Bind b) t) d0 u0 (S i1) (getl_head (Bind b) i1 c0
+(CHead d0 (Bind Abbr) u0) H12 t) t4 t5 H6 t6 H11)))) (\lambda (f: F).(\lambda
+(H12: (getl (r (Flat f) i1) c0 (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead
+c0 (Flat f) t) t4 t6 (pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1)
+H12 t4 t5 H6 t6 H11) f t)))) k (getl_gen_S k c0 (CHead d0 (Bind Abbr) u0) t1
+i1 H10)))))) i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u2 u1
+H)))).
theorem pc3_pr2_pr3_t:
\forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
-include "pr3/subst1.ma".
+include "LambdaDelta-1/pr3/subst1.ma".
theorem pc3_gen_cabbr:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to (\forall
(* This file was automatically generated: do not edit *********************)
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
-include "wcpr0/getl.ma".
+include "LambdaDelta-1/wcpr0/getl.ma".
theorem pc3_wcpr0__pc3_wcpr0_t_aux:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall
K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3
(CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0:
T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t))
-(\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr2 (CHead c1 k u) t3
-t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c1 k u) t0 t4)).(\lambda (H3:
-(pc3 (CHead c2 k u) t0 t4)).(pc3_t t0 (CHead c2 k u) t3 (let H4 \def (match
-H1 in pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).(\lambda
-(_: (pr2 c t t5)).((eq C c (CHead c1 k u)) \to ((eq T t t3) \to ((eq T t5 t0)
-\to (pc3 (CHead c2 k u) t3 t0)))))))) with [(pr2_free c t5 t6 H4) \Rightarrow
-(\lambda (H5: (eq C c (CHead c1 k u))).(\lambda (H6: (eq T t5 t3)).(\lambda
-(H7: (eq T t6 t0)).(eq_ind C (CHead c1 k u) (\lambda (_: C).((eq T t5 t3) \to
-((eq T t6 t0) \to ((pr0 t5 t6) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda
-(H8: (eq T t5 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t6 t0) \to ((pr0 t t6)
-\to (pc3 (CHead c2 k u) t3 t0)))) (\lambda (H9: (eq T t6 t0)).(eq_ind T t0
-(\lambda (t: T).((pr0 t3 t) \to (pc3 (CHead c2 k u) t3 t0))) (\lambda (H10:
-(pr0 t3 t0)).(pc3_pr2_r (CHead c2 k u) t3 t0 (pr2_free (CHead c2 k u) t3 t0
-H10))) t6 (sym_eq T t6 t0 H9))) t5 (sym_eq T t5 t3 H8))) c (sym_eq C c (CHead
-c1 k u) H5) H6 H7 H4)))) | (pr2_delta c d u0 i H4 t5 t6 H5 t H6) \Rightarrow
-(\lambda (H7: (eq C c (CHead c1 k u))).(\lambda (H8: (eq T t5 t3)).(\lambda
-(H9: (eq T t t0)).(eq_ind C (CHead c1 k u) (\lambda (c0: C).((eq T t5 t3) \to
-((eq T t t0) \to ((getl i c0 (CHead d (Bind Abbr) u0)) \to ((pr0 t5 t6) \to
-((subst0 i u0 t6 t) \to (pc3 (CHead c2 k u) t3 t0))))))) (\lambda (H10: (eq T
-t5 t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t t0) \to ((getl i (CHead c1 k
-u) (CHead d (Bind Abbr) u0)) \to ((pr0 t7 t6) \to ((subst0 i u0 t6 t) \to
-(pc3 (CHead c2 k u) t3 t0)))))) (\lambda (H11: (eq T t t0)).(eq_ind T t0
-(\lambda (t7: T).((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0
-t3 t6) \to ((subst0 i u0 t6 t7) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda
-(H12: (getl i (CHead c1 k u) (CHead d (Bind Abbr) u0))).(\lambda (H13: (pr0
-t3 t6)).(\lambda (H14: (subst0 i u0 t6 t0)).(ex3_2_ind C T (\lambda (e2:
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr2 (CHead c1 k u) t4
+t3)).(\lambda (t5: T).(\lambda (_: (pr3 (CHead c1 k u) t3 t5)).(\lambda (H3:
+(pc3 (CHead c2 k u) t3 t5)).(pc3_t t3 (CHead c2 k u) t4 (insert_eq C (CHead
+c1 k u) (\lambda (c: C).(pr2 c t4 t3)) (\lambda (_: C).(pc3 (CHead c2 k u) t4
+t3)) (\lambda (y: C).(\lambda (H4: (pr2 y t4 t3)).(pr2_ind (\lambda (c:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CHead c1 k u)) \to (pc3 (CHead
+c2 k u) t t0))))) (\lambda (c: C).(\lambda (t6: T).(\lambda (t0: T).(\lambda
+(H5: (pr0 t6 t0)).(\lambda (_: (eq C c (CHead c1 k u))).(pc3_pr2_r (CHead c2
+k u) t6 t0 (pr2_free (CHead c2 k u) t6 t0 H5))))))) (\lambda (c: C).(\lambda
+(d: C).(\lambda (u0: T).(\lambda (i: nat).(\lambda (H5: (getl i c (CHead d
+(Bind Abbr) u0))).(\lambda (t6: T).(\lambda (t0: T).(\lambda (H6: (pr0 t6
+t0)).(\lambda (t: T).(\lambda (H7: (subst0 i u0 t0 t)).(\lambda (H8: (eq C c
+(CHead c1 k u))).(let H9 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead
+d (Bind Abbr) u0))) H5 (CHead c1 k u) H8) in (ex3_2_ind C T (\lambda (e2:
C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2))))
(\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2:
-T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t3 t0) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H15: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda
-(_: (wcpr0 d x0)).(\lambda (H17: (pr0 u0 x1)).(ex2_ind T (\lambda (t7:
-T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t0 t7)) (pc3 (CHead c2 k u) t3
-t0) (\lambda (x: T).(\lambda (H18: (subst0 i x1 t6 x)).(\lambda (H19: (pr0 t0
-x)).(pc3_pr2_u (CHead c2 k u) x t3 (pr2_delta (CHead c2 k u) x0 x1 i H15 t3
-t6 H13 x H18) t0 (pc3_pr2_x (CHead c2 k u) x t0 (pr2_free (CHead c2 k u) t0 x
-H19)))))) (pr0_subst0_fwd u0 t6 t0 i H14 x1 H17))))))) (wcpr0_getl (CHead c1
-k u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind
-Abbr) H12))))) t (sym_eq T t t0 H11))) t5 (sym_eq T t5 t3 H10))) c (sym_eq C
-c (CHead c1 k u) H7) H8 H9 H4 H5 H6))))]) in (H4 (refl_equal C (CHead c1 k
-u)) (refl_equal T t3) (refl_equal T t0))) t4 H3))))))) t1 t2 H0)))))))).
+T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t6 t) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H10: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda
+(_: (wcpr0 d x0)).(\lambda (H12: (pr0 u0 x1)).(ex2_ind T (\lambda (t7:
+T).(subst0 i x1 t0 t7)) (\lambda (t7: T).(pr0 t t7)) (pc3 (CHead c2 k u) t6
+t) (\lambda (x: T).(\lambda (H13: (subst0 i x1 t0 x)).(\lambda (H14: (pr0 t
+x)).(pc3_pr2_u (CHead c2 k u) x t6 (pr2_delta (CHead c2 k u) x0 x1 i H10 t6
+t0 H6 x H13) t (pc3_pr2_x (CHead c2 k u) x t (pr2_free (CHead c2 k u) t x
+H14)))))) (pr0_subst0_fwd u0 t0 t i H7 x1 H12))))))) (wcpr0_getl (CHead c1 k
+u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr)
+H9)))))))))))))) y t4 t3 H4))) H1) t5 H3))))))) t1 t2 H0)))))))).
theorem pc3_wcpr0_t:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1:
(* This file was automatically generated: do not edit *********************)
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
-include "subst0/dec.ma".
+include "LambdaDelta-1/subst0/dec.ma".
-include "T/dec.ma".
+include "LambdaDelta-1/T/dec.ma".
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
theorem nf0_dec:
\forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T
(* This file was automatically generated: do not edit *********************)
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
inductive pr0: T \to (T \to Prop) \def
| pr0_refl: \forall (t: T).(pr0 t t)
(* This file was automatically generated: do not edit *********************)
-include "pr0/props.ma".
-
-theorem pr0_inv_coq:
- \forall (t1: T).(\forall (t2: T).(\forall (P: ((T \to (T \to
-Prop)))).((((pr0 t1 t2) \to (\forall (t: T).((eq T t t1) \to ((eq T t t2) \to
-(P t1 t2)))))) \to ((((pr0 t1 t2) \to (\forall (u1: T).(\forall (u2:
-T).(\forall (t0: T).(\forall (t3: T).(\forall (k: K).((eq T (THead k u1 t0)
-t1) \to ((eq T (THead k u2 t3) t2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (P
-t1 t2)))))))))))) \to ((((pr0 t1 t2) \to (\forall (u: T).(\forall (v1:
-T).(\forall (v2: T).(\forall (t0: T).(\forall (t3: T).((eq T (THead (Flat
-Appl) v1 (THead (Bind Abst) u t0)) t1) \to ((eq T (THead (Bind Abbr) v2 t3)
-t2) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (P t1 t2)))))))))))) \to ((((pr0 t1
-t2) \to (\forall (b: B).(\forall (v1: T).(\forall (v2: T).(\forall (u1:
-T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).((eq T (THead (Flat
-Appl) v1 (THead (Bind b) u1 t0)) t1) \to ((eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t3)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
-v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (P t1 t2)))))))))))))))) \to
-((((pr0 t1 t2) \to (\forall (u1: T).(\forall (u2: T).(\forall (t0:
-T).(\forall (t3: T).(\forall (w: T).((eq T (THead (Bind Abbr) u1 t0) t1) \to
-((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to
-((subst0 O u2 t3 w) \to (P t1 t2))))))))))))) \to ((((pr0 t1 t2) \to (\forall
-(b: B).(\forall (t0: T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Bind
-b) u (lift (S O) O t0)) t1) \to ((eq T t3 t2) \to ((not (eq B b Abst)) \to
-((pr0 t0 t3) \to (P t1 t2))))))))))) \to ((((pr0 t1 t2) \to (\forall (t0:
-T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Flat Cast) u t0) t1) \to
-((eq T t3 t2) \to ((pr0 t0 t3) \to (P t1 t2))))))))) \to ((pr0 t1 t2) \to (P
-t1 t2)))))))))))
-\def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (P: ((T \to (T \to
-Prop)))).(\lambda (H: (((pr0 t1 t2) \to (\forall (t: T).((eq T t t1) \to ((eq
-T t t2) \to (P t1 t2))))))).(\lambda (H0: (((pr0 t1 t2) \to (\forall (u1:
-T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).(\forall (k: K).((eq T
-(THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) \to ((pr0 u1 u2) \to ((pr0
-t0 t3) \to (P t1 t2))))))))))))).(\lambda (H1: (((pr0 t1 t2) \to (\forall (u:
-T).(\forall (v1: T).(\forall (v2: T).(\forall (t0: T).(\forall (t3: T).((eq T
-(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t1) \to ((eq T (THead (Bind
-Abbr) v2 t3) t2) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (P t1
-t2))))))))))))).(\lambda (H2: (((pr0 t1 t2) \to (\forall (b: B).(\forall (v1:
-T).(\forall (v2: T).(\forall (u1: T).(\forall (u2: T).(\forall (t0:
-T).(\forall (t3: T).((eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t1)
-\to ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t2)
-\to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3)
-\to (P t1 t2))))))))))))))))).(\lambda (H3: (((pr0 t1 t2) \to (\forall (u1:
-T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).(\forall (w: T).((eq T
-(THead (Bind Abbr) u1 t0) t1) \to ((eq T (THead (Bind Abbr) u2 w) t2) \to
-((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 w) \to (P t1
-t2)))))))))))))).(\lambda (H4: (((pr0 t1 t2) \to (\forall (b: B).(\forall
-(t0: T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Bind b) u (lift (S O)
-O t0)) t1) \to ((eq T t3 t2) \to ((not (eq B b Abst)) \to ((pr0 t0 t3) \to (P
-t1 t2)))))))))))).(\lambda (H5: (((pr0 t1 t2) \to (\forall (t0: T).(\forall
-(t3: T).(\forall (u: T).((eq T (THead (Flat Cast) u t0) t1) \to ((eq T t3 t2)
-\to ((pr0 t0 t3) \to (P t1 t2)))))))))).(\lambda (H6: (pr0 t1 t2)).(let H7
-\def (match H6 in pr0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_:
-(pr0 t t0)).((eq T t t1) \to ((eq T t0 t2) \to (P t1 t2)))))) with [(pr0_refl
-t) \Rightarrow (\lambda (H7: (eq T t t1)).(\lambda (H8: (eq T t t2)).(H H6 t
-H7 H8))) | (pr0_comp u1 u2 H7 t0 t3 H8 k) \Rightarrow (\lambda (H9: (eq T
-(THead k u1 t0) t1)).(\lambda (H10: (eq T (THead k u2 t3) t2)).(H0 H6 u1 u2
-t0 t3 k H9 H10 H7 H8))) | (pr0_beta u v1 v2 H7 t0 t3 H8) \Rightarrow (\lambda
-(H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t1)).(\lambda
-(H10: (eq T (THead (Bind Abbr) v2 t3) t2)).(H1 H6 u v1 v2 t0 t3 H9 H10 H7
-H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t0 t3 H10) \Rightarrow (\lambda
-(H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t1)).(\lambda (H12:
-(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t2)).(H2
-H6 b v1 v2 u1 u2 t0 t3 H11 H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t0 t3 H8
-w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t0)
-t1)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(H3 H6 u1 u2 t0 t3 w
-H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t0 t3 H8 u) \Rightarrow (\lambda (H9:
-(eq T (THead (Bind b) u (lift (S O) O t0)) t1)).(\lambda (H10: (eq T t3
-t2)).(H4 H6 b t0 t3 u H9 H10 H7 H8))) | (pr0_epsilon t0 t3 H7 u) \Rightarrow
-(\lambda (H8: (eq T (THead (Flat Cast) u t0) t1)).(\lambda (H9: (eq T t3
-t2)).(H5 H6 t0 t3 u H8 H9 H7)))]) in (H7 (refl_equal T t1) (refl_equal T
-t2))))))))))))).
+include "LambdaDelta-1/pr0/props.ma".
theorem pr0_gen_sort:
\forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
\def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n)
-x)).(pr0_inv_coq (TSort n) x (\lambda (t: T).(\lambda (t0: T).(eq T t0 t)))
-(\lambda (H0: (pr0 (TSort n) x)).(\lambda (t: T).(\lambda (H1: (eq T t (TSort
-n))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda (t0: T).(eq
-T t0 (TSort n))) H1 x H2) in (let H4 \def (eq_ind T x (\lambda (t0: T).(pr0
-(TSort n) t0)) H0 (TSort n) H3) in (let H5 \def (eq_ind T x (\lambda (t0:
-T).(pr0 (TSort n) t0)) H (TSort n) H3) in (eq_ind_r T (TSort n) (\lambda (t0:
-T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H3)))))))) (\lambda (H0:
-(pr0 (TSort n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u1 t0)
-(TSort n))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda (_: (pr0 u1
-u2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t:
-T).(pr0 (TSort n) t)) H0 (THead k u2 t3) H3) in (let H6 \def (eq_ind_r T x
-(\lambda (t: T).(pr0 (TSort n) t)) H (THead k u2 t3) H3) in (eq_ind T (THead
-k u2 t3) (\lambda (t: T).(eq T t (TSort n))) (let H7 \def (eq_ind T (THead k
-u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(insert_eq
+T (TSort n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0:
+T).((eq T t (TSort n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T
+t (TSort n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TSort n) H1) in
+(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TSort n))
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TSort n))).(let
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H5) in
+(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u:
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_:
+(((eq T v1 (TSort n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
+t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1))
+(TSort n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort n) H2) in (False_ind (eq T (THead k u2 t3)
-(TSort n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TSort n) x)).(\lambda
-(u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (TSort
-n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3) x)).(\lambda (_: (pr0 v1
-v2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t:
-T).(pr0 (TSort n) t)) H0 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def
-(eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H (THead (Bind Abbr) v2 t3)
-H3) in (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(eq T t (TSort
-n))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TSort n) H2) in (False_ind (eq T (THead (Bind Abbr) v2 t3) (TSort
-n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TSort n) x)).(\lambda (b:
-B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl)
-v1 (THead (Bind b) u1 t0)) (TSort n))).(\lambda (H5: (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
-t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H0
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8
-\def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(eq T t (TSort n)))
-(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TSort n) H4) in (False_ind (eq T (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t3)) (TSort n)) H9)) x H5))))))))))))))))) (\lambda
-(H0: (pr0 (TSort n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u1
-t0) (TSort n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).(\lambda (_:
-(pr0 u1 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 O u2 t3 w)).(let
-H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H0 (THead (Bind
-Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n)
-t)) H (THead (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w)
-(\lambda (t: T).(eq T t (TSort n))) (let H8 \def (eq_ind T (THead (Bind Abbr)
-u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+\Rightarrow True])) I (TSort n) H5) in (False_ind (eq T (THead (Bind Abbr) v2
+t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TSort n)) \to (eq T v2
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (TSort n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq T t2
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1))
+(TSort n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort n) H3) in (False_ind (eq T (THead (Bind Abbr) u2
-w) (TSort n)) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (TSort n)
-x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u:
-T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (TSort
-n))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq B b Abst))).(\lambda
-(H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x
-H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TSort n) H2) in (False_ind (eq T x (TSort n)) H6))))))))))))
-(\lambda (_: (pr0 (TSort n) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda
-(u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (TSort n))).(\lambda (H2:
-(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda
-(t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T (THead (Flat Cast) u
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort n) H1) in (False_ind (eq T x (TSort n))
-H5)))))))))) H))).
+\Rightarrow True])) I (TSort n) H8) in (False_ind (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind
+b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to (eq T u2
+u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda
+(_: (((eq T t1 (TSort n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_:
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TSort
+n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
+H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1))
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1
+(TSort n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead
+(Bind b) u (lift (S O) O t1)) (TSort n))).(let H5 \def (eq_ind T (THead (Bind
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind (eq T t2
+(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TSort n)) \to (eq
+T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1)
+(TSort n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x
+H0))) H))).
theorem pr0_gen_lref:
\forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
\def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n)
-x)).(pr0_inv_coq (TLRef n) x (\lambda (t: T).(\lambda (t0: T).(eq T t0 t)))
-(\lambda (H0: (pr0 (TLRef n) x)).(\lambda (t: T).(\lambda (H1: (eq T t (TLRef
-n))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda (t0: T).(eq
-T t0 (TLRef n))) H1 x H2) in (let H4 \def (eq_ind T x (\lambda (t0: T).(pr0
-(TLRef n) t0)) H0 (TLRef n) H3) in (let H5 \def (eq_ind T x (\lambda (t0:
-T).(pr0 (TLRef n) t0)) H (TLRef n) H3) in (eq_ind_r T (TLRef n) (\lambda (t0:
-T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H3)))))))) (\lambda (H0:
-(pr0 (TLRef n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u1 t0)
-(TLRef n))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda (_: (pr0 u1
-u2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t:
-T).(pr0 (TLRef n) t)) H0 (THead k u2 t3) H3) in (let H6 \def (eq_ind_r T x
-(\lambda (t: T).(pr0 (TLRef n) t)) H (THead k u2 t3) H3) in (eq_ind T (THead
-k u2 t3) (\lambda (t: T).(eq T t (TLRef n))) (let H7 \def (eq_ind T (THead k
-u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(insert_eq
+T (TLRef n) (\lambda (t: T).(pr0 t x)) (\lambda (t: T).(eq T x t)) (\lambda
+(y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0:
+T).((eq T t (TLRef n)) \to (eq T t0 t)))) (\lambda (t: T).(\lambda (H1: (eq T
+t (TLRef n))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (TLRef n) H1) in
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 t0)) (refl_equal T (TLRef n))
+t H2)))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
+t1)))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u1 t1) (TLRef n))).(let
+H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H5) in
+(False_ind (eq T (THead k u2 t2) (THead k u1 t1)) H6)))))))))))) (\lambda (u:
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_:
+(((eq T v1 (TLRef n)) \to (eq T v2 v1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
+t1)))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1))
+(TLRef n))).(let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H2) in (False_ind (eq T (THead k u2 t3)
-(TLRef n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TLRef n) x)).(\lambda
-(u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (TLRef
-n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3) x)).(\lambda (_: (pr0 v1
-v2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t:
-T).(pr0 (TLRef n) t)) H0 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def
-(eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H (THead (Bind Abbr) v2 t3)
-H3) in (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(eq T t (TLRef
-n))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0))
-(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TLRef n) H2) in (False_ind (eq T (THead (Bind Abbr) v2 t3) (TLRef
-n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TLRef n) x)).(\lambda (b:
-B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl)
-v1 (THead (Bind b) u1 t0)) (TLRef n))).(\lambda (H5: (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
-t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H0
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8
-\def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(eq T t (TLRef n)))
-(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TLRef n) H4) in (False_ind (eq T (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t3)) (TLRef n)) H9)) x H5))))))))))))))))) (\lambda
-(H0: (pr0 (TLRef n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u1
-t0) (TLRef n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).(\lambda (_:
-(pr0 u1 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 O u2 t3 w)).(let
-H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H0 (THead (Bind
-Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n)
-t)) H (THead (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w)
-(\lambda (t: T).(eq T t (TLRef n))) (let H8 \def (eq_ind T (THead (Bind Abbr)
-u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+\Rightarrow True])) I (TLRef n) H5) in (False_ind (eq T (THead (Bind Abbr) v2
+t2) (THead (Flat Appl) v1 (THead (Bind Abst) u t1))) H6)))))))))))) (\lambda
+(b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (TLRef n)) \to (eq T v2
+v1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (((eq T u1 (TLRef n)) \to (eq T u2 u1)))).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq T t2
+t1)))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1))
+(TLRef n))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1
+t1)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H3) in (False_ind (eq T (THead (Bind Abbr) u2
-w) (TLRef n)) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (TLRef n)
-x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u:
-T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (TLRef
-n))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq B b Abst))).(\lambda
-(H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x
-H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
-True])) I (TLRef n) H2) in (False_ind (eq T x (TLRef n)) H6))))))))))))
-(\lambda (_: (pr0 (TLRef n) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda
-(u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (TLRef n))).(\lambda (H2:
-(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda
-(t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T (THead (Flat Cast) u
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H1) in (False_ind (eq T x (TLRef n))
-H5)))))))))) H))).
+\Rightarrow True])) I (TLRef n) H8) in (False_ind (eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) v1 (THead (Bind
+b) u1 t1))) H9))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda
+(_: (pr0 u1 u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (eq T u2
+u1)))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda
+(_: (((eq T t1 (TLRef n)) \to (eq T t2 t1)))).(\lambda (w: T).(\lambda (_:
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u1 t1) (TLRef
+n))).(let H7 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (ee: T).(match
+ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
+(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n)
+H6) in (False_ind (eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u1 t1))
+H7))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda
+(t1: T).(\lambda (t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1
+(TLRef n)) \to (eq T t2 t1)))).(\lambda (u: T).(\lambda (H4: (eq T (THead
+(Bind b) u (lift (S O) O t1)) (TLRef n))).(let H5 \def (eq_ind T (THead (Bind
+b) u (lift (S O) O t1)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind (eq T t2
+(THead (Bind b) u (lift (S O) O t1))) H5)))))))))) (\lambda (t1: T).(\lambda
+(t2: T).(\lambda (_: (pr0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (eq
+T t2 t1)))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t1)
+(TLRef n))).(let H4 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TLRef n) H3) in (False_ind (eq T t2 (THead (Flat Cast) u t1)) H4)))))))) y x
+H0))) H))).
theorem pr0_gen_abst:
\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1
T).(\lambda (t2: T).(pr0 t1 t2)))))))
\def
\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Bind Abst) u1 t1) x)).(pr0_inv_coq (THead (Bind Abst) u1 t1) x (\lambda (_:
-T).(\lambda (t0: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
-(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))))) (\lambda (H0: (pr0 (THead
-(Bind Abst) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t (THead (Bind
-Abst) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda
-(t0: T).(eq T t0 (THead (Bind Abst) u1 t1))) H1 x H2) in (let H4 \def (eq_ind
-T x (\lambda (t0: T).(pr0 (THead (Bind Abst) u1 t1) t0)) H0 (THead (Bind
-Abst) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0: T).(pr0 (THead
-(Bind Abst) u1 t1) t0)) H (THead (Bind Abst) u1 t1) H3) in (eq_ind_r T (THead
-(Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+(Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (\lambda (y:
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T
+t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abst)
-u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T
-(THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) x H3)))))))) (\lambda
-(H0: (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (u0: T).(\lambda (u2:
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T
-(THead k u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2
-t3) x)).(\lambda (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 (THead k
-u2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind
-Abst) u1 t1) t)) H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda
-(t: T).(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind
-Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2))))) (let H7 \def (f_equal T K (\lambda (e:
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k |
-(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0)
-(THead (Bind Abst) u1 t1) H2) in ((let H8 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 |
-(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0)
-(THead (Bind Abst) u1 t1) H2) in ((let H9 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
-(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0)
-(THead (Bind Abst) u1 t1) H2) in (\lambda (H10: (eq T u0 u1)).(\lambda (H11:
-(eq K k (Bind Abst))).(let H12 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead
-(Bind Abst) u1 t1) (THead k0 u2 t3))) H6 (Bind Abst) H11) in (let H13 \def
-(eq_ind K k (\lambda (k0: K).(pr0 (THead (Bind Abst) u1 t1) (THead k0 u2
-t3))) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2
-T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 t3) (THead (Bind
-Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2))))) (let H14 \def (eq_ind T t0 (\lambda (t:
-T).(pr0 t t3)) H4 t1 H9) in (let H15 \def (eq_ind T u0 (\lambda (t: T).(pr0 t
-u2)) H1 u1 H10) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: T).(eq T
-(THead (Bind Abst) u2 t3) (THead (Bind Abst) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2))) u2 t3 (refl_equal T (THead (Bind Abst) u2 t3)) H15 H14))) k H11))))))
-H8)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Bind Abst) u1 t1)
-x)).(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead (Bind
-Abbr) v2 t3) x)).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t0 t3)).(let H5
-\def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0
-(THead (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t:
-T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in
-(eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))) (\lambda
+(t: T).(\lambda (H1: (eq T t (THead (Bind Abst) u1 t1))).(let H2 \def
+(f_equal T T (\lambda (e: T).e) t (THead (Bind Abst) u1 t1) H1) in (eq_ind_r
+T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2:
T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2))))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u
-t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1
-t1) H2) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
-(THead (Bind Abbr) v2 t3) (THead (Bind Abst) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2)))) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Bind Abst) u1 t1)
-x)).(\lambda (b: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u0:
-T).(\lambda (u2: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T
-(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abst) u1
-t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t3)) x)).(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1
-v2)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(let H7 \def
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8 \def
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(ex3_2
-T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u3 t2))))
+t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind
+Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1
+(refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) t
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda
+(H2: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0)
+(THead (Bind Abst) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind
+Abst) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind
+Abst) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind
+Abst) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind
+Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead
+(Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H4 t1 H8) in (let H12 \def
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T
+u0 (\lambda (t: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
+T).(pr0 t u2)) H1 u1 H9) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H14 H12))))) k H10))))
+H7)) H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2:
+T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind
+Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1
+t1)) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3))))))).(\lambda (H5: (eq T (THead (Flat Appl)
+v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(let H6 \def (eq_ind
+T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u1 t1) H5) in (False_ind (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) H6)))))))))))) (\lambda (b: B).(\lambda
+(_: (not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0
+v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abst) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2))))
(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2))))) (let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
-b) u0 t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1
-t1) H4) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind
-Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) H9)) x H5))))))))))))))))) (\lambda (H0:
-(pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda
-(t0: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind
-Abbr) u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H4: (eq T (THead (Bind
-Abbr) u2 w) x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda
-(_: (subst0 O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0
-(THead (Bind Abst) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead
-(Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t:
-T).(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u3
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2))))) (let H8 \def (eq_ind T (THead (Bind Abbr)
-u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (THead (Bind Abst) u1 t1) H3) in (False_ind (ex3_2 T
-T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead
-(Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2)))) H8)) x H4)))))))))))))) (\lambda (H0:
-(pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (b: B).(\lambda (t0: T).(\lambda
-(t3: T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O
-t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t3 x)).(\lambda (H1:
-(not (eq B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3
-(\lambda (t: T).(pr0 t0 t)) H4 x H3) in (let H6 \def (f_equal T B (\lambda
-(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b
-| (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind
+b) u0 t0)) (THead (Bind Abst) u1 t1))).(let H9 \def (eq_ind T (THead (Flat
+Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Abst) u1 t1) H8) in (False_ind (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t2)) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Abst) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2
+w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1
+t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind
+Abst) u1 t1) H6) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abst) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b
+Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
+(H3: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O
+t0)) (THead (Bind Abst) u1 t1))).(let H5 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
-b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in
-((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H4) in
+((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
-t1) H2) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return
+t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
-\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t2) \Rightarrow
-(THead k (lref_map f d u0) (lref_map f (s k d) t2))]) in lref_map) (\lambda
+\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow
+(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda
(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map
(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
-[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t2)
-\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t2))]) in
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in
lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1
-t1) H2) in (\lambda (_: (eq T u u1)).(\lambda (H10: (eq B b Abst)).(let H11
-\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H10) in (let
-H12 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t) x)) H0
-(lift (S O) O t0) H8) in (let H13 \def (eq_ind_r T t1 (\lambda (t: T).(pr0
-(THead (Bind Abst) u1 t) x)) H (lift (S O) O t0) H8) in (eq_ind T (lift (S O)
-O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x
-(THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr0 t t2))))) (let H14 \def (match (H11
-(refl_equal B Abst)) in False return (\lambda (_: False).(ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 (lift
-(S O) O t0) t2))))) with []) in H14) t1 H8))))))) H7)) H6))))))))))))
-(\lambda (_: (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (t0: T).(\lambda
-(t3: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead
-(Bind Abst) u1 t1))).(\lambda (H2: (eq T t3 x)).(\lambda (H3: (pr0 t0
-t3)).(let H4 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H3 x H2) in (let
-H5 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abst) u1 t1) H1) in (False_ind (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2)))) H5)))))))))) H)))).
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abst)).(let H10
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H9) in (let
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abst) u1 t))
+\to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))))) H3 (lift (S O) O t0) H7) in (eq_ind T
+(lift (S O) O t0) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3))))) (let H12
+\def (match (H10 (refl_equal B Abst)) in False return (\lambda (_:
+False).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))) with []) in H12) t1
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_:
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abst) u1 t1)) \to (ex3_2 T
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u
+t0) (THead (Bind Abst) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1
+t1) H3) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) H4)))))))) y x H0))) H)))).
theorem pr0_gen_appl:
\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1
(t2: T).(pr0 z1 t2))))))))))))
\def
\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Flat Appl) u1 t1) x)).(pr0_inv_coq (THead (Flat Appl) u1 t1) x (\lambda (_:
-T).(\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
-t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
-u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+(Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x
+(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (\lambda (y:
+T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda (t0: T).((eq T
+t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))))) (\lambda (H0: (pr0
-(THead (Flat Appl) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t (THead
-(Flat Appl) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t
-(\lambda (t0: T).(eq T t0 (THead (Flat Appl) u1 t1))) H1 x H2) in (let H4
-\def (eq_ind T x (\lambda (t0: T).(pr0 (THead (Flat Appl) u1 t1) t0)) H0
-(THead (Flat Appl) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0:
-T).(pr0 (THead (Flat Appl) u1 t1) t0)) H (THead (Flat Appl) u1 t1) H3) in
-(eq_ind_r T (THead (Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))) (\lambda (t:
+T).(\lambda (H1: (eq T t (THead (Flat Appl) u1 t1))).(let H2 \def (f_equal T
+T (\lambda (e: T).e) t (THead (Flat Appl) u1 t1) H1) in (eq_ind_r T (THead
+(Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T
+t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T
+T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
+T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl
+t1))) t H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0
+u2)).(\lambda (H2: (((eq T u0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind
+Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda
-(u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0
-t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl)
-u1 t1) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Flat
-Appl) u1 t1) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2)
-t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
-y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Appl)
-u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H3)))))))) (\lambda (H0: (pr0 (THead
-(Flat Appl) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u0 t0)
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda
-(H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x
-(\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H0 (THead k u2 t3) H3) in
-(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t))
-H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda (t: T).(or3
-(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u3
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1 t1)) \to
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl)
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(eq T t (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t (THead (Bind b)
-v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0
-u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (k: K).(\lambda (H5: (eq
+T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(let H6 \def (f_equal T K
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
+(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H7 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in ((let H8 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u0 t0) (THead (Flat Appl) u1 t1) H5) in (\lambda (H9: (eq T u0
+u1)).(\lambda (H10: (eq K k (Flat Appl))).(eq_ind_r K (Flat Appl) (\lambda
+(k0: K).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2
+t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(v2: T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2)))))))))) (let H7 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat
-Appl) u1 t1) H2) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat
-Appl) u1 t1) H2) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat
-Appl) u1 t1) H2) in (\lambda (H10: (eq T u0 u1)).(\lambda (H11: (eq K k (Flat
-Appl))).(let H12 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead (Flat Appl) u1
-t1) (THead k0 u2 t3))) H6 (Flat Appl) H11) in (let H13 \def (eq_ind K k
-(\lambda (k0: K).(pr0 (THead (Flat Appl) u1 t1) (THead k0 u2 t3))) H5 (Flat
-Appl) H11) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(or3 (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 t3) (THead (Flat Appl)
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(eq T (THead k0 u2 t3) (THead (Bind Abbr) u3 t2)))))) (\lambda (_:
+(t3: T).(pr0 z1 t3)))))))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq
+T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T
-(THead k0 u2 t3) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3)
-t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
-y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (let H14 \def
-(eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H4 t1 H9) in (let H15 \def (eq_ind T
-u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H10) in (or3_intro0 (ex3_2 T T (\lambda
-(u3: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u2 t3) (THead (Flat Appl)
-u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
+t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
+T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
+v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H4 t1 H8) in (let
+H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def
+(eq_ind T u0 (\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(eq T (THead (Flat Appl) u2 t3) (THead (Bind Abbr) u3 t2)))))) (\lambda
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
+T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T u2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H2 u1 H9) in (let H14 \def (eq_ind
+T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in (or3_intro0 (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl)
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
+T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda
(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
-t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda
-(t2: T).(eq T (THead (Flat Appl) u2 t3) (THead (Bind b) v2 (THead (Flat Appl)
-(lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+(t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl)
+(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
-t2)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead
-(Flat Appl) u2 t3) (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u2 t3
-(refl_equal T (THead (Flat Appl) u2 t3)) H15 H14)))) k H11)))))) H8)) H7)) x
-H3))))))))))))) (\lambda (H0: (pr0 (THead (Flat Appl) u1 t1) x)).(\lambda (u:
-T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead
-(Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3)
-x)).(\lambda (H1: (pr0 v1 v2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H0 (THead
-(Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0
-(THead (Flat Appl) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in (eq_ind T
-(THead (Bind Abbr) v2 t3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t (THead (Bind
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead
+(Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2
+(refl_equal T (THead (Flat Appl) u2 t2)) H14 H12)))))) k H10)))) H7))
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda
+(H1: (pr0 v1 v2)).(\lambda (H2: (((eq T v1 (THead (Flat Appl) u1 t1)) \to
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl)
+u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
+T).(eq T v2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H5: (eq T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(let H6 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat
+Appl) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u
+t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat
+Appl) u1 t1) H5) in (\lambda (H8: (eq T v1 u1)).(let H9 \def (eq_ind T v1
+(\lambda (t: T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T t (THead (Bind b) v3 (THead
-(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b) v3 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2)))))))))) (let H7 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
-\Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead
-(Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in ((let H8 \def (f_equal T
-T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow (THead (Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind
-Abst) u t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead
-(Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in (\lambda (H9: (eq T v1
-u1)).(let H10 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H1 u1 H9) in (let
-H11 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) (THead
-(Bind Abbr) v2 t3))) H6 (THead (Bind Abst) u t0) H8) in (let H12 \def
-(eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) (THead (Bind
-Abbr) v2 t3))) H5 (THead (Bind Abst) u t0) H8) in (eq_ind T (THead (Bind
-Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (THead (Bind Abbr) v2 t3) (THead (Flat Appl) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t
-t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr)
-v2 t3) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+(t3: T).(pr0 z1 t3))))))))))) H2 u1 H8) in (let H10 \def (eq_ind T v1
+(\lambda (t: T).(pr0 t v2)) H1 u1 H8) in (let H11 \def (eq_ind_r T t1
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v3 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))) H4 (THead (Bind Abst) u t0) H7) in (let H12
+\def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1 t)) \to
+(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T v2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind
-Abbr) v2 t3) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2)
-t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
-(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0
-y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro1 (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead
-(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t2: T).(pr0 (THead (Bind Abst) u t0) t2)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr)
-v2 t3) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda
-(t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead (Bind b) v3 (THead (Flat Appl)
-(lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
-t2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
-T).(eq T (THead (Bind Abbr) v2 t3) (THead (Bind Abbr) u2 t2)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
-t2))))) u t0 v2 t3 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T
-(THead (Bind Abbr) v2 t3)) H10 H4)) t1 H8)))))) H7)) x H3)))))))))))))
-(\lambda (H0: (pr0 (THead (Flat Appl) u1 t1) x)).(\lambda (b: B).(\lambda
-(v1: T).(\lambda (v2: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b)
-u0 t0)) (THead (Flat Appl) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (H1: (not (eq B b
-Abst))).(\lambda (H2: (pr0 v1 v2)).(\lambda (H3: (pr0 u0 u2)).(\lambda (H6:
-(pr0 t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
-Appl) u1 t1) t)) H0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-t3)) H5) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
-Appl) u1 t1) t)) H (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-t3)) H5) in (eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-t3)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
-t (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
+b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H9 (THead (Bind Abst) u t0) H7) in
+(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda
+(t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl)
+(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead
+(Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
+b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind
+Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2)
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0
+y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
+v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2
+(refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2
+t2)) H10 H3)) t1 H7))))))) H6)))))))))))) (\lambda (b: B).(\lambda (H1: (not
+(eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (H2: (pr0 v1
+v2)).(\lambda (H3: (((eq T v1 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Flat Appl) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T v2 (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t2: T).(pr0 z1 t2)))))))))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda
+(H4: (pr0 u0 u2)).(\lambda (H5: (((eq T u0 (THead (Flat Appl) u1 t1)) \to
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Appl)
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T u2 (THead (Bind
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H6: (pr0 t0 t2)).(\lambda (H7: (((eq T t0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (H8: (eq T (THead (Flat
+Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(let H9 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _)
+\Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat
+Appl) u1 t1) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0)
+| (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow
+t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1)
+H8) in (\lambda (H11: (eq T v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t:
+T).((eq T t (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T v2 (THead (Flat Appl) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))) H3 u1 H11) in (let H13 \def (eq_ind T v1
+(\lambda (t: T).(pr0 t v2)) H2 u1 H11) in (let H14 \def (eq_ind_r T t1
+(\lambda (t: T).((eq T t0 (THead (Flat Appl) u1 t)) \to (or3 (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v3 (THead
+(Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))))) H7 (THead (Bind b) u0 t0) H10) in (let H15 \def
+(eq_ind_r T t1 (\lambda (t: T).((eq T u0 (THead (Flat Appl) u1 t)) \to (or3
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead (Flat Appl) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3:
+T).(eq T u2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead (Bind
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H5 (THead (Bind b) u0 t0) H10) in
+(let H16 \def (eq_ind_r T t1 (\lambda (t: T).((eq T u1 (THead (Flat Appl) u1
+t)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T v2 (THead
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(t2: T).(eq T t (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
+(t3: T).(eq T v2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T t (THead (Bind
-b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T v2 (THead (Bind
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3)))))))
(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (let H9 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1
-| (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat
-Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in ((let H10
-\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) | (TLRef _) \Rightarrow
-(THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1
-(THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in (\lambda (H11: (eq T
-v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H2 u1 H11) in
-(let H13 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t)
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)))) H8 (THead
-(Bind b) u0 t0) H10) in (let H14 \def (eq_ind_r T t1 (\lambda (t: T).(pr0
-(THead (Flat Appl) u1 t) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t3)))) H7 (THead (Bind b) u0 t0) H10) in (eq_ind T (THead (Bind b) u0 t0)
-(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat
-Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2:
-T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead
-(Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
-b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat Appl)
-(lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
+T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H12 (THead (Bind b) u0 t0) H10) in
+(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda
+(u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S
+O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T
+T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat
+Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
-t2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat
-Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 (THead (Bind b) u0 t0) t2)))) (ex4_4 T T T T
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat
+Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind Abbr) u3 t2))))))
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3))))))
(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2:
-T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
+u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead
(Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat
-Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
+T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat
+Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
-t2)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0
Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind
b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat Appl)
-(lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+(u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl)
+(lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1
-t2))))))) b u0 t0 v2 u2 t3 H1 (refl_equal T (THead (Bind b) u0 t0))
-(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)))
-H12 H3 H6)) t1 H10)))))) H9)) x H5))))))))))))))))) (\lambda (H0: (pr0 (THead
-(Flat Appl) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u0
-t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w)
-x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0
-O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
-Appl) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T
-x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H (THead (Bind Abbr) u2
-w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).(or3 (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind
-Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2)))))))))) (let H8 \def (eq_ind T (THead (Bind Abbr) u0 t0)
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))) b u0 t0 v2 u2 t2 H1 (refl_equal T (THead (Bind b) u0 t0))
+(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)))
+H13 H4 H6)) t1 H10)))))))) H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2:
+T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T u2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t2: T).(pr0 z1 t2)))))))))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (w: T).(\lambda (_:
+(subst0 O u2 t2 w)).(\lambda (H6: (eq T (THead (Bind Abbr) u0 t0) (THead
+(Flat Appl) u1 t1))).(let H7 \def (eq_ind T (THead (Bind Abbr) u0 t0)
(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
-(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind
-(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2
-w) (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H6) in (False_ind
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2
+w) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda
(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3:
-T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3
-t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_:
T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
+(t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda
-(v2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2
-(THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\lambda (_: B).(\lambda (_:
+(v2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2
+(THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2))))))))) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (THead
-(Flat Appl) u1 t1) x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0))
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq
-B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t:
-T).(pr0 t0 t)) H4 x H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S
-O) O t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1
-t1) H2) in (False_ind (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T
-x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1:
+(t3: T).(pr0 z1 t3))))))))) H7))))))))))))) (\lambda (b: B).(\lambda (_: (not
+(eq B b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1 t1)) \to (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H4: (eq T (THead
+(Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(let H5 \def
+(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) u1 t1) H4) in (False_ind (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b0) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
+u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(t3: T).(pr0 z1 t3))))))))) H5)))))))))) (\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Appl) u1
+t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T
-(\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1:
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind
-b0) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_:
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2))))))))) H6)))))))))))) (\lambda (_: (pr0
-(THead (Flat Appl) u1 t1) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u:
-T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1
-t1))).(\lambda (H2: (eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def
-(eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T
+T).(\lambda (t3: T).(pr0 z1 t3)))))))))))).(\lambda (u: T).(\lambda (H3: (eq
+T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(let H4 \def (eq_ind T
(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return
(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow
-True])])])) I (THead (Flat Appl) u1 t1) H1) in (False_ind (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
+True])])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda
(_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
-Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b:
+T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_:
+(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
+(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t2: T).(pr0 z1 t2))))))))) H5)))))))))) H)))).
+(t3: T).(pr0 z1 t3))))))))) H4)))))))) y x H0))) H)))).
theorem pr0_gen_cast:
\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1
(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
\def
\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Flat Cast) u1 t1) x)).(pr0_inv_coq (THead (Flat Cast) u1 t1) x (\lambda (_:
-T).(\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0
-(THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0)))) (\lambda (H0:
-(pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t
-(THead (Flat Cast) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T
-t (\lambda (t0: T).(eq T t0 (THead (Flat Cast) u1 t1))) H1 x H2) in (let H4
-\def (eq_ind T x (\lambda (t0: T).(pr0 (THead (Flat Cast) u1 t1) t0)) H0
-(THead (Flat Cast) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0:
-T).(pr0 (THead (Flat Cast) u1 t1) t0)) H (THead (Flat Cast) u1 t1) H3) in
-(eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0: T).(or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead (Flat Cast) u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1))
+(Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
+u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))
+(\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t: T).(\lambda
+(t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
+t2)))) (pr0 t1 t0))))) (\lambda (t: T).(\lambda (H1: (eq T t (THead (Flat
+Cast) u1 t1))).(let H2 \def (f_equal T T (\lambda (e: T).e) t (THead (Flat
+Cast) u1 t1) H1) in (eq_ind_r T (THead (Flat Cast) u1 t1) (\lambda (t0:
+T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat
+Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1))
(ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast)
u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T
-(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H3))))))))
-(\lambda (H0: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (u0: T).(\lambda
-(u2: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T
-(THead k u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead k u2
-t3) x)).(\lambda (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def
-(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H0 (THead k
-u2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
-Cast) u1 t1) t)) H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda
-(t: T).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat
-Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t))) (let H7 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
-(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H8 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
-(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H9 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
-(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in (\lambda (H10: (eq T u0
-u1)).(\lambda (H11: (eq K k (Flat Cast))).(let H12 \def (eq_ind K k (\lambda
-(k0: K).(pr0 (THead (Flat Cast) u1 t1) (THead k0 u2 t3))) H6 (Flat Cast) H11)
-in (let H13 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead (Flat Cast) u1 t1)
-(THead k0 u2 t3))) H5 (Flat Cast) H11) in (eq_ind_r K (Flat Cast) (\lambda
-(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2
-t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead k0 u2
-t3)))) (let H14 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H4 t1 H9) in
-(let H15 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H10) in
-(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Flat
-Cast) u2 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_:
+(THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda
+(u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T u2 (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_:
T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
-(THead (Flat Cast) u2 t3)) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T (THead (Flat Cast) u2 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2))) u2 t3 (refl_equal T (THead (Flat Cast) u2 t3)) H15 H14)))) k H11))))))
-H8)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Flat Cast) u1 t1)
-x)).(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead (Bind
-Abbr) v2 t3) x)).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t0 t3)).(let H5
-\def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H0
-(THead (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t:
-T).(pr0 (THead (Flat Cast) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in
-(eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T t (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0 t2)).(\lambda
+(H4: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 t2))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0)
+(THead (Flat Cast) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat
+Cast) u1 t1) H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat
+Cast) u1 t1) H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat
+Cast) u1 t1) H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Flat
+Cast))).(eq_ind_r K (Flat Cast) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Flat Cast) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (THead k0 u2 t2)))) (let H11 \def (eq_ind T t0
+(\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 t2)))) H4 t1 H8) in (let H12 \def (eq_ind T t0
+(\lambda (t: T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda
+(t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T u2 (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 u2)))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
+T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0
+t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2
+t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
+v2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 t2))))).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind
+Abst) u t0)) (THead (Flat Cast) u1 t1))).(let H6 \def (eq_ind T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee: T).(match ee in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
+in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
+\Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H5) in (False_ind (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind Abbr) v2
+t2))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b
+Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda
+(_: (((eq T v1 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T v2 (THead (Flat Cast) u2 t2)))) (\lambda (u2:
T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2)))) (pr0 t1 t))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
-F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
-(Flat Cast) u1 t1) H2) in (False_ind (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead (Flat Cast) u2 t2)))) (\lambda
-(u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0
-t1 t2)))) (pr0 t1 (THead (Bind Abbr) v2 t3))) H7)) x H3))))))))))))) (\lambda
-(H0: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (b: B).(\lambda (v1:
-T).(\lambda (v2: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b)
-u0 t0)) (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0
-t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1
-t1) t)) H0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5)
-in (let H8 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1)
-t)) H (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in
-(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))
-(\lambda (t: T).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t
-(THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
-(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t))) (let H9 \def
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee:
+t2)))) (pr0 t1 v2))))).(\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
+u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Flat Cast) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (H8: (eq T (THead
+(Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(let H9
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee:
T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
-True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H4) in
-(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat Cast) u3
-t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t3)))) H9)) x H5))))))))))))))))) (\lambda (H0: (pr0
-(THead (Flat Cast) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u0
-t0) (THead (Flat Cast) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w)
-x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0
-O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat
-Cast) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T
-x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H (THead (Bind Abbr) u2
-w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).(or (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Cast) u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 t))) (let H8 \def (eq_ind T (THead (Bind Abbr) u0
-t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1
-t1) H3) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T
-(THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1
-t2)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H8)) x H4)))))))))))))) (\lambda
-(_: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (b: B).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u
-(lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t3
-x)).(\lambda (_: (not (eq B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def
-(eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x H3) in (let H6 \def (eq_ind T
-(THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
-False])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 x)) H6)))))))))))) (\lambda (_: (pr0 (THead (Flat
-Cast) u1 t1) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: T).(\lambda
-(H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(\lambda (H2:
-(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda
-(t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
-(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u
-t0) (THead (Flat Cast) u1 t1) H1) in ((let H6 \def (f_equal T T (\lambda (e:
+True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H8) in
+(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Cast) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t2)))) H9))))))))))))))))) (\lambda (u0: T).(\lambda
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Flat Cast)
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 u2))))).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead
+(Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq
+T t2 (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+t2))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T
+(THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(let H7 \def (eq_ind T
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) u1 t1) H6) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H7))))))))))))) (\lambda (b:
+B).(\lambda (_: (not (eq B b Abst))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Flat Cast) u1
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2))))).(\lambda (u:
+T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat
+Cast) u1 t1))).(let H5 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
+(Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)) H5)))))))))) (\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (((eq T t0
+(THead (Flat Cast) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+t2))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
+(Flat Cast) u1 t1))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
+\Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0)
+(THead (Flat Cast) u1 t1) H3) in ((let H5 \def (f_equal T T (\lambda (e:
T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 |
(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast)
-u t0) (THead (Flat Cast) u1 t1) H1) in (\lambda (_: (eq T u u1)).(let H8 \def
-(eq_ind T t0 (\lambda (t: T).(pr0 t x)) H4 t1 H6) in (or_intror (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 x) H8)))) H5)))))))))) H)))).
+u t0) (THead (Flat Cast) u1 t1) H3) in (\lambda (_: (eq T u u1)).(let H7 \def
+(eq_ind T t0 (\lambda (t: T).((eq T t (THead (Flat Cast) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2)))) H2 t1 H5) in (let H8 \def
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 t1 H5) in (or_intror (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 t2) H8))))) H4)))))))) y x H0))) H)))).
theorem pr0_gen_abbr:
\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1
(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
\def
\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Abbr) u1
-t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+(Bind Abbr) u1 t1) x)).(insert_eq T (THead (Bind Abbr) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda
(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S
-O) O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead
-(Bind Abbr) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abbr)
-u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0
-t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
-t2))))))) (pr0 t1 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Abbr)
-u1 t1) x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
+O) O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2))))
(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2:
-T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
-O u2 y t2))))))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr)
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O (THead (Bind
-Abbr) u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T
-(THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t:
+T).(\lambda (H1: (eq T t (THead (Bind Abbr) u1 t1))).(let H2 \def (f_equal T
+T (\lambda (e: T).e) t (THead (Bind Abbr) u1 t1) H1) in (eq_ind_r T (THead
+(Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0
+t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2:
T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0
-t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
-t2)))))) u1 t1 (refl_equal T (THead (Bind Abbr) u1 t1)) (pr0_refl u1)
-(or_introl (pr0 t1 t1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y:
-T).(subst0 O u1 y t1))) (pr0_refl t1)))) x H2)) t (sym_eq T t (THead (Bind
-Abbr) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda
-(H2: (eq T (THead k u0 t0) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T
-(THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _)
-\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind
-Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind
-Abbr) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e in T
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
-\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind
-Abbr) u1 t1) H2) in (eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u1) \to
-((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0
-t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind
-Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))))
-(\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to
-((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
+t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead
+(Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t2)))))) u1 t1 (refl_equal T (THead (Bind
+Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u1 y0 t1))) (pr0_refl t1)))) t
+H2)))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda
+(H2: (((eq T u0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t2: T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0
+t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0
+t2))))))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H3: (pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (k: K).(\lambda (H5: (eq T (THead k u0 t0) (THead (Bind
+Abbr) u1 t1))).(let H6 \def (f_equal T K (\lambda (e: T).(match e in T return
+(\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k |
+(THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
+H5) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0
+| (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
+H5) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0
+| (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1)
+H5) in (\lambda (H9: (eq T u0 u1)).(\lambda (H10: (eq K k (Bind
+Abbr))).(eq_ind_r K (Bind Abbr) (\lambda (k0: K).(or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k0 u2 t2) (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead k0 u2 t2))))) (let
+H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t1)) \to
+(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr)
+u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2))))) H4
+t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H3 t1 H8) in
+(let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead (Bind Abbr) u1
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O u2))))) H2
+u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H9) in
+(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind
+Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
+t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3
t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))
-(\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind
-Abbr) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda
-(u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
-t3))))))) (pr0 t1 (lift (S O) O x))))))) (\lambda (H9: (eq T (THead (Bind
-Abbr) u2 t2) x)).(eq_ind T (THead (Bind Abbr) u2 t2) (\lambda (t: T).((pr0 u1
-u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq
-T t (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
-u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y:
-T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O
-t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2)
-(THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
-(\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
-t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead
-(Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T
-(THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
-t3)))))) u2 t2 (refl_equal T (THead (Bind Abbr) u2 t2)) H10 (or_introl (pr0
-t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
-t2))) H11))))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k
-(sym_eq K k (Bind Abbr) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0
-t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead (Bind
-Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t0)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
-t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2)
-\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 t2 (refl_equal T (THead (Bind
+Abbr) u2 t2)) H14 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t2))) H12))))))) k H10)))) H7))
+H6)))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_:
+(pr0 v1 v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O
+v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
+t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead (Flat
+Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(let H6 \def
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H5) in (False_ind (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
-t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))))
-H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow
-(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead
-(Bind Abbr) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl)
-v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-Abbr) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0
-u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0
-u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda
-(y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S
-O) O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2)
-\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr)
-u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def
+(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S
+O) O (THead (Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_:
+(not (eq B b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Abbr) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2:
+T).(or (pr0 t1 t2) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u2 y0 t2))))))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0:
+T).(\lambda (u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead
+(Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq
+T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0 t1 (lift (S
+O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq
+T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1
+t1))).(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H8) in (False_ind
+(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3:
+T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0:
+T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t2))))) H9))))))))))))))))) (\lambda (u0:
+T).(\lambda (u2: T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0
+(THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
+T).(eq T u2 (THead (Bind Abbr) u3 t2)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t2))))))) (pr0
+t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3:
+(pr0 t0 t2)).(\lambda (H4: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t2 w)).(\lambda (H6: (eq
+T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(let H7 \def (f_equal
+T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
+(THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in ((let H8 \def
(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in
+\Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H6) in
+(\lambda (H9: (eq T u0 u1)).(let H10 \def (eq_ind T t0 (\lambda (t: T).((eq T
+t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
+t1 (lift (S O) O t2))))) H4 t1 H8) in (let H11 \def (eq_ind T t0 (\lambda (t:
+T).(pr0 t t2)) H3 t1 H8) in (let H12 \def (eq_ind T u0 (\lambda (t: T).((eq T
+t (THead (Bind Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T u2 (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
+(\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u3 y0 t3))))))) (pr0
+t1 (lift (S O) O u2))))) H2 u1 H9) in (let H13 \def (eq_ind T u0 (\lambda (t:
+T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or
+(pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O
+u3 y0 t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro
+T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead
+(Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u3 y0 t3)))))) u2 w (refl_equal T (THead (Bind
+Abbr) u2 w)) H13 (or_intror (pr0 t1 w) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 w))) (ex_intro2 T (\lambda (y0: T).(pr0 t1
+y0)) (\lambda (y0: T).(subst0 O u2 y0 w)) t2 H11 H5)))))))))) H7)))))))))))))
+(\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda (t0: T).(\lambda
+(t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (H3: (((eq T t0 (THead (Bind
+Abbr) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
+(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0:
+T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S
+O) O t2)))))).(\lambda (u: T).(\lambda (H4: (eq T (THead (Bind b) u (lift (S
+O) O t0)) (THead (Bind Abbr) u1 t1))).(let H5 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H4) in
((let H6 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t
-_) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3)
-in (eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr)
-u2 w) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or
-(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3
-t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))))
-(\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind
-Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to ((subst0 O u2 t2 w) \to
-(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr)
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))
-(\lambda (H8: (eq T (THead (Bind Abbr) u2 w) x)).(eq_ind T (THead (Bind Abbr)
-u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w)
-\to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind
-Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t)))))))
-(\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t1 t2)).(\lambda (H11: (subst0
-O u2 t2 w)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T
-(THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y
-t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T
-(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind
-Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u3 y t3)))))) u2 w (refl_equal T (THead (Bind Abbr)
-u2 w)) H9 (or_intror (pr0 t1 w) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda
-(y: T).(subst0 O u2 y w))) (ex_intro2 T (\lambda (y: T).(pr0 t1 y)) (\lambda
-(y: T).(subst0 O u2 y w)) t2 H10 H11))))))) x H8)) t0 (sym_eq T t0 t1 H7)))
-u0 (sym_eq T u0 u1 H6))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u)
-\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead
-(Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (TLRef _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
-\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
-t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
-t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead
-(Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e
-in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead
-(Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in (eq_ind B Abbr
-(\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2
-x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
-t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind
-T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not
-(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
-t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O
-t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not
-(eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t
-t3) (ex2 T (\lambda (y: T).(pr0 t y)) (\lambda (y: T).(subst0 O u2 y
-t3))))))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T
-x (\lambda (t: T).((not (eq B Abbr Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3:
-T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O
-t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift
-(S O) O x)))))) (\lambda (_: (not (eq B Abbr Abst))).(\lambda (H11: (pr0 t0
-x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y:
-T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0
-(lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq
-T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abbr H6))) H5))
-H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T
-(THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t1))).(\lambda (H2: (eq T t2
-x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e
-in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
-_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
-(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abbr) u1 t1) H1) in (False_ind ((eq T t2 x) \to
-((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
-t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))
-H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T
-x)))))).
+t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
+(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
+\Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3) \Rightarrow
+(THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in lref_map) (\lambda
+(x0: nat).(plus x0 (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map
+(f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t3)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t3))]) in
+lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Abbr)).(let H10
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abbr H9) in (let
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Abbr) u1 t))
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))) H3
+(lift (S O) O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y0: T).(pr0 t y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t (lift (S O) O t2))))
+(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y0:
+T).(pr0 (lift (S O) O t0) y0)) (\lambda (y0: T).(subst0 O u2 y0 t3)))))))
+(pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2 H2 (S O) O)) t1
+H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2: T).(\lambda (_:
+(pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Abbr) u1 t1)) \to (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
+(Bind Abbr) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H3) in (False_ind
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
+T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
+(\lambda (y0: T).(subst0 O u2 y0 t3))))))) (pr0 t1 (lift (S O) O t2)))
+H4)))))))) y x H0))) H)))).
theorem pr0_gen_void:
\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1
(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
\def
\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead
-(Bind Void) u1 t1) x)).(let H0 \def (match H in pr0 return (\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((eq T t (THead (Bind Void) u1
-t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+(Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1 t1) (\lambda (t:
+T).(pr0 t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0
u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O)
-O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind
-Void) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Void) u1 t1)
-(\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
-(lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Void) u1 t1)
-x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t0: T).(or (ex3_2 T T
+O x)))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(pr0_ind (\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2))))
(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda
-(u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void)
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O (THead (Bind Void)
-u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))))) (\lambda (t: T).(\lambda
+(H1: (eq T t (THead (Bind Void) u1 t1))).(let H2 \def (f_equal T T (\lambda
+(e: T).e) t (THead (Bind Void) u1 t1) H1) in (eq_ind_r T (THead (Bind Void)
+u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq
+T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1
+u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
+t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead
(Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1
-(refl_equal T (THead (Bind Void) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2))
-t (sym_eq T t (THead (Bind Void) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2
-H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Void) u1
-t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
-(THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
+(lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2
+t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1
+t1)) (pr0_refl u1) (pr0_refl t1))) t H2)))) (\lambda (u0: T).(\lambda (u2:
+T).(\lambda (H1: (pr0 u0 u2)).(\lambda (H2: (((eq T u0 (THead (Bind Void) u1
+t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H3: (pr0 t0
+t2)).(\lambda (H4: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (k: K).(\lambda
+(H5: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(let H6 \def (f_equal
+T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
+(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H7 \def (f_equal T T
(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t]))
-(THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
-(THead k u0 t0) (THead (Bind Void) u1 t1) H2) in (eq_ind K (Bind Void)
-(\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2)
-x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T
-u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Void) u2 t2) x) \to
-((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
-(t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in ((let H8 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t]))
+(THead k u0 t0) (THead (Bind Void) u1 t1) H5) in (\lambda (H9: (eq T u0
+u1)).(\lambda (H10: (eq K k (Bind Void))).(eq_ind_r K (Bind Void) (\lambda
+(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k0 u2
+t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1
+u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+(THead k0 u2 t2))))) (let H11 \def (eq_ind T t0 (\lambda (t: T).((eq T t
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
-(lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t:
-T).((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to
-(or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
-u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) (\lambda
-(H9: (eq T (THead (Bind Void) u2 t2) x)).(eq_ind T (THead (Bind Void) u2 t2)
-(\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda
-(u3: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 (lift (S O) O t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda
-(H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T
+(lift (S O) O t2))))) H4 t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t:
+T).(pr0 t t2)) H3 t1 H8) in (let H13 \def (eq_ind T u0 (\lambda (t: T).((eq T
+t (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T u2 (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_:
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+(lift (S O) O u2))))) H2 u1 H9) in (let H14 \def (eq_ind T u0 (\lambda (t:
+T).(pr0 t u2)) H1 u1 H9) in (or_introl (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda
+(u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0
+t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T
(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead
(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
(_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void)
-u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7)))
-k (sym_eq K k (Bind Void) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0
-t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead (Bind
-Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t0)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1
-t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2)
-\to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
+u2 t2)) H14 H12)))))) k H10)))) H7)) H6)))))))))))) (\lambda (u: T).(\lambda
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (((eq T v1
+(THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T v2 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1
+(lift (S O) O v2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0
+t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H5: (eq T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(let H6
+\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind Void) u1 t1) H5) in (False_ind (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2)
(THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))
-H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow
-(\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead
-(Bind Void) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl)
-v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e in T return (\lambda (_:
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead
+(Bind Abbr) v2 t2)))) H6)))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B
+b Abst))).(\lambda (v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1
+v2)).(\lambda (_: (((eq T v1 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T v2 (THead (Bind Void) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O v2)))))).(\lambda (u0: T).(\lambda
+(u2: T).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (((eq T u0 (THead (Bind Void)
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead
+(Bind Void) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O
+u2)))))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda
+(_: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (H8: (eq T (THead (Flat Appl)
+v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(let H9 \def (eq_ind T
+(THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind Void) u1 t1) H8) in (False_ind (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t2)) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1
+(lift (S O) O (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)))))
+H9))))))))))))))))) (\lambda (u0: T).(\lambda (u2: T).(\lambda (_: (pr0 u0
+u2)).(\lambda (_: (((eq T u0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead (Bind Void) u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O u2)))))).(\lambda (t0: T).(\lambda
+(t2: T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void)
+u1 t1)) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (w: T).(\lambda (_: (subst0 O u2 t2 w)).(\lambda (H6: (eq T
+(THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(let H7 \def (eq_ind T
+(THead (Bind Abbr) u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-Void) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0
-u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3:
-T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0
-u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O)
-O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2)
-\Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void)
-u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def
-(eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_:
-B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void
-\Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Void) u1
-t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to
-((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3:
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3:
-T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 (lift (S O) O x))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0
-t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0))
-(THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+[(Bind b) \Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr
+\Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat
+_) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H6) in (False_ind (or
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w)
+(THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead
+(Bind Abbr) u2 w)))) H7))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B
+b Abst))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
+t2)).(\lambda (H3: (((eq T t0 (THead (Bind Void) u1 t1)) \to (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)))))).(\lambda (u: T).(\lambda
+(H4: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1
+t1))).(let H5 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
+(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k
+_ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O
+t0)) (THead (Bind Void) u1 t1) H4) in ((let H6 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
+(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u
+(lift (S O) O t0)) (THead (Bind Void) u1 t1) H4) in ((let H7 \def (f_equal T
+T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T
\def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
| (THead k u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t)
\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1
-t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead
-(Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e
-in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead
-(Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in (eq_ind B Void
-(\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2
-x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T
-u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not
-(eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0)
-t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B
-Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift
-(S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not
-(eq B Void Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift
-(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_:
-(not (eq B Void Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+t1) H4) in (\lambda (_: (eq T u u1)).(\lambda (H9: (eq B b Void)).(let H10
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Void H9) in (let
+H11 \def (eq_ind_r T t1 (\lambda (t: T).((eq T t0 (THead (Bind Void) u1 t))
+\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O t2))))) H3 (lift (S O)
+O t0) H7) in (eq_ind T (lift (S O) O t0) (\lambda (t: T).(or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3))))
(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x))
-(pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u
-u1 H7))) b (sym_eq B b Void H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2
-H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind
-Void) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead
-(Flat Cast) u t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1
-t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0
-(refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x)))))).
+T).(pr0 t t3)))) (pr0 t (lift (S O) O t2)))) (or_intror (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift
+(S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O t2)) (pr0_lift t0 t2
+H2 (S O) O)) t1 H7)))))) H6)) H5)))))))))) (\lambda (t0: T).(\lambda (t2:
+T).(\lambda (_: (pr0 t0 t2)).(\lambda (_: (((eq T t0 (THead (Bind Void) u1
+t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O
+t2)))))).(\lambda (u: T).(\lambda (H3: (eq T (THead (Flat Cast) u t0) (THead
+(Bind Void) u1 t1))).(let H4 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H3) in (False_ind
+(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2))) H4)))))))) y x
+H0))) H)))).
theorem pr0_gen_lift:
\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0
(* This file was automatically generated: do not edit *********************)
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
-include "lift/tlt.ma".
+include "LambdaDelta-1/lift/tlt.ma".
theorem pr0_confluence__pr0_cong_upsilon_refl:
\forall (b: B).((not (eq B b Abst)) \to (\forall (u0: T).(\forall (u3:
((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda
(t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t5
t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7))
-H4 (THead k0 u0 t5) H11) in (let H17 \def (match H16 in eq return (\lambda
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k0 u0 t5)) \to (ex2 T
-(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 (THead k0 u3
-t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead k u1
-t3) (THead k0 u0 t5))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead k0
-u0 t5) H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0
-u0 t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+H4 (THead k0 u0 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u1 t3) (THead k0 u0
-t5) H17) in (eq_ind K k0 (\lambda (k1: K).((eq T u1 u0) \to ((eq T t3 t5) \to
-(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7)) (\lambda (t7: T).(pr0
-(THead k0 u3 t6) t7)))))) (\lambda (H21: (eq T u1 u0)).(eq_ind T u0 (\lambda
-(_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead k0 u2 t4) t8))
-(\lambda (t8: T).(pr0 (THead k0 u3 t6) t8))))) (\lambda (H22: (eq T t3
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead k0 u2
-t4) t8)) (\lambda (t8: T).(pr0 (THead k0 u3 t6) t8)))) (let H23 \def
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8:
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10:
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5)
-H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H22)
-in (let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u0 H21) in
-(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T
-(\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0
-u3 t6) t7))) (\lambda (x: T).(\lambda (H26: (pr0 u2 x)).(\lambda (H27: (pr0
-u3 x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7))
-(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0
-(THead k0 u3 t6) t7))) (\lambda (x0: T).(\lambda (H28: (pr0 t4 x0)).(\lambda
-(H29: (pr0 t6 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7))
-(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x x0) (pr0_comp u2 x
-H26 t4 x0 H28 k0) (pr0_comp u3 x H27 t6 x0 H29 k0))))) (H23 t5 (tlt_head_dx
-k0 u0 t5) t4 H24 t6 H15))))) (H23 u0 (tlt_head_sx k0 u0 t5) u2 H25 u3
-H14))))) t3 (sym_eq T t3 t5 H22))) u1 (sym_eq T u1 u0 H21))) k (sym_eq K k k0
-H20))) H19)) H18)))]) in (H17 (refl_equal T (THead k0 u0 t5))))))) t2 H13)) t
-H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda
-(H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) t)).(\lambda
-(H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1
-(THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6)
-t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead
-(Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7:
-T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2
-t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v1
-v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7:
-T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u
-t5)) H11) in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda
-(_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)))
-\to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0
-(THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda
-(H17: (eq T (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u
-t5)))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
-| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1
-(THead (Bind Abst) u t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3)
-(THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H17) in ((let H20 \def
-(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
-[(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
-\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u
-t5)) H17) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u1 v1) \to ((eq T
-t3 (THead (Bind Abst) u t5)) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2
-t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))))) (\lambda
-(H21: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t3 (THead (Bind Abst)
-u t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 t4) t8))
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8))))) (\lambda (H22: (eq T
-t3 (THead (Bind Abst) u t5))).(eq_ind T (THead (Bind Abst) u t5) (\lambda (_:
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 t4) t8)) (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) v2 t6) t8)))) (let H23 \def (eq_ind_r T t (\lambda
-(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to
-(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind
-Abst) u t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4))
-H8 (THead (Bind Abst) u t5) H22) in (let H25 \def (match H24 in pr0 return
-(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead
-(Bind Abst) u t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6)
-t9)))))))) with [(pr0_refl t7) \Rightarrow (\lambda (H25: (eq T t7 (THead
-(Bind Abst) u t5))).(\lambda (H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst)
-u t5) (\lambda (t8: T).((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6)
-t9))))) (\lambda (H27: (eq T (THead (Bind Abst) u t5) t4)).(eq_ind T (THead
-(Bind Abst) u t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat
-Appl) u2 t8) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (let
-H28 \def (eq_ind T u1 (\lambda (t8: T).(pr0 t8 u2)) H7 v1 H21) in (ex2_ind T
-(\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda
-(t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H29:
-(pr0 u2 x)).(\lambda (H30: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0
+t5) H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1
+| (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in
+((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _
+t7) \Rightarrow t7])) (THead k u1 t3) (THead k0 u0 t5) H16) in (\lambda (H20:
+(eq T u1 u0)).(\lambda (H21: (eq K k k0)).(let H22 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead k0 u0 t5) H11) in (eq_ind_r
+K k0 (\lambda (k1: K).(ex2 T (\lambda (t7: T).(pr0 (THead k1 u2 t4) t7))
+(\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H23 \def (eq_ind T u1
+(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H20) in (let H24 \def (eq_ind T t3
+(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead k0
+u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))) (\lambda (x:
+T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T (\lambda
+(t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7:
+T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))
+(\lambda (x0: T).(\lambda (H27: (pr0 u2 x0)).(\lambda (H28: (pr0 u3
+x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7:
+T).(pr0 (THead k0 u3 t6) t7)) (THead k0 x0 x) (pr0_comp u2 x0 H27 t4 x H25
+k0) (pr0_comp u3 x0 H28 t6 x H26 k0))))) (H22 u0 (tlt_head_sx k0 u0 t5) u2
+H23 u3 H14))))) (H22 t5 (tlt_head_dx k0 u0 t5) t4 H24 t6 H15)))) k H21)))))
+H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u v1 v2 H9 t5 t6 H10)
+\Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u
+t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind
+Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
+(H13: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2
+t6) (\lambda (t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14:
+(pr0 v1 v2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda
+(t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind
+Abst) u t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat
+Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let H18 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
+(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H16) in ((let
+H19 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
+\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t5)) H16) in (\lambda (H20: (eq T u1 v1)).(\lambda (H21: (eq K k (Flat
+Appl))).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H11) in
+(eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead
+k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))) (let
+H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 v1 H20) in (let H24
+\def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (THead (Bind Abst) u t5)
+H19) in (let H25 \def (match H24 in pr0 return (\lambda (t7: T).(\lambda (t8:
+T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t5)) \to ((eq T
+t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))))))) with [(pr0_refl
+t7) \Rightarrow (\lambda (H25: (eq T t7 (THead (Bind Abst) u t5))).(\lambda
+(H26: (eq T t7 t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda (t8: T).((eq
+T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) (\lambda (H27: (eq T
+(THead (Bind Abst) u t5) t4)).(eq_ind T (THead (Bind Abst) u t5) (\lambda
+(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)) (\lambda
+(t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)))) (ex2_ind T (\lambda (t8:
+T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0
(THead (Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0
-(THead (Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H29
-t5 t6 H15) (pr0_comp v2 x H30 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H23 v1
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H28 v2 H14))) t4
+(THead (Bind Abbr) v2 t6) t8))) (\lambda (x: T).(\lambda (H28: (pr0 u2
+x)).(\lambda (H29: (pr0 v2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead
+(Flat Appl) u2 (THead (Bind Abst) u t5)) t8)) (\lambda (t8: T).(pr0 (THead
+(Bind Abbr) v2 t6) t8)) (THead (Bind Abbr) x t6) (pr0_beta u u2 x H28 t5 t6
+H15) (pr0_comp v2 x H29 t6 t6 (pr0_refl t6) (Bind Abbr)))))) (H22 v1
+(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H23 v2 H14)) t4
H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t5) H25) H26))) | (pr0_comp u0 u3
H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead
(Bind Abst) u t5))).(\lambda (H28: (eq T (THead k0 u3 t8) t4)).((let H29 \def
t4)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to
((pr0 t5 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9)
t10)) (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t6) t10)))))) (\lambda (_:
-(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(let H37 \def (eq_ind T u1 (\lambda
-(t9: T).(pr0 t9 u2)) H7 v1 H21) in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9))
-(\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl)
-u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr)
-v2 t6) t9))) (\lambda (x: T).(\lambda (H38: (pr0 u2 x)).(\lambda (H39: (pr0
-v2 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9))
-(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8))
-t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0:
-T).(\lambda (H40: (pr0 t8 x0)).(\lambda (H41: (pr0 t6 x0)).(ex_intro2 T
+(pr0 u u3)).(\lambda (H36: (pr0 t5 t8)).(ex2_ind T (\lambda (t9: T).(pr0 t8
+t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat
+Appl) u2 (THead (Bind Abst) u3 t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind
+Abbr) v2 t6) t9))) (\lambda (x: T).(\lambda (H37: (pr0 t8 x)).(\lambda (H38:
+(pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2
+t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3
+t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))) (\lambda (x0:
+T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 v2 x0)).(ex_intro2 T
(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x x0)
-(pr0_beta u3 u2 x H38 t8 x0 H40) (pr0_comp v2 x H39 t6 x0 H41 (Bind
-Abbr)))))) (H23 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat Appl)
-v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5) (tlt_head_dx
-(Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15))))) (H23 v1
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2 H37 v2 H14))))) t4
+(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9)) (THead (Bind Abbr) x0 x)
+(pr0_beta u3 u2 x0 H39 t8 x H37) (pr0_comp v2 x0 H40 t6 x H38 (Bind
+Abbr)))))) (H22 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t5)) u2
+H23 v2 H14))))) (H22 t5 (tlt_trans (THead (Bind Abst) u t5) t5 (THead (Flat
+Appl) v1 (THead (Bind Abst) u t5)) (tlt_head_dx (Bind Abst) u t5)
+(tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t5))) t8 H36 t6 H15)))) t4
H34)) t7 (sym_eq T t7 t5 H33))) u0 (sym_eq T u0 u H32))) k0 (sym_eq K k0
(Bind Abst) H31))) H30)) H29)) H28 H25 H26))) | (pr0_beta u0 v0 v3 H25 t7 t8
H26) \Rightarrow (\lambda (H27: (eq T (THead (Flat Appl) v0 (THead (Bind
t4) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2
t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t6) t9))))) H28)) H27
H25)))]) in (H25 (refl_equal T (THead (Bind Abst) u t5)) (refl_equal T
-t4))))) t3 (sym_eq T t3 (THead (Bind Abst) u t5) H22))) u1 (sym_eq T u1 v1
-H21))) k (sym_eq K k (Flat Appl) H20))) H19)) H18)))]) in (H17 (refl_equal T
-(THead (Flat Appl) v1 (THead (Bind Abst) u t5)))))))) t2 H13)) t H11 H12 H9
-H10))) | (pr0_upsilon b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) \Rightarrow
-(\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t5))
-t)).(\lambda (H14: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
-v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5))
-(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
-v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda
-(t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to
+t4))))) k H21))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon
+b H9 v1 v2 H10 u0 u3 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead
+(Flat Appl) v1 (THead (Bind b) u0 t5)) t)).(\lambda (H14: (eq T (THead (Bind
+b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat
+Appl) v1 (THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to
((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda
-(H16: (not (eq B b Abst))).(\lambda (H17: (pr0 v1 v2)).(\lambda (H18: (pr0 u0
-u3)).(\lambda (H19: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7:
-T).(eq T (THead k u1 t3) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0
-t5)) H13) in (let H21 \def (match H20 in eq return (\lambda (t7: T).(\lambda
-(_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b) u0 t5))) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with
-[refl_equal \Rightarrow (\lambda (H21: (eq T (THead k u1 t3) (THead (Flat
-Appl) v1 (THead (Bind b) u0 t5)))).(let H22 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3)
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H21) in ((let H23 \def (f_equal
-T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
-(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H21) in ((let
-H24 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
-with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _)
-\Rightarrow k0])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0
-t5)) H21) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u1 v1) \to ((eq T
-t3 (THead (Bind b) u0 t5)) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4)
-t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
-O v2) t6)) t7)))))) (\lambda (H25: (eq T u1 v1)).(eq_ind T v1 (\lambda (_:
-T).((eq T t3 (THead (Bind b) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Flat Appl) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat
-Appl) (lift (S O) O v2) t6)) t8))))) (\lambda (H26: (eq T t3 (THead (Bind b)
-u0 t5))).(eq_ind T (THead (Bind b) u0 t5) (\lambda (_: T).(ex2 T (\lambda
-(t8: T).(pr0 (THead (Flat Appl) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead
-(Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))) (let H27 \def
-(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8:
-T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10:
-T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl)
-v1 (THead (Bind b) u0 t5)) H13) in (let H28 \def (eq_ind T t3 (\lambda (t7:
-T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H26) in (let H29 \def (match H28 in
+T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda
+(H15: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
+t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
+(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda
+(t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17:
+(pr0 v1 v2)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5 t6)).(let H20
+\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead
+(Flat Appl) v1 (THead (Bind b) u0 t5)) H13) in (let H21 \def (f_equal T K
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
+(THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H20) in ((let
+H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _)
+\Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1 (THead (Bind b) u0
+t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
+| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Flat Appl) v1
+(THead (Bind b) u0 t5)) H20) in (\lambda (H24: (eq T u1 v1)).(\lambda (H25:
+(eq K k (Flat Appl))).(let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall
+(v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9:
+T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10:
+T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H13)
+in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0
+(THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead (Flat
+Appl) (lift (S O) O v2) t6)) t7)))) (let H27 \def (eq_ind T u1 (\lambda (t7:
+T).(pr0 t7 u2)) H7 v1 H24) in (let H28 \def (eq_ind T t3 (\lambda (t7:
+T).(pr0 t7 t4)) H8 (THead (Bind b) u0 t5) H23) in (let H29 \def (match H28 in
pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T
t7 (THead (Bind b) u0 t5)) \to ((eq T t8 t4) \to (ex2 T (\lambda (t9: T).(pr0
(THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3
(\lambda (H31: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0
t5) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t8)
t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
-O v2) t6)) t9)))) (let H32 \def (eq_ind T u1 (\lambda (t8: T).(pr0 t8 u2)) H7
-v1 H25) in (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8: T).(pr0 v2
-t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0
-t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift
-(S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H33: (pr0 u2 x)).(\lambda
-(H34: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16 u0 u3 H18 t5
-t6 H19 u2 v2 x H33 H34)))) (H27 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind
-b) u0 t5)) u2 H32 v2 H17))) t4 H31)) t7 (sym_eq T t7 (THead (Bind b) u0 t5)
-H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow (\lambda (H31:
-(eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32: (eq T (THead
-k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 | (TLRef _)
-\Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7) (THead
-(Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 | (TLRef _)
-\Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7) (THead
-(Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e: T).(match e
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _)
-\Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7) (THead
-(Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4 u0) \to
-((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to ((pr0 t7
-t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda
-(t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9: T).((eq T
-t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to ((pr0 t7
-t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10))
-(\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
-v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda (t9:
-T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8) \to
-(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda (t10:
-T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
+O v2) t6)) t9)))) (ex2_ind T (\lambda (t8: T).(pr0 u2 t8)) (\lambda (t8:
+T).(pr0 v2 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead
+(Bind b) u0 t5)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u3 (THead (Flat
+Appl) (lift (S O) O v2) t6)) t8))) (\lambda (x: T).(\lambda (H32: (pr0 u2
+x)).(\lambda (H33: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H16
+u0 u3 H18 t5 t6 H19 u2 v2 x H32 H33)))) (H26 v1 (tlt_head_sx (Flat Appl) v1
+(THead (Bind b) u0 t5)) u2 H27 v2 H17)) t4 H31)) t7 (sym_eq T t7 (THead (Bind
+b) u0 t5) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow
+(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u0 t5))).(\lambda (H32:
+(eq T (THead k0 u5 t8) t4)).((let H33 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
+(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7)
+(THead (Bind b) u0 t5) H31) in ((let H34 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 |
+(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7)
+(THead (Bind b) u0 t5) H31) in ((let H35 \def (f_equal T K (\lambda (e:
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
+(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7)
+(THead (Bind b) u0 t5) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4
+u0) \to ((eq T t7 t5) \to ((eq T (THead k1 u5 t8) t4) \to ((pr0 u4 u5) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t6)) t9))))))))) (\lambda (H36: (eq T u4 u0)).(eq_ind T u0 (\lambda (t9:
+T).((eq T t7 t5) \to ((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 t9 u5) \to
+((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4)
+t10)) (\lambda (t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S
+O) O v2) t6)) t10)))))))) (\lambda (H37: (eq T t7 t5)).(eq_ind T t5 (\lambda
+(t9: T).((eq T (THead (Bind b) u5 t8) t4) \to ((pr0 u0 u5) \to ((pr0 t9 t8)
+\to (ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda
+(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t4)).(eq_ind T (THead
(Bind b) u5 t8) (\lambda (t9: T).((pr0 u0 u5) \to ((pr0 t5 t8) \to (ex2 T
(\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)) (\lambda (t10: T).(pr0
(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t10))))))
-(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(let H41 \def (eq_ind
-T u1 (\lambda (t9: T).(pr0 t9 u2)) H7 v1 H25) in (ex2_ind T (\lambda (t9:
-T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9: T).(pr0
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda
-(x: T).(\lambda (H42: (pr0 u2 x)).(\lambda (H43: (pr0 v2 x)).(ex2_ind T
-(\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda
-(t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9:
+(\lambda (H39: (pr0 u0 u5)).(\lambda (H40: (pr0 t5 t8)).(ex2_ind T (\lambda
+(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8)) t9)) (\lambda (t9:
T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))
-(\lambda (x0: T).(\lambda (H44: (pr0 t8 x0)).(\lambda (H45: (pr0 t6
-x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9))
+(\lambda (x: T).(\lambda (H41: (pr0 t8 x)).(\lambda (H42: (pr0 t6
+x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u3 t9))
(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t8))
t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O)
-O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H46: (pr0 u5 x1)).(\lambda (H47:
-(pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x H42 H43 t8
-t6 x0 H44 H45 u5 u3 x1 H46 H47)))) (H27 u0 (tlt_trans (THead (Bind b) u0 t5)
-u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b) u0 t5)
-(tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3 H18))))) (H27
-t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1 (THead (Bind b)
-u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind
-b) u0 t5))) t8 H40 t6 H19))))) (H27 v1 (tlt_head_sx (Flat Appl) v1 (THead
-(Bind b) u0 t5)) u2 H41 v2 H17))))) t4 H38)) t7 (sym_eq T t7 t5 H37))) u4
-(sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33)) H32 H29
-H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31: (eq T
-(THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0
+O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H43: (pr0 u5 x0)).(\lambda (H44:
+(pr0 u3 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0
+v2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5
+t8)) t9)) (\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift
+(S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda (H45: (pr0 u2 x1)).(\lambda
+(H46: (pr0 v2 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H16 u2 v2 x1 H45
+H46 t8 t6 x H41 H42 u5 u3 x0 H43 H44)))) (H26 v1 (tlt_head_sx (Flat Appl) v1
+(THead (Bind b) u0 t5)) u2 H27 v2 H17))))) (H26 u0 (tlt_trans (THead (Bind b)
+u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) (tlt_head_sx (Bind b)
+u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t5))) u5 H39 u3
+H18))))) (H26 t5 (tlt_trans (THead (Bind b) u0 t5) t5 (THead (Flat Appl) v1
+(THead (Bind b) u0 t5)) (tlt_head_dx (Bind b) u0 t5) (tlt_head_dx (Flat Appl)
+v1 (THead (Bind b) u0 t5))) t8 H40 t6 H19)))) t4 H38)) t7 (sym_eq T t7 t5
+H37))) u4 (sym_eq T u4 u0 H36))) k0 (sym_eq K k0 (Bind b) H35))) H34)) H33))
+H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30) \Rightarrow (\lambda (H31:
+(eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (THead (Bind b) u0
t5))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8) t4)).((let H33 \def
(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7)) (\lambda (e:
T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
\def (eq_ind_r B b (\lambda (b0: B).(\forall (v: T).((tlt v (THead (Flat
Appl) v1 (THead (Bind b0) u0 t5))) \to (\forall (t9: T).((pr0 v t9) \to
(\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11))
-(\lambda (t11: T).(pr0 t10 t11)))))))))) H27 Abbr H36) in (let H44 \def
-(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H26 Abbr
+(\lambda (t11: T).(pr0 t10 t11)))))))))) H26 Abbr H36) in (let H44 \def
+(eq_ind_r B b (\lambda (b0: B).(eq T t3 (THead (Bind b0) u0 t5))) H23 Abbr
H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst)))
-H16 Abbr H36) in (let H46 \def (eq_ind T u1 (\lambda (t9: T).(pr0 t9 u2)) H7
-v1 H25) in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2
-t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5
-w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift
-(S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H47: (pr0 u2 x)).(\lambda
-(H48: (pr0 v2 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9:
+H16 Abbr H36) in (ex2_ind T (\lambda (t9: T).(pr0 t8 t9)) (\lambda (t9:
T).(pr0 t6 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
(Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda (H49: (pr0
-t8 x0)).(\lambda (H50: (pr0 t6 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9))
+(Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x: T).(\lambda (H46: (pr0
+t8 x)).(\lambda (H47: (pr0 t6 x)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9))
(\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl)
u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0 (THead (Bind Abbr) u3
-(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x1: T).(\lambda
-(H51: (pr0 u5 x1)).(\lambda (H52: (pr0 u3
-x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x H47 H48
-t6 x0 H49 H50 u3 x1 H51 H52)))) (H43 u0 (tlt_trans (THead (Bind Abbr) u0 t5)
-u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind Abbr)
-u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40 u3
-H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat Appl)
-v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5) (tlt_head_dx
-(Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19))))) (H43 v1
-(tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) u2 H46 v2 H17)))))))))
+(THead (Flat Appl) (lift (S O) O v2) t6)) t9))) (\lambda (x0: T).(\lambda
+(H48: (pr0 u5 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T (\lambda (t9:
+T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9)) (ex2 T (\lambda (t9: T).(pr0
+(THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t9)) (\lambda (t9: T).(pr0
+(THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9)))
+(\lambda (x1: T).(\lambda (H50: (pr0 u2 x1)).(\lambda (H51: (pr0 v2
+x1)).(pr0_confluence__pr0_cong_upsilon_delta H45 u5 t8 w H42 u2 v2 x1 H50 H51
+t6 x H46 H47 u3 x0 H48 H49)))) (H43 v1 (tlt_head_sx (Flat Appl) v1 (THead
+(Bind Abbr) u0 t5)) u2 H27 v2 H17))))) (H43 u0 (tlt_trans (THead (Bind Abbr)
+u0 t5) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_sx (Bind
+Abbr) u0 t5) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) u5 H40
+u3 H18))))) (H43 t5 (tlt_trans (THead (Bind Abbr) u0 t5) t5 (THead (Flat
+Appl) v1 (THead (Bind Abbr) u0 t5)) (tlt_head_dx (Bind Abbr) u0 t5)
+(tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t5))) t8 H41 t6 H19))))))))
t4 H39)) t7 (sym_eq T t7 t5 H38))) u4 (sym_eq T u4 u0 H37))) b H36)) H35))
H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u) \Rightarrow (\lambda
(H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u0
(t9: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0
t9))) \to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11)
\to (ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11
-t12)))))))))) H27 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5
-(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H26 (lift (S O) O t7) H37)
+t12)))))))))) H26 (lift (S O) O t7) H37) in (let H42 \def (eq_ind_r T t5
+(\lambda (t9: T).(eq T t3 (THead (Bind b) u0 t9))) H23 (lift (S O) O t7) H37)
in (let H43 \def (eq_ind_r T t5 (\lambda (t9: T).(pr0 t9 t6)) H19 (lift (S O)
O t7) H37) in (ex2_ind T (\lambda (t9: T).(eq T t6 (lift (S O) O t9)))
(\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl)
x))).(\lambda (H45: (pr0 t7 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9:
T).(ex2 T (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t4) t10)) (\lambda
(t10: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t9))
-t10)))) (let H46 \def (eq_ind T u1 (\lambda (t9: T).(pr0 t9 u2)) H7 v1 H25)
-in (ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9))
+t10)))) (ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9))
(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
-x))) t9))) (\lambda (x0: T).(\lambda (H47: (pr0 u2 x0)).(\lambda (H48: (pr0
-v2 x0)).(ex2_ind T (\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t4 t9))
+x))) t9))) (\lambda (x0: T).(\lambda (H46: (pr0 x x0)).(\lambda (H47: (pr0 t4
+x0)).(ex2_ind T (\lambda (t9: T).(pr0 u2 t9)) (\lambda (t9: T).(pr0 v2 t9))
(ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9:
T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
-x))) t9))) (\lambda (x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t4
-x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x0 H47 H48
-x t4 x1 H49 H50)))) (H41 t7 (tlt_trans (THead (Bind b) u0 (lift (S O) O t7))
-t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t7))) (lift_tlt_dx
-(Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 (lift
-(S O) O t7)))) x H45 t4 H40))))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead
-(Bind b) u0 (lift (S O) O t7))) u2 H46 v2 H17))) t6 H44)))) (pr0_gen_lift t7
-t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u (sym_eq T u u0
-H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) | (pr0_epsilon t7
-t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast) u t7) (THead
-(Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def (eq_ind T
-(THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return (\lambda (_:
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
-(THead k0 _ _) \Rightarrow (match k0 in K return (\lambda (_: K).Prop) with
-[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
-b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0 t7 t8) \to (ex2 T
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t9))))) H32))
-H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5)) (refl_equal T
-t4))))) t3 (sym_eq T t3 (THead (Bind b) u0 t5) H26))) u1 (sym_eq T u1 v1
-H25))) k (sym_eq K k (Flat Appl) H24))) H23)) H22)))]) in (H21 (refl_equal T
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)))))))))) t2 H15)) t H13 H14 H9
-H10 H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda
-(H12: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind
-Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T
-(THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O
-u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8:
+x))) t9))) (\lambda (x1: T).(\lambda (H48: (pr0 u2 x1)).(\lambda (H49: (pr0
+v2 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H39 u0 u3 H18 u2 v2 x1 H48
+H49 x t4 x0 H46 H47)))) (H41 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b)
+u0 (lift (S O) O t7))) u2 H27 v2 H17))))) (H41 t7 (tlt_trans (THead (Bind b)
+u0 (lift (S O) O t7)) t7 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O)
+O t7))) (lift_tlt_dx (Bind b) u0 t7 (S O) O) (tlt_head_dx (Flat Appl) v1
+(THead (Bind b) u0 (lift (S O) O t7)))) x H45 t4 H40)) t6 H44))))
+(pr0_gen_lift t7 t6 (S O) O H43))))))) t8 (sym_eq T t8 t4 H38))) t5 H37)) u
+(sym_eq T u u0 H36))) b0 (sym_eq B b0 b H35))) H34)) H33)) H32 H29 H30))) |
+(pr0_epsilon t7 t8 H29 u) \Rightarrow (\lambda (H30: (eq T (THead (Flat Cast)
+u t7) (THead (Bind b) u0 t5))).(\lambda (H31: (eq T t8 t4)).((let H32 \def
+(eq_ind T (THead (Flat Cast) u t7) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u0 t5) H30) in (False_ind ((eq T t8 t4) \to ((pr0
+t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2)
+t6)) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead (Bind b) u0 t5))
+(refl_equal T t4))))) k H25))))) H22)) H21))))))) t2 H15)) t H13 H14 H9 H10
+H11 H12))) | (pr0_delta u0 u3 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12:
+(eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr)
+u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead
+(Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6
+w) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8:
T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind Abbr) u3 w)
t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to
((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead
k u2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (H15: (pr0 u0
u3)).(\lambda (H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u3 t6 w)).(let H18
\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead
-(Bind Abbr) u0 t5) H12) in (let H19 \def (match H18 in eq return (\lambda
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u0 t5)) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0
-(THead (Bind Abbr) u3 w) t8)))))) with [refl_equal \Rightarrow (\lambda (H19:
-(eq T (THead k u1 t3) (THead (Bind Abbr) u0 t5))).(let H20 \def (f_equal T T
+(Bind Abbr) u0 t5) H12) in (let H19 \def (f_equal T K (\lambda (e: T).(match
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind
+Abbr) u0 t5) H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead
+(Bind Abbr) u0 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead
+(Bind Abbr) u0 t5) H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K
+k (Bind Abbr))).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Bind Abbr) u0 t5) H12) in (eq_ind_r K (Bind Abbr)
+(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))) (let H25 \def (eq_ind T u1
+(\lambda (t7: T).(pr0 t7 u2)) H7 u0 H22) in (let H26 \def (eq_ind T t3
+(\lambda (t7: T).(pr0 t7 t4)) H8 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7)))
+(\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28: (pr0 t6
+x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7))
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H29: (pr0
+u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w
+H17 u2 x0 H29 H30 t4 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2
+H25 u3 H15))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16)))) k
+H23))))) H20)) H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5
+t6 H10 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O
+t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O)
+O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5
+t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8:
+T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7:
+T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0
+(THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not
+(eq B b Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t
+(\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead (Bind b) u (lift (S O)
+O t5)) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e in T
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Bind
+b) u (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
+(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3)
+(THead (Bind b) u (lift (S O) O t5)) H16) in ((let H19 \def (f_equal T T
(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7]))
-(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in ((let H21 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
-(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in ((let H22 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0]))
-(THead k u1 t3) (THead (Bind Abbr) u0 t5) H19) in (eq_ind K (Bind Abbr)
-(\lambda (k0: K).((eq T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7:
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w)
-t7)))))) (\lambda (H23: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3
-t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 t4) t8)) (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) u3 w) t8))))) (\lambda (H24: (eq T t3
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind
-Abbr) u2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w) t8)))) (let
-H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
-(THead (Bind Abbr) u0 t5) H12) in (let H26 \def (eq_ind T t3 (\lambda (t7:
-T).(pr0 t7 t4)) H8 t5 H24) in (let H27 \def (eq_ind T u1 (\lambda (t7:
-T).(pr0 t7 u2)) H7 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0 u2 t7))
-(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr)
-u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) (\lambda (x:
-T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: (pr0 u3 x)).(ex2_ind T (\lambda
-(t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) u3 w) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 t4 x0)).(\lambda (H31:
-(pr0 t6 x0)).(pr0_confluence__pr0_cong_delta u3 t6 w H17 u2 x H28 H29 t4 x0
-H30 H31)))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6 H16))))) (H25
-u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H27 u3 H15))))) t3 (sym_eq T t3 t5
-H24))) u1 (sym_eq T u1 u0 H23))) k (sym_eq K k (Bind Abbr) H22))) H21))
-H20)))]) in (H19 (refl_equal T (THead (Bind Abbr) u0 t5)))))))) t2 H14)) t
-H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u) \Rightarrow (\lambda
-(H11: (eq T (THead (Bind b) u (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6
-t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6
-t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
-(H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to
-((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda
-(t8: T).(pr0 t2 t8)))))) (\lambda (H14: (not (eq B b Abst))).(\lambda (H15:
-(pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1
-t3) t7)) H4 (THead (Bind b) u (lift (S O) O t5)) H11) in (let H17 \def (match
-H16 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead
-(Bind b) u (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2
-t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow
-(\lambda (H17: (eq T (THead k u1 t3) (THead (Bind b) u (lift (S O) O
-t5)))).(let H18 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
-| (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead (Bind b) u (lift (S
-O) O t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead
-(Bind b) u (lift (S O) O t5)) H17) in ((let H20 \def (f_equal T K (\lambda
-(e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k
-| (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3)
-(THead (Bind b) u (lift (S O) O t5)) H17) in (eq_ind K (Bind b) (\lambda (k0:
-K).((eq T u1 u) \to ((eq T t3 (lift (S O) O t5)) \to (ex2 T (\lambda (t7:
-T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))))) (\lambda
-(H21: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3 (lift (S O) O t5))
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t4) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))) (\lambda (H22: (eq T t3 (lift (S O) O t5))).(eq_ind T
-(lift (S O) O t5) (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind
-b) u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H23 \def (eq_ind_r T t
+(THead k u1 t3) (THead (Bind b) u (lift (S O) O t5)) H16) in (\lambda (H20:
+(eq T u1 u)).(\lambda (H21: (eq K k (Bind b))).(let H22 \def (eq_ind_r T t
(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b) u (lift (S O) O
-t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8
-(lift (S O) O t5) H22) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O
-t7))) (\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind
-b) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25:
-(eq T t4 (lift (S O) O x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S
-O) O x) (\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7)
-t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H27 \def (eq_ind T u1 (\lambda (t7:
-T).(pr0 t7 u2)) H7 u H21) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda
-(t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S
-O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H28:
-(pr0 x x0)).(\lambda (H29: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0
-(THead (Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0
-(pr0_zeta b H14 x x0 H28 u2) H29)))) (H23 t5 (lift_tlt_dx (Bind b) u t5 (S O)
-O) x H26 t2 H15))) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) t3 (sym_eq
-T t3 (lift (S O) O t5) H22))) u1 (sym_eq T u1 u H21))) k (sym_eq K k (Bind b)
-H20))) H19)) H18)))]) in (H17 (refl_equal T (THead (Bind b) u (lift (S O) O
-t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6
+t5)) H11) in (eq_ind_r K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7:
+T).(pr0 (THead k0 u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H23 \def
+(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H20) in (let H24 \def (eq_ind
+T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 (lift (S O) O t5) H19) in (ex2_ind T
+(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7))
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 t4) t7)) (\lambda (t7:
+T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H25: (eq T t4 (lift (S O) O
+x))).(\lambda (H26: (pr0 t5 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t7:
+T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 t7) t8)) (\lambda (t8:
+T).(pr0 t2 t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7:
+T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O
+x)) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0: T).(\lambda (H27: (pr0
+x x0)).(\lambda (H28: (pr0 t2 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead
+(Bind b) u2 (lift (S O) O x)) t7)) (\lambda (t7: T).(pr0 t2 t7)) x0 (pr0_zeta
+b H14 x x0 H27 u2) H28)))) (H22 t5 (lift_tlt_dx (Bind b) u t5 (S O) O) x H26
+t2 H15)) t4 H25)))) (pr0_gen_lift t5 t4 (S O) O H24)))) k H21))))) H18))
+H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6
H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u t5) t)).(\lambda
(H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T
t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2 t4) t8))
(\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead k u2
t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H13: (pr0 t5 t2)).(let
H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead k u1 t3) t7)) H4 (THead
-(Flat Cast) u t5) H10) in (let H15 \def (match H14 in eq return (\lambda (t7:
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u t5)) \to (ex2 T
-(\lambda (t8: T).(pr0 (THead k u2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))
-with [refl_equal \Rightarrow (\lambda (H15: (eq T (THead k u1 t3) (THead
-(Flat Cast) u t5))).(let H16 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead
-(Flat Cast) u t5) H15) in ((let H17 \def (f_equal T T (\lambda (e: T).(match
-e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead
-(Flat Cast) u t5) H15) in ((let H18 \def (f_equal T K (\lambda (e: T).(match
-e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
+(Flat Cast) u t5) H10) in (let H15 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _)
\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u1 t3) (THead (Flat
-Cast) u t5) H15) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 u) \to
-((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda
-(t7: T).(pr0 t2 t7)))))) (\lambda (H19: (eq T u1 u)).(eq_ind T u (\lambda (_:
-T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t4)
-t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H20: (eq T t3 t5)).(eq_ind T
-t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t4)
-t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H21 \def (eq_ind_r T t (\lambda
-(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to
-(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u t5) H10) in
-(let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H20) in (let
-H23 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H7 u H19) in (ex2_ind T
-(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda
-(t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))
-(\lambda (x: T).(\lambda (H24: (pr0 t4 x)).(\lambda (H25: (pr0 t2
-x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7))
-(\lambda (t7: T).(pr0 t2 t7)) x (pr0_epsilon t4 x H24 u2) H25)))) (H21 t5
-(tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13))))) t3 (sym_eq T t3 t5 H20)))
-u1 (sym_eq T u1 u H19))) k (sym_eq K k (Flat Cast) H18))) H17)) H16)))]) in
-(H15 (refl_equal T (THead (Flat Cast) u t5)))))) t6 (sym_eq T t6 t2 H12))) t
-H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5
-H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4 H3) \Rightarrow (\lambda (H4: (eq T
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t)).(\lambda (H5: (eq T
-(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t3)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) t1) \to ((pr0
-v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda
-(t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T (THead (Bind Abbr) v2 t4)
-t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t5: T).((pr0 v1 v2) \to
-((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0
-t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (pr0 t3 t4)).(let H9
-\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_:
-(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0
-(THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
-[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5
-t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))))
-(\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let
-H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H11 (THead (Flat Appl)
-v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead (Flat Appl) v1 (THead
-(Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t
-(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat Appl) v1 (THead (Bind Abst) u
-t3)) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v:
-T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v
-t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8
-t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in
-(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2 t4) t6)) (\lambda
-(t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t6)) (THead
-(Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4)) (pr0_beta u v1 v2 H7
-t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) |
-(pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1
-t5) t)).(\lambda (H12: (eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5)
-(\lambda (_: T).((eq T (THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6)
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6) t2)).(eq_ind T
-(THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7
-t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def (match H16 in eq
-return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1 t5))
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 (THead k u2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17:
-(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5))).(let
-H18 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow
-(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat
-Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H17) in ((let H19 \def
+Cast) u t5) H14) in ((let H16 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead k u1 t3) (THead
+(Flat Cast) u t5) H14) in ((let H17 \def (f_equal T T (\lambda (e: T).(match
+e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead k u1 t3) (THead
+(Flat Cast) u t5) H14) in (\lambda (H18: (eq T u1 u)).(\lambda (H19: (eq K k
+(Flat Cast))).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
+T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
+t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Flat Cast) u t5) H10) in (eq_ind_r K (Flat Cast)
+(\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead k0 u2 t4) t7)) (\lambda
+(t7: T).(pr0 t2 t7)))) (let H21 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7
+u2)) H7 u H18) in (let H22 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8
+t5 H17) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t2
+t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t4) t7)) (\lambda
+(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H23: (pr0 t4 x)).(\lambda
+(H24: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead (Flat Cast) u2
+t4) t7)) (\lambda (t7: T).(pr0 t2 t7)) x (pr0_epsilon t4 x H23 u2) H24))))
+(H20 t5 (tlt_head_dx (Flat Cast) u t5) t4 H22 t2 H13)))) k H19))))) H16))
+H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t)
+(refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t3 t4
+H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst)
+u t3)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t4) t1)).(eq_ind T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead
+(Bind Abbr) v2 t4) t1) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda
+(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H6: (eq T
+(THead (Bind Abbr) v2 t4) t1)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda
+(t5: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5
+t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda
+(H8: (pr0 t3 t4)).(let H9 \def (match H1 in pr0 return (\lambda (t5:
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
+T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5
+t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
+\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
+T).(pr0 t2 t7))))) (\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_:
+T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
+T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2))
+H11 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H4) in (eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t6: T).(ex2 T (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t6 t7))))
+(let H13 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Flat
+Appl) v1 (THead (Bind Abst) u t3)) H4) in (let H14 \def (eq_ind_r T t
+(\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7)
+\to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9))
+(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind
+Abst) u t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) v2
+t4) t6)) (\lambda (t6: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) t6)) (THead (Bind Abbr) v2 t4) (pr0_refl (THead (Bind Abbr) v2 t4))
+(pr0_beta u v1 v2 H7 t3 t4 H8)))) t2 H12)) t (sym_eq T t t2 H11))) t5 (sym_eq
+T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow (\lambda
+(H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6)
+t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to
+((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind
+Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T
+(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2)
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4)
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 u1 u2)).(\lambda
+(H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
+(Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead k u1 t5) H11) in (let
+H17 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) |
+(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
+(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat
+Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in ((let H19 \def
(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _)
-\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead k u1
-t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T return
-(\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _)
-\Rightarrow (Flat Appl) | (THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl)
-v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H17) in (eq_ind K (Flat Appl)
-(\lambda (k0: K).((eq T v1 u1) \to ((eq T (THead (Bind Abst) u t3) t5) \to
-(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7:
-T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda (H21: (eq T v1 u1)).(eq_ind T u1
-(\lambda (_: T).((eq T (THead (Bind Abst) u t3) t5) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat
-Appl) u2 t6) t8))))) (\lambda (H22: (eq T (THead (Bind Abst) u t3)
-t5)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (_: T).(ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat
-Appl) u2 t6) t8)))) (let H23 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead
-k0 u1 t5) t)) H11 (Flat Appl) H20) in (let H24 \def (eq_ind_r T t5 (\lambda
-(t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3) H22) in (let H25 \def
-(match H24 in pr0 return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0
-t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T t8 t6) \to (ex2 T
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow (\lambda
-(H25: (eq T t7 (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t7
-t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq T t8 t6) \to
-(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H27: (eq T (THead (Bind
-Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).(ex2 T
-(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 t8) t9)))) (let H28 \def (eq_ind_r T t5 (\lambda (t8:
-T).(eq T (THead (Flat Appl) u1 t8) t)) H23 (THead (Bind Abst) u t3) H22) in
-(let H29 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to
-(\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T
-(\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H
-(THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H28) in (let H30 \def (eq_ind
-T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H21) in (ex2_ind T (\lambda (t8:
-T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda (t8: T).(pr0
+[(TSort _) \Rightarrow (THead (Bind Abst) u t3) | (TLRef _) \Rightarrow
+(THead (Bind Abst) u t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat
+Appl) v1 (THead (Bind Abst) u t3)) (THead k u1 t5) H16) in (\lambda (H20: (eq
+T v1 u1)).(\lambda (H21: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda
+(k0: K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda
+(t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r K k (\lambda
+(k0: K).(eq T (THead k0 u1 t5) t)) H11 (Flat Appl) H21) in (let H23 \def
+(eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (THead (Bind Abst) u t3)
+H19) in (let H24 \def (match H23 in pr0 return (\lambda (t7: T).(\lambda (t8:
+T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead (Bind Abst) u t3)) \to ((eq T
+t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9)))))))) with [(pr0_refl
+t7) \Rightarrow (\lambda (H24: (eq T t7 (THead (Bind Abst) u t3))).(\lambda
+(H25: (eq T t7 t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda (t8: T).((eq
+T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9))
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) (\lambda (H26: (eq T
+(THead (Bind Abst) u t3) t6)).(eq_ind T (THead (Bind Abst) u t3) (\lambda
+(t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda
+(t9: T).(pr0 (THead (Flat Appl) u2 t8) t9)))) (let H27 \def (eq_ind_r T t5
+(\lambda (t8: T).(eq T (THead (Flat Appl) u1 t8) t)) H22 (THead (Bind Abst) u
+t3) H19) in (let H28 \def (eq_ind_r T t (\lambda (t8: T).(\forall (v:
+T).((tlt v t8) \to (\forall (t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v
+t10) \to (ex2 T (\lambda (t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10
+t11)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H27) in (let
+H29 \def (eq_ind T v1 (\lambda (t8: T).(pr0 t8 v2)) H7 u1 H20) in (ex2_ind T
+(\lambda (t8: T).(pr0 v2 t8)) (\lambda (t8: T).(pr0 u2 t8)) (ex2 T (\lambda
+(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead
+(Flat Appl) u2 (THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H30:
+(pr0 v2 x)).(\lambda (H31: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0
(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2
-(THead (Bind Abst) u t3)) t8))) (\lambda (x: T).(\lambda (H31: (pr0 v2
-x)).(\lambda (H32: (pr0 u2 x)).(ex_intro2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u2 (THead
-(Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H31 t4 t4
-(pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H32 t3 t4 H8))))) (H29 u1
-(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H30 u2 H14))))) t6
-H27)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H25) H26))) | (pr0_comp u0 u3
-H25 t7 t8 H26 k0) \Rightarrow (\lambda (H27: (eq T (THead k0 u0 t7) (THead
-(Bind Abst) u t3))).(\lambda (H28: (eq T (THead k0 u3 t8) t6)).((let H29 \def
+(THead (Bind Abst) u t3)) t8)) (THead (Bind Abbr) x t4) (pr0_comp v2 x H30 t4
+t4 (pr0_refl t4) (Bind Abbr)) (pr0_beta u u2 x H31 t3 t4 H8))))) (H28 u1
+(tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t3)) v2 H29 u2 H14))))) t6
+H26)) t7 (sym_eq T t7 (THead (Bind Abst) u t3) H24) H25))) | (pr0_comp u0 u3
+H24 t7 t8 H25 k0) \Rightarrow (\lambda (H26: (eq T (THead k0 u0 t7) (THead
+(Bind Abst) u t3))).(\lambda (H27: (eq T (THead k0 u3 t8) t6)).((let H28 \def
(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
[(TSort _) \Rightarrow t7 | (TLRef _) \Rightarrow t7 | (THead _ _ t9)
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H27) in ((let H30
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H29
\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t9 _)
-\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H27) in ((let H31
+\Rightarrow t9])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in ((let H30
\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
-\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H27) in (eq_ind K
+\Rightarrow k1])) (THead k0 u0 t7) (THead (Bind Abst) u t3) H26) in (eq_ind K
(Bind Abst) (\lambda (k1: K).((eq T u0 u) \to ((eq T t7 t3) \to ((eq T (THead
k1 u3 t8) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat
-Appl) u2 t6) t9))))))))) (\lambda (H32: (eq T u0 u)).(eq_ind T u (\lambda
+Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0 u)).(eq_ind T u (\lambda
(t9: T).((eq T t7 t3) \to ((eq T (THead (Bind Abst) u3 t8) t6) \to ((pr0 t9
u3) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2
t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))))
-(\lambda (H33: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead
+(\lambda (H32: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq T (THead
(Bind Abst) u3 t8) t6) \to ((pr0 u u3) \to ((pr0 t9 t8) \to (ex2 T (\lambda
(t10: T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead
-(Flat Appl) u2 t6) t10))))))) (\lambda (H34: (eq T (THead (Bind Abst) u3 t8)
+(Flat Appl) u2 t6) t10))))))) (\lambda (H33: (eq T (THead (Bind Abst) u3 t8)
t6)).(eq_ind T (THead (Bind Abst) u3 t8) (\lambda (t9: T).((pr0 u u3) \to
((pr0 t3 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t9) t10)))))) (\lambda (_:
-(pr0 u u3)).(\lambda (H36: (pr0 t3 t8)).(let H37 \def (eq_ind_r T t5 (\lambda
-(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H23 (THead (Bind Abst) u t3) H22)
-in (let H38 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9)
+(pr0 u u3)).(\lambda (H35: (pr0 t3 t8)).(let H36 \def (eq_ind_r T t5 (\lambda
+(t9: T).(eq T (THead (Flat Appl) u1 t9) t)) H22 (THead (Bind Abst) u t3) H19)
+in (let H37 \def (eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9)
\to (\forall (t10: T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to
(ex2 T (\lambda (t12: T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11
-t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H37) in (let
-H39 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H21) in (ex2_ind T
+t12)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t3)) H36) in (let
+H38 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H7 u1 H20) in (ex2_ind T
(\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda
(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead
(Flat Appl) u2 (THead (Bind Abst) u3 t8)) t9))) (\lambda (x: T).(\lambda
-(H40: (pr0 v2 x)).(\lambda (H41: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0
+(H39: (pr0 v2 x)).(\lambda (H40: (pr0 u2 x)).(ex2_ind T (\lambda (t9: T).(pr0
t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead
(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
-(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H42: (pr0 t8
-x0)).(\lambda (H43: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead
+(Bind Abst) u3 t8)) t9))) (\lambda (x0: T).(\lambda (H41: (pr0 t8
+x0)).(\lambda (H42: (pr0 t4 x0)).(ex_intro2 T (\lambda (t9: T).(pr0 (THead
(Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 (THead
-(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H40 t4 x0
-H43 (Bind Abbr)) (pr0_beta u3 u2 x H41 t8 x0 H42))))) (H38 t3 (tlt_trans
+(Bind Abst) u3 t8)) t9)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H39 t4 x0
+H42 (Bind Abbr)) (pr0_beta u3 u2 x H40 t8 x0 H41))))) (H37 t3 (tlt_trans
(THead (Bind Abst) u t3) t3 (THead (Flat Appl) u1 (THead (Bind Abst) u t3))
(tlt_head_dx (Bind Abst) u t3) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst)
-u t3))) t8 H36 t4 H8))))) (H38 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind
-Abst) u t3)) v2 H39 u2 H14))))))) t6 H34)) t7 (sym_eq T t7 t3 H33))) u0
-(sym_eq T u0 u H32))) k0 (sym_eq K k0 (Bind Abst) H31))) H30)) H29)) H28 H25
-H26))) | (pr0_beta u0 v0 v3 H25 t7 t8 H26) \Rightarrow (\lambda (H27: (eq T
+u t3))) t8 H35 t4 H8))))) (H37 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind
+Abst) u t3)) v2 H38 u2 H14))))))) t6 H33)) t7 (sym_eq T t7 t3 H32))) u0
+(sym_eq T u0 u H31))) k0 (sym_eq K k0 (Bind Abst) H30))) H29)) H28)) H27 H24
+H25))) | (pr0_beta u0 v0 v3 H24 t7 t8 H25) \Rightarrow (\lambda (H26: (eq T
(THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (THead (Bind Abst) u
-t3))).(\lambda (H28: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H29 \def
+t3))).(\lambda (H27: (eq T (THead (Bind Abbr) v3 t8) t6)).((let H28 \def
(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t7)) (\lambda (e:
T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T
+\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T
(THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to (ex2 T
(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 t6) t9)))))) H29)) H28 H25 H26))) | (pr0_upsilon b H25
-v0 v3 H26 u0 u3 H27 t7 t8 H28) \Rightarrow (\lambda (H29: (eq T (THead (Flat
-Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H30:
+(THead (Flat Appl) u2 t6) t9)))))) H28)) H27 H24 H25))) | (pr0_upsilon b H24
+v0 v3 H25 u0 u3 H26 t7 t8 H27) \Rightarrow (\lambda (H28: (eq T (THead (Flat
+Appl) v0 (THead (Bind b) u0 t7)) (THead (Bind Abst) u t3))).(\lambda (H29:
(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6)).((let
-H31 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e:
+H30 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t7)) (\lambda (e:
T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H29) in (False_ind ((eq T
+\Rightarrow True])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T
(THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not
(eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to (ex2 T
(\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0
-(THead (Flat Appl) u2 t6) t9)))))))) H31)) H30 H25 H26 H27 H28))) |
-(pr0_delta u0 u3 H25 t7 t8 H26 w H27) \Rightarrow (\lambda (H28: (eq T (THead
-(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H29: (eq T (THead
-(Bind Abbr) u3 w) t6)).((let H30 \def (eq_ind T (THead (Bind Abbr) u0 t7)
+(THead (Flat Appl) u2 t6) t9)))))))) H30)) H29 H24 H25 H26 H27))) |
+(pr0_delta u0 u3 H24 t7 t8 H25 w H26) \Rightarrow (\lambda (H27: (eq T (THead
+(Bind Abbr) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T (THead
+(Bind Abbr) u3 w) t6)).((let H29 \def (eq_ind T (THead (Bind Abbr) u0 t7)
(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow
(match k0 in K return (\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match
b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst
\Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow
-False])])) I (THead (Bind Abst) u t3) H28) in (False_ind ((eq T (THead (Bind
+False])])) I (THead (Bind Abst) u t3) H27) in (False_ind ((eq T (THead (Bind
Abbr) u3 w) t6) \to ((pr0 u0 u3) \to ((pr0 t7 t8) \to ((subst0 O u3 t8 w) \to
(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H30)) H29 H25 H26 H27))) |
-(pr0_zeta b H25 t7 t8 H26 u0) \Rightarrow (\lambda (H27: (eq T (THead (Bind
-b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H28: (eq T t8
-t6)).((let H29 \def (f_equal T T (\lambda (e: T).(match e in T return
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))))) H29)) H28 H24 H25 H26))) |
+(pr0_zeta b H24 t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead (Bind
+b) u0 (lift (S O) O t7)) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8
+t6)).((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return
(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
\to nat))) (d: nat) (t9: T) on t9: T \def (match t9 with [(TSort n)
\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
(THead k0 u3 t10) \Rightarrow (THead k0 (lref_map f d u3) (lref_map f (s k0
d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
t9) \Rightarrow t9])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind
-Abst) u t3) H27) in ((let H30 \def (f_equal T T (\lambda (e: T).(match e in T
+Abst) u t3) H26) in ((let H29 \def (f_equal T T (\lambda (e: T).(match e in T
return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
\Rightarrow u0 | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b) u0 (lift (S
-O) O t7)) (THead (Bind Abst) u t3) H27) in ((let H31 \def (f_equal T B
+O) O t7)) (THead (Bind Abst) u t3) H26) in ((let H30 \def (f_equal T B
(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
\Rightarrow b | (TLRef _) \Rightarrow b | (THead k0 _ _) \Rightarrow (match
k0 in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
\Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t7)) (THead (Bind Abst) u
-t3) H27) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S
+t3) H26) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S
O) O t7) t3) \to ((eq T t8 t6) \to ((not (eq B b0 Abst)) \to ((pr0 t7 t8) \to
(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9:
-T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H32: (eq T u0
+T).(pr0 (THead (Flat Appl) u2 t6) t9))))))))) (\lambda (H31: (eq T u0
u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t7) t3) \to ((eq T t8
t6) \to ((not (eq B Abst Abst)) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t10:
T).(pr0 (THead (Bind Abbr) v2 t4) t10)) (\lambda (t10: T).(pr0 (THead (Flat
-Appl) u2 t6) t10)))))))) (\lambda (H33: (eq T (lift (S O) O t7) t3)).(eq_ind
+Appl) u2 t6) t10)))))))) (\lambda (H32: (eq T (lift (S O) O t7) t3)).(eq_ind
T (lift (S O) O t7) (\lambda (_: T).((eq T t8 t6) \to ((not (eq B Abst Abst))
\to ((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10))))))) (\lambda
-(H34: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to
+(H33: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not (eq B Abst Abst)) \to
((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) v2 t4)
t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u2 t6) t10)))))) (\lambda
-(H35: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H37 \def (match
-(H35 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
+(H34: (not (eq B Abst Abst))).(\lambda (_: (pr0 t7 t6)).(let H36 \def (match
+(H34 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
(t9: T).(pr0 (THead (Bind Abbr) v2 t4) t9)) (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u2 t6) t9)))) with []) in H37))) t8 (sym_eq T t8 t6 H34))) t3
-H33)) u0 (sym_eq T u0 u H32))) b (sym_eq B b Abst H31))) H30)) H29)) H28 H25
-H26))) | (pr0_epsilon t7 t8 H25 u0) \Rightarrow (\lambda (H26: (eq T (THead
-(Flat Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H27: (eq T t8
-t6)).((let H28 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e:
+(Flat Appl) u2 t6) t9)))) with []) in H36))) t8 (sym_eq T t8 t6 H33))) t3
+H32)) u0 (sym_eq T u0 u H31))) b (sym_eq B b Abst H30))) H29)) H28)) H27 H24
+H25))) | (pr0_epsilon t7 t8 H24 u0) \Rightarrow (\lambda (H25: (eq T (THead
+(Flat Cast) u0 t7) (THead (Bind Abst) u t3))).(\lambda (H26: (eq T t8
+t6)).((let H27 \def (eq_ind T (THead (Flat Cast) u0 t7) (\lambda (e:
T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind Abst) u t3) H26) in (False_ind ((eq T t8
+\Rightarrow True])])) I (THead (Bind Abst) u t3) H25) in (False_ind ((eq T t8
t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) v2
-t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H28)) H27
-H25)))]) in (H25 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T
-t6))))) t5 H22)) v1 (sym_eq T v1 u1 H21))) k H20)) H19)) H18)))]) in (H17
-(refl_equal T (THead k u1 t5))))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta
-u0 v0 v3 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0
+t4) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u2 t6) t9))))) H27)) H26
+H24)))]) in (H24 (refl_equal T (THead (Bind Abst) u t3)) (refl_equal T
+t6))))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta u0
+v0 v3 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0
(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v3 t6)
t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (\lambda (_:
T).((eq T (THead (Bind Abbr) v3 t6) t2) \to ((pr0 v0 v3) \to ((pr0 t5 t6) \to
(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H14: (pr0 v0 v3)).(\lambda (H15:
(pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind
-Abst) u0 t5)) H11) in (let H17 \def (match H16 in eq return (\lambda (t7:
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v0 (THead (Bind
-Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4)
-t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))))) with
-[refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v1 (THead
-(Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)))).(let
-H18 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
-with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
-\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow
-t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0
-(THead (Bind Abst) u0 t5)) H17) in ((let H19 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u |
-(TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match t7 in T return
+Abst) u0 t5)) H11) in (let H17 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1
+(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5))
+H16) in ((let H18 \def (f_equal T T (\lambda (e: T).(match e in T return
(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u
-t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H17) in ((let H20 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _)
-\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead
-(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H17) in (eq_ind T v0 (\lambda (_:
-T).((eq T u u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6)
-t8)))))) (\lambda (H21: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3
-t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8))))) (\lambda (H22: (eq T t3
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind
-Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))) (let
-H23 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
-(THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) H11) in (let H24 \def
-(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H8 t5 H22) in (let H25 \def
-(eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H7 v0 H20) in (ex2_ind T (\lambda
-(t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) v3 t6) t7))) (\lambda (x: T).(\lambda (H26: (pr0 v2 x)).(\lambda (H27:
-(pr0 v3 x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6
-t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda
-(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda (x0: T).(\lambda (H28:
-(pr0 t4 x0)).(\lambda (H29: (pr0 t6 x0)).(ex_intro2 T (\lambda (t7: T).(pr0
-(THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3
-t6) t7)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H26 t4 x0 H28 (Bind Abbr))
-(pr0_comp v3 x H27 t6 x0 H29 (Bind Abbr)))))) (H23 t5 (tlt_trans (THead (Bind
-Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx
-(Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4
-H24 t6 H15))))) (H23 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0
-t5)) v2 H25 v3 H14))))) t3 (sym_eq T t3 t5 H22))) u (sym_eq T u u0 H21))) v1
-(sym_eq T v1 v0 H20))) H19)) H18)))]) in (H17 (refl_equal T (THead (Flat
-Appl) v0 (THead (Bind Abst) u0 t5)))))))) t2 H13)) t H11 H12 H9 H10))) |
-(pr0_upsilon b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13:
-(eq T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T
-(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b
-Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2
-t8))))))))) (\lambda (H15: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v3) t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v3) t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind
-Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq
-B b Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_:
-(pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
-Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind
-b) u1 t5)) H13) in (let H21 \def (match H20 in eq return (\lambda (t7:
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v0 (THead (Bind b)
-u1 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8))
-(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3)
-t6)) t8)))))) with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat
-Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1
-t5)))).(let H22 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
-| (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8)
+(THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _)
\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead
-(Flat Appl) v0 (THead (Bind b) u1 t5)) H21) in ((let H23 \def (f_equal T T
+(Flat Appl) v0 (THead (Bind Abst) u0 t5)) H16) in ((let H19 \def (f_equal T T
(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7) \Rightarrow (match
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
-\Rightarrow u | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1
-(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21)
-in ((let H24 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
-(_: T).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst |
-(THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with
-[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1
-(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21)
-in ((let H25 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead
-_ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3))
-(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H21) in (eq_ind T v0 (\lambda
-(_: T).((eq B Abst b) \to ((eq T u u1) \to ((eq T t3 t5) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t8))))))) (\lambda
-(H26: (eq B Abst b)).(eq_ind B Abst (\lambda (b0: B).((eq T u u1) \to ((eq T
-t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7))
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match
+t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1
+(THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5))
+H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq T v1 v0)).(let H22 \def
+(eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8:
+T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10:
+T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl)
+v0 (THead (Bind Abst) u0 t5)) H11) in (let H23 \def (eq_ind T v1 (\lambda
+(t7: T).(pr0 t7 v2)) H7 v0 H21) in (let H24 \def (eq_ind T t3 (\lambda (t7:
+T).(pr0 t7 t4)) H8 t5 H19) in (ex2_ind T (\lambda (t7: T).(pr0 t4 t7))
+(\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr)
+v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))) (\lambda
+(x: T).(\lambda (H25: (pr0 t4 x)).(\lambda (H26: (pr0 t6 x)).(ex2_ind T
+(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) v3 t6) t7))) (\lambda (x0: T).(\lambda (H27: (pr0 v2
+x0)).(\lambda (H28: (pr0 v3 x0)).(ex_intro2 T (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7))
+(THead (Bind Abbr) x0 x) (pr0_comp v2 x0 H27 t4 x H25 (Bind Abbr)) (pr0_comp
+v3 x0 H28 t6 x H26 (Bind Abbr)))))) (H22 v0 (tlt_head_sx (Flat Appl) v0
+(THead (Bind Abst) u0 t5)) v2 H23 v3 H14))))) (H22 t5 (tlt_trans (THead (Bind
+Abst) u0 t5) t5 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t5)) (tlt_head_dx
+(Bind Abst) u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t5))) t4
+H24 t6 H15)))))))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon
+b H9 v0 v3 H10 u1 u2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead
+(Flat Appl) v0 (THead (Bind b) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind
+b) u2 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Flat
+Appl) v0 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b Abst)) \to
+((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))))
+(\lambda (H15: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3)
+t6)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3)
+t6)) (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2)
+\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4)
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (H16: (not (eq B b
+Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
+t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl)
+v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1
+t5)) H13) in (let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1
+| (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H22 \def
+(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
+[(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t7)
+\Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 |
+(Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u
+t3)) (THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in ((let H23 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t7)
+\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t8 _) \Rightarrow t8])]))
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Appl) v0 (THead
+(Bind b) u1 t5)) H20) in ((let H24 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead
+_ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t3))
+(THead (Flat Appl) v0 (THead (Bind b) u1 t5)) H20) in (\lambda (_: (eq T u
+u1)).(\lambda (H26: (eq B Abst b)).(\lambda (_: (eq T v1 v0)).(eq_ind B Abst
+(\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7))
(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O
-v3) t6)) t7)))))) (\lambda (H27: (eq T u u1)).(eq_ind T u1 (\lambda (_:
-T).((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4)
-t8)) (\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S
-O) O v3) t6)) t8))))) (\lambda (H28: (eq T t3 t5)).(eq_ind T t5 (\lambda (_:
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t6))
-t8)))) (let H29 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16
-Abst H26) in (let H30 \def (match (H29 (refl_equal B Abst)) in False return
-(\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4)
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S
-O) O v3) t6)) t7)))) with []) in H30)) t3 (sym_eq T t3 t5 H28))) u (sym_eq T
-u u1 H27))) b H26)) v1 (sym_eq T v1 v0 H25))) H24)) H23)) H22)))]) in (H21
-(refl_equal T (THead (Flat Appl) v0 (THead (Bind b) u1 t5)))))))))) t2 H15))
-t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6 H10 w H11)
-\Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5) t)).(\lambda (H13:
-(eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t5)
-(\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to
-((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14:
-(eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w)
+v3) t6)) t7)))) (let H28 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
+Abst))) H16 Abst H26) in (let H29 \def (match (H28 (refl_equal B Abst)) in
+False return (\lambda (_: False).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat
+Appl) (lift (S O) O v3) t6)) t7)))) with []) in H29)) b H26))))) H23)) H22))
+H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6
+H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5)
+t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
+Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
+(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda
+(H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w)
(\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to
(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t5
t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda
(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead
-(Bind Abbr) u1 t5) H12) in (let H19 \def (match H18 in eq return (\lambda
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) u2 w) t8)))))) with [refl_equal \Rightarrow
-(\lambda (H19: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead
-(Bind Abbr) u1 t5))).(let H20 \def (eq_ind T (THead (Flat Appl) v1 (THead
-(Bind Abst) u t3)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
-t5) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4)
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))) H20)))]) in (H19
-(refl_equal T (THead (Bind Abbr) u1 t5)))))))) t2 H14)) t H12 H13 H9 H10
-H11))) | (pr0_zeta b H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead
-(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T
-(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not
-(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind
-Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6
-t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5
-t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
-(THead (Bind Abst) u t3)) t7)) H4 (THead (Bind b) u0 (lift (S O) O t5)) H11)
-in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda (_: (eq ?
-? t7)).((eq T t7 (THead (Bind b) u0 (lift (S O) O t5))) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))
-with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v1
-(THead (Bind Abst) u t3)) (THead (Bind b) u0 (lift (S O) O t5)))).(let H18
-\def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H17) in
-(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7))
-(\lambda (t7: T).(pr0 t2 t7))) H18)))]) in (H17 (refl_equal T (THead (Bind b)
-u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) |
-(pr0_epsilon t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast)
-u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5)
-(\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8))))))
-(\lambda (H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t
+(Bind Abbr) u1 t5) H12) in (let H19 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+Abbr) u1 t5) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)))
+H19)))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b H9 t5 t6 H10 u0)
+\Rightarrow (\lambda (H11: (eq T (THead (Bind b) u0 (lift (S O) O t5))
+t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O
+t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6)
+\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8:
+T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7:
+T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0
+(THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_:
+(not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t
(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7))
-H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (match H14 in eq return
-(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u0
-t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda
-(t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T
-(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (THead (Flat Cast) u0
-t5))).(let H16 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3))
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
-(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
-\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u0 t5)
-H15) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4)
-t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15 (refl_equal T (THead
-(Flat Cast) u0 t5)))))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in (H9
-(refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) |
-(pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5) \Rightarrow (\lambda (H6: (eq T
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t)).(\lambda (H7: (eq T (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T (THead
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1) \to ((not (eq B b Abst)) \to
-((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6:
-T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))))) (\lambda (H8: (eq T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)).(eq_ind T
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t5:
-T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4)
-\to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))))
-(\lambda (H9: (not (eq B b Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11:
-(pr0 u1 u2)).(\lambda (H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0
-return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5
-t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2
-t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H13: (eq T t5
-t)).(\lambda (H14: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
+H4 (THead (Bind b) u0 (lift (S O) O t5)) H11) in (let H17 \def (eq_ind T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (ee: T).(match ee in
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
+True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H16) in (False_ind (ex2 T
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2
+t7))) H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon
+t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5)
+t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda
+(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq
+T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) v2 t4) t8)) (\lambda (t8: T).(pr0 t2 t8)))))
+(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T
+(THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t7)) H4 (THead (Flat Cast) u0
+t5) H10) in (let H15 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst)
+u t3)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
+F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead
+(Flat Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead
+(Bind Abbr) v2 t4) t7)) (\lambda (t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T
+t6 t2 H12))) t H10 H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2)))))
+t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t3 t4 H5)
+\Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to
+((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0
+t2 t6))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) (\lambda (t5: T).((not (eq B b Abst)) \to ((pr0 v1 v2)
+\to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6))
+(\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H9: (not (eq B b
+Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda
+(H12: (pr0 t3 t4)).(let H13 \def (match H1 in pr0 return (\lambda (t5:
+T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H15: (eq T t
-t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind
-b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2
-t7)))) (let H16 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead
-(Flat Appl) v1 (THead (Bind b) u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with [(pr0_refl t5)
+\Rightarrow (\lambda (H13: (eq T t5 t)).(\lambda (H14: (eq T t5 t2)).(eq_ind
+T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 (THead
(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7:
-T).(pr0 t6 t7)))) (let H17 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6))
-H13 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6) in (let H18 \def
-(eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7:
-T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9:
-T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) H6) in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead
-(Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1
-v2 v2 H10 (pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5
-t H13) H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15:
-(eq T (THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T
+T).(pr0 t2 t7))))) (\lambda (H15: (eq T t t2)).(eq_ind T t2 (\lambda (_:
+T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H16 \def (eq_ind_r
+T t (\lambda (t6: T).(eq T t6 t2)) H15 (THead (Flat Appl) v1 (THead (Bind b)
+u1 t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H17
+\def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H13 (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) H6) in (let H18 \def (eq_ind_r T t (\lambda (t6:
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
+T).(pr0 t8 t9)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H6)
+in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t3))) (pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)))
+(pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t3 t4 H12 v1 v2 v2 H10
+(pr0_refl v2))))) t2 H16)) t (sym_eq T t t2 H15))) t5 (sym_eq T t5 t H13)
+H14))) | (pr0_comp u0 u3 H13 t5 t6 H14 k) \Rightarrow (\lambda (H15: (eq T
+(THead k u0 t5) t)).(\lambda (H16: (eq T (THead k u3 t6) t2)).(eq_ind T
(THead k u0 t5) (\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3)
\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead
(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))
(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
T).(pr0 t7 t8)))))) (\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t5
t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def (match
-H20 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead
-k u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead k u3 t6)
-t8)))))) with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat Appl)
-v1 (THead (Bind b) u1 t3)) (THead k u0 t5))).(let H22 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead (Bind b) u1
-t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b)
-u1 t3)) (THead k u0 t5) H21) in ((let H23 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
-(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat
-Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H21) in ((let H24 \def
+(THead (Bind b) u1 t3)) t7)) H6 (THead k u0 t5) H15) in (let H21 \def
(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
[(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) |
(THead k0 _ _) \Rightarrow k0])) (THead (Flat Appl) v1 (THead (Bind b) u1
-t3)) (THead k u0 t5) H21) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v1
-u0) \to ((eq T (THead (Bind b) u1 t3) t5) \to (ex2 T (\lambda (t7: T).(pr0
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda
-(t7: T).(pr0 (THead k0 u3 t6) t7)))))) (\lambda (H25: (eq T v1 u0)).(eq_ind T
-u0 (\lambda (_: T).((eq T (THead (Bind b) u1 t3) t5) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8))
-(\lambda (t8: T).(pr0 (THead (Flat Appl) u3 t6) t8))))) (\lambda (H26: (eq T
-(THead (Bind b) u1 t3) t5)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (_:
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Flat Appl) u3 t6) t8))))
-(let H27 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15
-(Flat Appl) H24) in (let H28 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7
-t6)) H19 (THead (Bind b) u1 t3) H26) in (let H29 \def (match H28 in pr0
-return (\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7
-(THead (Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda
-(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7)
-\Rightarrow (\lambda (H29: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H30:
-(eq T t7 t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6)
-\to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))
-(\lambda (H31: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1
+t3)) (THead k u0 t5) H20) in ((let H22 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 |
+(TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat
+Appl) v1 (THead (Bind b) u1 t3)) (THead k u0 t5) H20) in ((let H23 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow (THead (Bind b) u1 t3) | (TLRef _) \Rightarrow (THead
+(Bind b) u1 t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (THead k u0 t5) H20) in (\lambda (H24: (eq T v1
+u0)).(\lambda (H25: (eq K (Flat Appl) k)).(eq_ind K (Flat Appl) (\lambda (k0:
+K).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
+(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7)))) (let H26
+\def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H15 (Flat
+Appl) H25) in (let H27 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H19
+(THead (Bind b) u1 t3) H23) in (let H28 \def (match H27 in pr0 return
+(\lambda (t7: T).(\lambda (t8: T).(\lambda (_: (pr0 t7 t8)).((eq T t7 (THead
+(Bind b) u1 t3)) \to ((eq T t8 t6) \to (ex2 T (\lambda (t9: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9:
+T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) with [(pr0_refl t7) \Rightarrow
+(\lambda (H28: (eq T t7 (THead (Bind b) u1 t3))).(\lambda (H29: (eq T t7
+t6)).(eq_ind T (THead (Bind b) u1 t3) (\lambda (t8: T).((eq T t8 t6) \to (ex2
+T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
+v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))
+(\lambda (H30: (eq T (THead (Bind b) u1 t3) t6)).(eq_ind T (THead (Bind b) u1
t3) (\lambda (t8: T).(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead
(Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat
-Appl) u3 t8) t9)))) (let H32 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T
-(THead (Flat Appl) u0 t8) t)) H27 (THead (Bind b) u1 t3) H26) in (let H33
+Appl) u3 t8) t9)))) (let H31 \def (eq_ind_r T t5 (\lambda (t8: T).(eq T
+(THead (Flat Appl) u0 t8) t)) H26 (THead (Bind b) u1 t3) H23) in (let H32
\def (eq_ind_r T t (\lambda (t8: T).(\forall (v: T).((tlt v t8) \to (\forall
(t9: T).((pr0 v t9) \to (\forall (t10: T).((pr0 v t10) \to (ex2 T (\lambda
(t11: T).(pr0 t9 t11)) (\lambda (t11: T).(pr0 t10 t11)))))))))) H (THead
-(Flat Appl) u0 (THead (Bind b) u1 t3)) H32) in (let H34 \def (eq_ind T v1
-(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H25) in (ex2_ind T (\lambda (t8: T).(pr0
+(Flat Appl) u0 (THead (Bind b) u1 t3)) H31) in (let H33 \def (eq_ind T v1
+(\lambda (t8: T).(pr0 t8 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t8: T).(pr0
v2 t8)) (\lambda (t8: T).(pr0 u3 t8)) (ex2 T (\lambda (t8: T).(pr0 (THead
(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t3)) t8))) (\lambda (x:
-T).(\lambda (H35: (pr0 v2 x)).(\lambda (H36: (pr0 u3 x)).(ex2_sym T (pr0
+T).(\lambda (H34: (pr0 v2 x)).(\lambda (H35: (pr0 u3 x)).(ex2_sym T (pr0
(THead (Flat Appl) u3 (THead (Bind b) u1 t3))) (pr0 (THead (Bind b) u2 (THead
(Flat Appl) (lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_refl b
-H9 u1 u2 H11 t3 t4 H12 u3 v2 x H36 H35))))) (H33 u0 (tlt_head_sx (Flat Appl)
-u0 (THead (Bind b) u1 t3)) v2 H34 u3 H18))))) t6 H31)) t7 (sym_eq T t7 (THead
-(Bind b) u1 t3) H29) H30))) | (pr0_comp u4 u5 H29 t7 t8 H30 k0) \Rightarrow
-(\lambda (H31: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32:
-(eq T (THead k0 u5 t8) t6)).((let H33 \def (f_equal T T (\lambda (e:
+H9 u1 u2 H11 t3 t4 H12 u3 v2 x H35 H34))))) (H32 u0 (tlt_head_sx (Flat Appl)
+u0 (THead (Bind b) u1 t3)) v2 H33 u3 H18))))) t6 H30)) t7 (sym_eq T t7 (THead
+(Bind b) u1 t3) H28) H29))) | (pr0_comp u4 u5 H28 t7 t8 H29 k0) \Rightarrow
+(\lambda (H30: (eq T (THead k0 u4 t7) (THead (Bind b) u1 t3))).(\lambda (H31:
+(eq T (THead k0 u5 t8) t6)).((let H32 \def (f_equal T T (\lambda (e:
T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead k0 u4 t7)
-(THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T (\lambda (e:
+(THead (Bind b) u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e:
T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u4 |
(TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9])) (THead k0 u4 t7)
-(THead (Bind b) u1 t3) H31) in ((let H35 \def (f_equal T K (\lambda (e:
+(THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T K (\lambda (e:
T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u4 t7)
-(THead (Bind b) u1 t3) H31) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4
+(THead (Bind b) u1 t3) H30) in (eq_ind K (Bind b) (\lambda (k1: K).((eq T u4
u1) \to ((eq T t7 t3) \to ((eq T (THead k1 u5 t8) t6) \to ((pr0 u4 u5) \to
((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat
Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3
-t6) t9))))))))) (\lambda (H36: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9:
+t6) t9))))))))) (\lambda (H35: (eq T u4 u1)).(eq_ind T u1 (\lambda (t9:
T).((eq T t7 t3) \to ((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 t9 u5) \to
((pr0 t7 t8) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead
(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat
-Appl) u3 t6) t10)))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda
+Appl) u3 t6) t10)))))))) (\lambda (H36: (eq T t7 t3)).(eq_ind T t3 (\lambda
(t9: T).((eq T (THead (Bind b) u5 t8) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8)
\to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift
(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
-t10))))))) (\lambda (H38: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead
+t10))))))) (\lambda (H37: (eq T (THead (Bind b) u5 t8) t6)).(eq_ind T (THead
(Bind b) u5 t8) (\lambda (t9: T).((pr0 u1 u5) \to ((pr0 t3 t8) \to (ex2 T
(\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t9) t10))))))
-(\lambda (H39: (pr0 u1 u5)).(\lambda (H40: (pr0 t3 t8)).(let H41 \def
-(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H27
-(THead (Bind b) u1 t3) H26) in (let H42 \def (eq_ind_r T t (\lambda (t9:
+(\lambda (H38: (pr0 u1 u5)).(\lambda (H39: (pr0 t3 t8)).(let H40 \def
+(eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t)) H26
+(THead (Bind b) u1 t3) H23) in (let H41 \def (eq_ind_r T t (\lambda (t9:
T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v t10) \to
(\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10 t12))
(\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead (Bind
-b) u1 t3)) H41) in (let H43 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2))
-H10 u0 H25) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0
+b) u1 t3)) H40) in (let H42 \def (eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2))
+H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0
u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
(lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3
-(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H44: (pr0 v2
-x)).(\lambda (H45: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9))
+(THead (Bind b) u5 t8)) t9))) (\lambda (x: T).(\lambda (H43: (pr0 v2
+x)).(\lambda (H44: (pr0 u3 x)).(ex2_ind T (\lambda (t9: T).(pr0 t8 t9))
(\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2
(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H46:
-(pr0 t8 x0)).(\lambda (H47: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5
+(Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x0: T).(\lambda (H45:
+(pr0 t8 x0)).(\lambda (H46: (pr0 t4 x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5
t9)) (\lambda (t9: T).(pr0 u2 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead (Bind
b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0
(THead (Flat Appl) u3 (THead (Bind b) u5 t8)) t9))) (\lambda (x1: T).(\lambda
-(H48: (pr0 u5 x1)).(\lambda (H49: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat
+(H47: (pr0 u5 x1)).(\lambda (H48: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat
Appl) u3 (THead (Bind b) u5 t8))) (pr0 (THead (Bind b) u2 (THead (Flat Appl)
(lift (S O) O v2) t4))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x
-H45 H44 t8 t4 x0 H46 H47 u5 u2 x1 H48 H49))))) (H42 u1 (tlt_trans (THead
+H44 H43 t8 t4 x0 H45 H46 u5 u2 x1 H47 H48))))) (H41 u1 (tlt_trans (THead
(Bind b) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t3)) (tlt_head_sx
-(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H39
-u2 H11))))) (H42 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl)
+(Bind b) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t3))) u5 H38
+u2 H11))))) (H41 t3 (tlt_trans (THead (Bind b) u1 t3) t3 (THead (Flat Appl)
u0 (THead (Bind b) u1 t3)) (tlt_head_dx (Bind b) u1 t3) (tlt_head_dx (Flat
-Appl) u0 (THead (Bind b) u1 t3))) t8 H40 t4 H12))))) (H42 u0 (tlt_head_sx
-(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H43 u3 H18))))))) t6 H38)) t7
-(sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) k0 (sym_eq K k0 (Bind b)
-H35))) H34)) H33)) H32 H29 H30))) | (pr0_beta u v0 v3 H29 t7 t8 H30)
-\Rightarrow (\lambda (H31: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u
-t7)) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T (THead (Bind Abbr) v3 t8)
-t6)).((let H33 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7))
+Appl) u0 (THead (Bind b) u1 t3))) t8 H39 t4 H12))))) (H41 u0 (tlt_head_sx
+(Flat Appl) u0 (THead (Bind b) u1 t3)) v2 H42 u3 H18))))))) t6 H37)) t7
+(sym_eq T t7 t3 H36))) u4 (sym_eq T u4 u1 H35))) k0 (sym_eq K k0 (Bind b)
+H34))) H33)) H32)) H31 H28 H29))) | (pr0_beta u v0 v3 H28 t7 t8 H29)
+\Rightarrow (\lambda (H30: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u
+t7)) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T (THead (Bind Abbr) v3 t8)
+t6)).((let H32 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t7))
(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
\Rightarrow False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow
(match k0 in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False
-| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H31) in (False_ind
+| (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind
((eq T (THead (Bind Abbr) v3 t8) t6) \to ((pr0 v0 v3) \to ((pr0 t7 t8) \to
(ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))))))
-H33)) H32 H29 H30))) | (pr0_upsilon b0 H29 v0 v3 H30 u4 u5 H31 t7 t8 H32)
-\Rightarrow (\lambda (H33: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4
-t7)) (THead (Bind b) u1 t3))).(\lambda (H34: (eq T (THead (Bind b0) u5 (THead
-(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H35 \def (eq_ind T (THead (Flat
+H32)) H31 H28 H29))) | (pr0_upsilon b0 H28 v0 v3 H29 u4 u5 H30 t7 t8 H31)
+\Rightarrow (\lambda (H32: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4
+t7)) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T (THead (Bind b0) u5 (THead
+(Flat Appl) (lift (S O) O v3) t8)) t6)).((let H34 \def (eq_ind T (THead (Flat
Appl) v0 (THead (Bind b0) u4 t7)) (\lambda (e: T).(match e in T return
(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
\Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in K return (\lambda
(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u1 t3) H33) in (False_ind ((eq T (THead (Bind b0)
+True])])) I (THead (Bind b) u1 t3) H32) in (False_ind ((eq T (THead (Bind b0)
u5 (THead (Flat Appl) (lift (S O) O v3) t8)) t6) \to ((not (eq B b0 Abst))
\to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9:
T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9))
-(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H35)) H34 H29 H30
-H31 H32))) | (pr0_delta u4 u5 H29 t7 t8 H30 w H31) \Rightarrow (\lambda (H32:
-(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H33: (eq T
-(THead (Bind Abbr) u5 w) t6)).((let H34 \def (f_equal T T (\lambda (e:
+(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))) H34)) H33 H28 H29
+H30 H31))) | (pr0_delta u4 u5 H28 t7 t8 H29 w H30) \Rightarrow (\lambda (H31:
+(eq T (THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3))).(\lambda (H32: (eq T
+(THead (Bind Abbr) u5 w) t6)).((let H33 \def (f_equal T T (\lambda (e:
T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t7 |
(TLRef _) \Rightarrow t7 | (THead _ _ t9) \Rightarrow t9])) (THead (Bind
-Abbr) u4 t7) (THead (Bind b) u1 t3) H32) in ((let H35 \def (f_equal T T
+Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H34 \def (f_equal T T
(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
\Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t9 _) \Rightarrow t9]))
-(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H32) in ((let H36 \def
+(THead (Bind Abbr) u4 t7) (THead (Bind b) u1 t3) H31) in ((let H35 \def
(f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B) with
[(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k0 _ _)
\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t7)
-(THead (Bind b) u1 t3) H32) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1)
+(THead (Bind b) u1 t3) H31) in (eq_ind B Abbr (\lambda (b0: B).((eq T u4 u1)
\to ((eq T t7 t3) \to ((eq T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u4 u5)
\to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t9: T).(pr0
(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda
-(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H37: (eq T u4
+(t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))) (\lambda (H36: (eq T u4
u1)).(eq_ind T u1 (\lambda (t9: T).((eq T t7 t3) \to ((eq T (THead (Bind
Abbr) u5 w) t6) \to ((pr0 t9 u5) \to ((pr0 t7 t8) \to ((subst0 O u5 t8 w) \to
(ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift
(S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
-t10))))))))) (\lambda (H38: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq
+t10))))))))) (\lambda (H37: (eq T t7 t3)).(eq_ind T t3 (\lambda (t9: T).((eq
T (THead (Bind Abbr) u5 w) t6) \to ((pr0 u1 u5) \to ((pr0 t9 t8) \to ((subst0
O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead (Bind Abbr) u2 (THead
(Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat
-Appl) u3 t6) t10)))))))) (\lambda (H39: (eq T (THead (Bind Abbr) u5 w)
+Appl) u3 t6) t10)))))))) (\lambda (H38: (eq T (THead (Bind Abbr) u5 w)
t6)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t9: T).((pr0 u1 u5) \to
((pr0 t3 t8) \to ((subst0 O u5 t8 w) \to (ex2 T (\lambda (t10: T).(pr0 (THead
(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10:
-T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H40: (pr0 u1
-u5)).(\lambda (H41: (pr0 t3 t8)).(\lambda (H42: (subst0 O u5 t8 w)).(let H43
-\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H26
-Abbr H36) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
-Abst))) H9 Abbr H36) in (let H45 \def (eq_ind_r B b (\lambda (b0: B).(eq T
+T).(pr0 (THead (Flat Appl) u3 t9) t10))))))) (\lambda (H39: (pr0 u1
+u5)).(\lambda (H40: (pr0 t3 t8)).(\lambda (H41: (subst0 O u5 t8 w)).(let H42
+\def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u1 t3) t5)) H23
+Abbr H35) in (let H43 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0
+Abst))) H9 Abbr H35) in (let H44 \def (eq_ind_r B b (\lambda (b0: B).(eq T
(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t1)) H8 Abbr
-H36) in (let H46 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat
-Appl) u0 t9) t)) H27 (THead (Bind Abbr) u1 t3) H43) in (let H47 \def
+H35) in (let H45 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat
+Appl) u0 t9) t)) H26 (THead (Bind Abbr) u1 t3) H42) in (let H46 \def
(eq_ind_r T t (\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10:
T).((pr0 v t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12:
T).(pr0 t10 t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat
-Appl) u0 (THead (Bind Abbr) u1 t3)) H46) in (let H48 \def (eq_ind T v1
-(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H25) in (ex2_ind T (\lambda (t9: T).(pr0
+Appl) u0 (THead (Bind Abbr) u1 t3)) H45) in (let H47 \def (eq_ind T v1
+(\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda (t9: T).(pr0
v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9: T).(pr0 (THead
(Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9:
T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9))) (\lambda (x:
-T).(\lambda (H49: (pr0 v2 x)).(\lambda (H50: (pr0 u3 x)).(ex2_ind T (\lambda
+T).(\lambda (H48: (pr0 v2 x)).(\lambda (H49: (pr0 u3 x)).(ex2_ind T (\lambda
(t9: T).(pr0 t8 t9)) (\lambda (t9: T).(pr0 t4 t9)) (ex2 T (\lambda (t9:
T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t9))
(\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t9)))
-(\lambda (x0: T).(\lambda (H51: (pr0 t8 x0)).(\lambda (H52: (pr0 t4
+(\lambda (x0: T).(\lambda (H50: (pr0 t8 x0)).(\lambda (H51: (pr0 t4
x0)).(ex2_ind T (\lambda (t9: T).(pr0 u5 t9)) (\lambda (t9: T).(pr0 u2 t9))
(ex2 T (\lambda (t9: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift
(S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 (THead
-(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H53: (pr0 u5
-x1)).(\lambda (H54: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead
+(Bind Abbr) u5 w)) t9))) (\lambda (x1: T).(\lambda (H52: (pr0 u5
+x1)).(\lambda (H53: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead
(Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O)
-O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H44 u5 t8 w H42 u3 v2 x
-H50 H49 t4 x0 H51 H52 u2 x1 H53 H54))))) (H47 u1 (tlt_trans (THead (Bind
+O v2) t4))) (pr0_confluence__pr0_cong_upsilon_delta H43 u5 t8 w H41 u3 v2 x
+H49 H48 t4 x0 H50 H51 u2 x1 H52 H53))))) (H46 u1 (tlt_trans (THead (Bind
Abbr) u1 t3) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_sx
(Bind Abbr) u1 t3) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) u5
-H40 u2 H11))))) (H47 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat
+H39 u2 H11))))) (H46 t3 (tlt_trans (THead (Bind Abbr) u1 t3) t3 (THead (Flat
Appl) u0 (THead (Bind Abbr) u1 t3)) (tlt_head_dx (Bind Abbr) u1 t3)
-(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H41 t4 H12)))))
-(H47 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H48 u3
-H18))))))))))) t6 H39)) t7 (sym_eq T t7 t3 H38))) u4 (sym_eq T u4 u1 H37))) b
-H36)) H35)) H34)) H33 H29 H30 H31))) | (pr0_zeta b0 H29 t7 t8 H30 u)
-\Rightarrow (\lambda (H31: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead
-(Bind b) u1 t3))).(\lambda (H32: (eq T t8 t6)).((let H33 \def (f_equal T T
+(tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t3))) t8 H40 t4 H12)))))
+(H46 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t3)) v2 H47 u3
+H18))))))))))) t6 H38)) t7 (sym_eq T t7 t3 H37))) u4 (sym_eq T u4 u1 H36))) b
+H35)) H34)) H33)) H32 H28 H29 H30))) | (pr0_zeta b0 H28 t7 t8 H29 u)
+\Rightarrow (\lambda (H30: (eq T (THead (Bind b0) u (lift (S O) O t7)) (THead
+(Bind b) u1 t3))).(\lambda (H31: (eq T t8 t6)).((let H32 \def (f_equal T T
(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t9: T) on t9: T
\def (match t9 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
| (THead k0 u4 t10) \Rightarrow (THead k0 (lref_map f d u4) (lref_map f (s k0
d) t10))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t7) | (THead _ _
t9) \Rightarrow t9])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b)
-u1 t3) H31) in ((let H34 \def (f_equal T T (\lambda (e: T).(match e in T
+u1 t3) H30) in ((let H33 \def (f_equal T T (\lambda (e: T).(match e in T
return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
\Rightarrow u | (THead _ t9 _) \Rightarrow t9])) (THead (Bind b0) u (lift (S
-O) O t7)) (THead (Bind b) u1 t3) H31) in ((let H35 \def (f_equal T B (\lambda
+O) O t7)) (THead (Bind b) u1 t3) H30) in ((let H34 \def (f_equal T B (\lambda
(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b0
| (TLRef _) \Rightarrow b0 | (THead k0 _ _) \Rightarrow (match k0 in K return
(\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _) \Rightarrow
-b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H31) in
+b0])])) (THead (Bind b0) u (lift (S O) O t7)) (THead (Bind b) u1 t3) H30) in
(eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t7) t3)
\to ((eq T t8 t6) \to ((not (eq B b1 Abst)) \to ((pr0 t7 t8) \to (ex2 T
(\lambda (t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
t4)) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9)))))))))
-(\lambda (H36: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O
+(\lambda (H35: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O
t7) t3) \to ((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to (ex2
T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O
v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10))))))))
-(\lambda (H37: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7)
+(\lambda (H36: (eq T (lift (S O) O t7) t3)).(eq_ind T (lift (S O) O t7)
(\lambda (_: T).((eq T t8 t6) \to ((not (eq B b Abst)) \to ((pr0 t7 t8) \to
(ex2 T (\lambda (t10: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
O) O v2) t4)) t10)) (\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6)
-t10))))))) (\lambda (H38: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not
+t10))))))) (\lambda (H37: (eq T t8 t6)).(eq_ind T t6 (\lambda (t9: T).((not
(eq B b Abst)) \to ((pr0 t7 t9) \to (ex2 T (\lambda (t10: T).(pr0 (THead
(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t10)) (\lambda (t10:
-T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H39: (not (eq B b
-Abst))).(\lambda (H40: (pr0 t7 t6)).(let H41 \def (eq_ind_r T t3 (\lambda
-(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H26 (lift (S O) O t7) H37) in (let
-H42 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t))
-H27 (THead (Bind b) u1 (lift (S O) O t7)) H41) in (let H43 \def (eq_ind_r T t
+T).(pr0 (THead (Flat Appl) u3 t6) t10)))))) (\lambda (H38: (not (eq B b
+Abst))).(\lambda (H39: (pr0 t7 t6)).(let H40 \def (eq_ind_r T t3 (\lambda
+(t9: T).(eq T (THead (Bind b) u1 t9) t5)) H23 (lift (S O) O t7) H36) in (let
+H41 \def (eq_ind_r T t5 (\lambda (t9: T).(eq T (THead (Flat Appl) u0 t9) t))
+H26 (THead (Bind b) u1 (lift (S O) O t7)) H40) in (let H42 \def (eq_ind_r T t
(\lambda (t9: T).(\forall (v: T).((tlt v t9) \to (\forall (t10: T).((pr0 v
t10) \to (\forall (t11: T).((pr0 v t11) \to (ex2 T (\lambda (t12: T).(pr0 t10
t12)) (\lambda (t12: T).(pr0 t11 t12)))))))))) H (THead (Flat Appl) u0 (THead
-(Bind b) u1 (lift (S O) O t7))) H42) in (let H44 \def (eq_ind_r T t3 (\lambda
-(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H37) in (ex2_ind T (\lambda (t9:
+(Bind b) u1 (lift (S O) O t7))) H41) in (let H43 \def (eq_ind_r T t3 (\lambda
+(t9: T).(pr0 t9 t4)) H12 (lift (S O) O t7) H36) in (ex2_ind T (\lambda (t9:
T).(eq T t4 (lift (S O) O t9))) (\lambda (t9: T).(pr0 t7 t9)) (ex2 T (\lambda
(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x:
-T).(\lambda (H45: (eq T t4 (lift (S O) O x))).(\lambda (H46: (pr0 t7
+T).(\lambda (H44: (eq T t4 (lift (S O) O x))).(\lambda (H45: (pr0 t7
x)).(eq_ind_r T (lift (S O) O x) (\lambda (t9: T).(ex2 T (\lambda (t10:
T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t9)) t10))
-(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H47 \def
-(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H25) in (ex2_ind T (\lambda
+(\lambda (t10: T).(pr0 (THead (Flat Appl) u3 t6) t10)))) (let H46 \def
+(eq_ind T v1 (\lambda (t9: T).(pr0 t9 v2)) H10 u0 H24) in (ex2_ind T (\lambda
(t9: T).(pr0 v2 t9)) (\lambda (t9: T).(pr0 u3 t9)) (ex2 T (\lambda (t9:
T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O
x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda (x0:
-T).(\lambda (H48: (pr0 v2 x0)).(\lambda (H49: (pr0 u3 x0)).(ex2_ind T
+T).(\lambda (H47: (pr0 v2 x0)).(\lambda (H48: (pr0 u3 x0)).(ex2_ind T
(\lambda (t9: T).(pr0 x t9)) (\lambda (t9: T).(pr0 t6 t9)) (ex2 T (\lambda
(t9: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S
O) O x))) t9)) (\lambda (t9: T).(pr0 (THead (Flat Appl) u3 t6) t9))) (\lambda
-(x1: T).(\lambda (H50: (pr0 x x1)).(\lambda (H51: (pr0 t6 x1)).(ex2_sym T
+(x1: T).(\lambda (H49: (pr0 x x1)).(\lambda (H50: (pr0 t6 x1)).(ex2_sym T
(pr0 (THead (Flat Appl) u3 t6)) (pr0 (THead (Bind b) u2 (THead (Flat Appl)
(lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta
-b H39 u1 u2 H11 u3 v2 x0 H49 H48 x t6 x1 H50 H51))))) (H43 t7 (tlt_trans
+b H38 u1 u2 H11 u3 v2 x0 H48 H47 x t6 x1 H49 H50))))) (H42 t7 (tlt_trans
(THead (Bind b) u1 (lift (S O) O t7)) t7 (THead (Flat Appl) u0 (THead (Bind
b) u1 (lift (S O) O t7))) (lift_tlt_dx (Bind b) u1 t7 (S O) O) (tlt_head_dx
-(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H46 t6 H40))))) (H43
-u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H47
-u3 H18))) t4 H45)))) (pr0_gen_lift t7 t4 (S O) O H44)))))))) t8 (sym_eq T t8
-t6 H38))) t3 H37)) u (sym_eq T u u1 H36))) b0 (sym_eq B b0 b H35))) H34))
-H33)) H32 H29 H30))) | (pr0_epsilon t7 t8 H29 u) \Rightarrow (\lambda (H30:
-(eq T (THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H31: (eq T
-t8 t6)).((let H32 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e:
+(Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7)))) x H45 t6 H39))))) (H42
+u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t7))) v2 H46
+u3 H18))) t4 H44)))) (pr0_gen_lift t7 t4 (S O) O H43)))))))) t8 (sym_eq T t8
+t6 H37))) t3 H36)) u (sym_eq T u u1 H35))) b0 (sym_eq B b0 b H34))) H33))
+H32)) H31 H28 H29))) | (pr0_epsilon t7 t8 H28 u) \Rightarrow (\lambda (H29:
+(eq T (THead (Flat Cast) u t7) (THead (Bind b) u1 t3))).(\lambda (H30: (eq T
+t8 t6)).((let H31 \def (eq_ind T (THead (Flat Cast) u t7) (\lambda (e:
T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
False | (TLRef _) \Rightarrow False | (THead k0 _ _) \Rightarrow (match k0 in
K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b) u1 t3) H30) in (False_ind ((eq T t8
+\Rightarrow True])])) I (THead (Bind b) u1 t3) H29) in (False_ind ((eq T t8
t6) \to ((pr0 t7 t8) \to (ex2 T (\lambda (t9: T).(pr0 (THead (Bind b) u2
(THead (Flat Appl) (lift (S O) O v2) t4)) t9)) (\lambda (t9: T).(pr0 (THead
-(Flat Appl) u3 t6) t9))))) H32)) H31 H29)))]) in (H29 (refl_equal T (THead
-(Bind b) u1 t3)) (refl_equal T t6))))) t5 H26)) v1 (sym_eq T v1 u0 H25))) k
-H24)) H23)) H22)))]) in (H21 (refl_equal T (THead k u0 t5))))))) t2 H17)) t
+(Flat Appl) u3 t6) t9))))) H31)) H30 H28)))]) in (H28 (refl_equal T (THead
+(Bind b) u1 t3)) (refl_equal T t6))))) k H25)))) H22)) H21))))) t2 H17)) t
H15 H16 H13 H14))) | (pr0_beta u v0 v3 H13 t5 t6 H14) \Rightarrow (\lambda
(H15: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) t)).(\lambda
(H16: (eq T (THead (Bind Abbr) v3 t6) t2)).(eq_ind T (THead (Flat Appl) v0
Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda
(_: (pr0 v0 v3)).(\lambda (_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t
(\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6
-(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def (match
-H20 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead
-(Flat Appl) v0 (THead (Bind Abst) u t5))) \to (ex2 T (\lambda (t8: T).(pr0
-(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))))) with [refl_equal \Rightarrow
-(\lambda (H21: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead
-(Flat Appl) v0 (THead (Bind Abst) u t5)))).(let H22 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow (match
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t8) \Rightarrow t8])])) (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21)
-in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H21 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _)
+\Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat
+Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H22 \def (f_equal T B
+(\lambda (e: T).(match e in T return (\lambda (_: T).B) with [(TSort _)
+\Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match
+t7 in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
+\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
+K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead
+(Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind
+Abst) u t5)) H20) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda
(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead
-_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t8 _) \Rightarrow
+_ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H20) in ((let H24 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
+\Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow
t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0
-(THead (Bind Abst) u t5)) H21) in ((let H24 \def (f_equal T B (\lambda (e:
-T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
-(TLRef _) \Rightarrow b | (THead _ _ t7) \Rightarrow (match t7 in T return
-(\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
-b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])])) (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21)
-in ((let H25 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
-(_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead
-_ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
-(THead (Flat Appl) v0 (THead (Bind Abst) u t5)) H21) in (eq_ind T v0 (\lambda
-(_: T).((eq B b Abst) \to ((eq T u1 u) \to ((eq T t3 t5) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8))))))) (\lambda (H26:
-(eq B b Abst)).(eq_ind B Abst (\lambda (b0: B).((eq T u1 u) \to ((eq T t3 t5)
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6)
-t7)))))) (\lambda (H27: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3
-t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6)
-t8))))) (\lambda (H28: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T
-(\lambda (t8: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O
-v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) v3 t6) t8)))) (let H29
-\def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall
-(t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda
-(t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat
-Appl) v0 (THead (Bind Abst) u t5)) H15) in (let H30 \def (eq_ind T t3
-(\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in (let H31 \def (eq_ind T u1
-(\lambda (t7: T).(pr0 t7 u2)) H11 u H27) in (let H32 \def (eq_ind B b
-(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H33 \def (match
-(H32 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
+(THead (Bind Abst) u t5)) H20) in (\lambda (_: (eq T u1 u)).(\lambda (H26:
+(eq B b Abst)).(\lambda (H27: (eq T v1 v0)).(let H28 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
+Abst) u t5)) H15) in (let H29 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2))
+H10 v0 H27) in (eq_ind_r B Abst (\lambda (b0: B).(ex2 T (\lambda (t7: T).(pr0
+(THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) (let H30 \def (eq_ind B b
+(\lambda (b0: B).(not (eq B b0 Abst))) H9 Abst H26) in (let H31 \def (match
+(H30 (refl_equal B Abst)) in False return (\lambda (_: False).(ex2 T (\lambda
(t7: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in
-H33))))) t3 (sym_eq T t3 t5 H28))) u1 (sym_eq T u1 u H27))) b (sym_eq B b
-Abst H26))) v1 (sym_eq T v1 v0 H25))) H24)) H23)) H22)))]) in (H21
-(refl_equal T (THead (Flat Appl) v0 (THead (Bind Abst) u t5)))))))) t2 H17))
-t H15 H16 H13 H14))) | (pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16)
-\Rightarrow (\lambda (H17: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u0
-t5)) t)).(\lambda (H18: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S
-O) O v3) t6)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5))
-(\lambda (_: T).((eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O
-v3) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))))
-(\lambda (H19: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3)
-t6)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3)
-t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3)
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t7 t8))))))))
-(\lambda (_: (not (eq B b0 Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22:
-(pr0 u0 u3)).(\lambda (H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda
-(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H17) in (let H25 \def (match H24 in
-eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat
-Appl) v0 (THead (Bind b0) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
-T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6))
-t8)))))) with [refl_equal \Rightarrow (\lambda (H25: (eq T (THead (Flat Appl)
-v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0
-t5)))).(let H26 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
-| (THead _ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8)
-\Rightarrow t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H25) in ((let H27 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match
-t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H25)
-in ((let H28 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
+t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v3 t6) t7)))) with []) in H31))
+b H26))))))) H23)) H22)) H21))))) t2 H17)) t H15 H16 H13 H14))) |
+(pr0_upsilon b0 H13 v0 v3 H14 u0 u3 H15 t5 t6 H16) \Rightarrow (\lambda (H17:
+(eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) t)).(\lambda (H18: (eq T
+(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T
+(THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (\lambda (_: T).((eq T (THead
+(Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2) \to ((not (eq B b0
+Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda
+(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
+t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H19: (eq T (THead (Bind
+b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t2)).(eq_ind T (THead (Bind
+b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) (\lambda (t7: T).((not (eq B
+b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
+t4)) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0
+Abst))).(\lambda (H21: (pr0 v0 v3)).(\lambda (H22: (pr0 u0 u3)).(\lambda
+(H23: (pr0 t5 t6)).(let H24 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
+(Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Appl) v0 (THead
+(Bind b0) u0 t5)) H17) in (let H25 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow v1 | (TLRef _)
+\Rightarrow v1 | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24)
+in ((let H26 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda
(_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _
_ t7) \Rightarrow (match t7 in T return (\lambda (_: T).B) with [(TSort _)
\Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k
in K return (\lambda (_: K).B) with [(Bind b1) \Rightarrow b1 | (Flat _)
\Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead
-(Flat Appl) v0 (THead (Bind b0) u0 t5)) H25) in ((let H29 \def (f_equal T T
+(Flat Appl) v0 (THead (Bind b0) u0 t5)) H24) in ((let H27 \def (f_equal T T
(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t7 _) \Rightarrow t7]))
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead
-(Bind b0) u0 t5)) H25) in (eq_ind T v0 (\lambda (_: T).((eq B b b0) \to ((eq
-T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b)
-u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0
-(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t8)))))))
-(\lambda (H30: (eq B b b0)).(eq_ind B b0 (\lambda (b1: B).((eq T u1 u0) \to
-((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b1) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind
-b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7)))))) (\lambda (H31: (eq
-T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8))
-(\lambda (t8: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O
-v3) t6)) t8))))) (\lambda (H32: (eq T t3 t5)).(eq_ind T t5 (\lambda (_:
-T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift
-(S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind b0) u3 (THead (Flat
-Appl) (lift (S O) O v3) t6)) t8)))) (let H33 \def (eq_ind_r T t (\lambda (t7:
-T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall
-(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10:
-T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind b0) u0 t5))
-H17) in (let H34 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H32)
-in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H31) in
-(let H36 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0 H30)
-in (let H37 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2)) H10 v0 H29) in
-(ex2_ind T (\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T
+\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t7) \Rightarrow (match
+t7 in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
+\Rightarrow u1 | (THead _ t8 _) \Rightarrow t8])])) (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) H24)
+in ((let H28 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead
+_ _ t7) \Rightarrow (match t7 in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t8) \Rightarrow
+t8])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (THead (Flat Appl) v0
+(THead (Bind b0) u0 t5)) H24) in (\lambda (H29: (eq T u1 u0)).(\lambda (H30:
+(eq B b b0)).(\lambda (H31: (eq T v1 v0)).(let H32 \def (eq_ind_r T t
+(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
+\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Flat Appl) v0 (THead (Bind
+b0) u0 t5)) H17) in (let H33 \def (eq_ind T v1 (\lambda (t7: T).(pr0 t7 v2))
+H10 v0 H31) in (eq_ind_r B b0 (\lambda (b1: B).(ex2 T (\lambda (t7: T).(pr0
+(THead (Bind b1) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda
+(t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6))
+t7)))) (let H34 \def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H9 b0
+H30) in (let H35 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H11 u0 H29)
+in (let H36 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H12 t5 H28) in
+(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T
(\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O
v2) t4)) t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl)
-(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H38: (pr0 v2
-x)).(\lambda (H39: (pr0 v3 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7))
+(lift (S O) O v3) t6)) t7))) (\lambda (x: T).(\lambda (H37: (pr0 t4
+x)).(\lambda (H38: (pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7))
(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b0)
u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0
(THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t7))) (\lambda
-(x0: T).(\lambda (H40: (pr0 u2 x0)).(\lambda (H41: (pr0 u3 x0)).(ex2_ind T
-(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda
+(x0: T).(\lambda (H39: (pr0 u2 x0)).(\lambda (H40: (pr0 u3 x0)).(ex2_ind T
+(\lambda (t7: T).(pr0 v2 t7)) (\lambda (t7: T).(pr0 v3 t7)) (ex2 T (\lambda
(t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O)
-O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H42: (pr0 t4 x1)).(\lambda (H43:
-(pr0 t6 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H36 v2 v3 x H38 H39 u2
-u3 x0 H40 H41 t4 t6 x1 H42 H43)))) (H33 t5 (tlt_trans (THead (Bind b0) u0 t5)
-t5 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0
-t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) t4 H34 t6 H23)))))
-(H33 u0 (tlt_trans (THead (Bind b0) u0 t5) u0 (THead (Flat Appl) v0 (THead
-(Bind b0) u0 t5)) (tlt_head_sx (Bind b0) u0 t5) (tlt_head_dx (Flat Appl) v0
-(THead (Bind b0) u0 t5))) u2 H35 u3 H22))))) (H33 v0 (tlt_head_sx (Flat Appl)
-v0 (THead (Bind b0) u0 t5)) v2 H37 v3 H21))))))) t3 (sym_eq T t3 t5 H32))) u1
-(sym_eq T u1 u0 H31))) b (sym_eq B b b0 H30))) v1 (sym_eq T v1 v0 H29)))
-H28)) H27)) H26)))]) in (H25 (refl_equal T (THead (Flat Appl) v0 (THead (Bind
-b0) u0 t5)))))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3
-H13 t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0
-t5) t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead
-(Bind Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to
-((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind
-Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0
+O v3) t6)) t7))) (\lambda (x1: T).(\lambda (H41: (pr0 v2 x1)).(\lambda (H42:
+(pr0 v3 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H34 v2 v3 x1 H41 H42 u2
+u3 x0 H39 H40 t4 t6 x H37 H38)))) (H32 v0 (tlt_head_sx (Flat Appl) v0 (THead
+(Bind b0) u0 t5)) v2 H33 v3 H21))))) (H32 u0 (tlt_trans (THead (Bind b0) u0
+t5) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t5)) (tlt_head_sx (Bind b0)
+u0 t5) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t5))) u2 H35 u3
+H22))))) (H32 t5 (tlt_trans (THead (Bind b0) u0 t5) t5 (THead (Flat Appl) v0
+(THead (Bind b0) u0 t5)) (tlt_head_dx (Bind b0) u0 t5) (tlt_head_dx (Flat
+Appl) v0 (THead (Bind b0) u0 t5))) t4 H36 t6 H23))))) b H30))))))) H27))
+H26)) H25))))))) t2 H19)) t H17 H18 H13 H14 H15 H16))) | (pr0_delta u0 u3 H13
+t5 t6 H14 w H15) \Rightarrow (\lambda (H16: (eq T (THead (Bind Abbr) u0 t5)
+t)).(\lambda (H17: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind
+Abbr) u0 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0
u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda
-(t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5
+(t8: T).(pr0 t2 t8)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u3 w)
+t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t7: T).((pr0 u0 u3) \to
+((pr0 t5 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
+T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5
t6)).(\lambda (_: (subst0 O u3 t6 w)).(let H22 \def (eq_ind_r T t (\lambda
(t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead
-(Bind Abbr) u0 t5) H16) in (let H23 \def (match H22 in eq return (\lambda
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u0 t5)) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w) t8))))))
-with [refl_equal \Rightarrow (\lambda (H23: (eq T (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Bind Abbr) u0 t5))).(let H24 \def (eq_ind T
-(THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind Abbr) u0 t5) H23) in (False_ind (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7))
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w) t7))) H24)))]) in (H23
-(refl_equal T (THead (Bind Abbr) u0 t5)))))))) t2 H18)) t H16 H17 H13 H14
-H15))) | (pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T
-(THead (Bind b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6
-t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T
-t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8))
-(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2
-(\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4))
-t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0
-Abst))).(\lambda (_: (pr0 t5 t2)).(let H20 \def (eq_ind_r T t (\lambda (t7:
-T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Bind
-b0) u (lift (S O) O t5)) H15) in (let H21 \def (match H20 in eq return
-(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind b0) u (lift
-(S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))))
-with [refl_equal \Rightarrow (\lambda (H21: (eq T (THead (Flat Appl) v1
-(THead (Bind b) u1 t3)) (THead (Bind b0) u (lift (S O) O t5)))).(let H22 \def
-(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
-\Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H21) in
-(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H22)))]) in (H21
-(refl_equal T (THead (Bind b0) u (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2
-H17))) t H15 H16 H13 H14))) | (pr0_epsilon t5 t6 H13 u) \Rightarrow (\lambda
-(H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15: (eq T t6 t2)).(eq_ind
-T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S
-O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H16: (eq T t6
-t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8))
-(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H18 \def
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1
-t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in (let H19 \def (match H18 in eq
-return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat
-Cast) u t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with
-[refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Flat Appl) v1 (THead
-(Bind b) u1 t3)) (THead (Flat Cast) u t5))).(let H20 \def (eq_ind T (THead
-(Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f
-in F return (\lambda (_: F).Prop) with [Appl \Rightarrow True | Cast
-\Rightarrow False])])])) I (THead (Flat Cast) u t5) H19) in (False_ind (ex2 T
-(\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2)
-t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H20)))]) in (H19 (refl_equal T
-(THead (Flat Cast) u t5)))))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in
-(H13 (refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5)))
-| (pr0_delta u1 u2 H2 t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead
-(Bind Abbr) u1 t3) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w)
-t1)).(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind
-Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to
-(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))))
-(\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind
-Abbr) u2 w) (\lambda (t5: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2
-t4 w) \to (ex2 T (\lambda (t6: T).(pr0 t5 t6)) (\lambda (t6: T).(pr0 t2
-t6))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda
-(H10: (subst0 O u2 t4 w)).(let H11 \def (match H1 in pr0 return (\lambda (t5:
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
-T).(pr0 t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5
-t)).(\lambda (H12: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
-T).(pr0 t2 t7))))) (\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_:
-T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
-T).(pr0 t2 t7)))) (let H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2))
-H13 (THead (Bind Abbr) u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3)
-(\lambda (t6: T).(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))
-(\lambda (t7: T).(pr0 t6 t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6:
-T).(eq T t5 t6)) H11 (THead (Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r
-T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v
-t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9))
-(\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in
-(ex_intro2 T (\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda
-(t6: T).(pr0 (THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w)
-(pr0_refl (THead (Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2
-H14)) t (sym_eq T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3
-H11 t5 t6 H12 k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5)
-t)).(\lambda (H14: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u0 t5)
-(\lambda (_: T).((eq T (THead k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6)
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8:
-T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T
-(THead k u3 t6) (\lambda (t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7
-t8)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18
-\def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5
-(THead k u0 t5) H13) in (let H19 \def (match H18 in eq return (\lambda (t7:
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u0 t5)) \to (ex2 T (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead k u3
-t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Bind
-Abbr) u1 t3) (THead k u0 t5))).(let H20 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
-(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind
-Abbr) u1 t3) (THead k u0 t5) H19) in ((let H21 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 |
-(TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind
-Abbr) u1 t3) (THead k u0 t5) H19) in ((let H22 \def (f_equal T K (\lambda (e:
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind
-Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | (THead k0 _ _) \Rightarrow k0]))
-(THead (Bind Abbr) u1 t3) (THead k u0 t5) H19) in (eq_ind K (Bind Abbr)
-(\lambda (k0: K).((eq T u1 u0) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6)
-t7)))))) (\lambda (H23: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t3
-t5) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) u3 t6) t8))))) (\lambda (H24: (eq T t3
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind
-Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 t6) t8)))) (let
-H25 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13 (Bind
-Abbr) H22) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
+(Bind Abbr) u0 t5) H16) in (let H23 \def (eq_ind T (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind
+Abbr) u0 t5) H22) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u3 w) t7))) H23)))))) t2 H18)) t H16 H17 H13 H14 H15))) |
+(pr0_zeta b0 H13 t5 t6 H14 u) \Rightarrow (\lambda (H15: (eq T (THead (Bind
+b0) u (lift (S O) O t5)) t)).(\lambda (H16: (eq T t6 t2)).(eq_ind T (THead
+(Bind b0) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B
+b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b)
+u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2
+t8))))))) (\lambda (H17: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not
+(eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8:
+T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t5
+t2)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Appl) v1
+(THead (Bind b) u1 t3)) t7)) H6 (THead (Bind b0) u (lift (S O) O t5)) H15) in
+(let H21 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t5)) H20)
+in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat
+Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7))) H21))))) t6
+(sym_eq T t6 t2 H17))) t H15 H16 H13 H14))) | (pr0_epsilon t5 t6 H13 u)
+\Rightarrow (\lambda (H14: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H15:
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2
+(THead (Flat Appl) (lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2
+t8)))))) (\lambda (H16: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5
+t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl)
+(lift (S O) O v2) t4)) t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_:
+(pr0 t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
+Appl) v1 (THead (Bind b) u1 t3)) t7)) H6 (THead (Flat Cast) u t5) H14) in
+(let H19 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
+\Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t5)
+H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead
+(Flat Appl) (lift (S O) O v2) t4)) t7)) (\lambda (t7: T).(pr0 t2 t7)))
+H19)))) t6 (sym_eq T t6 t2 H16))) t H14 H15 H13)))]) in (H13 (refl_equal T t)
+(refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2
+t3 t4 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t3)
+t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind
+Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1
+u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0
+t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))) (\lambda (H7: (eq T (THead (Bind
+Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t5: T).((pr0 u1
+u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t6: T).(pr0
+t5 t6)) (\lambda (t6: T).(pr0 t2 t6))))))) (\lambda (H8: (pr0 u1
+u2)).(\lambda (H9: (pr0 t3 t4)).(\lambda (H10: (subst0 O u2 t4 w)).(let H11
+\def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6: T).(\lambda (_:
+(pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
+[(pr0_refl t5) \Rightarrow (\lambda (H11: (eq T t5 t)).(\lambda (H12: (eq T
+t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))))
+(\lambda (H13: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let
+H14 \def (eq_ind_r T t (\lambda (t6: T).(eq T t6 t2)) H13 (THead (Bind Abbr)
+u1 t3) H5) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t6: T).(ex2 T
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t6
+t7)))) (let H15 \def (eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H11 (THead
+(Bind Abbr) u1 t3) H5) in (let H16 \def (eq_ind_r T t (\lambda (t6:
+T).(\forall (v: T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall
+(t8: T).((pr0 v t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9:
+T).(pr0 t8 t9)))))))))) H (THead (Bind Abbr) u1 t3) H5) in (ex_intro2 T
+(\lambda (t6: T).(pr0 (THead (Bind Abbr) u2 w) t6)) (\lambda (t6: T).(pr0
+(THead (Bind Abbr) u1 t3) t6)) (THead (Bind Abbr) u2 w) (pr0_refl (THead
+(Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t3 t4 H9 w H10)))) t2 H14)) t (sym_eq
+T t t2 H13))) t5 (sym_eq T t5 t H11) H12))) | (pr0_comp u0 u3 H11 t5 t6 H12
+k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t5) t)).(\lambda (H14: (eq T
+(THead k u3 t6) t2)).(eq_ind T (THead k u0 t5) (\lambda (_: T).((eq T (THead
+k u3 t6) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))
+(\lambda (H15: (eq T (THead k u3 t6) t2)).(eq_ind T (THead k u3 t6) (\lambda
+(t7: T).((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
+(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (H16: (pr0
+u0 u3)).(\lambda (H17: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
+T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead k u0 t5) H13) in (let H19
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) |
+(THead k0 _ _) \Rightarrow k0])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
+H18) in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1
+| (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
+H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
+| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead k u0 t5)
+H18) in (\lambda (H22: (eq T u1 u0)).(\lambda (H23: (eq K (Bind Abbr)
+k)).(eq_ind K (Bind Abbr) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead k0 u3 t6) t7))))
+(let H24 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u0 t5) t)) H13
+(Bind Abbr) H23) in (let H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Bind Abbr) u0 t5) H25) in (let H27 \def (eq_ind T t3
-(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H24) in (let H28 \def (eq_ind T u1
-(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0
-u2 t7)) (\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead
+t10)))))))))) H (THead (Bind Abbr) u0 t5) H24) in (let H26 \def (eq_ind T u1
+(\lambda (t7: T).(pr0 t7 u2)) H8 u0 H22) in (let H27 \def (eq_ind T t3
+(\lambda (t7: T).(pr0 t7 t4)) H9 t5 H21) in (ex2_ind T (\lambda (t7: T).(pr0
+t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead
(Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 t6) t7)))
-(\lambda (x: T).(\lambda (H29: (pr0 u2 x)).(\lambda (H30: (pr0 u3
-x)).(ex2_ind T (\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7))
+(\lambda (x: T).(\lambda (H28: (pr0 t4 x)).(\lambda (H29: (pr0 t6
+x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3 t7))
(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H31: (pr0
-t4 x0)).(\lambda (H32: (pr0 t6 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3
+T).(pr0 (THead (Bind Abbr) u3 t6) t7))) (\lambda (x0: T).(\lambda (H30: (pr0
+u2 x0)).(\lambda (H31: (pr0 u3 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3
t6)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t4 w
-H10 u3 x H30 H29 t6 x0 H32 H31))))) (H26 t5 (tlt_head_dx (Bind Abbr) u0 t5)
-t4 H27 t6 H17))))) (H26 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H28 u3
-H16)))))) t3 (sym_eq T t3 t5 H24))) u1 (sym_eq T u1 u0 H23))) k H22)) H21))
-H20)))]) in (H19 (refl_equal T (THead k u0 t5))))))) t2 H15)) t H13 H14 H11
-H12))) | (pr0_beta u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T
-(THead (Flat Appl) v1 (THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T
-(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind
-Abst) u t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0
-v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr)
-u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead
-(Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7:
-T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1
-v2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
-T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind
-Abst) u t5)) H13) in (let H19 \def (match H18 in eq return (\lambda (t7:
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind
-Abst) u t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal
-\Rightarrow (\lambda (H19: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Appl)
-v1 (THead (Bind Abst) u t5)))).(let H20 \def (eq_ind T (THead (Bind Abbr) u1
-t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1
-(THead (Bind Abst) u t5)) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6)
-t7))) H20)))]) in (H19 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst)
-u t5)))))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0
-u3 H13 t5 t6 H14) \Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1
-(THead (Bind b) u0 t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1
-(THead (Bind b) u0 t5)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1
-v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17:
-(eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t6))
-(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
-(\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0
-t5 t6)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr)
-u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23
-\def (match H22 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq
-T t7 (THead (Flat Appl) v1 (THead (Bind b) u0 t5))) \to (ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind b)
-u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal
-\Rightarrow (\lambda (H23: (eq T (THead (Bind Abbr) u1 t3) (THead (Flat Appl)
-v1 (THead (Bind b) u0 t5)))).(let H24 \def (eq_ind T (THead (Bind Abbr) u1
-t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1
-(THead (Bind b) u0 t5)) H23) in (False_ind (ex2 T (\lambda (t7: T).(pr0
-(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u3 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t7))) H24)))]) in (H23 (refl_equal T
-(THead (Flat Appl) v1 (THead (Bind b) u0 t5)))))))))) t2 H17)) t H15 H16 H11
-H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow (\lambda
-(H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T (THead (Bind
-Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda (_: T).((eq T
-(THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0
-O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
-(\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead (Bind Abbr)
-u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7: T).((pr0 u0 u3)
-\to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0
-(THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda
-(H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19: (subst0 O u3 t6
-w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1
-t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21 \def (match H20 in eq
-return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind
-Abbr) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w0) t8)))))) with [refl_equal
-\Rightarrow (\lambda (H21: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind Abbr)
-u0 t5))).(let H22 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3
-| (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind
-Abbr) u0 t5) H21) in ((let H23 \def (f_equal T T (\lambda (e: T).(match e in
-T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _)
-\Rightarrow u1 | (THead _ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3)
-(THead (Bind Abbr) u0 t5) H21) in (eq_ind T u0 (\lambda (_: T).((eq T t3 t5)
-\to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) u3 w0) t8))))) (\lambda (H24: (eq T t3
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 (THead (Bind
-Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u3 w0) t8)))) (let
-H25 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
-(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
-(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
-(THead (Bind Abbr) u0 t5) H14) in (let H26 \def (eq_ind T t3 (\lambda (t7:
-T).(pr0 t7 t4)) H9 t5 H24) in (let H27 \def (eq_ind T u1 (\lambda (t7:
-T).(pr0 t7 u2)) H8 u0 H23) in (ex2_ind T (\lambda (t7: T).(pr0 u2 t7))
-(\lambda (t7: T).(pr0 u3 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr)
-u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x:
-T).(\lambda (H28: (pr0 u2 x)).(\lambda (H29: (pr0 u3 x)).(ex2_ind T (\lambda
-(t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H30: (pr0 t4 x0)).(\lambda
-(H31: (pr0 t6 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w H10 u3 t6 w0 H19
-x H28 H29 x0 H30 H31)))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6
-H18))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5) u2 H27 u3 H17))))) t3
-(sym_eq T t3 t5 H24))) u1 (sym_eq T u1 u0 H23))) H22)))]) in (H21 (refl_equal
-T (THead (Bind Abbr) u0 t5)))))))) t2 H16)) t H14 H15 H11 H12 H13))) |
-(pr0_zeta b H11 t5 t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b)
-u (lift (S O) O t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind
-b) u (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b
-Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr)
-u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6
-t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to
+H10 u3 x0 H31 H30 t6 x H29 H28))))) (H25 u0 (tlt_head_sx (Bind Abbr) u0 t5)
+u2 H26 u3 H16))))) (H25 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H27 t6
+H17)))))) k H23)))) H20)) H19))))) t2 H15)) t H13 H14 H11 H12))) | (pr0_beta
+u v1 v2 H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1
+(THead (Bind Abst) u t5)) t)).(\lambda (H14: (eq T (THead (Bind Abbr) v2 t6)
+t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) (\lambda (_:
+T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to
(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8:
-T).(pr0 t2 t8)))))) (\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0
-t5 t2)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr)
-u1 t3) t7)) H5 (THead (Bind b) u (lift (S O) O t5)) H13) in (let H19 \def
-(match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7
-(THead (Bind b) u (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 (THead
-(Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal
-\Rightarrow (\lambda (H19: (eq T (THead (Bind Abbr) u1 t3) (THead (Bind b) u
-(lift (S O) O t5)))).(let H20 \def (f_equal T T (\lambda (e: T).(match e in T
-return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3)
-(THead (Bind b) u (lift (S O) O t5)) H19) in ((let H21 \def (f_equal T T
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _) \Rightarrow t7]))
-(THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H19) in ((let
-H22 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_: T).B)
-with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t3)
-(THead (Bind b) u (lift (S O) O t5)) H19) in (eq_ind B Abbr (\lambda (_:
-B).((eq T u1 u) \to ((eq T t3 (lift (S O) O t5)) \to (ex2 T (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))))))
-(\lambda (H23: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t3 (lift (S O)
-O t5)) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
-(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H24: (eq T t3 (lift (S O) O
-t5))).(eq_ind T (lift (S O) O t5) (\lambda (_: T).(ex2 T (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let
-H25 \def (eq_ind_r B b (\lambda (b0: B).(not (eq B b0 Abst))) H16 Abbr H22)
-in (let H26 \def (eq_ind_r B b (\lambda (b0: B).(eq T (THead (Bind b0) u
-(lift (S O) O t5)) t)) H13 Abbr H22) in (let H27 \def (eq_ind_r T t (\lambda
+T).(pr0 t2 t8))))))) (\lambda (H15: (eq T (THead (Bind Abbr) v2 t6)
+t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
+(\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0
+t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr)
+u1 t3) t7)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H13) in (let
+H19 \def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t5)) H18) in
+(False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))
+(\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H19))))) t2 H15)) t H13
+H14 H11 H12))) | (pr0_upsilon b H11 v1 v2 H12 u0 u3 H13 t5 t6 H14)
+\Rightarrow (\lambda (H15: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0
+t5)) t)).(\lambda (H16: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S
+O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t5))
+(\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O
+v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to
+((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H17: (eq T (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u3
+(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t5 t6) \to (ex2 T (\lambda
+(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7
+t8)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1
+v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t5 t6)).(let H22 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead
+(Flat Appl) v1 (THead (Bind b) u0 t5)) H15) in (let H23 \def (eq_ind T (THead
+(Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Appl) v1 (THead (Bind b) u0 t5)) H22) in (False_ind (ex2 T (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind b)
+u3 (THead (Flat Appl) (lift (S O) O v2) t6)) t7))) H23))))))) t2 H17)) t H15
+H16 H11 H12 H13 H14))) | (pr0_delta u0 u3 H11 t5 t6 H12 w0 H13) \Rightarrow
+(\lambda (H14: (eq T (THead (Bind Abbr) u0 t5) t)).(\lambda (H15: (eq T
+(THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t5) (\lambda
+(_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t5 t6)
+\to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr)
+u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H16: (eq T (THead
+(Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t7:
+T).((pr0 u0 u3) \to ((pr0 t5 t6) \to ((subst0 O u3 t6 w0) \to (ex2 T (\lambda
+(t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t7 t8)))))))
+(\lambda (H17: (pr0 u0 u3)).(\lambda (H18: (pr0 t5 t6)).(\lambda (H19:
+(subst0 O u3 t6 w0)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T
+(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Bind Abbr) u0 t5) H14) in (let H21
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t7 _)
+\Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5) H20) in
+((let H22 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _
+t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind Abbr) u0 t5)
+H20) in (\lambda (H23: (eq T u1 u0)).(let H24 \def (eq_ind_r T t (\lambda
(t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to
(\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O
-t5)) H26) in (let H28 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9
-(lift (S O) O t5) H24) in (ex2_ind T (\lambda (t7: T).(eq T t4 (lift (S O) O
-t7))) (\lambda (t7: T).(pr0 t5 t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda
-(H29: (eq T t4 (lift (S O) O x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def
-(eq_ind T t4 (\lambda (t7: T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29)
-in (let H32 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H23) in
-(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T
-(\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2
-t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2
-x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H31 x (pr0_refl
-(lift (S O) O x)) t2)))) (H27 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30
-t2 H17))))))) (pr0_gen_lift t5 t4 (S O) O H28)))))) t3 (sym_eq T t3 (lift (S
-O) O t5) H24))) u1 (sym_eq T u1 u H23))) b H22)) H21)) H20)))]) in (H19
-(refl_equal T (THead (Bind b) u (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2
-H15))) t H13 H14 H11 H12))) | (pr0_epsilon t5 t6 H11 u) \Rightarrow (\lambda
-(H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13: (eq T t6 t2)).(eq_ind
-T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to
-(ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8:
-T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7:
-T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w)
-t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H16 \def
+(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind Abbr) u0 t5) H14) in
+(let H25 \def (eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u0 H23) in (let
+H26 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 t5 H22) in (ex2_ind T
+(\lambda (t7: T).(pr0 t4 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda
+(t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 (THead (Bind
+Abbr) u3 w0) t7))) (\lambda (x: T).(\lambda (H27: (pr0 t4 x)).(\lambda (H28:
+(pr0 t6 x)).(ex2_ind T (\lambda (t7: T).(pr0 u2 t7)) (\lambda (t7: T).(pr0 u3
+t7)) (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u3 w0) t7))) (\lambda (x0: T).(\lambda (H29: (pr0
+u2 x0)).(\lambda (H30: (pr0 u3 x0)).(pr0_confluence__pr0_delta_delta u2 t4 w
+H10 u3 t6 w0 H19 x0 H29 H30 x H27 H28)))) (H24 u0 (tlt_head_sx (Bind Abbr) u0
+t5) u2 H25 u3 H17))))) (H24 t5 (tlt_head_dx (Bind Abbr) u0 t5) t4 H26 t6
+H18))))))) H21)))))) t2 H16)) t H14 H15 H11 H12 H13))) | (pr0_zeta b H11 t5
+t6 H12 u) \Rightarrow (\lambda (H13: (eq T (THead (Bind b) u (lift (S O) O
+t5)) t)).(\lambda (H14: (eq T t6 t2)).(eq_ind T (THead (Bind b) u (lift (S O)
+O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5
+t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda
+(t8: T).(pr0 t2 t8))))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda
+(t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8))))))
+(\lambda (H16: (not (eq B b Abst))).(\lambda (H17: (pr0 t5 t2)).(let H18 \def
(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t3) t7)) H5 (THead
-(Flat Cast) u t5) H12) in (let H17 \def (match H16 in eq return (\lambda (t7:
-T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Cast) u t5)) \to (ex2 T
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2
-t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Bind Abbr)
-u1 t3) (THead (Flat Cast) u t5))).(let H18 \def (eq_ind T (THead (Bind Abbr)
-u1 t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
-[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t5)
-H17) in (False_ind (ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7))
-(\lambda (t7: T).(pr0 t2 t7))) H18)))]) in (H17 (refl_equal T (THead (Flat
-Cast) u t5)))))) t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11
-(refl_equal T t) (refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) |
-(pr0_zeta b H2 t3 t4 H3 u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u
-(lift (S O) O t3)) t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b)
-u (lift (S O) O t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst))
-\to ((pr0 t3 t4) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6:
-T).(pr0 t2 t6))))))) (\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5:
-T).((not (eq B b Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1
-t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b
-Abst))).(\lambda (H8: (pr0 t3 t1)).(let H9 \def (match H1 in pr0 return
-(\lambda (t5: T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to
-((eq T t6 t2) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
-t2 t7)))))))) with [(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5
-t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))
-(\lambda (H11: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t
-(\lambda (t6: T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4)
-in (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def
-(eq_ind_r T t (\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O)
-O t3)) H4) in (let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v:
-T).((tlt v t6) \to (\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v
-t8) \to (ex2 T (\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8
-t9)))))))))) H (THead (Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T
-(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift
-(S O) O t3)) t6)) t1 (pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t
-(sym_eq T t t2 H11))) t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6
-H10 k) \Rightarrow (\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12:
-(eq T (THead k u2 t6) t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T
-(THead k u2 t6) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T
-(THead k u2 t6) t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2)
-\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8:
-T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5
-t6)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u
-(lift (S O) O t3)) t7)) H4 (THead k u1 t5) H11) in (let H17 \def (match H16
-in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1
-t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead k
-u2 t6) t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead
-(Bind b) u (lift (S O) O t3)) (THead k u1 t5))).(let H18 \def (f_equal T T
+(Bind b) u (lift (S O) O t5)) H13) in (let H19 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow Abbr |
+(TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+Abbr])])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift (S O) O t5)) H18)
+in ((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead
+_ t7 _) \Rightarrow t7])) (THead (Bind Abbr) u1 t3) (THead (Bind b) u (lift
+(S O) O t5)) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in
+T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
+\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Bind Abbr) u1 t3)
+(THead (Bind b) u (lift (S O) O t5)) H18) in (\lambda (H22: (eq T u1
+u)).(\lambda (H23: (eq B Abbr b)).(let H24 \def (eq_ind_r B b (\lambda (b0:
+B).(not (eq B b0 Abst))) H16 Abbr H23) in (let H25 \def (eq_ind_r B b
+(\lambda (b0: B).(eq T (THead (Bind b0) u (lift (S O) O t5)) t)) H13 Abbr
+H23) in (let H26 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
+t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
+(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
+t10)))))))))) H (THead (Bind Abbr) u (lift (S O) O t5)) H25) in (let H27 \def
+(eq_ind T u1 (\lambda (t7: T).(pr0 t7 u2)) H8 u H22) in (let H28 \def (eq_ind
+T t3 (\lambda (t7: T).(pr0 t7 t4)) H9 (lift (S O) O t5) H21) in (ex2_ind T
+(\lambda (t7: T).(eq T t4 (lift (S O) O t7))) (\lambda (t7: T).(pr0 t5 t7))
+(ex2 T (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7:
+T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H29: (eq T t4 (lift (S O) O
+x))).(\lambda (H30: (pr0 t5 x)).(let H31 \def (eq_ind T t4 (\lambda (t7:
+T).(subst0 O u2 t7 w)) H10 (lift (S O) O x) H29) in (ex2_ind T (\lambda (t7:
+T).(pr0 x t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x0:
+T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2
+x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H31 x (pr0_refl
+(lift (S O) O x)) t2)))) (H26 t5 (lift_tlt_dx (Bind Abbr) u t5 (S O) O) x H30
+t2 H17)))))) (pr0_gen_lift t5 t4 (S O) O H28)))))))))) H20)) H19))))) t6
+(sym_eq T t6 t2 H15))) t H13 H14 H11 H12))) | (pr0_epsilon t5 t6 H11 u)
+\Rightarrow (\lambda (H12: (eq T (THead (Flat Cast) u t5) t)).(\lambda (H13:
+(eq T t6 t2)).(eq_ind T (THead (Flat Cast) u t5) (\lambda (_: T).((eq T t6
+t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2
+w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H14: (eq T t6
+t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8:
+T).(pr0 (THead (Bind Abbr) u2 w) t8)) (\lambda (t8: T).(pr0 t2 t8)))))
+(\lambda (_: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T
+(THead (Bind Abbr) u1 t3) t7)) H5 (THead (Flat Cast) u t5) H12) in (let H17
+\def (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow
+False])])) I (THead (Flat Cast) u t5) H16) in (False_ind (ex2 T (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7)) (\lambda (t7: T).(pr0 t2 t7))) H17))))
+t6 (sym_eq T t6 t2 H14))) t H12 H13 H11)))]) in (H11 (refl_equal T t)
+(refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t3 t4 H3
+u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t3))
+t)).(\lambda (H5: (eq T t4 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O
+t3)) (\lambda (_: T).((eq T t4 t1) \to ((not (eq B b Abst)) \to ((pr0 t3 t4)
+\to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))))
+(\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((not (eq B b
+Abst)) \to ((pr0 t3 t5) \to (ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda
+(t6: T).(pr0 t2 t6)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8:
+(pr0 t3 t1)).(let H9 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda
+(t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
+[(pr0_refl t5) \Rightarrow (\lambda (H9: (eq T t5 t)).(\lambda (H10: (eq T t5
+t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H11: (eq T t
+t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7))
+(\lambda (t7: T).(pr0 t2 t7)))) (let H12 \def (eq_ind_r T t (\lambda (t6:
+T).(eq T t6 t2)) H11 (THead (Bind b) u (lift (S O) O t3)) H4) in (eq_ind T
+(THead (Bind b) u (lift (S O) O t3)) (\lambda (t6: T).(ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)))) (let H13 \def (eq_ind_r T t
+(\lambda (t6: T).(eq T t5 t6)) H9 (THead (Bind b) u (lift (S O) O t3)) H4) in
+(let H14 \def (eq_ind_r T t (\lambda (t6: T).(\forall (v: T).((tlt v t6) \to
+(\forall (t7: T).((pr0 v t7) \to (\forall (t8: T).((pr0 v t8) \to (ex2 T
+(\lambda (t9: T).(pr0 t7 t9)) (\lambda (t9: T).(pr0 t8 t9)))))))))) H (THead
+(Bind b) u (lift (S O) O t3)) H4) in (ex_intro2 T (\lambda (t6: T).(pr0 t1
+t6)) (\lambda (t6: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t6)) t1
+(pr0_refl t1) (pr0_zeta b H7 t3 t1 H8 u)))) t2 H12)) t (sym_eq T t t2 H11)))
+t5 (sym_eq T t5 t H9) H10))) | (pr0_comp u1 u2 H9 t5 t6 H10 k) \Rightarrow
+(\lambda (H11: (eq T (THead k u1 t5) t)).(\lambda (H12: (eq T (THead k u2 t6)
+t2)).(eq_ind T (THead k u1 t5) (\lambda (_: T).((eq T (THead k u2 t6) t2) \to
+((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T (THead k u2 t6)
+t2)).(eq_ind T (THead k u2 t6) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6)
+\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))
+(\lambda (_: (pr0 u1 u2)).(\lambda (H15: (pr0 t5 t6)).(let H16 \def (eq_ind_r
+T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4
+(THead k u1 t5) H11) in (let H17 \def (f_equal T K (\lambda (e: T).(match e
+in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef
+_) \Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
+(lift (S O) O t3)) (THead k u1 t5) H16) in ((let H18 \def (f_equal T T
(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T
-\def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d u0) (lref_map f (s k0
-d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _)
-\Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T
-\def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow
-(TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)]))
-| (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d u0) (lref_map f (s k0
-d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _
-t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5)
-H17) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead
-k u1 t5) H17) in ((let H20 \def (f_equal T K (\lambda (e: T).(match e in T
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _)
-\Rightarrow (Bind b) | (THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u
-(lift (S O) O t3)) (THead k u1 t5) H17) in (eq_ind K (Bind b) (\lambda (k0:
-K).((eq T u u1) \to ((eq T (lift (S O) O t3) t5) \to (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda
-(H21: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t3) t5)
-\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind
-b) u2 t6) t8))))) (\lambda (H22: (eq T (lift (S O) O t3) t5)).(eq_ind T (lift
-(S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
-(t8: T).(pr0 (THead (Bind b) u2 t6) t8)))) (let H23 \def (eq_ind_r K k
-(\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H20) in (let H24
-\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H22)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7]))
+(THead (Bind b) u (lift (S O) O t3)) (THead k u1 t5) H16) in ((let H19 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7:
+T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map f d
+u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat)))
+(d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort
+n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
+false \Rightarrow (f i)])) | (THead k0 u0 t8) \Rightarrow (THead k0 (lref_map
+f d u0) (lref_map f (s k0 d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
+O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O
+t3)) (THead k u1 t5) H16) in (\lambda (_: (eq T u u1)).(\lambda (H21: (eq K
+(Bind b) k)).(eq_ind K (Bind b) (\lambda (k0: K).(ex2 T (\lambda (t7: T).(pr0
+t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))) (let H22 \def (eq_ind_r
+K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H11 (Bind b) H21) in (let H23
+\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H15 (lift (S O) O t3) H19)
in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7:
T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
-(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift (S
-O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda
-(t7: T).(eq T (THead (Bind b) u1 t7) t)) H23 (lift (S O) O t3) H22) in (let
-H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
+(THead (Bind b) u2 t6) t7))) (\lambda (x: T).(\lambda (H24: (eq T t6 (lift (S
+O) O x))).(\lambda (H25: (pr0 t3 x)).(let H26 \def (eq_ind_r T t5 (\lambda
+(t7: T).(eq T (THead (Bind b) u1 t7) t)) H22 (lift (S O) O t3) H19) in (let
+H27 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
-(THead (Bind b) u1 (lift (S O) O t3)) H27) in (eq_ind_r T (lift (S O) O x)
+(THead (Bind b) u1 (lift (S O) O t3)) H26) in (eq_ind_r T (lift (S O) O x)
(\lambda (t7: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
(THead (Bind b) u2 t7) t8)))) (ex2_ind T (\lambda (t7: T).(pr0 x t7))
(\lambda (t7: T).(pr0 t1 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
(t7: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t7))) (\lambda (x0:
-T).(\lambda (H29: (pr0 x x0)).(\lambda (H30: (pr0 t1 x0)).(ex_intro2 T
+T).(\lambda (H28: (pr0 x x0)).(\lambda (H29: (pr0 t1 x0)).(ex_intro2 T
(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (lift
-(S O) O x)) t7)) x0 H30 (pr0_zeta b H7 x x0 H29 u2))))) (H28 t3 (lift_tlt_dx
-(Bind b) u1 t3 (S O) O) x H26 t1 H8)) t6 H25)))))) (pr0_gen_lift t3 t6 (S O)
-O H24)))) t5 H22)) u (sym_eq T u u1 H21))) k H20)) H19)) H18)))]) in (H17
-(refl_equal T (THead k u1 t5))))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta
+(S O) O x)) t7)) x0 H29 (pr0_zeta b H7 x x0 H28 u2))))) (H27 t3 (lift_tlt_dx
+(Bind b) u1 t3 (S O) O) x H25 t1 H8)) t6 H24)))))) (pr0_gen_lift t3 t6 (S O)
+O H23)))) k H21)))) H18)) H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_beta
u0 v1 v2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1
(THead (Bind Abst) u0 t5)) t)).(\lambda (H12: (eq T (THead (Bind Abbr) v2 t6)
t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_:
(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_:
(pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H16 \def (eq_ind_r T t (\lambda
(t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl)
-v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (match H16 in eq return
-(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1
-(THead (Bind Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
-(t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal \Rightarrow
-(\lambda (H17: (eq T (THead (Bind b) u (lift (S O) O t3)) (THead (Flat Appl)
-v1 (THead (Bind Abst) u0 t5)))).(let H18 \def (eq_ind T (THead (Bind b) u
-(lift (S O) O t3)) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _
-_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1
-(THead (Bind Abst) u0 t5)) H17) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1
-t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H18)))]) in (H17
-(refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)))))))) t2 H13))
-t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2 H11 t5 t6 H12)
-\Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1
-t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S
-O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t5))
-(\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O
-v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
-t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2 (THead (Flat Appl)
-(lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2 (THead (Flat Appl)
-(lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0 Abst)) \to ((pr0 v1
-v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
-t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not (eq B b0
-Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0
-t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u
-(lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0) u1 t5))
-H13) in (let H21 \def (match H20 in eq return (\lambda (t7: T).(\lambda (_:
-(eq ? ? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b0) u1 t5))) \to
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind b0)
-u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal
-\Rightarrow (\lambda (H21: (eq T (THead (Bind b) u (lift (S O) O t3)) (THead
-(Flat Appl) v1 (THead (Bind b0) u1 t5)))).(let H22 \def (eq_ind T (THead
-(Bind b) u (lift (S O) O t3)) (\lambda (e: T).(match e in T return (\lambda
-(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
-| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+v1 (THead (Bind Abst) u0 t5)) H11) in (let H17 \def (eq_ind T (THead (Bind b)
+u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False |
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
-Appl) v1 (THead (Bind b0) u1 t5)) H21) in (False_ind (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl)
-(lift (S O) O v2) t6)) t7))) H22)))]) in (H21 (refl_equal T (THead (Flat
-Appl) v1 (THead (Bind b0) u1 t5)))))))))) t2 H15)) t H13 H14 H9 H10 H11
-H12))) | (pr0_delta u1 u2 H9 t5 t6 H10 w H11) \Rightarrow (\lambda (H12: (eq
-T (THead (Bind Abbr) u1 t5) t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w)
-t2)).(eq_ind T (THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind
-Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))))
-(\lambda (H14: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
-Abbr) u2 w) (\lambda (t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2
-t6 w) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7
-t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H16: (pr0 t5 t6)).(\lambda
-(H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T
-(THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Bind Abbr) u1 t5) H12)
-in (let H19 \def (match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ?
-? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to (ex2 T (\lambda (t8: T).(pr0
-t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)))))) with
-[refl_equal \Rightarrow (\lambda (H19: (eq T (THead (Bind b) u (lift (S O) O
-t3)) (THead (Bind Abbr) u1 t5))).(let H20 \def (f_equal T T (\lambda (e:
-T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let
-rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7
-with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
-(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0
-t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in
+Appl) v1 (THead (Bind Abst) u0 t5)) H16) in (False_ind (ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))
+H17))))) t2 H13)) t H11 H12 H9 H10))) | (pr0_upsilon b0 H9 v1 v2 H10 u1 u2
+H11 t5 t6 H12) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v1 (THead
+(Bind b0) u1 t5)) t)).(\lambda (H14: (eq T (THead (Bind b0) u2 (THead (Flat
+Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead
+(Bind b0) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat
+Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2)
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H15: (eq T (THead (Bind b0) u2
+(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b0) u2
+(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b0
+Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not
+(eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda
+(_: (pr0 t5 t6)).(let H20 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
+(Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Appl) v1 (THead (Bind b0)
+u1 t5)) H13) in (let H21 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3))
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True |
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1
+t5)) H20) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7)))
+H21))))))) t2 H15)) t H13 H14 H9 H10 H11 H12))) | (pr0_delta u1 u2 H9 t5 t6
+H10 w H11) \Rightarrow (\lambda (H12: (eq T (THead (Bind Abbr) u1 t5)
+t)).(\lambda (H13: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
+Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
+t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H14: (eq T (THead (Bind
+Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1
+u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
+t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda
+(H16: (pr0 t5 t6)).(\lambda (H17: (subst0 O u2 t6 w)).(let H18 \def (eq_ind_r
+T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4
+(THead (Bind Abbr) u1 t5) H12) in (let H19 \def (f_equal T B (\lambda (e:
+T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b |
+(TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow
+b])])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H18) in
+((let H20 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
+T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7
+_) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr)
+u1 t5) H18) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f:
+((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u0 t8)
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t8))]) in
lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow
((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match
t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
(THead k u0 t8) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d)
t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7)
\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1
-t5) H19) in ((let H21 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u |
-(THead _ t7 _) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead
-(Bind Abbr) u1 t5) H19) in ((let H22 \def (f_equal T B (\lambda (e: T).(match
-e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _)
-\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_:
-K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow b])])) (THead
-(Bind b) u (lift (S O) O t3)) (THead (Bind Abbr) u1 t5) H19) in (eq_ind B
-Abbr (\lambda (_: B).((eq T u u1) \to ((eq T (lift (S O) O t3) t5) \to (ex2 T
+t5) H18) in (\lambda (_: (eq T u u1)).(\lambda (H23: (eq B b Abbr)).(let H24
+\def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3) H21)
+in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda (t7:
+T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0
+(THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H25: (eq T t6 (lift
+(S O) O x))).(\lambda (H26: (pr0 t3 x)).(let H27 \def (eq_ind_r T t5 (\lambda
+(t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3) H21) in
+(let H28 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
+(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
+(THead (Bind Abbr) u1 (lift (S O) O t3)) H27) in (let H29 \def (eq_ind T t6
+(\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x) H25) in (let H30
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7 Abbr H23) in
+(ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1 t7)) (ex2 T
(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) u2 w)
-t7)))))) (\lambda (H23: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T
-(lift (S O) O t3) t5) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8:
-T).(pr0 (THead (Bind Abbr) u2 w) t8))))) (\lambda (H24: (eq T (lift (S O) O
-t3) t5)).(eq_ind T (lift (S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8)))) (let
-H25 \def (eq_ind_r T t5 (\lambda (t7: T).(pr0 t7 t6)) H16 (lift (S O) O t3)
-H24) in (ex2_ind T (\lambda (t7: T).(eq T t6 (lift (S O) O t7))) (\lambda
-(t7: T).(pr0 t3 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
-T).(pr0 (THead (Bind Abbr) u2 w) t7))) (\lambda (x: T).(\lambda (H26: (eq T
-t6 (lift (S O) O x))).(\lambda (H27: (pr0 t3 x)).(let H28 \def (eq_ind_r T t5
-(\lambda (t7: T).(eq T (THead (Bind Abbr) u1 t7) t)) H12 (lift (S O) O t3)
-H24) in (let H29 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v
-t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to
-(ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O t3)) H28) in (let H30
-\def (eq_ind T t6 (\lambda (t7: T).(subst0 O u2 t7 w)) H17 (lift (S O) O x)
-H26) in (let H31 \def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H7
-Abbr H22) in (ex2_ind T (\lambda (t7: T).(pr0 x t7)) (\lambda (t7: T).(pr0 t1
-t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind
-Abbr) u2 w) t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0
-t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1)
-(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H30 x (pr0_refl
-(lift (S O) O x)) t1))))) (H29 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x
-H27 t1 H8))))))))) (pr0_gen_lift t3 t6 (S O) O H25))) t5 H24)) u (sym_eq T u
-u1 H23))) b (sym_eq B b Abbr H22))) H21)) H20)))]) in (H19 (refl_equal T
-(THead (Bind Abbr) u1 t5)))))))) t2 H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta
-b0 H9 t5 t6 H10 u0) \Rightarrow (\lambda (H11: (eq T (THead (Bind b0) u0
-(lift (S O) O t5)) t)).(\lambda (H12: (eq T t6 t2)).(eq_ind T (THead (Bind
-b0) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not (eq B b0
-Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
-(t8: T).(pr0 t2 t8))))))) (\lambda (H13: (eq T t6 t2)).(eq_ind T t2 (\lambda
-(t7: T).((not (eq B b0 Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq B b0
-Abst))).(\lambda (H15: (pr0 t5 t2)).(let H16 \def (eq_ind_r T t (\lambda (t7:
-T).(eq T (THead (Bind b) u (lift (S O) O t3)) t7)) H4 (THead (Bind b0) u0
-(lift (S O) O t5)) H11) in (let H17 \def (match H16 in eq return (\lambda
-(t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind b0) u0 (lift (S O) O
-t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2
-t8)))))) with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Bind b) u
-(lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)))).(let H18 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7:
-T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
-\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
-\Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f d u1)
-(lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O
-t3) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat)
-(t7: T) on t7: T \def (match t7 with [(TSort n) \Rightarrow (TSort n) |
-(TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i |
-false \Rightarrow (f i)])) | (THead k u1 t8) \Rightarrow (THead k (lref_map f
-d u1) (lref_map f (s k d) t8))]) in lref_map) (\lambda (x: nat).(plus x (S
-O))) O t3) | (THead _ _ t7) \Rightarrow t7])) (THead (Bind b) u (lift (S O) O
-t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H17) in ((let H19 \def (f_equal T
+t7))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1
+x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1)
+(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H29 x (pr0_refl
+(lift (S O) O x)) t1))))) (H28 t3 (lift_tlt_dx (Bind Abbr) u1 t3 (S O) O) x
+H26 t1 H8))))))))) (pr0_gen_lift t3 t6 (S O) O H24)))))) H20)) H19)))))) t2
+H14)) t H12 H13 H9 H10 H11))) | (pr0_zeta b0 H9 t5 t6 H10 u0) \Rightarrow
+(\lambda (H11: (eq T (THead (Bind b0) u0 (lift (S O) O t5)) t)).(\lambda
+(H12: (eq T t6 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t5)) (\lambda
+(_: T).((eq T t6 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t5 t6) \to (ex2 T
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
+(H13: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b0 Abst)) \to
+((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
+t2 t8)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H15: (pr0 t5
+t2)).(let H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u
+(lift (S O) O t3)) t7)) H4 (THead (Bind b0) u0 (lift (S O) O t5)) H11) in
+(let H17 \def (f_equal T B (\lambda (e: T).(match e in T return (\lambda (_:
+T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b1)
+\Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O
+t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H16) in ((let H18 \def (f_equal T
T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7]))
(THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5))
-H17) in ((let H20 \def (f_equal T B (\lambda (e: T).(match e in T return
-(\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b |
-(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind
-b1) \Rightarrow b1 | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S
-O) O t3)) (THead (Bind b0) u0 (lift (S O) O t5)) H17) in (eq_ind B b0
-(\lambda (_: B).((eq T u u0) \to ((eq T (lift (S O) O t3) (lift (S O) O t5))
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))
-(\lambda (H21: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O
-t3) (lift (S O) O t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8:
-T).(pr0 t2 t8))))) (\lambda (H22: (eq T (lift (S O) O t3) (lift (S O) O
-t5))).(eq_ind T (lift (S O) O t3) (\lambda (_: T).(ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))) (let H23 \def (eq_ind_r T t
-(\lambda (t7: T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8)
-\to (\forall (t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10))
-(\lambda (t10: T).(pr0 t9 t10)))))))))) H (THead (Bind b0) u0 (lift (S O) O
-t5)) H11) in (let H24 \def (eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H8 t5
-(lift_inj t3 t5 (S O) O H22)) in (let H25 \def (eq_ind B b (\lambda (b1:
-B).(not (eq B b1 Abst))) H7 b0 H20) in (ex2_ind T (\lambda (t7: T).(pr0 t1
-t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7))
-(\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H26: (pr0 t1
-x)).(\lambda (H27: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7))
-(\lambda (t7: T).(pr0 t2 t7)) x H26 H27)))) (H23 t5 (lift_tlt_dx (Bind b0) u0
-t5 (S O) O) t1 H24 t2 H15))))) (lift (S O) O t5) H22)) u (sym_eq T u u0
-H21))) b (sym_eq B b b0 H20))) H19)) H18)))]) in (H17 (refl_equal T (THead
-(Bind b0) u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9
-H10))) | (pr0_epsilon t5 t6 H9 u0) \Rightarrow (\lambda (H10: (eq T (THead
-(Flat Cast) u0 t5) t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat
-Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda
-(H12: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_:
-(pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind
-b) u (lift (S O) O t3)) t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15
-\def (match H14 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq
-T t7 (THead (Flat Cast) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15:
-(eq T (THead (Bind b) u (lift (S O) O t3)) (THead (Flat Cast) u0 t5))).(let
-H16 \def (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (e:
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _)
-\Rightarrow False])])) I (THead (Flat Cast) u0 t5) H15) in (False_ind (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15
-(refl_equal T (THead (Flat Cast) u0 t5)))))) t6 (sym_eq T t6 t2 H12))) t H10
-H11 H9)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t4 (sym_eq T t4 t1
-H6))) t H4 H5 H2 H3))) | (pr0_epsilon t3 t4 H2 u) \Rightarrow (\lambda (H3:
-(eq T (THead (Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 t1)).(eq_ind T
-(THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to ((pr0 t3 t4) \to
-(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))))
-(\lambda (H5: (eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 t3 t5) \to
-(ex2 T (\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))
-(\lambda (H6: (pr0 t3 t1)).(let H7 \def (match H1 in pr0 return (\lambda (t5:
-T).(\lambda (t6: T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2)
-\to (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))))))
-with [(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq
-T t5 t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
+H16) in ((let H19 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t7: T) on t7: T \def (match t7 with [(TSort n)
+\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with
+[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u1 t8)
+\Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d) t8))]) in
+lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (TLRef _) \Rightarrow
+((let rec lref_map (f: ((nat \to nat))) (d: nat) (t7: T) on t7: T \def (match
+t7 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef
+(match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) |
+(THead k u1 t8) \Rightarrow (THead k (lref_map f d u1) (lref_map f (s k d)
+t8))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t3) | (THead _ _ t7)
+\Rightarrow t7])) (THead (Bind b) u (lift (S O) O t3)) (THead (Bind b0) u0
+(lift (S O) O t5)) H16) in (\lambda (_: (eq T u u0)).(\lambda (H21: (eq B b
+b0)).(let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7)
+\to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
+(THead (Bind b0) u0 (lift (S O) O t5)) H11) in (let H23 \def (eq_ind T t3
+(\lambda (t7: T).(pr0 t7 t1)) H8 t5 (lift_inj t3 t5 (S O) O H19)) in (let H24
+\def (eq_ind B b (\lambda (b1: B).(not (eq B b1 Abst))) H7 b0 H21) in
+(ex2_ind T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x:
+T).(\lambda (H25: (pr0 t1 x)).(\lambda (H26: (pr0 t2 x)).(ex_intro2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) x H25 H26))))
+(H22 t5 (lift_tlt_dx (Bind b0) u0 t5 (S O) O) t1 H23 t2 H15)))))))) H18))
+H17))))) t6 (sym_eq T t6 t2 H13))) t H11 H12 H9 H10))) | (pr0_epsilon t5 t6
+H9 u0) \Rightarrow (\lambda (H10: (eq T (THead (Flat Cast) u0 t5)
+t)).(\lambda (H11: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda
+(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
+t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H12: (eq T t6 t2)).(eq_ind T
+t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (_: (pr0 t5 t2)).(let H14 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Bind b) u (lift (S O) O t3))
+t7)) H4 (THead (Flat Cast) u0 t5) H10) in (let H15 \def (eq_ind T (THead
+(Bind b) u (lift (S O) O t3)) (\lambda (ee: T).(match ee in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with
+[(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat
+Cast) u0 t5) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
+(t7: T).(pr0 t2 t7))) H15)))) t6 (sym_eq T t6 t2 H12))) t H10 H11 H9)))]) in
+(H9 (refl_equal T t) (refl_equal T t2))))) t4 (sym_eq T t4 t1 H6))) t H4 H5
+H2 H3))) | (pr0_epsilon t3 t4 H2 u) \Rightarrow (\lambda (H3: (eq T (THead
+(Flat Cast) u t3) t)).(\lambda (H4: (eq T t4 t1)).(eq_ind T (THead (Flat
+Cast) u t3) (\lambda (_: T).((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T
+(\lambda (t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6)))))) (\lambda (H5:
+(eq T t4 t1)).(eq_ind T t1 (\lambda (t5: T).((pr0 t3 t5) \to (ex2 T (\lambda
+(t6: T).(pr0 t1 t6)) (\lambda (t6: T).(pr0 t2 t6))))) (\lambda (H6: (pr0 t3
+t1)).(let H7 \def (match H1 in pr0 return (\lambda (t5: T).(\lambda (t6:
+T).(\lambda (_: (pr0 t5 t6)).((eq T t5 t) \to ((eq T t6 t2) \to (ex2 T
+(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)))))))) with
+[(pr0_refl t5) \Rightarrow (\lambda (H7: (eq T t5 t)).(\lambda (H8: (eq T t5
+t2)).(eq_ind T t (\lambda (t6: T).((eq T t6 t2) \to (ex2 T (\lambda (t7:
T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))))) (\lambda (H9: (eq T t
t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t7: T).(pr0 t1 t7))
(\lambda (t7: T).(pr0 t2 t7)))) (let H10 \def (eq_ind_r T t (\lambda (t6:
\to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8:
T).(pr0 t7 t8)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t5
t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u
-t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (match H14 in eq return
-(\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead k u1 t5)) \to
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead k u2 t6)
-t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T (THead (Flat Cast)
-u t3) (THead k u1 t5))).(let H16 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat Cast) u t3)
-(THead k u1 t5) H15) in ((let H17 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
-\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3)
-(THead k u1 t5) H15) in ((let H18 \def (f_equal T K (\lambda (e: T).(match e
-in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat Cast) |
-(TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) \Rightarrow k0])) (THead
-(Flat Cast) u t3) (THead k u1 t5) H15) in (eq_ind K (Flat Cast) (\lambda (k0:
-K).((eq T u u1) \to ((eq T t3 t5) \to (ex2 T (\lambda (t7: T).(pr0 t1 t7))
-(\lambda (t7: T).(pr0 (THead k0 u2 t6) t7)))))) (\lambda (H19: (eq T u
-u1)).(eq_ind T u1 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t6) t8)))))
-(\lambda (H20: (eq T t3 t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Flat Cast) u2 t6) t8))))
-(let H21 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1 t5) t)) H9
-(Flat Cast) H18) in (let H22 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v:
-T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v
-t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9
-t10)))))))))) H (THead (Flat Cast) u1 t5) H21) in (let H23 \def (eq_ind T t3
-(\lambda (t7: T).(pr0 t7 t1)) H6 t5 H20) in (ex2_ind T (\lambda (t7: T).(pr0
-t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7))
-(\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7))) (\lambda (x:
-T).(\lambda (H24: (pr0 t1 x)).(\lambda (H25: (pr0 t6 x)).(ex_intro2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6)
-t7)) x H24 (pr0_epsilon t6 x H25 u2))))) (H22 t5 (tlt_head_dx (Flat Cast) u1
-t5) t1 H23 t6 H13))))) t3 (sym_eq T t3 t5 H20))) u (sym_eq T u u1 H19))) k
-H18)) H17)) H16)))]) in (H15 (refl_equal T (THead k u1 t5))))))) t2 H11)) t
-H9 H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow (\lambda (H9:
-(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) t)).(\lambda (H10: (eq
-T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind
-Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t6) t2) \to ((pr0
-v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
-(t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t6)
-t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda (t7: T).((pr0 v1 v2) \to
-((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
-t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t5 t6)).(let H14 \def
-(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead
-(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in (let H15 \def (match H14 in
-eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Flat
-Appl) v1 (THead (Bind Abst) u0 t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 (THead (Bind Abbr) v2 t6) t8)))))) with [refl_equal
-\Rightarrow (\lambda (H15: (eq T (THead (Flat Cast) u t3) (THead (Flat Appl)
-v1 (THead (Bind Abst) u0 t5)))).(let H16 \def (eq_ind T (THead (Flat Cast) u
-t3) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
-\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_:
-F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead
-(Flat Appl) v1 (THead (Bind Abst) u0 t5)) H15) in (False_ind (ex2 T (\lambda
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr) v2 t6) t7)))
-H16)))]) in (H15 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0
-t5)))))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9
-t5 t6 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead
-(Bind b) u1 t5)) t)).(\lambda (H12: (eq T (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead
-(Bind b) u1 t5)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl)
-(lift (S O) O v2) t6)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to
-((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7: T).((not (eq B b
+t3) t7)) H3 (THead k u1 t5) H9) in (let H15 \def (f_equal T K (\lambda (e:
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat
+Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k0 _ _) \Rightarrow k0]))
+(THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H16 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7 _) \Rightarrow t7]))
+(THead (Flat Cast) u t3) (THead k u1 t5) H14) in ((let H17 \def (f_equal T T
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _)
+\Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7]))
+(THead (Flat Cast) u t3) (THead k u1 t5) H14) in (\lambda (_: (eq T u
+u1)).(\lambda (H19: (eq K (Flat Cast) k)).(eq_ind K (Flat Cast) (\lambda (k0:
+K).(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead k0 u2
+t6) t7)))) (let H20 \def (eq_ind_r K k (\lambda (k0: K).(eq T (THead k0 u1
+t5) t)) H9 (Flat Cast) H19) in (let H21 \def (eq_ind_r T t (\lambda (t7:
+T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall
+(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10:
+T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u1 t5) H20) in (let H22 \def
+(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H17) in (ex2_ind T (\lambda
+(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t6 t7)) (ex2 T (\lambda (t7:
+T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Flat Cast) u2 t6) t7)))
+(\lambda (x: T).(\lambda (H23: (pr0 t1 x)).(\lambda (H24: (pr0 t6
+x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead
+(Flat Cast) u2 t6) t7)) x H23 (pr0_epsilon t6 x H24 u2))))) (H21 t5
+(tlt_head_dx (Flat Cast) u1 t5) t1 H22 t6 H13))))) k H19)))) H16)) H15)))))
+t2 H11)) t H9 H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t5 t6 H8) \Rightarrow
+(\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5))
+t)).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead
+(Flat Appl) v1 (THead (Bind Abst) u0 t5)) (\lambda (_: T).((eq T (THead (Bind
+Abbr) v2 t6) t2) \to ((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8:
+T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T
+(THead (Bind Abbr) v2 t6) t2)).(eq_ind T (THead (Bind Abbr) v2 t6) (\lambda
+(t7: T).((pr0 v1 v2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
+t8)) (\lambda (t8: T).(pr0 t7 t8)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_:
+(pr0 t5 t6)).(let H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat
+Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t5)) H9) in
+(let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return
+(\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow
+(match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow False |
+Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0
+t5)) H14) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) v2 t6) t7))) H15))))) t2 H11)) t H9 H10 H7 H8))) |
+(pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t5 t6 H10) \Rightarrow (\lambda (H11: (eq
+T (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) t)).(\lambda (H12: (eq T
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T
+(THead (Flat Appl) v1 (THead (Bind b) u1 t5)) (\lambda (_: T).((eq T (THead
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2) \to ((not (eq B b
Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6) \to (ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8)))))))) (\lambda (_: (not
-(eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda
-(_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead
-(Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind b) u1 t5)) H11)
-in (let H19 \def (match H18 in eq return (\lambda (t7: T).(\lambda (_: (eq ?
-? t7)).((eq T t7 (THead (Flat Appl) v1 (THead (Bind b) u1 t5))) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind b) u2 (THead
-(Flat Appl) (lift (S O) O v2) t6)) t8)))))) with [refl_equal \Rightarrow
-(\lambda (H19: (eq T (THead (Flat Cast) u t3) (THead (Flat Appl) v1 (THead
-(Bind b) u1 t5)))).(let H20 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda
-(e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
-False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
-return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f)
-\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow
-False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind b)
-u1 t5)) H19) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
-T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t7)))
-H20)))]) in (H19 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u1
-t5)))))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t5 t6
-H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t5)
-t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind
-Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
-t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T (THead (Bind
-Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t7: T).((pr0 u1
-u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda (t8: T).(pr0
-t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_: (pr0 u1 u2)).(\lambda
-(_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let H16 \def (eq_ind_r T t
-(\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind Abbr) u1
-t5) H10) in (let H17 \def (match H16 in eq return (\lambda (t7: T).(\lambda
-(_: (eq ? ? t7)).((eq T t7 (THead (Bind Abbr) u1 t5)) \to (ex2 T (\lambda
-(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 (THead (Bind Abbr) u2 w) t8))))))
-with [refl_equal \Rightarrow (\lambda (H17: (eq T (THead (Flat Cast) u t3)
-(THead (Bind Abbr) u1 t5))).(let H18 \def (eq_ind T (THead (Flat Cast) u t3)
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _)
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))))) (\lambda (H13: (eq
+T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) t2)).(eq_ind T
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t6)) (\lambda (t7:
+T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t5 t6)
+\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))))
+(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0
+u1 u2)).(\lambda (_: (pr0 t5 t6)).(let H18 \def (eq_ind_r T t (\lambda (t7:
+T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Appl) v1 (THead (Bind
+b) u1 t5)) H11) in (let H19 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow
(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False |
-(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t5) H17) in (False_ind
-(ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 (THead (Bind Abbr)
-u2 w) t7))) H18)))]) in (H17 (refl_equal T (THead (Bind Abbr) u1 t5))))))))
-t2 H12)) t H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow
-(\lambda (H9: (eq T (THead (Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10:
-(eq T t6 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_:
-T).((eq T t6 t2) \to ((not (eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T
-(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))) (\lambda
-(H11: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((not (eq B b Abst)) \to
-((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0
-t2 t8)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t5 t2)).(let
-H14 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7))
-H3 (THead (Bind b) u0 (lift (S O) O t5)) H9) in (let H15 \def (match H14 in
-eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq T t7 (THead (Bind
-b) u0 (lift (S O) O t5))) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda
-(t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H15: (eq T
-(THead (Flat Cast) u t3) (THead (Bind b) u0 (lift (S O) O t5)))).(let H16
-\def (eq_ind T (THead (Flat Cast) u t3) (\lambda (e: T).(match e in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda
-(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow
-True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H15) in (False_ind (ex2 T
-(\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) H16)))]) in (H15
-(refl_equal T (THead (Bind b) u0 (lift (S O) O t5)))))))) t6 (sym_eq T t6 t2
-H11))) t H9 H10 H7 H8))) | (pr0_epsilon t5 t6 H7 u0) \Rightarrow (\lambda
-(H8: (eq T (THead (Flat Cast) u0 t5) t)).(\lambda (H9: (eq T t6 t2)).(eq_ind
-T (THead (Flat Cast) u0 t5) (\lambda (_: T).((eq T t6 t2) \to ((pr0 t5 t6)
-\to (ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))))
-(\lambda (H10: (eq T t6 t2)).(eq_ind T t2 (\lambda (t7: T).((pr0 t5 t7) \to
-(ex2 T (\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))
-(\lambda (H11: (pr0 t5 t2)).(let H12 \def (eq_ind_r T t (\lambda (t7: T).(eq
-T (THead (Flat Cast) u t3) t7)) H3 (THead (Flat Cast) u0 t5) H8) in (let H13
-\def (match H12 in eq return (\lambda (t7: T).(\lambda (_: (eq ? ? t7)).((eq
-T t7 (THead (Flat Cast) u0 t5)) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8)))))) with [refl_equal \Rightarrow (\lambda (H13:
-(eq T (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5))).(let H14 \def
-(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
-[(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t7)
-\Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5) H13) in
-((let H15 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t7
-_) \Rightarrow t7])) (THead (Flat Cast) u t3) (THead (Flat Cast) u0 t5) H13)
-in (eq_ind T u0 (\lambda (_: T).((eq T t3 t5) \to (ex2 T (\lambda (t8:
-T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H16: (eq T t3
-t5)).(eq_ind T t5 (\lambda (_: T).(ex2 T (\lambda (t8: T).(pr0 t1 t8))
-(\lambda (t8: T).(pr0 t2 t8)))) (let H17 \def (eq_ind_r T t (\lambda (t7:
-T).(\forall (v: T).((tlt v t7) \to (\forall (t8: T).((pr0 v t8) \to (\forall
-(t9: T).((pr0 v t9) \to (ex2 T (\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10:
-T).(pr0 t9 t10)))))))))) H (THead (Flat Cast) u0 t5) H8) in (let H18 \def
-(eq_ind T t3 (\lambda (t7: T).(pr0 t7 t1)) H6 t5 H16) in (ex2_ind T (\lambda
-(t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7:
-T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H19:
-(pr0 t1 x)).(\lambda (H20: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1
-t7)) (\lambda (t7: T).(pr0 t2 t7)) x H19 H20)))) (H17 t5 (tlt_head_dx (Flat
-Cast) u0 t5) t1 H18 t2 H11)))) t3 (sym_eq T t3 t5 H16))) u (sym_eq T u u0
-H15))) H14)))]) in (H13 (refl_equal T (THead (Flat Cast) u0 t5)))))) t6
-(sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7 (refl_equal T t) (refl_equal T
-t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t)
-(refl_equal T t1))))))))) t0).
+(Flat f) \Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl
+\Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1
+(THead (Bind b) u1 t5)) H18) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1
+t7)) (\lambda (t7: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O)
+O v2) t6)) t7))) H19))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta
+u1 u2 H7 t5 t6 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr)
+u1 t5) t)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T
+(THead (Bind Abbr) u1 t5) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2)
+\to ((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T (\lambda
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))))) (\lambda (H12: (eq T
+(THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda
+(t7: T).((pr0 u1 u2) \to ((pr0 t5 t6) \to ((subst0 O u2 t6 w) \to (ex2 T
+(\lambda (t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t7 t8))))))) (\lambda (_:
+(pr0 u1 u2)).(\lambda (_: (pr0 t5 t6)).(\lambda (_: (subst0 O u2 t6 w)).(let
+H16 \def (eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7))
+H3 (THead (Bind Abbr) u1 t5) H10) in (let H17 \def (eq_ind T (THead (Flat
+Cast) u t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _)
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _)
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1
+t5) H16) in (False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 (THead (Bind Abbr) u2 w) t7))) H17)))))) t2 H12)) t H10 H11 H7 H8
+H9))) | (pr0_zeta b H7 t5 t6 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead
+(Bind b) u0 (lift (S O) O t5)) t)).(\lambda (H10: (eq T t6 t2)).(eq_ind T
+(THead (Bind b) u0 (lift (S O) O t5)) (\lambda (_: T).((eq T t6 t2) \to ((not
+(eq B b Abst)) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t2 t8))))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2
+(\lambda (t7: T).((not (eq B b Abst)) \to ((pr0 t5 t7) \to (ex2 T (\lambda
+(t8: T).(pr0 t1 t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (_: (not (eq
+B b Abst))).(\lambda (_: (pr0 t5 t2)).(let H14 \def (eq_ind_r T t (\lambda
+(t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead (Bind b) u0 (lift (S O)
+O t5)) H9) in (let H15 \def (eq_ind T (THead (Flat Cast) u t3) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _)
+\Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t5)) H14) in
+(False_ind (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7: T).(pr0 t2
+t7))) H15))))) t6 (sym_eq T t6 t2 H11))) t H9 H10 H7 H8))) | (pr0_epsilon t5
+t6 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t5)
+t)).(\lambda (H9: (eq T t6 t2)).(eq_ind T (THead (Flat Cast) u0 t5) (\lambda
+(_: T).((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t8: T).(pr0 t1
+t8)) (\lambda (t8: T).(pr0 t2 t8)))))) (\lambda (H10: (eq T t6 t2)).(eq_ind T
+t2 (\lambda (t7: T).((pr0 t5 t7) \to (ex2 T (\lambda (t8: T).(pr0 t1 t8))
+(\lambda (t8: T).(pr0 t2 t8))))) (\lambda (H11: (pr0 t5 t2)).(let H12 \def
+(eq_ind_r T t (\lambda (t7: T).(eq T (THead (Flat Cast) u t3) t7)) H3 (THead
+(Flat Cast) u0 t5) H8) in (let H13 \def (f_equal T T (\lambda (e: T).(match e
+in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _)
+\Rightarrow u | (THead _ t7 _) \Rightarrow t7])) (THead (Flat Cast) u t3)
+(THead (Flat Cast) u0 t5) H12) in ((let H14 \def (f_equal T T (\lambda (e:
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 |
+(TLRef _) \Rightarrow t3 | (THead _ _ t7) \Rightarrow t7])) (THead (Flat
+Cast) u t3) (THead (Flat Cast) u0 t5) H12) in (\lambda (_: (eq T u u0)).(let
+H16 \def (eq_ind_r T t (\lambda (t7: T).(\forall (v: T).((tlt v t7) \to
+(\forall (t8: T).((pr0 v t8) \to (\forall (t9: T).((pr0 v t9) \to (ex2 T
+(\lambda (t10: T).(pr0 t8 t10)) (\lambda (t10: T).(pr0 t9 t10)))))))))) H
+(THead (Flat Cast) u0 t5) H8) in (let H17 \def (eq_ind T t3 (\lambda (t7:
+T).(pr0 t7 t1)) H6 t5 H14) in (ex2_ind T (\lambda (t7: T).(pr0 t1 t7))
+(\lambda (t7: T).(pr0 t2 t7)) (ex2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda
+(t7: T).(pr0 t2 t7))) (\lambda (x: T).(\lambda (H18: (pr0 t1 x)).(\lambda
+(H19: (pr0 t2 x)).(ex_intro2 T (\lambda (t7: T).(pr0 t1 t7)) (\lambda (t7:
+T).(pr0 t2 t7)) x H18 H19)))) (H16 t5 (tlt_head_dx (Flat Cast) u0 t5) t1 H17
+t2 H11)))))) H13)))) t6 (sym_eq T t6 t2 H10))) t H8 H9 H7)))]) in (H7
+(refl_equal T t) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H5))) t H3 H4
+H2)))]) in (H2 (refl_equal T t) (refl_equal T t1))))))))) t0).
(* This file was automatically generated: do not edit *********************)
-include "pr0/defs.ma".
+include "LambdaDelta-1/pr0/defs.ma".
-include "subst0/subst0.ma".
+include "LambdaDelta-1/subst0/subst0.ma".
theorem pr0_lift:
\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall
(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 (t0: T).(\lambda (t3: T).(\lambda
-(_: (pr0 t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0
-(lift h d t0) (lift h d t3)))))).(\lambda (k: K).(\lambda (h: nat).(\lambda
-(d: nat).(eq_ind_r T (THead k (lift h d u1) (lift h (s k d) t0)) (\lambda (t:
-T).(pr0 t (lift h d (THead k u2 t3)))) (eq_ind_r T (THead k (lift h d u2)
-(lift h (s k d) t3)) (\lambda (t: T).(pr0 (THead k (lift h d u1) (lift h (s k
-d) t0)) t)) (pr0_comp (lift h d u1) (lift h d u2) (H1 h d) (lift h (s k d)
-t0) (lift h (s k d) t3) (H3 h (s k d)) k) (lift h d (THead k u2 t3))
-(lift_head k u2 t3 h d)) (lift h d (THead k u1 t0)) (lift_head k u1 t0 h
+(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 (t0: T).(\lambda (t3: T).(\lambda (_: (pr0
-t0 t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0)
-(lift h d t3)))))).(\lambda (h: nat).(\lambda (d: nat).(eq_ind_r T (THead
+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
-t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r
+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)) t0)) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) t)
-(lift h d (THead (Bind Abbr) v2 t3)))) (eq_ind_r T (THead (Bind Abbr) (lift h
-d v2) (lift h (s (Bind Abbr) d) t3)) (\lambda (t: T).(pr0 (THead (Flat Appl)
+(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)) t0))) t)) (pr0_beta (lift h (s (Flat Appl) d)
+(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)) t0) (lift h (s (Bind Abbr) d) t3) (H3 h (s (Bind Abbr) d))) (lift h d
-(THead (Bind Abbr) v2 t3)) (lift_head (Bind Abbr) v2 t3 h d)) (lift h (s
-(Flat Appl) d) (THead (Bind Abst) u t0)) (lift_head (Bind Abst) u t0 h (s
-(Flat Appl) d))) (lift h d (THead (Flat Appl) v1 (THead (Bind Abst) u t0)))
-(lift_head (Flat Appl) v1 (THead (Bind Abst) u t0) h d))))))))))))) (\lambda
+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 (t0: T).(\lambda (t3:
-T).(\lambda (_: (pr0 t0 t3)).(\lambda (H6: ((\forall (h: nat).(\forall (d:
-nat).(pr0 (lift h d t0) (lift h d t3)))))).(\lambda (h: nat).(\lambda (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 t0))) (\lambda (t: T).(pr0 t (lift h d (THead (Bind b) u2
-(THead (Flat Appl) (lift (S O) O v2) t3))))) (eq_ind_r T (THead (Bind b)
-(lift h (s (Flat Appl) d) u1) (lift h (s (Bind b) (s (Flat Appl) d)) t0))
+(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) t3))))) (eq_ind_r T (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) t3))) (\lambda (t: T).(pr0 (THead (Flat Appl) (lift h d v1) (THead
+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))
-t0))) 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)) t3)) (\lambda (t: T).(pr0 (THead
+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)) t0))) (THead (Bind b) (lift h d u2) t)))
+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 t0))) (THead (Bind b) (lift h d
-u2) (THead (Flat Appl) (lift h n (lift (S O) O v2)) (lift h n t3)))))
+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)
-t0))) (THead (Bind b) (lift h d u2) (THead (Flat Appl) t (lift h (plus (S O)
-d) t3))))) (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) t0) (lift h (plus (S O) d)
-t3) (H6 h (plus (S O) d))) (lift h (plus (S O) d) (lift (S O) O v2)) (lift_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) t3)) (lift_head (Flat Appl) (lift (S
-O) O v2) t3 h (s (Bind b) d))) (lift h d (THead (Bind b) u2 (THead (Flat
-Appl) (lift (S O) O v2) t3))) (lift_head (Bind b) u2 (THead (Flat Appl) (lift
-(S O) O v2) t3) h d)) (lift h (s (Flat Appl) d) (THead (Bind b) u1 t0))
-(lift_head (Bind b) u1 t0 h (s (Flat Appl) d))) (lift h d (THead (Flat Appl)
-v1 (THead (Bind b) u1 t0))) (lift_head (Flat Appl) v1 (THead (Bind b) u1 t0)
+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 (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0
-t3)).(\lambda (H3: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0)
-(lift h d t3)))))).(\lambda (w: T).(\lambda (H4: (subst0 O u2 t3 w)).(\lambda
+(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) t0)) (\lambda (t: T).(pr0 t (lift h d (THead (Bind Abbr)
+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) t0)) t)) (pr0_delta (lift h d u1) (lift h d u2) (H1 h d) (lift h (S
-d) t0) (lift h (S d) t3) (H3 h (S d)) (lift h (S d) w) (let d' \def (S d) in
+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' t3) (lift h d' w))) (subst0_lift_lt t3 w u2 O H4 (S d) (le_n_S O d
+(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 t0))
-(lift_head (Bind Abbr) u1 t0 h d)))))))))))))) (\lambda (b: B).(\lambda (H0:
-(not (eq B b Abst))).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (pr0 t0
-t3)).(\lambda (H2: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t0)
-(lift h d t3)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d:
+(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 t0))) (\lambda (t: T).(pr0 t (lift h d t3))) (eq_ind nat (plus (S O) d)
+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
-t0))) (lift h d t3))) (eq_ind_r T (lift (S O) O (lift h d t0)) (\lambda (t:
-T).(pr0 (THead (Bind b) (lift h d u) t) (lift h d t3))) (pr0_zeta b H0 (lift
-h d t0) (lift h d t3) (H2 h d) (lift h d u)) (lift h (plus (S O) d) (lift (S
-O) O t0)) (lift_d t0 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 t0))) (lift_head (Bind b) u (lift
-(S O) O t0) h d))))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (_:
-(pr0 t0 t3)).(\lambda (H1: ((\forall (h: nat).(\forall (d: nat).(pr0 (lift h
-d t0) (lift h d t3)))))).(\lambda (u: T).(\lambda (h: nat).(\lambda (d:
+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)
-t0)) (\lambda (t: T).(pr0 t (lift h d t3))) (pr0_epsilon (lift h (s (Flat
-Cast) d) t0) (lift h d t3) (H1 h d) (lift h d u)) (lift h d (THead (Flat
-Cast) u t0)) (lift_head (Flat Cast) u t0 h d))))))))) t1 t2 H))).
+t3)) (\lambda (t: T).(pr0 t (lift h d t4))) (pr0_epsilon (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
(\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 (u0: T).(\lambda
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1:
-((\forall (u3: T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t))
-(\lambda (t: T).(pr0 t u0))))))).(\lambda (t: T).(\lambda (k: K).(\lambda
-(u3: T).(\lambda (H2: (pr0 u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3
-u1 t0)) (\lambda (t0: T).(pr0 t0 u0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0 (THead k u0 t)))) (\lambda (x:
-T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 x u0)).(ex_intro2 T
-(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0 t0
-(THead k u0 t))) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp x u0
-H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\lambda (k: K).(\lambda (v:
-T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (_: (subst0
-(s k i0) v t3 t0)).(\lambda (H1: ((\forall (u1: T).((pr0 u1 v) \to (ex2 T
-(\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t
-t0))))))).(\lambda (u: T).(\lambda (u1: T).(\lambda (H2: (pr0 u1 v)).(ex2_ind
-T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t t0)) (ex2
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0 t
-(THead k u t0)))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3
-x)).(\lambda (H4: (pr0 x t0)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1
-(THead k u t3) t)) (\lambda (t: T).(pr0 t (THead k u t0))) (THead k u x)
-(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) x t0 H4 k))))) (H1
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u3:
-T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) (\lambda (t:
-T).(pr0 t u0))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u3:
-T).((pr0 u3 v) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda
-(t: T).(pr0 t t3))))))).(\lambda (u3: T).(\lambda (H4: (pr0 u3 v)).(ex2_ind T
-(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t t3)) (ex2 T
-(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t
-(THead k u0 t3)))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0
-x)).(\lambda (H6: (pr0 x t3)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t))
-(\lambda (t: T).(pr0 t u0)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1
-t0) t)) (\lambda (t: T).(pr0 t (THead k u0 t3)))) (\lambda (x0: T).(\lambda
-(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 x0 u0)).(ex_intro2 T (\lambda
-(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 t (THead k u0
-t3))) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp x0 u0
-H8 x t3 H6 k))))) (H1 u3 H4))))) (H3 u3 H4))))))))))))))) i u2 t1 t2 H))))).
+(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
(\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 (u0: T).(\lambda
-(u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1:
-((\forall (u3: T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t))
-(\lambda (t: T).(pr0 u0 t))))))).(\lambda (t: T).(\lambda (k: K).(\lambda
-(u3: T).(\lambda (H2: (pr0 v u3)).(ex2_ind T (\lambda (t0: T).(subst0 i0 u3
-u1 t0)) (\lambda (t0: T).(pr0 u0 t0)) (ex2 T (\lambda (t0: T).(subst0 i0 u3
-(THead k u1 t) t0)) (\lambda (t0: T).(pr0 (THead k u0 t) t0))) (\lambda (x:
-T).(\lambda (H3: (subst0 i0 u3 u1 x)).(\lambda (H4: (pr0 u0 x)).(ex_intro2 T
-(\lambda (t0: T).(subst0 i0 u3 (THead k u1 t) t0)) (\lambda (t0: T).(pr0
-(THead k u0 t) t0)) (THead k x t) (subst0_fst u3 x u1 i0 H3 t k) (pr0_comp u0
-x H4 t t (pr0_refl t) k))))) (H1 u3 H2)))))))))))) (\lambda (k: K).(\lambda
-(v: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (_:
-(subst0 (s k i0) v t3 t0)).(\lambda (H1: ((\forall (u1: T).((pr0 v u1) \to
-(ex2 T (\lambda (t: T).(subst0 (s k i0) u1 t3 t)) (\lambda (t: T).(pr0 t0
+(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 t3 t)) (\lambda (t: T).(pr0 t0 t)) (ex2
-T (\lambda (t: T).(subst0 i0 u1 (THead k u t3) t)) (\lambda (t: T).(pr0
-(THead k u t0) t))) (\lambda (x: T).(\lambda (H3: (subst0 (s k i0) u1 t3
-x)).(\lambda (H4: (pr0 t0 x)).(ex_intro2 T (\lambda (t: T).(subst0 i0 u1
-(THead k u t3) t)) (\lambda (t: T).(pr0 (THead k u t0) t)) (THead k u x)
-(subst0_snd k u1 x t3 i0 H3 u) (pr0_comp u u (pr0_refl u) t0 x H4 k))))) (H1
-u1 H2)))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda
-(i0: nat).(\lambda (_: (subst0 i0 v u1 u0)).(\lambda (H1: ((\forall (u3:
-T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 i0 u3 u1 t)) (\lambda (t:
-T).(pr0 u0 t))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (_: (subst0 (s k i0) v t0 t3)).(\lambda (H3: ((\forall (u3:
-T).((pr0 v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda
-(t: T).(pr0 t3 t))))))).(\lambda (u3: T).(\lambda (H4: (pr0 v u3)).(ex2_ind T
-(\lambda (t: T).(subst0 (s k i0) u3 t0 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T
-(\lambda (t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead
-k u0 t3) t))) (\lambda (x: T).(\lambda (H5: (subst0 (s k i0) u3 t0
-x)).(\lambda (H6: (pr0 t3 x)).(ex2_ind T (\lambda (t: T).(subst0 i0 u3 u1 t))
-(\lambda (t: T).(pr0 u0 t)) (ex2 T (\lambda (t: T).(subst0 i0 u3 (THead k u1
-t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3) t))) (\lambda (x0: T).(\lambda
-(H7: (subst0 i0 u3 u1 x0)).(\lambda (H8: (pr0 u0 x0)).(ex_intro2 T (\lambda
-(t: T).(subst0 i0 u3 (THead k u1 t0) t)) (\lambda (t: T).(pr0 (THead k u0 t3)
-t)) (THead k x0 x) (subst0_both u3 u1 x0 i0 H7 k t0 x H5) (pr0_comp u0 x0 H8
-t3 x H6 k))))) (H1 u3 H4))))) (H3 u3 H4))))))))))))))) i u2 t1 t2 H))))).
+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
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)).(eq_ind_r T (THead
-(Flat Appl) v1 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))))) (eq_ind_r T (THead (Bind Abst) x0 t3) (\lambda (t:
-T).(or (pr0 (THead (Flat Appl) v1 t) (THead (Bind Abbr) v2 t4)) (ex2 T
-(\lambda (w2: T).(pr0 (THead (Flat Appl) v1 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)) x H10) w1 H7)))) 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
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-(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 (H13: (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 H13 t3 t4 H2))) (\lambda (H13: (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:
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-(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 H14 t3 t4 H2) (subst0_fst v3 x2 v2 i H15
-t4 (Bind Abbr))))))) H13)) (H1 v0 x0 i H8 v3 H5)) x1 H11) w1 H7)))) 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))
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-T).(\lambda (H11: (eq T x1 (THead (Bind Abst) u x))).(\lambda (H12: (subst0
-(s (Bind Abst) (s (Flat Appl) i)) v0 t3 x)).(eq_ind_r T (THead (Flat Appl) 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))))) (eq_ind_r T (THead (Bind Abst) u x) (\lambda (t: T).(or (pr0 (THead
-(Flat Appl) x0 t) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2: T).(pr0
-(THead (Flat Appl) x0 t) w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind
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-(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))))
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-(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 x0
+(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 H14 x t4 H13))) (\lambda (H14: (ex2 T (\lambda
+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 (H15: (pr0 x0 x2)).(\lambda (H16: (subst0 i v3 v2
+(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 H15 x t4 H13)
-(subst0_fst v3 x2 v2 i H16 t4 (Bind Abbr))))))) H14)) (H1 v0 x0 i H8 v3 H5)))
-(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0
+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 (H14: (pr0 x x2)).(\lambda (H15: (subst0 (s
+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 (H16:
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(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
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-(subst0_snd (Bind Abbr) v3 x2 t4 i H15 v2)))) (\lambda (H16: (ex2 T (\lambda
+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 (H17: (pr0 x0 x3)).(\lambda (H18: (subst0 i v3 v2
+(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 H17 x x2 H14)
-(subst0_both v3 v2 x3 i H18 (Bind Abbr) t4 x2 H15)))))) H16)) (H1 v0 x0 i H8
-v3 H5))))) H13)) (H3 v0 x (s (Bind Abst) (s (Flat Appl) i)) H12 v3 H5)) x1
-H11) w1 H7)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda (t5:
+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
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)).(eq_ind_r T (THead (Flat
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-(THead (Flat Appl) x0 t) (THead (Bind Abbr) v2 t4)) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Flat Appl) x0 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
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-w2)))) (\lambda (H14: (pr0 x3 t4)).(or_ind (pr0 x0 v2) (ex2 T (\lambda (w2:
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+(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 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 H15 x3 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) x2 x3)) (THead
+(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
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+Appl) (lift (S O) O v2) t4))) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Appl)
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+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)
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+(x0: T).(\lambda (x1: T).(\lambda (H10: (eq T w1 (THead (Flat Appl) x0
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+(\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v3 (THead (Bind b)
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+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)
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-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
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(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
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(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))
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(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
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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
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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 (H17: (pr0 x x2)).(\lambda
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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)
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v2)).(or_intror (pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) (THead
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T).(pr0 (THead (Flat Appl) x0 (THead (Bind b) x t3)) w2)) (\lambda (w2:
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 H19 x x2 H17 t3 t4 H5) (subst0_fst
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-(H19: (ex2 T (\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v3 v2
+(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))))
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+(\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:
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 H20 x x2 H17 t3 t4 H5) (subst0_both
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Appl) (lift (S O) O x3) t4) (subst0_fst v3 (lift (S O) O x3) (lift (S O) O
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-(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
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-(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)).(eq_ind_r T (THead (Flat Appl) x0
-x1) (\lambda (t: T).(or (pr0 t (THead (Bind b) u2 (THead (Flat Appl) (lift (S
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-v3 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) w2)))))
-(eq_ind_r T (THead (Bind b) u1 x) (\lambda (t: T).(or (pr0 (THead (Flat Appl)
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-(\lambda (w2: T).(pr0 (THead (Flat Appl) x0 t) w2)) (\lambda (w2: T).(subst0
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-(or_ind (pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2:
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+i) H15 v3 H8)) w1 H16))))) H13)) (\lambda (H13: (ex2 T (\lambda (t5: T).(eq T
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+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 (H16: (pr0 x t4)).(or_ind
+(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
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-(H9: (subst0 (minus (s (Bind b) i) (S O)) v1 t3 x0)).(eq_ind_r T (THead (Bind
-b) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2))
-(\lambda (w2: T).(subst0 i v2 t4 w2))))) (eq_ind_r T (lift (S O) O x0)
-(\lambda (t: T).(or (pr0 (THead (Bind b) u t) t4) (ex2 T (\lambda (w2:
-T).(pr0 (THead (Bind b) u t) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))))
-(let H10 \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 (H11: (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 H11 u))) (\lambda (H11: (ex2 T
-(\lambda (w2: T).(pr0 x0 w2)) (\lambda (w2: T).(subst0 i v2 t4
+(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 (H12: (pr0 x0
-x1)).(\lambda (H13: (subst0 i v2 t4 x1)).(or_intror (pr0 (THead (Bind b) u
+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 H12 u) H13))))) H11)) (H2 v1
-x0 i H10 v2 H4))) x H8) w1 H6)))) (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 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:
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)).(eq_ind_r
-T (THead (Bind b) x0 x1) (\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2:
-T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (eq_ind_r T (lift (S
-O) O x) (\lambda (t: T).(or (pr0 (THead (Bind b) x0 t) t4) (ex2 T (\lambda
-(w2: T).(pr0 (THead (Bind b) x0 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 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: T).(pr0 (THead (Bind b) x0
-(lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H12:
-(pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T
+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: T).(pr0
+(THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4
+w2)))) (\lambda (H13: (pr0 x t4)).(or_introl (pr0 (THead (Bind b) x0 (lift (S
+O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O
+x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H13 x0)))
+(\lambda (H13: (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2)))).(ex2_ind 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: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2))) (pr0_zeta b H0 x t4 H12 x0))) (\lambda (H12: (ex2 T
-(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))).(ex2_ind
-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: T).(pr0 (THead
-(Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))
-(\lambda (x2: T).(\lambda (H13: (pr0 x x2)).(\lambda (H14: (subst0 i v2 t4
-x2)).(or_intror (pr0 (THead (Bind b) x0 (lift (S O) O x)) t4) (ex2 T (\lambda
+T).(subst0 i v2 t4 w2)))) (\lambda (x2: T).(\lambda (H14: (pr0 x
+x2)).(\lambda (H15: (subst0 i v2 t4 x2)).(or_intror (pr0 (THead (Bind b) x0
+(lift (S O) O x)) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Bind b) x0 (lift
+(S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda
(w2: T).(pr0 (THead (Bind b) x0 (lift (S O) O x)) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Bind b)
-x0 (lift (S O) O x)) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)) x2 (pr0_zeta
-b H0 x x2 H13 x0) H14))))) H12)) (H2 v1 x i H11 v2 H4))) x1 H9) w1 H6))))
-(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 (t5:
-T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s (Flat
-Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda (w2: T).(pr0 w1 w2))
-(\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4: (ex2 T (\lambda (u2:
-T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2: T).(subst0 i v1 u
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 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 (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda (_: (subst0 i
-v1 u x)).(eq_ind_r T (THead (Flat Cast) x 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 (Flat Cast) x t3) t4) (ex2 T (\lambda (w2: T).(pr0
-(THead (Flat Cast) x t3) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
-(pr0_epsilon t3 t4 H0 x)) w1 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t5:
-T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat
-Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T w1 (THead (Flat Cast)
-u t5))) (\lambda (t5: T).(subst0 (s (Flat Cast) i) 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 (H5: (eq T w1 (THead (Flat Cast) u x))).(\lambda
-(H6: (subst0 (s (Flat Cast) i) v1 t3 x)).(eq_ind_r T (THead (Flat Cast) u x)
-(\lambda (t: T).(or (pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda
-(w2: T).(subst0 i v2 t4 w2))))) (or_ind (pr0 x t4) (ex2 T (\lambda (w2:
-T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or
-(pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat
-Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H7: (pr0 x
-t4)).(or_introl (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0
-(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
-(pr0_epsilon x t4 H7 u))) (\lambda (H7: (ex2 T (\lambda (w2: T).(pr0 x w2))
-(\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)))).(ex2_ind T (\lambda
-(w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2)) (or
-(pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat
-Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (x0:
-T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0 (s (Flat Cast) i) v2 t4
-x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2 T (\lambda (w2: T).(pr0
-(THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))
-(ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2:
-T).(subst0 i v2 t4 w2)) x0 (pr0_epsilon x x0 H8 u) H9))))) H7)) (H1 v1 x (s
-(Flat Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda
-(u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T (\lambda (u2:
+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 (t5: T).(eq T w1 (THead (Flat Cast) u2 t5)))) (\lambda (u2:
T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_: T).(\lambda (t5:
-T).(subst0 (s (Flat Cast) i) 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 (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast) x0
-x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat Cast)
-i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (pr0
-t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4
-w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
+T).(subst0 (s (Flat Cast) i) v1 t3 t5)))) (or (pr0 w1 t4) (ex2 T (\lambda
+(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst0 i v2 t4 w2)))) (\lambda (H4:
+(ex2 T (\lambda (u2: T).(eq T w1 (THead (Flat Cast) u2 t3))) (\lambda (u2:
+T).(subst0 i v1 u u2)))).(ex2_ind T (\lambda (u2: T).(eq T w1 (THead (Flat
+Cast) u2 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 (H5: (eq T w1 (THead (Flat Cast) x t3))).(\lambda
+(_: (subst0 i v1 u x)).(eq_ind_r T (THead (Flat Cast) x 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 (Flat Cast) x t3) t4) (ex2
+T (\lambda (w2: T).(pr0 (THead (Flat Cast) x t3) w2)) (\lambda (w2:
+T).(subst0 i v2 t4 w2))) (pr0_epsilon t3 t4 H0 x)) w1 H5)))) H4)) (\lambda
+(H4: (ex2 T (\lambda (t5: T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda
+(t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5)))).(ex2_ind T (\lambda (t5:
+T).(eq T w1 (THead (Flat Cast) u t5))) (\lambda (t5: T).(subst0 (s (Flat
+Cast) i) 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 (H5: (eq T
+w1 (THead (Flat Cast) u x))).(\lambda (H6: (subst0 (s (Flat Cast) i) v1 t3
+x)).(eq_ind_r T (THead (Flat Cast) u x) (\lambda (t: T).(or (pr0 t t4) (ex2 T
+(\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2 t4 w2))))) (or_ind
+(pr0 x t4) (ex2 T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s
+(Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2)))) (\lambda (H7: (pr0 x t4)).(or_introl (pr0 (THead (Flat Cast) u
+x) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda
+(w2: T).(subst0 i v2 t4 w2))) (pr0_epsilon x t4 H7 u))) (\lambda (H7: (ex2 T
+(\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s (Flat Cast) i) v2 t4
+w2)))).(ex2_ind T (\lambda (w2: T).(pr0 x w2)) (\lambda (w2: T).(subst0 (s
+(Flat Cast) i) v2 t4 w2)) (or (pr0 (THead (Flat Cast) u x) t4) (ex2 T
+(\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0 i
+v2 t4 w2)))) (\lambda (x0: T).(\lambda (H8: (pr0 x x0)).(\lambda (H9: (subst0
+(s (Flat Cast) i) v2 t4 x0)).(or_intror (pr0 (THead (Flat Cast) u x) t4) (ex2
+T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2)) (\lambda (w2: T).(subst0
+i v2 t4 w2))) (ex_intro2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) u x) w2))
+(\lambda (w2: T).(subst0 i v2 t4 w2)) x0 (pr0_epsilon x x0 H8 u) H9))))) H7))
+(H1 v1 x (s (Flat Cast) i) H6 v2 H3)) w1 H5)))) H4)) (\lambda (H4: (ex3_2 T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s (Flat Cast) i) v1 t3 t5))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t5: T).(eq T w1 (THead (Flat Cast) u2 t5))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i v1 u u2))) (\lambda (_:
+T).(\lambda (t5: T).(subst0 (s (Flat Cast) i) 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 (x1: T).(\lambda (H5: (eq T w1 (THead (Flat Cast)
+x0 x1))).(\lambda (_: (subst0 i v1 u x0)).(\lambda (H7: (subst0 (s (Flat
+Cast) i) v1 t3 x1)).(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or
+(pr0 t t4) (ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst0 i v2
+t4 w2))))) (or_ind (pr0 x1 t4) (ex2 T (\lambda (w2: T).(pr0 x1 w2)) (\lambda
(w2: T).(subst0 (s (Flat Cast) i) v2 t4 w2))) (or (pr0 (THead (Flat Cast) x0
x1) t4) (ex2 T (\lambda (w2: T).(pr0 (THead (Flat Cast) x0 x1) w2)) (\lambda
(w2: T).(subst0 i v2 t4 w2)))) (\lambda (H8: (pr0 x1 t4)).(or_introl (pr0
(* This file was automatically generated: do not edit *********************)
-include "pr0/props.ma".
+include "LambdaDelta-1/pr0/props.ma".
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
theorem pr0_delta1:
\forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "pr0/defs.ma".
+include "LambdaDelta-1/pr0/defs.ma".
inductive pr1: T \to (T \to Prop) \def
| pr1_refl: \forall (t: T).(pr1 t t)
(* This file was automatically generated: do not edit *********************)
-include "pr1/props.ma".
+include "LambdaDelta-1/pr1/props.ma".
-include "pr0/pr0.ma".
+include "LambdaDelta-1/pr0/pr0.ma".
theorem pr1_strip:
\forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr0 t0
(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5:
T).(\lambda (H3: (pr0 t3 t5)).(ex2_ind T (\lambda (t: T).(pr0 t5 t)) (\lambda
(t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
-t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2
-x)).(ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 x t)) (ex2 T
-(\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x0:
-T).(\lambda (H6: (pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda
-(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_sing x t5 H4 x0
-H7))))) (H2 x H5))))) (pr0_confluence t3 t5 H3 t2 H0)))))))))) t0 t1 H))).
+t))) (\lambda (x: T).(\lambda (H4: (pr0 t5 x)).(\lambda (H5: (pr0 t2 x)).(let
+H6 \def (H2 x H5) in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda (t:
+T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)))
+(\lambda (x0: T).(\lambda (H7: (pr1 t4 x0)).(\lambda (H8: (pr1 x
+x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0
+H7 (pr1_t x t5 (pr1_pr0 t5 x H4) x0 H8))))) H6))))) (pr0_confluence t3 t5 H3
+t2 H0)))))))))) t0 t1 H))).
theorem pr1_confluence:
\forall (t0: T).(\forall (t1: T).((pr1 t0 t1) \to (\forall (t2: T).((pr1 t0
T).(\lambda (t3: T).(\lambda (H0: (pr0 t3 t2)).(\lambda (t4: T).(\lambda (_:
(pr1 t2 t4)).(\lambda (H2: ((\forall (t5: T).((pr1 t2 t5) \to (ex2 T (\lambda
(t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))))))).(\lambda (t5: T).(\lambda
-(H3: (pr1 t3 t5)).(ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1
-t2 t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)))
-(\lambda (x: T).(\lambda (H4: (pr1 t5 x)).(\lambda (H5: (pr1 t2 x)).(ex2_ind
-T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 x t)) (ex2 T (\lambda (t:
-T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x0: T).(\lambda (H6:
-(pr1 t4 x0)).(\lambda (H7: (pr1 x x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4
-t)) (\lambda (t: T).(pr1 t5 t)) x0 H6 (pr1_t x t5 H4 x0 H7))))) (H2 x H5)))))
-(pr1_strip t3 t5 H3 t2 H0)))))))))) t0 t1 H))).
+(H3: (pr1 t3 t5)).(let H_x \def (pr1_strip t3 t5 H3 t2 H0) in (let H4 \def
+H_x in (ex2_ind T (\lambda (t: T).(pr1 t5 t)) (\lambda (t: T).(pr1 t2 t))
+(ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t))) (\lambda (x:
+T).(\lambda (H5: (pr1 t5 x)).(\lambda (H6: (pr1 t2 x)).(let H_x0 \def (H2 x
+H6) in (let H7 \def H_x0 in (ex2_ind T (\lambda (t: T).(pr1 t4 t)) (\lambda
+(t: T).(pr1 x t)) (ex2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5
+t))) (\lambda (x0: T).(\lambda (H8: (pr1 t4 x0)).(\lambda (H9: (pr1 x
+x0)).(ex_intro2 T (\lambda (t: T).(pr1 t4 t)) (\lambda (t: T).(pr1 t5 t)) x0
+H8 (pr1_t x t5 H5 x0 H9))))) H7)))))) H4))))))))))) t0 t1 H))).
(* This file was automatically generated: do not edit *********************)
-include "pr1/defs.ma".
+include "LambdaDelta-1/pr1/defs.ma".
-include "pr0/subst1.ma".
+include "LambdaDelta-1/pr0/subst1.ma".
-include "subst1/props.ma".
+include "LambdaDelta-1/subst1/props.ma".
-include "T/props.ma".
+include "LambdaDelta-1/T/props.ma".
theorem pr1_pr0:
\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr1 t1 t2)))
(* This file was automatically generated: do not edit *********************)
-include "pr2/props.ma".
+include "LambdaDelta-1/pr2/props.ma".
-include "clen/getl.ma".
+include "LambdaDelta-1/clen/getl.ma".
theorem pr2_gen_ctail:
\forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall
(THead (Bind b) v t2)))))))
\def
\lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let H0 \def (match H
-in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_:
-(pr2 c0 t t0)).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0
-t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))))) with
-[(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b)
-v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead
-c (Bind b) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0
-t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4:
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to
-(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v
-t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead
-(Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6
-(Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0
-(CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2)
-\Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T
-t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda
-(c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr)
-u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1)
-(THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1
-(\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead
-(Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind
-T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u))
-\to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1)
-(THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead
-d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3
-t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let
-H11 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (Bind Abbr) u)
-(CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda
-(j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead (Bind b) v t1)
-(THead (Bind b) v t2)) (\lambda (H12: (land (eq nat i O) (eq C (CHead d (Bind
-Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind
-Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v
-t2)) (\lambda (H13: (eq nat i O)).(\lambda (H14: (eq C (CHead d (Bind Abbr)
-u) (CHead c (Bind b) v))).(let H15 \def (f_equal C C (\lambda (e: C).(match e
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _)
-\Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H14) in ((let
-H16 \def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
-with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k in K
-return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H14) in
-((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead
-d (Bind Abbr) u) (CHead c (Bind b) v) H14) in (\lambda (H18: (eq B Abbr
-b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind nat i (\lambda (n:
-nat).(subst0 n u t3 t2)) H10 O H13) in (let H21 \def (eq_ind T u (\lambda
-(t4: T).(subst0 O t4 t3 t2)) H20 v H17) in (eq_ind B Abbr (\lambda (b0:
-B).(pr2 c (THead (Bind b0) v t1) (THead (Bind b0) v t2))) (pr2_free c (THead
-(Bind Abbr) v t1) (THead (Bind Abbr) v t2) (pr0_delta v v (pr0_refl v) t1 t3
-H9 t2 H21)) b H18)))))) H16)) H15)))) H12)) (\lambda (H12: (ex2 nat (\lambda
-(j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr)
-u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j:
-nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) v t1) (THead
-(Bind b) v t2)) (\lambda (x: nat).(\lambda (H13: (eq nat i (S x))).(\lambda
-(H14: (getl x c (CHead d (Bind Abbr) u))).(let H15 \def (f_equal nat nat
-(\lambda (e: nat).e) i (S x) H13) in (let H16 \def (eq_ind nat i (\lambda (n:
-nat).(subst0 n u t3 t2)) H10 (S x) H15) in (pr2_head_2 c v t1 t2 (Bind b)
-(pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead c (Bind
-b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H14) t1 t3 H9 t2
-H16))))))) H12)) H11)))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0
-(sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal
-C (CHead c (Bind b) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+(t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(insert_eq C (CHead c
+(Bind b) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead
+(Bind b) v t1) (THead (Bind b) v t2))) (\lambda (y: C).(\lambda (H0: (pr2 y
+t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0
+(CHead c (Bind b) v)) \to (pr2 c (THead (Bind b) v t) (THead (Bind b) v
+t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Bind b) v))).(pr2_free c (THead
+(Bind b) v t3) (THead (Bind b) v t4) (pr0_comp v v (pr0_refl v) t3 t4 H1
+(Bind b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) v))).(let H5 \def
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
+c (Bind b) v) H4) in (let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u)
+v i H5) in (let H6 \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d
+(Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S
+j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead
+(Bind b) v t3) (THead (Bind b) v t)) (\lambda (H7: (land (eq nat i O) (eq C
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C
+(CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t3)
+(THead (Bind b) v t)) (\lambda (H8: (eq nat i O)).(\lambda (H9: (eq C (CHead
+d (Bind Abbr) u) (CHead c (Bind b) v))).(let H10 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d
+| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
+v) H9) in ((let H11 \def (f_equal C B (\lambda (e: C).(match e in C return
+(\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _)
+\Rightarrow (match k in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
+(CHead c (Bind b) v) H9) in ((let H12 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v)
+H9) in (\lambda (H13: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H15 \def
+(eq_ind nat i (\lambda (n: nat).(subst0 n u t4 t)) H3 O H8) in (let H16 \def
+(eq_ind T u (\lambda (t0: T).(subst0 O t0 t4 t)) H15 v H12) in (eq_ind B Abbr
+(\lambda (b0: B).(pr2 c (THead (Bind b0) v t3) (THead (Bind b0) v t)))
+(pr2_free c (THead (Bind Abbr) v t3) (THead (Bind Abbr) v t) (pr0_delta v v
+(pr0_refl v) t3 t4 H2 t H16)) b H13)))))) H11)) H10)))) H7)) (\lambda (H7:
+(ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c
+(CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j)))
+(\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b)
+v t3) (THead (Bind b) v t)) (\lambda (x: nat).(\lambda (H8: (eq nat i (S
+x))).(\lambda (H9: (getl x c (CHead d (Bind Abbr) u))).(let H10 \def (f_equal
+nat nat (\lambda (e: nat).e) i (S x) H8) in (let H11 \def (eq_ind nat i
+(\lambda (n: nat).(subst0 n u t4 t)) H3 (S x) H10) in (pr2_head_2 c v t3 t
+(Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead
+c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H9) t3 t4 H2 t
+H11))))))) H7)) H6))))))))))))))) y t1 t2 H0))) H)))))).
theorem pr2_gen_cflat:
\forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall
(t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
\def
\lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda
-(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let H0 \def (match H
-in pr2 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_:
-(pr2 c0 t t0)).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0
-t2) \to (pr2 c t1 t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow
-(\lambda (H1: (eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_:
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2)))))
-(\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to
-((pr0 t t3) \to (pr2 c t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1
-t2)).(pr2_free c t1 t2 H6)) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4)))
-c0 (sym_eq C c0 (CHead c (Flat f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i
-H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f)
-v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead
-c (Flat f) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1
-(CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c
-t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T
-t t2) \to ((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4
-t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t
-t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) v) (CHead d
-(Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c t1
-t2))))) (\lambda (H8: (getl i (CHead c (Flat f) v) (CHead d (Bind Abbr)
-u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y
-\def (getl_gen_flat f c (CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i
-H_y t1 t3 H9 t2 H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0
-(sym_eq C c0 (CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal
-C (CHead c (Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+(t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(insert_eq C (CHead c
+(Flat f) v) (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c t1 t2))
+(\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to (pr2
+c t t0))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c (Flat f) v))).(pr2_free c t3 t4
+H1)))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3:
+(subst0 i u t4 t)).(\lambda (H4: (eq C c0 (CHead c (Flat f) v))).(let H5 \def
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H1 (CHead
+c (Flat f) v) H4) in (let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u)
+v i H5) in (pr2_delta c d u i H_y t3 t4 H2 t H3)))))))))))))) y t1 t2 H0)))
+H)))))).
(* This file was automatically generated: do not edit *********************)
-include "pr0/defs.ma".
+include "LambdaDelta-1/pr0/defs.ma".
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
inductive pr2: C \to (T \to (T \to Prop)) \def
| pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to
(* This file was automatically generated: do not edit *********************)
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
theorem pr2_gen_sort:
\forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to
(eq T x (TSort n)))))
\def
\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort
-n) x)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t
-(TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))))) with [(pr2_free c0
-t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1
-(TSort n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1
-(TSort n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n))))))
-(\lambda (H4: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t: T).((eq T
-t2 x) \to ((pr0 t t2) \to (eq T x (TSort n))))) (\lambda (H5: (eq T t2
-x)).(eq_ind T x (\lambda (t: T).((pr0 (TSort n) t) \to (eq T x (TSort n))))
-(\lambda (H6: (pr0 (TSort n) x)).(let H7 \def (eq_ind T x (\lambda (t:
-T).(pr2 c (TSort n) t)) H (TSort n) (pr0_gen_sort x n H6)) in (eq_ind_r T
-(TSort n) (\lambda (t: T).(eq T t (TSort n))) (refl_equal T (TSort n)) x
-(pr0_gen_sort x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 (TSort n)
-H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t
-H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 (TSort
-n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 (TSort
-n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2)
-\to ((subst0 i u t2 t) \to (eq T x (TSort n)))))))) (\lambda (H6: (eq T t1
-(TSort n))).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t x) \to ((getl i c
-(CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (eq T x
-(TSort n))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl
-i c (CHead d (Bind Abbr) u)) \to ((pr0 (TSort n) t2) \to ((subst0 i u t2 t0)
-\to (eq T x (TSort n)))))) (\lambda (_: (getl i c (CHead d (Bind Abbr)
-u))).(\lambda (H9: (pr0 (TSort n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let
-H11 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 x)) H10 (TSort n)
-(pr0_gen_sort t2 n H9)) in (subst0_gen_sort u x i n H11 (eq T x (TSort
-n))))))) t (sym_eq T t x H7))) t1 (sym_eq T t1 (TSort n) H6))) c0 (sym_eq C
-c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TSort
-n)) (refl_equal T x)))))).
+n) x)).(insert_eq T (TSort n) (\lambda (t: T).(pr2 c t x)) (\lambda (t:
+T).(eq T x t)) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda
+(_: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to (eq T t0
+t))))) (\lambda (_: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0
+t1 t2)).(\lambda (H2: (eq T t1 (TSort n))).(let H3 \def (eq_ind T t1 (\lambda
+(t: T).(pr0 t t2)) H1 (TSort n) H2) in (eq_ind_r T (TSort n) (\lambda (t:
+T).(eq T t2 t)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n)))
+(refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H3)) t1 H2))))))) (\lambda
+(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (_: (getl
+i c0 (CHead d (Bind Abbr) u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(H2: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda
+(H4: (eq T t1 (TSort n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0
+t2)) H2 (TSort n) H4) in (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t t0))
+(let H6 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 t)) H3 (TSort n)
+(pr0_gen_sort t2 n H5)) in (subst0_gen_sort u t i n H6 (eq T t (TSort n))))
+t1 H4))))))))))))) c y x H0))) H)))).
theorem pr2_gen_lref:
\forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to
n) O u)))))))))
\def
\lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef
-n) x)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t
-(TLRef n)) \to ((eq T t0 x) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda
-(d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(eq T x (lift (S n) O u))))))))))))) with [(pr2_free c0 t1
-t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TLRef
-n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TLRef
-n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (or (eq T x (TLRef n)) (ex2_2 C T
-(\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda
-(_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))))) (\lambda (H4: (eq T
-t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t
-t2) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u:
-T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T
-x (lift (S n) O u))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda
-(t: T).((pr0 (TLRef n) t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))) (\lambda (H6: (pr0 (TLRef
-n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (TLRef
-n) (pr0_gen_lref x n H6)) in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T
-t (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d
-(Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O
-u))))))) (or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d:
-C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_:
-C).(\lambda (u: T).(eq T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef
-n))) x (pr0_gen_lref x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1
-(TLRef n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0
-t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1
-(TLRef n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1
-(TLRef n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0
-t1 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda
-(d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
-C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))))) (\lambda (H6: (eq T
-t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t x) \to ((getl i
-c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or
-(eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift
-(S n) O u0))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0:
-T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TLRef n) t2) \to ((subst0 i
-u t2 t0) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0:
-T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0:
-T).(eq T x (lift (S n) O u0)))))))))) (\lambda (H8: (getl i c (CHead d (Bind
-Abbr) u))).(\lambda (H9: (pr0 (TLRef n) t2)).(\lambda (H10: (subst0 i u t2
-x)).(let H11 \def (eq_ind T t2 (\lambda (t0: T).(subst0 i u t0 x)) H10 (TLRef
-n) (pr0_gen_lref t2 n H9)) in (and_ind (eq nat n i) (eq T x (lift (S n) O u))
-(or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift
-(S n) O u0)))))) (\lambda (H12: (eq nat n i)).(\lambda (H13: (eq T x (lift (S
-n) O u))).(let H14 \def (eq_ind_r nat i (\lambda (n0: nat).(getl n0 c (CHead
-d (Bind Abbr) u))) H8 n H12) in (let H15 \def (eq_ind T x (\lambda (t0:
-T).(pr2 c (TLRef n) t0)) H (lift (S n) O u) H13) in (eq_ind_r T (lift (S n) O
-u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0:
-C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
-C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (or_intror (eq T (lift
-(S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c
-(CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S
-n) O u) (lift (S n) O u0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0:
-T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0:
-T).(eq T (lift (S n) O u) (lift (S n) O u0)))) d u H14 (refl_equal T (lift (S
-n) O u)))) x H13))))) (subst0_gen_lref u x i n H11)))))) t (sym_eq T t x
-H7))) t1 (sym_eq T t1 (TLRef n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1
-H2))))]) in (H0 (refl_equal C c) (refl_equal T (TLRef n)) (refl_equal T
-x)))))).
+n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(pr2 c t x)) (\lambda (t:
+T).(or (eq T x t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead
+d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O
+u))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to (or (eq T t0 t)
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr)
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t0 (lift (S n) O u))))))))))
+(\lambda (c0: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H1: (pr0 t1
+t2)).(\lambda (H2: (eq T t1 (TLRef n))).(let H3 \def (eq_ind T t1 (\lambda
+(t: T).(pr0 t t2)) H1 (TLRef n) H2) in (eq_ind_r T (TLRef n) (\lambda (t:
+T).(or (eq T t2 t) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0
+(CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t2 (lift (S
+n) O u))))))) (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n))
+(ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c0 (CHead d (Bind Abbr)
+u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u)))))))
+(or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u:
+T).(getl n c0 (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq
+T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) t2 (pr0_gen_lref
+t2 n H3)) t1 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H2: (pr0 t1 t2)).(\lambda
+(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t1 (TLRef
+n))).(let H5 \def (eq_ind T t1 (\lambda (t0: T).(pr0 t0 t2)) H2 (TLRef n) H4)
+in (eq_ind_r T (TLRef n) (\lambda (t0: T).(or (eq T t t0) (ex2_2 C T (\lambda
+(d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
+C).(\lambda (u0: T).(eq T t (lift (S n) O u0))))))) (let H6 \def (eq_ind T t2
+(\lambda (t0: T).(subst0 i u t0 t)) H3 (TLRef n) (pr0_gen_lref t2 n H5)) in
+(and_ind (eq nat n i) (eq T t (lift (S n) O u)) (or (eq T t (TLRef n)) (ex2_2
+C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (_: C).(\lambda (u0: T).(eq T t (lift (S n) O u0)))))) (\lambda (H7:
+(eq nat n i)).(\lambda (H8: (eq T t (lift (S n) O u))).(eq_ind_r T (lift (S
+n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
+C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (let H9 \def (eq_ind_r
+nat i (\lambda (n0: nat).(getl n0 c0 (CHead d (Bind Abbr) u))) H1 n H7) in
+(or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0:
+C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0)))) (\lambda (_:
+C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) (ex2_2_intro
+C T (\lambda (d0: C).(\lambda (u0: T).(getl n c0 (CHead d0 (Bind Abbr) u0))))
+(\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))
+d u H9 (refl_equal T (lift (S n) O u))))) t H8))) (subst0_gen_lref u t i n
+H6))) t1 H4))))))))))))) c y x H0))) H)))).
theorem pr2_gen_abst:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H in pr2 return
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
-t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 x)
-\to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst)
-u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-t1 t2))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq
-C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq
-T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to
-((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 t3)))))))))) (\lambda (H4: (eq T t0 (THead
-(Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t: T).((eq
-T t2 x) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) t1 t3))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x
-(\lambda (t: T).((pr0 (THead (Bind Abst) u1 t1) t) \to (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))) (\lambda (H6:
-(pr0 (THead (Bind Abst) u1 t1) x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
-(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0
-x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def
-(eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead
-(Bind Abst) x0 x1) H7) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t:
-T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
-(Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind
-Abst) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free
-(CHead c (Bind b) u) t1 x1 H9)))) x H7))))))) (pr0_gen_abst u1 t1 x H6))) t2
-(sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H4))) c0
-(sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2)
-\Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind
-Abst) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T
-t0 (THead (Bind Abst) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind
-Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))))))))) (\lambda
-(H6: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1)
-(\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to
-((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda
-(t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))))))) (\lambda (H7: (eq T t
-x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
-((pr0 (THead (Bind Abst) u1 t1) t2) \to ((subst0 i u t2 t3) \to (ex3_2 T T
-(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
-t4)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
-(pr0 (THead (Bind Abst) u1 t1) t2)).(\lambda (H10: (subst0 i u t2
-x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
-Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+(H: (pr2 c (THead (Bind Abst) u1 t1) x)).(insert_eq T (THead (Bind Abst) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))) (\lambda (y:
+T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to (ex3_2 T T
+(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1
+t2)))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1:
+(pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(let H3 \def
+(eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Bind Abst) u1 t1) H2) in
+(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
-T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (pr0 u1
-x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t3:
-T).(subst0 i u t3 x)) H10 (THead (Bind Abst) x0 x1) H11) in (or3_ind (ex2 T
-(\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0
-i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind Abst) i) u x1 t3)))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (H15: (ex2 T (\lambda
-(u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1)))
-(\lambda (u2: T).(subst0 i u x0 u2)) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda
-(H16: (eq T x (THead (Bind Abst) x2 x1))).(\lambda (H17: (subst0 i u x0
-x2)).(let H18 \def (eq_ind T x (\lambda (t3: T).(pr2 c (THead (Bind Abst) u1
-t1) t3)) H (THead (Bind Abst) x2 x1) H16) in (eq_ind_r T (THead (Bind Abst)
-x2 x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
-(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c0 (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H4: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H5: (pr0 u1
+x0)).(\lambda (H6: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abst) x0 x1)
+(\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead
+c0 (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))))) x0 x1
+(refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c0 u1 x0 H5) (\lambda (b:
+B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 x1 H6)))) t2 H4))))))
+(pr0_gen_abst u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda
+(t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Bind
+Abst) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2
+(THead (Bind Abst) u1 t1) H4) in (ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead
+(Bind Abst) x0 x1))).(\lambda (H7: (pr0 u1 x0)).(\lambda (H8: (pr0 t1
+x1)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead
+(Bind Abst) x0 x1) H6) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead
+(Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda
+(t3: T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3)))))) (\lambda (H10: (ex2 T (\lambda (u2: T).(eq T t (THead (Bind
+Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda
+(u2: T).(eq T t (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x2
+x1))).(\lambda (H12: (subst0 i u x0 x2)).(eq_ind_r T (THead (Bind Abst) x2
+x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
+(THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c0 d
+u i H1 u1 x0 H7 x2 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0
+(Bind b) u0) t1 x1 H8)))) t H11)))) H10)) (\lambda (H10: (ex2 T (\lambda (t3:
+T).(eq T t (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abst)
+x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\lambda (x2: T).(\lambda (H11: (eq T t (THead (Bind Abst) x0
+x2))).(\lambda (H12: (subst0 (s (Bind Abst) i) u x1 x2)).(eq_ind_r T (THead
+(Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c0 u1
+x0 H7) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
+(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x2
+H12)))) t H11)))) H10)) (\lambda (H10: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind
+Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
+x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1
+t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H11: (eq T t
+(THead (Bind Abst) x2 x3))).(\lambda (H12: (subst0 i u x0 x2)).(\lambda (H13:
+(subst0 (s (Bind Abst) i) u x1 x3)).(eq_ind_r T (THead (Bind Abst) x2 x3)
+(\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
+(Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
-c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1
-(refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2
-H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1
-H13)))) x H16))))) H15)) (\lambda (H15: (ex2 T (\lambda (t3: T).(eq T x
-(THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3)))
-(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2:
-T).(\lambda (H16: (eq T x (THead (Bind Abst) x0 x2))).(\lambda (H17: (subst0
-(s (Bind Abst) i) u x1 x2)).(let H18 \def (eq_ind T x (\lambda (t3: T).(pr2 c
-(THead (Bind Abst) u1 t1) t3)) H (THead (Bind Abst) x0 x2) H16) in (eq_ind_r
-T (THead (Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead
-(Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead
-c (Bind b) u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2))
-(pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c
-(Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0)
-t1 x1 H13 x2 H17)))) x H16))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind Abst) i) u x1 t3))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_:
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-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (H17: (subst0
-i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Abst) i) u x1 x3)).(let H19 \def
-(eq_ind T x (\lambda (t3: T).(pr2 c (THead (Bind Abst) u1 t1) t3)) H (THead
-(Bind Abst) x2 x3) H16) in (eq_ind_r T (THead (Bind Abst) x2 x3) (\lambda
-(t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
-Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
-u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3
-(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2
-H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S
-i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3
-H18)))) x H16))))))) H15)) (subst0_gen_head (Bind Abst) u x0 x1 x i
-H14)))))))) (pr0_gen_abst u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T
-t0 (THead (Bind Abst) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))])
-in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal
-T x))))))).
+c0 (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))))) x2 x3
+(refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H7 x2
+H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
+(S i) (getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H8 x3
+H13)))) t H11)))))) H10)) (subst0_gen_head (Bind Abst) u x0 x1 t i H9))))))))
+(pr0_gen_abst u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
theorem pr2_gen_cast:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
t1 x))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H in pr2 return
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
-t0)).((eq C c0 c) \to ((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 x)
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat
-Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x))))))))) with [(pr2_free c0
-t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0
-(THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda
-(_: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2)
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))) (\lambda (H4: (eq T t0
-(THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t:
-T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c
-t1 x))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead
-(Flat Cast) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x))))
-(\lambda (H6: (pr0 (THead (Flat Cast) u1 t1) x)).(or_ind (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (pr0 t1 x) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H7:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1:
-T).(\lambda (H8: (eq T x (THead (Flat Cast) x0 x1))).(\lambda (H9: (pr0 u1
-x0)).(\lambda (H10: (pr0 t1 x1)).(let H11 \def (eq_ind T x (\lambda (t:
-T).(pr2 c (THead (Flat Cast) u1 t1) t)) H (THead (Flat Cast) x0 x1) H8) in
-(eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3)))) (pr2 c t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3)))) (pr2 c t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Cast) x0
-x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8))))))) H7)) (\lambda
-(H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)
-(pr2_free c t1 x H7))) (pr0_gen_cast u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0
-(sym_eq T t0 (THead (Flat Cast) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3
-H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq
-C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq
-T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to
-((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to
-((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x))))))))
-(\lambda (H6: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat
-Cast) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr)
-u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1
-t4)))) (pr2 c t1 x))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3:
-T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Cast) u1 t1)
-t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
-T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1
-x)))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0
-(THead (Flat Cast) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind
+(H: (pr2 c (THead (Flat Cast) u1 t1) x)).(insert_eq T (THead (Flat Cast) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1
+t2)))) (pr2 c t1 x))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast)
+u1 t1)) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
+(Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (pr2 c0 t1 t0))))))
+(\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H1: (pr0 t0
+t2)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(let H3 \def (eq_ind
+T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Cast) u1 t1) H2) in (or_ind
(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2
t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3)))) (pr2 c t1 x)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t2)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
-(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0:
-T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Cast) x0
-x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def
-(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Flat Cast) x0
-x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2
-x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x
-(THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
-t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t2))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t2 (THead (Flat Cast)
+x0 x1))).(\lambda (H6: (pr0 u1 x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T
+(THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda
+(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr2 c0 t1 t3)))) (pr2 c0 t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x1 (refl_equal T (THead
+(Flat Cast) x0 x1)) (pr2_free c0 u1 x0 H6) (pr2_free c0 t1 x1 H7))) t2
+H5)))))) H4)) (\lambda (H4: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t2) (pr2_free c0 t1 t2 H4))) (pr0_gen_cast u1 t1 t2
+H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i:
+nat).(\lambda (H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3:
+(subst0 i u t2 t)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(let H5
+\def (eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Cast) u1 t1)
+H4) in (or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t))
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T t2 (THead (Flat Cast)
+x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def
+(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Flat Cast) x0 x1)
+H7) in (or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1)))
+(\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead
+(Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H16: (ex2 T (\lambda (u2:
-T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0
-u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1)))
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (H11: (ex2 T (\lambda (u2:
+T).(eq T t (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Cast) u2 x1)))
(\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c
-t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2
-x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 H17 (pr2_delta c
-d u i H8 u1 x0 H13 x2 H18) (pr2_free c t1 x1 H14)))))) H16)) (\lambda (H16:
-(ex2 T (\lambda (t3: T).(eq T x (THead (Flat Cast) x0 t3))) (\lambda (t3:
-T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x
-(THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
-t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
-Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda
-(H17: (eq T x (THead (Flat Cast) x0 x2))).(\lambda (H18: (subst0 (s (Flat
-Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 H17 (pr2_free c u1 x0 H13)
-(pr2_delta c d u i H8 t1 x1 H14 x2 H18)))))) H16)) (\lambda (H16: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (x3:
-T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x3))).(\lambda (H18: (subst0
-i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Cast) i) u x1 x3)).(or_introl
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8
-t1 x1 H14 x3 H19)))))))) H16)) (subst0_gen_head (Flat Cast) u x0 x1 x i
-H15)))))))) H11)) (\lambda (H11: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3)))) (pr2 c t1 x) (pr2_delta c d u i H8 t1 t2 H11 x H10))) (pr0_gen_cast u1
-t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1)
-H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
-(refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x))))))).
+(t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2
+c0 t1 t)) (\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2
+x1))).(\lambda (H13: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x1 H12
+(pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9)))))) H11))
+(\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Cast) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))).(ex2_ind T (\lambda
+(t3: T).(eq T t (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat
+Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t))
+(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Flat Cast) x0 x2))).(\lambda
+(H13: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (pr2 c0 t1 t) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 H12
+(pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13)))))) H11))
+(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
+Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)) (\lambda (x2:
+T).(\lambda (x3: T).(\lambda (H12: (eq T t (THead (Flat Cast) x2
+x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat
+Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0
+H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1 H9 x3 H14)))))))) H11))
+(subst0_gen_head (Flat Cast) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
+t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (pr2 c0 t1 t)
+(pr2_delta c0 d u i H1 t1 t2 H6 t H3))) (pr0_gen_cast u1 t1 t2
+H5)))))))))))))) c y x H0))) H))))).
theorem pr2_gen_csort:
\forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2)
\to (pr0 t1 t2))))
\def
\lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort
-n) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c: C).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr2 c t t0)).((eq C c (CSort n)) \to ((eq T
-t t1) \to ((eq T t0 t2) \to (pr0 t1 t2)))))))) with [(pr2_free c t0 t3 H0)
-\Rightarrow (\lambda (H1: (eq C c (CSort n))).(\lambda (H2: (eq T t0
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CSort n) (\lambda (_: C).((eq T
-t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr0 t1 t2))))) (\lambda (H4:
-(eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to
-(pr0 t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0
-t1 t) \to (pr0 t1 t2))) (\lambda (H6: (pr0 t1 t2)).H6) t3 (sym_eq T t3 t2
-H5))) t0 (sym_eq T t0 t1 H4))) c (sym_eq C c (CSort n) H1) H2 H3 H0)))) |
-(pr2_delta c d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c (CSort
-n))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CSort
-n) (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d
-(Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2)))))))
-(\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to
-((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u
-t3 t) \to (pr0 t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda
-(t4: T).((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
-((subst0 i u t3 t4) \to (pr0 t1 t2))))) (\lambda (H8: (getl i (CSort n)
-(CHead d (Bind Abbr) u))).(\lambda (_: (pr0 t1 t3)).(\lambda (_: (subst0 i u
-t3 t2)).(getl_gen_sort n i (CHead d (Bind Abbr) u) H8 (pr0 t1 t2))))) t
-(sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c (sym_eq C c (CSort n) H3) H4
-H5 H0 H1 H2))))]) in (H0 (refl_equal C (CSort n)) (refl_equal T t1)
-(refl_equal T t2)))))).
+n) t1 t2)).(insert_eq C (CSort n) (\lambda (c: C).(pr2 c t1 t2)) (\lambda (_:
+C).(pr0 t1 t2)) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind
+(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to (pr0
+t t0))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (_: (eq C c (CSort n))).H1))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (pr0
+t3 t4)).(\lambda (t: T).(\lambda (_: (subst0 i u t4 t)).(\lambda (H4: (eq C c
+(CSort n))).(let H5 \def (eq_ind C c (\lambda (c0: C).(getl i c0 (CHead d
+(Bind Abbr) u))) H1 (CSort n) H4) in (getl_gen_sort n i (CHead d (Bind Abbr)
+u) H5 (pr0 t3 t)))))))))))))) y t1 t2 H0))) H)))).
theorem pr2_gen_appl:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H in pr2 return
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
-t0)).((eq C c0 c) \to ((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 x)
-\to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat
-Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2:
-T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))))) with [(pr2_free c0
-t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0
-(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda
-(_: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2)
-\to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H4: (eq T t0
-(THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t:
-T).((eq T t2 x) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T
-T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
-(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))) (\lambda
-(H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Appl) u1 t1)
-t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))) (\lambda (H6: (pr0 (THead
-(Flat Appl) u1 t1) x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))
-(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x
-(THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H7: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3:
-T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
-(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+(H: (pr2 c (THead (Flat Appl) u1 t1) x)).(insert_eq T (THead (Flat Appl) u1
+t1) (\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1
+t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
-Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
+Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
+z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x
-(THead (Flat Appl) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10: (pr0 t1
-x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
+y2) z1 z2)))))))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind
+(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl)
+u1 t1)) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t2: T).(pr2 c0 t1 t2)))) (ex4_4 T T T T (\lambda
(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind Abbr) u2
-t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t2)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t0 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (t2: T).(\lambda
+(H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(let H3
+\def (eq_ind T t0 (\lambda (t: T).(pr0 t t2)) H1 (THead (Flat Appl) u1 t1)
+H2) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H4: (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t2 (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0
-x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8)))))) H7)) (\lambda
-(H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H5: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H6: (pr0 u1
+x0)).(\lambda (H7: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1)
+(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl)
+x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat
+Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+(THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c0 u1 x0
+H6) (pr2_free c0 t1 x1 H7))) t2 H5)))))) H4)) (\lambda (H4: (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
+(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (H5: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H6: (eq T t2
+(THead (Bind Abbr) x2 x3))).(\lambda (H7: (pr0 u1 x2)).(\lambda (H8: (pr0 x1
+x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H8: (eq T t1 (THead (Bind
-Abst) x0 x1))).(\lambda (H9: (eq T x (THead (Bind Abbr) x2 x3))).(\lambda
-(H10: (pr0 u1 x2)).(\lambda (H11: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr)
-x2 x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead
-(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r
-T (THead (Bind Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind
+Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
+x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
+(u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
-(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1
-z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2:
-T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x3) (THead
-(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr)
-x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T
-(THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free
-c u1 x2 H10) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u)
-x1 x3 H11))))) t1 H8) x H9))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T
-(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b)
-v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0
-u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
+Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2
+t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b)
+u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1))
+(refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c0 u1 x2 H7) (\lambda (b:
+B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) x1 x3 H8))))) t1 H5) t2
+H6))))))))) H4)) (\lambda (H4: (ex6_6 B T T T T T (\lambda (b: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not
(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b)
y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat
+T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat
Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
-t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T
-T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
-(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq
-T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda
-(_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead
-(Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
-(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda
-(x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4:
-T).(\lambda (x5: T).(\lambda (H8: (not (eq B x0 Abst))).(\lambda (H9: (eq T
-t1 (THead (Bind x0) x1 x2))).(\lambda (H10: (eq T x (THead (Bind x0) x4
-(THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H11: (pr0 u1
-x3)).(\lambda (H12: (pr0 x1 x4)).(\lambda (H13: (pr0 x2 x5)).(eq_ind_r T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t:
-T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
+t3))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift
+(S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1
+t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda
+(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
+(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+t2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3:
+T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H5: (not (eq B x0
+Abst))).(\lambda (H6: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H7: (eq T
+t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda
+(H8: (pr0 u1 x3)).(\lambda (H9: (pr0 x1 x4)).(\lambda (H10: (pr0 x2
+x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3)
+x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
+t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T
+(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
+(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T
+T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t
+(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
+(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2
+T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
+Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
+x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind
-x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat
+Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
-(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
-O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
-(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+(CHead c0 (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b)
y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))
(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
-x0 x1 x2 x5 x3 x4 H8 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c u1
-x3 H11) (pr2_free c x1 x4 H12) (pr2_free (CHead c (Bind x0) x4) x2 x5 H13)))
-t1 H9) x H10))))))))))))) H7)) (pr0_gen_appl u1 t1 x H6))) t2 (sym_eq T t2 x
-H5))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H4))) c0 (sym_eq C c0 c H1)
-H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda
-(H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda
-(H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Appl) u1
-t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0
-t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
-z2))))))))))))))) (\lambda (H6: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind
-T (THead (Flat Appl) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c
-(CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x5 x3 x4 H5 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c0
+u1 x3 H8) (pr2_free c0 x1 x4 H9) (pr2_free (CHead c0 (Bind x0) x4) x2 x5
+H10))) t1 H6) t2 H7))))))))))))) H4)) (pr0_gen_appl u1 t1 t2 H3))))))))
+(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H1: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2:
+T).(\lambda (H2: (pr0 t0 t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2
+t)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(let H5 \def (eq_ind T
+t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (THead (Flat Appl) u1 t1) H4) in (or3_ind
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))) (\lambda (H7: (eq T t
-x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
-((pr0 (THead (Flat Appl) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or3 (ex3_2 T
-T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x
-(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H8: (getl i c
-(CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Appl) u1 t1)
-t2)).(\lambda (H10: (subst0 i u t2 x)).(or3_ind (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda
+(_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
+(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead
-(Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x
-(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
-(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))
-(\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T (\lambda (u2:
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H6: (ex3_2 T T (\lambda (u2:
T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1
-t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
-(x1: T).(\lambda (H12: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H13:
-(pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def (eq_ind T t2 (\lambda
-(t3: T).(subst0 i u t3 x)) H10 (THead (Flat Appl) x0 x1) H12) in (or3_ind
-(ex2 T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2:
-T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0
-t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x
-(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
-b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
-(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H16: (ex2 T
-(\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0
-i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1)))
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1
+x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2 (\lambda (t3:
+T).(subst0 i u t3 t)) H3 (THead (Flat Appl) x0 x1) H7) in (or3_ind (ex2 T
+(\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0
+i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3)))
+(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
+Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
+Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
+(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
+(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
+(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex2 T (\lambda (u2:
+T).(eq T t (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0
+u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Flat Appl) u2 x1)))
(\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H17: (eq T x
-(THead (Flat Appl) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(eq_ind_r T
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H12: (eq T t
+(THead (Flat Appl) x2 x1))).(\lambda (H13: (subst0 i u x0 x2)).(eq_ind_r T
(THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1
-t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
+t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind
Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
(u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift
(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2:
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
-(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
+(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
(THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O
u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c d u i H8 u1
-x0 H13 x2 H18) (pr2_free c t1 x1 H14))) x H17)))) H16)) (\lambda (H16: (ex2 T
-(\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0
-(s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead
-(Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c0 d u i
+H1 u1 x0 H8 x2 H13) (pr2_free c0 t1 x1 H9))) t H12)))) H11)) (\lambda (H11:
+(ex2 T (\lambda (t3: T).(eq T t (THead (Flat Appl) x0 t3))) (\lambda (t3:
+T).(subst0 (s (Flat Appl) i) u x1 t3)))).(ex2_ind T (\lambda (t3: T).(eq T t
+(THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
-(H17: (eq T x (THead (Flat Appl) x0 x2))).(\lambda (H18: (subst0 (s (Flat
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
+(H12: (eq T t (THead (Flat Appl) x0 x2))).(\lambda (H13: (subst0 (s (Flat
Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3:
T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind
b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2)))))))
+(_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2)))))))
(\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda
-(_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))
+(_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))))
(or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat
Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T
-T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
-t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4
+T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
+T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr)
u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2
(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) x0
-x2)) (pr2_free c u1 x0 H13) (pr2_delta c d u i H8 t1 x1 H14 x2 H18))) x
-H17)))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
-T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl)
+x0 x2)) (pr2_free c0 u1 x0 H8) (pr2_delta c0 d u i H1 t1 x1 H9 x2 H13))) t
+H12)))) H11)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u
x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1
-t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
+t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2)))
(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
-(x3: T).(\lambda (H17: (eq T x (THead (Flat Appl) x2 x3))).(\lambda (H18:
-(subst0 i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Appl) i) u x1
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda
+(x3: T).(\lambda (H12: (eq T t (THead (Flat Appl) x2 x3))).(\lambda (H13:
+(subst0 i u x0 x2)).(\lambda (H14: (subst0 (s (Flat Appl) i) u x1
x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T
(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3
(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T
(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda
(y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead
(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
(u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2
(THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda
(u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) x2
-x3)) (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 t1 x1 H14
-x3 H19))) x H17)))))) H16)) (subst0_gen_head (Flat Appl) u x0 x1 x i
-H15)))))))) H11)) (\lambda (H11: (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl)
+x2 x3)) (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (pr2_delta c0 d u i H1 t1 x1
+H9 x3 H14))) t H12)))))) H11)) (subst0_gen_head (Flat Appl) u x0 x1 t i
+H10)))))))) H6)) (\lambda (H6: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda
+(_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))).(ex4_4_ind T T T T
-(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda
-(_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+(z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t
+(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
-(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t1 (THead
-(Bind Abst) x0 x1))).(\lambda (H13: (eq T t2 (THead (Bind Abbr) x2
-x3))).(\lambda (H14: (pr0 u1 x2)).(\lambda (H15: (pr0 x1 x3)).(let H16 \def
-(eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Abbr) x2
-x3) H13) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3
-(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
-y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
-(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda
+(x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq T t1 (THead (Bind
+Abst) x0 x1))).(\lambda (H8: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda
+(H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(let H11 \def (eq_ind T t2
+(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x2 x3) H8) in
+(eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(or3 (ex3_2 T T
+(\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2 t4))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1 z1))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t
+(THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
+(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind
+b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
-(u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2
-u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
+(u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2
+u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3))) (\lambda
(t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3:
T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))) (\lambda (H17: (ex2 T (\lambda (u2: T).(eq T x (THead
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (H12: (ex2 T (\lambda (u2: T).(eq T t (THead
(Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T
-(\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0
-i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3))))
+(\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0
+i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3))))
(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
(Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda
-(H18: (eq T x (THead (Bind Abbr) x4 x3))).(\lambda (H19: (subst0 i u x2
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda
+(H13: (eq T t (THead (Bind Abbr) x4 x3))).(\lambda (H14: (subst0 i u x2
x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t3: T).(or3 (ex3_2 T T
(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1:
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4))))))))
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4))))))))
(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1))
-(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c d u i H8 u1 x2 H14 x4
-H19) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) x1 x3
-H15))))) x H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x
+(refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c0 d u i H1 u1 x2 H9 x4
+H14) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) x1 x3
+H10))))) t H13)))) H12)) (\lambda (H12: (ex2 T (\lambda (t3: T).(eq T t
(THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3)))
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Abbr) x2 t3)))
(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H18: (eq T x (THead (Bind Abbr)
-x2 x4))).(\lambda (H19: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H13: (eq T t (THead (Bind Abbr)
+x2 x4))).(\lambda (H14: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T
(THead (Bind Abbr) x2 x4) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
(THead (Bind Abst) x0 x1) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1))
-(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c u1 x2 H14) (\lambda (b:
-B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i)
-(getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d
-(Bind Abbr) u) i H8) x1 x3 H15 x4 H19))))) x H18)))) H17)) (\lambda (H17:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2
+(refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c0 u1 x2 H9) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
+(getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0) (CHead d
+(Bind Abbr) u) i H1) x1 x3 H10 x4 H14))))) t H13)))) H12)) (\lambda (H12:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2
t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
(\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_:
T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1:
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst)
x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))))
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
-T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
+T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O)
O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H18: (eq T x
-(THead (Bind Abbr) x4 x5))).(\lambda (H19: (subst0 i u x2 x4)).(\lambda (H20:
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq T t
+(THead (Bind Abbr) x4 x5))).(\lambda (H14: (subst0 i u x2 x4)).(\lambda (H15:
(subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5)
(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind Abst) x0 x1) t4))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead
-(Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
-T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
-Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
-t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
-z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-(THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind Abst) x0 x1)
+t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
+T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
+(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
+B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1)
(THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind
Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1))
-(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c d u i H8 u1 x2 H14 x4
-H19) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S
-i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d
-(Bind Abbr) u) i H8) x1 x3 H15 x5 H20))))) x H18)))))) H17)) (subst0_gen_head
-(Bind Abbr) u x2 x3 x i H16)) t1 H12)))))))))) H11)) (\lambda (H11: (ex6_6 B
-T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
-B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T
-t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3)))))))))
-(\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1
-v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+(refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c0 d u i H1 u1 x2 H9 x4
+H14) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u
+(S i) (getl_clear_bind b (CHead c0 (Bind b) u0) c0 u0 (clear_bind b c0 u0)
+(CHead d (Bind Abbr) u) i H1) x1 x3 H10 x5 H15))))) t H13)))))) H12))
+(subst0_gen_head (Bind Abbr) u x2 x3 t i H11)) t1 H7)))))))))) H6)) (\lambda
+(H6: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst))))))))
+(\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda
+(_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda
+(t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2)
+t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
+T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda
+(y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0
+y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))).(ex6_6_ind B T T T T
T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda
T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2)))))))
(\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1
-t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1))))))))
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda
-(u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift
+(u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift
(S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1:
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1:
T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5:
-T).(\lambda (H12: (not (eq B x0 Abst))).(\lambda (H13: (eq T t1 (THead (Bind
-x0) x1 x2))).(\lambda (H14: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl)
-(lift (S O) O x3) x5)))).(\lambda (H15: (pr0 u1 x3)).(\lambda (H16: (pr0 x1
-x4)).(\lambda (H17: (pr0 x2 x5)).(let H18 \def (eq_ind T t2 (\lambda (t3:
-T).(subst0 i u t3 x)) H10 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
-O x3) x5)) H14) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3
-(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c t3 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
-T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (H7: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind
+x0) x1 x2))).(\lambda (H9: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl)
+(lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1
+x4)).(\lambda (H12: (pr0 x2 x5)).(let H13 \def (eq_ind T t2 (\lambda (t3:
+T).(subst0 i u t3 t)) H3 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O
+x3) x5)) H9) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(or3
+(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t (THead (Flat Appl) u2
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr2 c0 t3 t4)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t3 (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t4: T).(eq T t (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t3
(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b)
+T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b)
y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
-(u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3)
-x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T x
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda
+(u2: T).(eq T t (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3)
+x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T t
(THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead
(Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+(t3: T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))) (\lambda (H19: (ex2 T (\lambda (u2: T).(eq T x (THead
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (H14: (ex2 T (\lambda (u2: T).(eq T t (THead
(Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2:
-T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind x0)
+T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T t (THead (Bind x0)
u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u
-x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
-(H20: (eq T x (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3)
-x5)))).(\lambda (H21: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
+(H15: (eq T t (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3)
+x5)))).(\lambda (H16: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6
(THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(or3 (ex3_2 T T
(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
(Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H12 (refl_equal T (THead
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H7 (refl_equal T (THead
(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
-(S O) O x3) x5))) (pr2_free c u1 x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6
-H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x H20)))) H19)) (\lambda
-(H19: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda (t3:
+(S O) O x3) x5))) (pr2_free c0 u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6
+H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) t H15)))) H14)) (\lambda
+(H14: (ex2 T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda (t3:
T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
-t3)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind x0) x4 t3))) (\lambda
(t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
-t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead
+(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
-(H20: (eq T x (THead (Bind x0) x4 x6))).(\lambda (H21: (subst0 (s (Bind x0)
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda
+(H15: (eq T t (THead (Bind x0) x4 x6))).(\lambda (H16: (subst0 (s (Bind x0)
i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind
x0) x4 x6) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0)
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0)
x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
(y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat
Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3)
O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H22: (ex2 T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H17: (ex2 T
(\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2:
T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
(u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H23: (eq T x6
-(THead (Flat Appl) x7 x5))).(\lambda (H24: (subst0 (s (Bind x0) i) u (lift (S
-O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (H18: (eq T
+x6 (THead (Flat Appl) x7 x5))).(\lambda (H19: (subst0 (s (Bind x0) i) u (lift
+(S O) O x3) x7)).(eq_ind_r T (THead (Flat Appl) x7 x5) (\lambda (t3: T).(or3
(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3)
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7
(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
x0) x4 (THead (Flat Appl) x7 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x5)) (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
(Flat Appl) x7 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H25: (eq T x7 (lift (S O) O
-x8))).(\lambda (H26: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H27
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H20: (eq T x7 (lift (S O) O
+x8))).(\lambda (H21: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H22
\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
-x3 x8)) H26 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3:
+x3 x8)) H21 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t3:
T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
x4 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x5)) (THead (Bind
Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
x4 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
(Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H12 (refl_equal T (THead
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x8 x4 H7 (refl_equal T (THead
(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
-(S O) O x8) x5))) (pr2_delta c d u i H8 u1 x3 H15 x8 H27) (pr2_free c x1 x4
-H16) (pr2_free (CHead c (Bind x0) x4) x2 x5 H17))) x7 H25)))))
-(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H24 (le_n_S O i (le_O_n
-i)))) x6 H23)))) H22)) (\lambda (H22: (ex2 T (\lambda (t3: T).(eq T x6 (THead
+(S O) O x8) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x8 H22) (pr2_free c0 x1 x4
+H11) (pr2_free (CHead c0 (Bind x0) x4) x2 x5 H12))) x7 H20)))))
+(subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H19 (le_n_S O i (le_O_n
+i)))) x6 H18)))) H17)) (\lambda (H17: (ex2 T (\lambda (t3: T).(eq T x6 (THead
(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
(s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x6 (THead
(Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl)
(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda
-(H23: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H24:
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda
+(H18: (eq T x6 (THead (Flat Appl) (lift (S O) O x3) x7))).(\lambda (H19:
(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x7)).(eq_ind_r T (THead (Flat
Appl) (lift (S O) O x3) x7) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
T).(\lambda (t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Flat Appl) u2
-t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
+t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
(y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
O x3) x7)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2)
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2)
t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead
(Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7)) (THead (Bind b)
y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
(y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))
(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
-x0 x1 x2 x7 x3 x4 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
-(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c u1
-x3 H15) (pr2_free c x1 x4 H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i)
-(getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d
-(Bind Abbr) u) i H8) x2 x5 H17 x7 H24))) x6 H23)))) H22)) (\lambda (H22:
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O)
-O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
-x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s
-(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3:
-T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x7 x3 x4 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x7))) (pr2_free c0
+u1 x3 H10) (pr2_free c0 x1 x4 H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S
+i) (getl_clear_bind x0 (CHead c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4)
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x7 H19))) x6 H18)))) H17)) (\lambda
+(H17: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u
+(lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat
+Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda
+(t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T
(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
(y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8:
-T).(\lambda (H23: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H24: (subst0
-(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H25: (subst0 (s (Flat
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8:
+T).(\lambda (H18: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H19: (subst0
+(s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H20: (subst0 (s (Flat
Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8)
(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
(THead (Bind x0) x4 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t4: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t3) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8))
(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
(THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda
-(H26: (eq T x7 (lift (S O) O x9))).(\lambda (H27: (subst0 (minus (s (Bind x0)
-i) (S O)) u x3 x9)).(let H28 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
-(\lambda (n: nat).(subst0 n u x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x9: T).(\lambda
+(H21: (eq T x7 (lift (S O) O x9))).(\lambda (H22: (subst0 (minus (s (Bind x0)
+i) (S O)) u x3 x9)).(let H23 \def (eq_ind nat (minus (s (Bind x0) i) (S O))
+(\lambda (n: nat).(subst0 n u x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T
(lift (S O) O x9) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda
(t4: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8)) (THead (Flat
-Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
+Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
x4 (THead (Flat Appl) t3 x8)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t3 x8))
(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
x4 (THead (Flat Appl) (lift (S O) O x9) x8)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O)
O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead
(Flat Appl) (lift (S O) O x9) x8)) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H12 (refl_equal T (THead
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x9 x4 H7 (refl_equal T (THead
(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift
-(S O) O x9) x8))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_free c x1 x4
-H16) (pr2_delta (CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c
-(Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5
-H17 x8 H25))) x7 H26))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O
-H24 (le_n_S O i (le_O_n i)))) x6 H23)))))) H22)) (subst0_gen_head (Flat Appl)
-u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H21)) x H20)))) H19)) (\lambda
-(H19: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O)
-O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
-(THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4
-u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat
-Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0)
-x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
-z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+(S O) O x9) x8))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_free c0 x1 x4
+H11) (pr2_delta (CHead c0 (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead
+c0 (Bind x0) x4) c0 x4 (clear_bind x0 c0 x4) (CHead d (Bind Abbr) u) i H1) x2
+x5 H12 x8 H20))) x7 H21))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S
+O) O H19 (le_n_S O i (le_O_n i)))) x6 H18)))))) H17)) (subst0_gen_head (Flat
+Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H16)) t H15)))) H14))
+(\lambda (H14: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl)
+(lift (S O) O x3) x5) t3))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0)
+i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
+(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
+T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
+Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
+(y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H20: (eq T x
-(THead (Bind x0) x6 x7))).(\lambda (H21: (subst0 i u x4 x6)).(\lambda (H22:
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq T t
+(THead (Bind x0) x6 x7))).(\lambda (H16: (subst0 i u x4 x6)).(\lambda (H17:
(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5)
x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t3: T).(or3 (ex3_2 T T
(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
(u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S
O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda
-(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+(_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat
Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3)
O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind
x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H23: (ex2 T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T
(\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2:
T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda
(u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s
(Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2:
T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H24: (eq T x7
-(THead (Flat Appl) x8 x5))).(\lambda (H25: (subst0 (s (Bind x0) i) u (lift (S
-O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3
+(CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T
+x7 (THead (Flat Appl) x8 x5))).(\lambda (H20: (subst0 (s (Bind x0) i) u (lift
+(S O) O x3) x8)).(eq_ind_r T (THead (Flat Appl) x8 x5) (\lambda (t3: T).(or3
(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 t3)
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
+(u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
(y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8
+(CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8
(lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u
x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind
x0) x6 (THead (Flat Appl) x8 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x5)) (THead (Bind
Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
(Flat Appl) x8 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
-y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H26: (eq T x8 (lift (S O) O
-x9))).(\lambda (H27: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H28
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
+y2) z1 z2))))))))) (\lambda (x9: T).(\lambda (H21: (eq T x8 (lift (S O) O
+x9))).(\lambda (H22: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x9)).(let H23
\def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u
-x3 x9)) H27 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3:
+x3 x9)) H22 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x9) (\lambda (t3:
T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
x6 (THead (Flat Appl) t3 x5)) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x5)) (THead (Bind
Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
-(_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+(_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
(Flat Appl) t3 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
x6 (THead (Flat Appl) (lift (S O) O x9) x5)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
(Flat Appl) (lift (S O) O x9) x5)) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H12 (refl_equal T (THead
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x9 x6 H7 (refl_equal T (THead
(Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift
-(S O) O x9) x5))) (pr2_delta c d u i H8 u1 x3 H15 x9 H28) (pr2_delta c d u i
-H8 x1 x4 H16 x6 H21) (pr2_free (CHead c (Bind x0) x6) x2 x5 H17))) x8
-H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H25 (le_n_S O i
-(le_O_n i)))) x7 H24)))) H23)) (\lambda (H23: (ex2 T (\lambda (t3: T).(eq T
+(S O) O x9) x5))) (pr2_delta c0 d u i H1 u1 x3 H10 x9 H23) (pr2_delta c0 d u
+i H1 x1 x4 H11 x6 H16) (pr2_free (CHead c0 (Bind x0) x6) x2 x5 H12))) x8
+H21))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i
+(le_O_n i)))) x7 H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t3: T).(eq T
x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s
(Flat Appl) (s (Bind x0) i)) u x5 t3)))).(ex2_ind T (\lambda (t3: T).(eq T x7
(THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat
Appl) (s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda
(t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda
-(u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2
-c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda
-(z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead
-(Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2
+(u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3:
+T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
+x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
+(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2
t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
+T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b
Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7)
(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))
-(\lambda (x8: T).(\lambda (H24: (eq T x7 (THead (Flat Appl) (lift (S O) O x3)
-x8))).(\lambda (H25: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
+z2))))))))) (\lambda (x8: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) (lift
+(S O) O x3) x8))).(\lambda (H20: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5
x8)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x8) (\lambda (t3:
T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
-x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c
-u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2)
+x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2)
t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4:
T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T
(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat
Appl) (lift (S O) O x3) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0
(THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8)) (THead (Bind b)
y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T
(\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
(y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))
(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))
-x0 x1 x2 x8 x3 x6 H12 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
-(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c u1
-x3 H15) (pr2_delta c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0)
-x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0
-c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H17 x8 H25))) x7 H24)))) H23))
-(\lambda (H23: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead
-(Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0)
-i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s
-(Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda
-(_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))
-(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6
-x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3))))
-(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
-T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
-x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda
-(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1:
-T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T
-(THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b:
-B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda
-(y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl)
-(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
-(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
-T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (x9:
-T).(\lambda (H24: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H25: (subst0
-(s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H26: (subst0 (s (Flat
-Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9)
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))
+x0 x1 x2 x8 x3 x6 H7 (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T
+(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x8))) (pr2_free c0
+u1 x3 H10) (pr2_delta c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0
+(Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6
+(clear_bind x0 c0 x6) (CHead d (Bind Abbr) u) i H1) x2 x5 H12 x8 H20))) x7
+H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s
+(Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3)
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0)
+i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead
+(Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0)
+x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
+T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1
+z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3:
+T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_:
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
+(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
+B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
+(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
+(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
+T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead
+(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda
+(x9: T).(\lambda (H19: (eq T x7 (THead (Flat Appl) x8 x9))).(\lambda (H20:
+(subst0 (s (Bind x0) i) u (lift (S O) O x3) x8)).(\lambda (H21: (subst0 (s
+(Flat Appl) (s (Bind x0) i)) u x5 x9)).(eq_ind_r T (THead (Flat Appl) x8 x9)
(\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T
(THead (Bind x0) x6 t3) (THead (Flat Appl) u2 t4)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0
(THead (Bind x0) x1 x2) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind
Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda
(t4: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind Abbr) u2 t4)))))) (\lambda
-(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+(_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
(b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2:
T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t3) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3:
T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0)
i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1:
T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1
x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda
(u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9))
(THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1:
+T).(\lambda (_: T).(pr2 c0 u1 u2))))) (\lambda (_: T).(\lambda (z1:
T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
+(CHead c0 (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b:
B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda
(z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind
T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T
(THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead
(Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0
+u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b:
B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda
-(H27: (eq T x8 (lift (S O) O x10))).(\lambda (H28: (subst0 (minus (s (Bind
-x0) i) (S O)) u x3 x10)).(let H29 \def (eq_ind nat (minus (s (Bind x0) i) (S
-O)) (\lambda (n: nat).(subst0 n u x3 x10)) H28 i (s_arith1 x0 i)) in
+(y2: T).(pr2 (CHead c0 (Bind b) y2) z1 z2))))))))) (\lambda (x10: T).(\lambda
+(H22: (eq T x8 (lift (S O) O x10))).(\lambda (H23: (subst0 (minus (s (Bind
+x0) i) (S O)) u x3 x10)).(let H24 \def (eq_ind nat (minus (s (Bind x0) i) (S
+O)) (\lambda (n: nat).(subst0 n u x3 x10)) H23 i (s_arith1 x0 i)) in
(eq_ind_r T (lift (S O) O x10) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2:
T).(\lambda (t4: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
-(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Bind x0) x1 x2) t4)))) (ex4_4
-T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq
-T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
+(THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
+u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c0 (THead (Bind x0) x1 x2) t4))))
+(ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_:
+T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T (THead (Bind x0)
x6 (THead (Flat Appl) t3 x9)) (THead (Bind Abbr) u2 t4)))))) (\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))
(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T
-T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t4)))))))) (ex6_6 B T
+T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda
T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t3 x9))
(THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda
(_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1:
-T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1
-y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
-T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1
+T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_: B).(\lambda (y1:
+T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c0
+y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2:
+T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b) y2) z1
z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
(THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Flat
-Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
+Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda
+(_: T).(\lambda (t3: T).(pr2 c0 (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T
(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T
(THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_:
T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0)
x6 (THead (Flat Appl) (lift (S O) O x10) x9)) (THead (Bind Abbr) u2 t3))))))
-(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1
+(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1
u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3:
-T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))))
+T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) z1 t3))))))))
(ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda
(_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b:
B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda
T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O)
O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2)
z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_:
+T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))))))) (\lambda (_:
B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda
-(y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
-T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b)
+(y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1:
+T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c0 (Bind b)
y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_:
T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B
b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_:
(z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead
(Flat Appl) (lift (S O) O x10) x9)) (THead (Bind b) y2 (THead (Flat Appl)
(lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_:
-T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))))))
+T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))))))
(\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_:
-T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_:
+T).(\lambda (y2: T).(pr2 c0 y1 y2))))))) (\lambda (b: B).(\lambda (_:
T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2
-(CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H12 (refl_equal T
+(CHead c0 (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x10 x6 H7 (refl_equal T
(THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl)
-(lift (S O) O x10) x9))) (pr2_delta c d u i H8 u1 x3 H15 x10 H29) (pr2_delta
-c d u i H8 x1 x4 H16 x6 H21) (pr2_delta (CHead c (Bind x0) x6) d u (S i)
-(getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d
-(Bind Abbr) u) i H8) x2 x5 H17 x9 H26))) x8 H27))))) (subst0_gen_lift_ge u x3
-x8 (s (Bind x0) i) (S O) O H25 (le_n_S O i (le_O_n i)))) x7 H24)))))) H23))
-(subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s (Bind x0) i) H22))
-x H20)))))) H19)) (subst0_gen_head (Bind x0) u x4 (THead (Flat Appl) (lift (S
-O) O x3) x5) x i H18)) t1 H13)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2
-H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6)))
-c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
-(refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x))))))).
+(lift (S O) O x10) x9))) (pr2_delta c0 d u i H1 u1 x3 H10 x10 H24) (pr2_delta
+c0 d u i H1 x1 x4 H11 x6 H16) (pr2_delta (CHead c0 (Bind x0) x6) d u (S i)
+(getl_clear_bind x0 (CHead c0 (Bind x0) x6) c0 x6 (clear_bind x0 c0 x6)
+(CHead d (Bind Abbr) u) i H1) x2 x5 H12 x9 H21))) x8 H22)))))
+(subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H20 (le_n_S O i (le_O_n
+i)))) x7 H19)))))) H18)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5
+x7 (s (Bind x0) i) H17)) t H15)))))) H14)) (subst0_gen_head (Bind x0) u x4
+(THead (Flat Appl) (lift (S O) O x3) x5) t i H13)) t1 H8)))))))))))))) H6))
+(pr0_gen_appl u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
theorem pr2_gen_abbr:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H in pr2 return
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
-t0)).((eq C c0 c) \to ((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 x)
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
-Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
-b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
-c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2))))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
-x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0
-c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2
-x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 t1)) \to ((eq
-T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
-(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
-(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c
-(Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind
-Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t: T).((eq T t2
-x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T
-x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1
-u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
-(\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y:
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
-(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c
-(Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x
-(\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) t) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
-t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind
-Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c
-(Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda
-(_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))
-(\lambda (H6: (pr0 (THead (Bind Abbr) u1 t1) x)).(or_ind (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
-t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y
-t3))))))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u:
-T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T
-T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3:
-T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0
-O u2 y t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2)))
-(\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0
-t1 y)) (\lambda (y: T).(subst0 O u2 y t3)))))) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
-t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda
-(x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x (THead (Bind Abbr) x0
-x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T
-(\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))))).(or_ind
-(pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y
-x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind
-b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead
-c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
-x))))) (\lambda (H10: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1)
-(\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t
-(THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
-(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u:
-T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda
-(_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z:
-T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1)
-z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
-(lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T
-(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1
-t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind
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-z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
-(lift (S O) O x)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind
-b) u) t1 (lift (S O) O x) H7))))) (pr0_gen_abbr u1 t1 x H6))) t2 (sym_eq T t2
-x H5))) t0 (sym_eq T t0 (THead (Bind Abbr) u1 t1) H4))) c0 (sym_eq C c0 c H1)
-H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda
-(H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda
-(H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Bind Abbr) u1
-t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0
-t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1
-u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T
-(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0:
-T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))))) (\lambda (H6: (eq
-T t0 (THead (Bind Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda
-(t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2)
-\to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4:
-T).(eq T x (THead (Bind Abbr) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(or3 (\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))) (ex2 T (\lambda (u0: T).(pr0 u1
-u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t4))) (ex3_2 T T
-(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c (Bind Abbr) u1) z t4)))))))) (\forall (b: B).(\forall (u0:
-T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))))))) (\lambda (H7: (eq
-T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to
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-T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abbr) u2 t4))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t4: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
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-(Bind Abbr) u0) t1 t4))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t4))))))))
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O
-x))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
-(pr0 (THead (Bind Abbr) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind
+T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u))
+(\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda
+(y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1))
+(pr2_free c0 u1 x0 H6) (or3_intro1 (\forall (b: B).(\forall (u: T).(pr2
+(CHead c0 (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
+(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x1)))) (ex_intro2 T (\lambda (u: T).(pr0 u1 u)) (\lambda
+(u: T).(pr2 (CHead c0 (Bind Abbr) u) t1 x1)) x0 H6 (pr2_delta (CHead c0 (Bind
+Abbr) x0) c0 x0 O (getl_refl Abbr c0 x0) t1 x2 H7 x1 H8))))))))
+(pr0_subst0_back x0 x2 x1 O H8 u1 H6)) t2 H5)))) H_x0)) H_x)))))) H4))
+(\lambda (H4: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 t3))) (ex2 T
+(\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c0 (Bind Abbr) u) t1
+t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr)
+u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u: T).(pr2 (CHead c0 (Bind b) u) t1 (lift (S O) O t2))))
+(\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c0 (Bind b) u) t1 (lift (S
+O) O t2) H4))))) (pr0_gen_abbr u1 t1 t2 H3)))))))) (\lambda (c0: C).(\lambda
+(d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d
+(Bind Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
+t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0
+(THead (Bind Abbr) u1 t1))).(let H5 \def (eq_ind T t0 (\lambda (t3: T).(pr0
+t3 t2)) H2 (THead (Bind Abbr) u1 t1) H4) in (or_ind (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
+t3))))))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda
+(t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0:
+T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3)))
+(ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1
+y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda
+(z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (H6:
(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2
t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2:
-T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
-(\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O t2)) (or
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
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-(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_:
-T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0
-y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
-(lift (S O) O x))))) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T
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-t3)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda
-(u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y))
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-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
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-(pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y
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-(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_:
-T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0
-y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
-(lift (S O) O x))))) (\lambda (H14: (pr0 t1 x1)).(let H15 \def (eq_ind T t2
-(\lambda (t3: T).(subst0 i u t3 x)) H10 (THead (Bind Abbr) x0 x1) H12) in
-(or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x1))) (\lambda
-(u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind
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+(\lambda (y0: T).(subst0 O u2 y0 t3)))))))).(ex3_2_ind T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0
+t1 t3) (ex2 T (\lambda (y0: T).(pr0 t1 y0)) (\lambda (y0: T).(subst0 O u2 y0
+t3)))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c0 (Bind b) u0) t1 (lift (S O) O t))))) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H7: (eq T t2 (THead (Bind Abbr) x0 x1))).(\lambda (H8: (pr0 u1
+x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T (\lambda (y0: T).(pr0 t1 y0))
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+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
+(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
+t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T t2
+(\lambda (t3: T).(subst0 i u t3 t)) H3 (THead (Bind Abbr) x0 x1) H7) in
+(or3_ind (ex2 T (\lambda (u2: T).(eq T t (THead (Bind Abbr) u2 x1))) (\lambda
+(u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead (Bind
Abbr) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (ex3_2 T
-T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
+T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3)))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O
-x))))) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2
-x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq
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-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2
-t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
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-(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_:
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-y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
-t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
-(lift (S O) O x))))) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind
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-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
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-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O
-x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
-Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind
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-(CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_:
-T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0
-y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z
-t3))))))) x2 x1 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (or3_intro0
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x1))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
-t1 x1))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (\lambda (b:
-B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 H14))))))))) H16))
-(\lambda (H16: (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x0 t3)))
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+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x3: T).(\lambda (H13: (eq T t (THead (Bind Abbr) x0 x3))).(\lambda
+(H14: (subst0 (s (Bind Abbr) i) u x1 x3)).(ex2_ind T (\lambda (t3: T).(subst0
+O u1 x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x4: T).(\lambda (H15: (subst0 O u1 x2 x4)).(\lambda (H16: (pr0 x4
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
+(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x0 x3 H13
+(pr2_free c0 u1 x0 H8) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2
+(CHead c0 (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y0: T).(\lambda (_:
+T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z:
+T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr)
+u1) z x3))) x4 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O (getl_refl
+Abbr c0 u1) t1 x2 H9 x4 H15) H16 (pr2_delta (CHead c0 (Bind Abbr) u1) d u (S
+i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1 x1 (pr0_refl
+x1) x3 H14)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8))))) H12)) (\lambda
+(H12: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))))).(ex3_2_ind T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0
+(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2
+(CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0
+z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z
+t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H13: (eq T t
+(THead (Bind Abbr) x3 x4))).(\lambda (H14: (subst0 i u x0 x3)).(\lambda (H15:
(subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t3: T).(subst0 O u1
x2 t3)) (\lambda (t3: T).(pr0 t3 x1)) (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
-(\lambda (x5: T).(\lambda (H21: (subst0 O u1 x2 x5)).(\lambda (H22: (pr0 x5
-x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead
-(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2)))
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x5: T).(\lambda (H16: (subst0 O u1 x2 x5)).(\lambda (H17: (pr0 x5
+x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead
+(Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2)))
(\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2
-(CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
-(u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y:
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
-(Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c
-(Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T
-(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0)
-t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr)
-u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_:
-T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x3 x4 H18
-(pr2_delta c d u i H8 u1 x0 H13 x3 H19) (or3_intro2 (\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
-u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (ex3_2 T T
-(\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y)))
-(\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z:
-T).(pr2 (CHead c (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y:
-T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y:
-T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c
-(Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O
-(getl_refl Abbr c u1) t1 x2 H14 x5 H21) H22 (pr2_delta (CHead c (Bind Abbr)
-u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1
-(pr0_refl x1) x4 H20)))))))) (pr0_subst0_back x0 x2 x1 O H15 u1 H13)))))))
-H17)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H16)))))) H_x0)) H_x))))))
-H11)) (\lambda (H11: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
-t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c
-(Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2
-(CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z)))
-(\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))))
-(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O
-x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u
-(S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O
-t2) H11 (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i)))))))
-(pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead
-(Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
-(refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T
-x))))))).
+(CHead c0 (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda
+(u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead
+c0 (Bind b) u0) t1 (lift (S O) O t)))) (ex3_2_intro T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3))))))) x3 x4 H13
+(pr2_delta c0 d u i H1 u1 x0 H8 x3 H14) (or3_intro2 (\forall (b: B).(\forall
+(u0: T).(pr2 (CHead c0 (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1
+u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0) t1 x4))) (ex3_2 T T
+(\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0)))
+(\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z:
+T).(pr2 (CHead c0 (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y0:
+T).(\lambda (_: T).(pr2 (CHead c0 (Bind Abbr) u1) t1 y0))) (\lambda (y0:
+T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead
+c0 (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c0 (Bind Abbr) u1) c0 u1 O
+(getl_refl Abbr c0 u1) t1 x2 H9 x5 H16) H17 (pr2_delta (CHead c0 (Bind Abbr)
+u1) d u (S i) (getl_head (Bind Abbr) i c0 (CHead d (Bind Abbr) u) H1 u1) x1
+x1 (pr0_refl x1) x4 H15)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8)))))))
+H12)) (subst0_gen_head (Bind Abbr) u x0 x1 t i H11)))))) H_x0)) H_x))))))
+H6)) (\lambda (H6: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3
+(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3))) (ex2 T
+(\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c0 (Bind Abbr) u0)
+t1 t3))) (ex3_2 T T (\lambda (y0: T).(\lambda (_: T).(pr2 (CHead c0 (Bind
+Abbr) u1) t1 y0))) (\lambda (y0: T).(\lambda (z: T).(pr0 y0 z))) (\lambda (_:
+T).(\lambda (z: T).(pr2 (CHead c0 (Bind Abbr) u1) z t3)))))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i)
+(getl_head (Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2)
+H6 (lift (S O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i)))))))
+(pr0_gen_abbr u1 t1 t2 H5)))))))))))))) c y x H0))) H))))).
theorem pr2_gen_void:
\forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c
T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda
-(H: (pr2 c (THead (Bind Void) u1 t1) x)).(let H0 \def (match H in pr2 return
-(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t
-t0)).((eq C c0 c) \to ((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 x)
-\to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind
-Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
-(lift (S O) O x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda
-(H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda
-(H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1
-t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b:
-B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
-(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind
-Void) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))
-(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O
-x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead
-(Bind Void) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
-c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u:
-T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2
-(CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead
-(Bind Void) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O)
-O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind
-Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u)
-t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1
-(lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
-T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0
-u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+(H: (pr2 c (THead (Bind Void) u1 t1) x)).(insert_eq T (THead (Bind Void) u1
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+(CHead c0 (Bind b) u0) t1 t3))))) x2 x1 H12 (pr2_delta c0 d u i H1 u1 x0 H8
+x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c0 (Bind b) u0) t1
+x1 H9)))))))) H11)) (\lambda (H11: (ex2 T (\lambda (t3: T).(eq T t (THead
+(Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1
+t3)))).(ex2_ind T (\lambda (t3: T).(eq T t (THead (Bind Void) x0 t3)))
(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda
-(u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))
-(\lambda (x2: T).(\lambda (H17: (eq T x (THead (Bind Void) x0 x2))).(\lambda
-(H18: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
-u0) t1 t3))))) x0 x2 H17 (pr2_free c u1 x0 H13) (\lambda (b: B).(\lambda (u0:
-T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead
-d (Bind Abbr) u) H8 u0) t1 x1 H14 x2 H18)))))))) H16)) (\lambda (H16: (ex3_2
-T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
+(u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))))
+(\lambda (x2: T).(\lambda (H12: (eq T t (THead (Bind Void) x0 x2))).(\lambda
+(H13: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x0 x2 H12 (pr2_free c0 u1 x0 H8) (\lambda (b: B).(\lambda (u0:
+T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head (Bind b) i c0
+(CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x2 H13)))))))) H11)) (\lambda (H11:
+(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2
+t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))))).(ex3_2_ind T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
(\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_:
T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T
-(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda
-(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1
-t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift
-(S O) O x))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq T x
-(THead (Bind Void) x2 x3))).(\lambda (H18: (subst0 i u x0 x2)).(\lambda (H19:
+(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda
+(t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1
+(lift (S O) O t))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H12: (eq T t
+(THead (Bind Void) x2 x3))).(\lambda (H13: (subst0 i u x0 x2)).(\lambda (H14:
(subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2:
-T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2:
-T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
-(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b:
-B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))
-(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void)
-u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b)
-u0) t1 t3))))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (\lambda (b:
-B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head
-(Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H14 x3 H19)))))))))) H16))
-(subst0_gen_head (Bind Void) u x0 x1 x i H15)))))))) H11)) (\lambda (H11:
-(pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda
-(t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_:
-T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall
-(u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
-T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda
-(u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c
-(CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H11 (lift (S O) O x)
-(subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2
-H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6)))
-c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c)
-(refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))).
+T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2:
+T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall
+(b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b:
+B).(\forall (u0: T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t))))
+(ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void)
+u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c0 (Bind b)
+u0) t1 t3))))) x2 x3 H12 (pr2_delta c0 d u i H1 u1 x0 H8 x2 H13) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 x1 H9 x3 H14)))))))))) H11))
+(subst0_gen_head (Bind Void) u x0 x1 t i H10)))))))) H6)) (\lambda (H6: (pr0
+t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3:
+T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2
+c0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c0 (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0:
+T).(pr2 (CHead c0 (Bind b) u0) t1 (lift (S O) O t)))) (\lambda (b:
+B).(\lambda (u0: T).(pr2_delta (CHead c0 (Bind b) u0) d u (S i) (getl_head
+(Bind b) i c0 (CHead d (Bind Abbr) u) H1 u0) t1 (lift (S O) O t2) H6 (lift (S
+O) O t) (subst0_lift_ge_S t2 t u i H3 O (le_O_n i))))))) (pr0_gen_void u1 t1
+t2 H5)))))))))))))) c y x H0))) H))))).
theorem pr2_gen_lift:
\forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall
t2))))))))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda
-(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(\lambda (e: C).(\lambda (H0:
-(drop h d c e)).(let H1 \def (match H in pr2 return (\lambda (c0: C).(\lambda
-(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t
-(lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d
-t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) with [(pr2_free c0 t0 t2 H1)
-\Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d
-t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift
-h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T
-x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0
-(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to
-((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3:
-T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t:
-T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3)))
-(\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1)
-x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0
-t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2
-e t1 t3))) (\lambda (x0: T).(\lambda (H8: (eq T x (lift h d x0))).(\lambda
-(H9: (pr0 t1 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda
-(t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2
-T (\lambda (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2
-e t1 t3)) x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H9)) x H8))))
-(pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h
-d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0
-t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0
-(lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T
-t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c1 (CHead d0 (Bind Abbr) u))
-\to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x
-(lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0
-(lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to
-((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t)
-\to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e
-t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i
-c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2
-t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2
-e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda
-(H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(ex2_ind T
+(d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(insert_eq T (lift h d t1)
+(\lambda (t: T).(pr2 c t x)) (\lambda (_: T).(\forall (e: C).((drop h d c e)
+\to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e
+t1 t2)))))) (\lambda (y: T).(\lambda (H0: (pr2 c y x)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq T t (lift h d t1)) \to (\forall (e:
+C).((drop h d c0 e) \to (ex2 T (\lambda (t2: T).(eq T t0 (lift h d t2)))
+(\lambda (t2: T).(pr2 e t1 t2))))))))) (\lambda (c0: C).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t0 t2)).(\lambda (H2: (eq T t0 (lift h
+d t1))).(\lambda (e: C).(\lambda (_: (drop h d c0 e)).(let H4 \def (eq_ind T
+t0 (\lambda (t: T).(pr0 t t2)) H1 (lift h d t1) H2) in (ex2_ind T (\lambda
+(t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T
+(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (x0: T).(\lambda (H5: (eq T t2 (lift h d x0))).(\lambda (H6: (pr0 t1
+x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T
+t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda
+(t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))
+x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H6)) t2 H5))))
+(pr0_gen_lift t1 t2 h d H4)))))))))) (\lambda (c0: C).(\lambda (d0:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H1: (getl i c0 (CHead d0 (Bind
+Abbr) u))).(\lambda (t0: T).(\lambda (t2: T).(\lambda (H2: (pr0 t0
+t2)).(\lambda (t: T).(\lambda (H3: (subst0 i u t2 t)).(\lambda (H4: (eq T t0
+(lift h d t1))).(\lambda (e: C).(\lambda (H5: (drop h d c0 e)).(let H6 \def
+(eq_ind T t0 (\lambda (t3: T).(pr0 t3 t2)) H2 (lift h d t1) H4) in (ex2_ind T
(\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2
-T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
-(\lambda (x0: T).(\lambda (H12: (eq T t2 (lift h d x0))).(\lambda (H13: (pr0
-t1 x0)).(let H14 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 x)) H11
-(lift h d x0) H12) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d
-t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H15: (lt i d)).(let H16 \def
-(eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S
-i)))) (lt_plus_minus i d H15)) in (let H17 \def (eq_ind nat d (\lambda (n:
-nat).(subst0 i u (lift h n x0) x)) H14 (S (plus i (minus d (S i))))
-(lt_plus_minus i d H15)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (x0: T).(\lambda (H7: (eq T t2 (lift h d x0))).(\lambda (H8: (pr0 t1
+x0)).(let H9 \def (eq_ind T t2 (\lambda (t3: T).(subst0 i u t3 t)) H3 (lift h
+d x0) H7) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T t (lift h d t3)))
+(\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H10: (lt i d)).(let H11 \def
+(eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) t)) H9 (S (plus i
+(minus d (S i)))) (lt_plus_minus i d H10)) in (let H12 \def (eq_ind nat d
+(\lambda (n: nat).(drop h n c0 e)) H5 (S (plus i (minus d (S i))))
+(lt_plus_minus i d H10)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0:
C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0:
-C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T x (lift h d
+C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T t (lift h d
t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2:
-C).(\lambda (H18: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H19: (getl
+C).(\lambda (H13: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H14: (getl
i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0
-x2)).(let H21 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus
-i (minus d (S i)))) x0) x)) H17 (lift h (minus d (S i)) x1) H18) in (ex2_ind
-T (\lambda (t3: T).(eq T x (lift h (S (plus i (minus d (S i)))) t3)))
-(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h
-d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H22: (eq
-T x (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H23: (subst0 i x1 x0
-x3)).(let H24 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n:
-nat).(eq T x (lift h n x3))) H22 d (lt_plus_minus i d H15)) in (ex_intro2 T
-(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3
-H24 (pr2_delta e x2 x1 i H19 t1 x0 H13 x3 H23)))))) (subst0_gen_lift_lt x1 x0
-x i h (minus d (S i)) H21)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h
-(minus d (S i)) H16))))) (\lambda (H15: (le d i)).(lt_le_e i (plus d h) (ex2
-T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
-(\lambda (H16: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H15 H16
-H14 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e
-t1 t3))))) (\lambda (H16: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq
-T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T
-(\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
-(\lambda (x1: T).(\lambda (H17: (eq T x (lift h d x1))).(\lambda (H18:
-(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d
-t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H17 (pr2_delta e d0 u (minus i h)
-(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H16) t1 x0 H13 x1
-H18))))) (subst0_gen_lift_ge u x0 x i h d H14 H16)))))))))) (pr0_gen_lift t1
-t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0
-(sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T
-(lift h d t1)) (refl_equal T x)))))))))).
+x2)).(let H16 \def (eq_ind T u (\lambda (t3: T).(subst0 i t3 (lift h (S (plus
+i (minus d (S i)))) x0) t)) H11 (lift h (minus d (S i)) x1) H13) in (ex2_ind
+T (\lambda (t3: T).(eq T t (lift h (S (plus i (minus d (S i)))) t3)))
+(\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T t (lift h
+d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H17: (eq
+T t (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H18: (subst0 i x1 x0
+x3)).(let H19 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n:
+nat).(eq T t (lift h n x3))) H17 d (lt_plus_minus i d H10)) in (ex_intro2 T
+(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3
+H19 (pr2_delta e x2 x1 i H14 t1 x0 H8 x3 H18)))))) (subst0_gen_lift_lt x1 x0
+t i h (minus d (S i)) H16)))))))) (getl_drop_conf_lt Abbr c0 d0 u i H1 e h
+(minus d (S i)) H12))))) (\lambda (H10: (le d i)).(lt_le_e i (plus d h) (ex2
+T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (H11: (lt i (plus d h))).(subst0_gen_lift_false x0 u t h d i H10 H11
+H9 (ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1
+t3))))) (\lambda (H11: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T t
+(lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T
+(\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))
+(\lambda (x1: T).(\lambda (H12: (eq T t (lift h d x1))).(\lambda (H13:
+(subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T t (lift h d
+t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H12 (pr2_delta e d0 u (minus i h)
+(getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c0 H1 e h d H5 H11) t1 x0 H8 x1
+H13))))) (subst0_gen_lift_ge u x0 t i h d H9 H11)))))))))) (pr0_gen_lift t1
+t2 h d H6)))))))))))))))) c y x H0))) H)))))).
(* This file was automatically generated: do not edit *********************)
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
-include "pr0/pr0.ma".
+include "LambdaDelta-1/pr0/pr0.ma".
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
theorem pr2_confluence__pr2_free_free:
\forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0
(* This file was automatically generated: do not edit *********************)
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
-include "pr0/props.ma".
+include "LambdaDelta-1/pr0/props.ma".
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
-include "getl/clear.ma".
+include "LambdaDelta-1/getl/clear.ma".
theorem pr2_thin_dx:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
t2)))))))
\def
\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead
-k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind
-b) u) t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda
-(t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Bind
-b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1)
-(THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow
-(\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_:
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind
-b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u
-t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b)
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead
-(Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1)
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda
-(H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda
-(H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b)
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T
-t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind
-b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4:
-T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3)
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u
-t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr)
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).(nat_ind
-(\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to
-((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u
-t2))))) (\lambda (H11: (getl O (CHead c (Bind b) u) (CHead d (Bind Abbr)
-u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(let H13 \def (f_equal C C (\lambda
-(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d
-| (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b)
-u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind
-b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H14 \def (f_equal C B (\lambda
-(e: C).(match e in C return (\lambda (_: C).B) with [(CSort _) \Rightarrow
-Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K return (\lambda (_: K).B)
-with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d
+(k: K).(\lambda (H: (pr2 (CHead c k u) t1 t2)).(insert_eq C (CHead c k u)
+(\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c (THead k u t1) (THead
+k u t2))) (\lambda (y: C).(\lambda (H0: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u)) \to (pr2 c
+(THead k u t) (THead k u t0)))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda (_: (eq C c0 (CHead c k
+u))).(pr2_free c (THead k u t3) (THead k u t4) (pr0_comp u u (pr0_refl u) t3
+t4 H1 k))))))) (K_ind (\lambda (k0: K).(\forall (c0: C).(\forall (d:
+C).(\forall (u0: T).(\forall (i: nat).((getl i c0 (CHead d (Bind Abbr) u0))
+\to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall (t:
+T).((subst0 i u0 t4 t) \to ((eq C c0 (CHead c k0 u)) \to (pr2 c (THead k0 u
+t3) (THead k0 u t)))))))))))))) (\lambda (b: B).(\lambda (c0: C).(\lambda (d:
+C).(\lambda (u0: T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0
+(CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4)
+\to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Bind b) u))
+\to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u t)))))))))) (\lambda (H1:
+(getl O c0 (CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H2: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4
+t)).(\lambda (H4: (eq C c0 (CHead c (Bind b) u))).(let H5 \def (eq_ind C c0
+(\lambda (c1: C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Bind b)
+u) H4) in (let H6 \def (f_equal C C (\lambda (e: C).(match e in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
+c1])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c
+(CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind
+Abbr) u0) H5))) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e in C
+return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
+\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
+\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0)
+(CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u
+(getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in ((let H8
+\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
+with [(CSort _) \Rightarrow u0 | (CHead _ _ t0) \Rightarrow t0])) (CHead d
(Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr)
-u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in
-((let H15 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_:
-C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t4) \Rightarrow t4]))
-(CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d
-(Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)
-H11))) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H18
-\def (eq_ind T u0 (\lambda (t4: T).(subst0 O t4 t3 t2)) H12 u H15) in (eq_ind
-B Abbr (\lambda (b0: B).(pr2 c (THead (Bind b0) u t1) (THead (Bind b0) u
-t2))) (pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2)
-(pr0_delta u u (pr0_refl u) t1 t3 H9 t2 H18)) b H16))))) H14)) H13))))
-(\lambda (i0: nat).(\lambda (_: (((getl i0 (CHead c (Bind b) u) (CHead d
-(Bind Abbr) u0)) \to ((subst0 i0 u0 t3 t2) \to (pr2 c (THead (Bind b) u t1)
-(THead (Bind b) u t2)))))).(\lambda (H11: (getl (S i0) (CHead c (Bind b) u)
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S i0) u0 t3
-t2)).(pr2_delta c d u0 (r (Bind b) i0) (getl_gen_S (Bind b) c (CHead d (Bind
-Abbr) u0) u i0 H11) (THead (Bind b) u t1) (THead (Bind b) u t3) (pr0_comp u u
-(pr0_refl u) t1 t3 H9 (Bind b)) (THead (Bind b) u t2) (subst0_snd (Bind b) u0
-t2 t3 (r (Bind b) i0) H12 u)))))) i H8 H10)))) t (sym_eq T t t2 H7))) t0
-(sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1
-H2))))]) in (H0 (refl_equal C (CHead c (Bind b) u)) (refl_equal T t1)
-(refl_equal T t2))))) (\lambda (f: F).(\lambda (H: (pr2 (CHead c (Flat f) u)
-t1 t2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 (CHead c (Flat f)
-u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Flat f) u t1)
-(THead (Flat f) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow
-(\lambda (H1: (eq C c0 (CHead c (Flat f) u))).(\lambda (H2: (eq T t0
-t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (_:
-C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Flat
-f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1
-(\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Flat f) u
-t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2
-(\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f)
-u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Flat f) u t1) (THead
-(Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Flat f)))) t3 (sym_eq T
-t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat f) u) H1)
-H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda
-(H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: (eq T t0 t1)).(\lambda
-(H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (c1: C).((eq T t0
-t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0
-t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f)
-u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T
-t t2) \to ((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0
-t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat
-f) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4:
-T).((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3)
-\to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u
-t2)))))) (\lambda (H8: (getl i (CHead c (Flat f) u) (CHead d (Bind Abbr)
-u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).(nat_ind
-(\lambda (n: nat).((getl n (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to
-((subst0 n u0 t3 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u
-t2))))) (\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr)
-u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c
-(CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind
-Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11)))
-(THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3
-H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u))))
-(\lambda (i0: nat).(\lambda (_: (((getl i0 (CHead c (Flat f) u) (CHead d
-(Bind Abbr) u0)) \to ((subst0 i0 u0 t3 t2) \to (pr2 c (THead (Flat f) u t1)
-(THead (Flat f) u t2)))))).(\lambda (H11: (getl (S i0) (CHead c (Flat f) u)
-(CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S i0) u0 t3
-t2)).(pr2_delta c d u0 (r (Flat f) i0) (getl_gen_S (Flat f) c (CHead d (Bind
-Abbr) u0) u i0 H11) (THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u
-(pr0_refl u) t1 t3 H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0
-t2 t3 (r (Flat f) i0) H12 u)))))) i H8 H10)))) t (sym_eq T t t2 H7))) t0
-(sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1
-H2))))]) in (H0 (refl_equal C (CHead c (Flat f) u)) (refl_equal T t1)
-(refl_equal T t2))))) k))))).
+u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H5))) in
+(\lambda (H9: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H11 \def (eq_ind T
+u0 (\lambda (t0: T).(subst0 O t0 t4 t)) H3 u H8) in (eq_ind B Abbr (\lambda
+(b0: B).(pr2 c (THead (Bind b0) u t3) (THead (Bind b0) u t))) (pr2_free c
+(THead (Bind Abbr) u t3) (THead (Bind Abbr) u t) (pr0_delta u u (pr0_refl u)
+t3 t4 H2 t H11)) b H9))))) H7)) H6)))))))))) (\lambda (n: nat).(\lambda (H1:
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
+c (Bind b) u)) \to (pr2 c (THead (Bind b) u t3) (THead (Bind b) u
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Bind b)
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
+Abbr) u0))) H2 (CHead c (Bind b) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
+(CHead c (Bind b) u)) \to (pr2 c (THead (Bind b) u t5) (THead (Bind b) u
+t0)))))))))) H1 (CHead c (Bind b) u) H5) in (pr2_delta c d u0 (r (Bind b) n)
+(getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H6) (THead (Bind b) u t3)
+(THead (Bind b) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Bind b)) (THead
+(Bind b) u t) (subst0_snd (Bind b) u0 t t4 (r (Bind b) n) H4 u)))))))))))))
+i)))))) (\lambda (f: F).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0:
+T).(\lambda (i: nat).(nat_ind (\lambda (n: nat).((getl n c0 (CHead d (Bind
+Abbr) u0)) \to (\forall (t3: T).(\forall (t4: T).((pr0 t3 t4) \to (\forall
+(t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead c (Flat f) u)) \to (pr2 c
+(THead (Flat f) u t3) (THead (Flat f) u t)))))))))) (\lambda (H1: (getl O c0
+(CHead d (Bind Abbr) u0))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H3: (subst0 O u0 t4 t)).(\lambda (H4:
+(eq C c0 (CHead c (Flat f) u))).(let H5 \def (eq_ind C c0 (\lambda (c1:
+C).(getl O c1 (CHead d (Bind Abbr) u0))) H1 (CHead c (Flat f) u) H4) in
+(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c)
+(clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f)
+u) (CHead d (Bind Abbr) u0) H5))) (THead (Flat f) u t3) (THead (Flat f) u t4)
+(pr0_comp u u (pr0_refl u) t3 t4 H2 (Flat f)) (THead (Flat f) u t)
+(subst0_snd (Flat f) u0 t t4 O H3 u)))))))))) (\lambda (n: nat).(\lambda (H1:
+(((getl n c0 (CHead d (Bind Abbr) u0)) \to (\forall (t3: T).(\forall (t4:
+T).((pr0 t3 t4) \to (\forall (t: T).((subst0 n u0 t4 t) \to ((eq C c0 (CHead
+c (Flat f) u)) \to (pr2 c (THead (Flat f) u t3) (THead (Flat f) u
+t))))))))))).(\lambda (H2: (getl (S n) c0 (CHead d (Bind Abbr) u0))).(\lambda
+(t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda
+(H4: (subst0 (S n) u0 t4 t)).(\lambda (H5: (eq C c0 (CHead c (Flat f)
+u))).(let H6 \def (eq_ind C c0 (\lambda (c1: C).(getl (S n) c1 (CHead d (Bind
+Abbr) u0))) H2 (CHead c (Flat f) u) H5) in (let H7 \def (eq_ind C c0 (\lambda
+(c1: C).((getl n c1 (CHead d (Bind Abbr) u0)) \to (\forall (t5: T).(\forall
+(t6: T).((pr0 t5 t6) \to (\forall (t0: T).((subst0 n u0 t6 t0) \to ((eq C c1
+(CHead c (Flat f) u)) \to (pr2 c (THead (Flat f) u t5) (THead (Flat f) u
+t0)))))))))) H1 (CHead c (Flat f) u) H5) in (pr2_delta c d u0 (r (Flat f) n)
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H6) (THead (Flat f) u t3)
+(THead (Flat f) u t4) (pr0_comp u u (pr0_refl u) t3 t4 H3 (Flat f)) (THead
+(Flat f) u t) (subst0_snd (Flat f) u0 t t4 (r (Flat f) n) H4 u)))))))))))))
+i)))))) k) y t1 t2 H0))) H)))))).
theorem clear_pr2_trans:
\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to
(\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
\def
\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1
-t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H in
-pr2 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2
-c t t0)).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1
-t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c
-c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1
-t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3
-t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind
-T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1
-t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1
-H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t
-H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1)
-\to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3)
-\to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d
-(Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1
-t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2
-(CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1
-t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10:
-(pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i
-(clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11))))
-t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1
-H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T
-t2)))))))).
+t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).(\forall (c1:
+C).((clear c1 c) \to (pr2 c1 t t0)))))) (\lambda (c: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c1: C).(\lambda (_:
+(clear c1 c)).(pr2_free c1 t3 t4 H0))))))) (\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
+Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3
+t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c1:
+C).(\lambda (H3: (clear c1 c)).(pr2_delta c1 d u i (clear_getl_trans i c
+(CHead d (Bind Abbr) u) H0 c1 H3) t3 t4 H1 t H2))))))))))))) c2 t1 t2 H)))).
theorem pr2_cflat:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
(f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
\def
\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1
-t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (f:
-F).(\forall (v: T).(pr2 (CHead c0 (Flat f) v) t t0)))))) (\lambda (c0:
-C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (f:
-F).(\lambda (v: T).(pr2_free (CHead c0 (Flat f) v) t3 t4 H0))))))) (\lambda
-(c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl
-i c0 (CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda
-(H1: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda
-(f: F).(\lambda (v: T).(pr2_delta (CHead c0 (Flat f) v) d u i (getl_flat c0
-(CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))))) c t1 t2 H)))).
+t2)).(\lambda (f: F).(\lambda (v: T).(pr2_ind (\lambda (c0: C).(\lambda (t:
+T).(\lambda (t0: T).(pr2 (CHead c0 (Flat f) v) t t0)))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(pr2_free
+(CHead c0 (Flat f) v) t3 t4 H0))))) (\lambda (c0: C).(\lambda (d: C).(\lambda
+(u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr)
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H2: (subst0 i u t4 t)).(pr2_delta (CHead c0 (Flat f) v) d u
+i (getl_flat c0 (CHead d (Bind Abbr) u) i H0 f v) t3 t4 H1 t H2))))))))))) c
+t1 t2 H)))))).
theorem pr2_ctail:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
\def
\lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda
(H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1
-t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 e) \to ((eq T t t1)
-\to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with
-[(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3:
-(eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T
-t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d
-t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3
-t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6:
-(eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d
-t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1)
-(lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T
-t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0
-t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0
-t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1)
-\to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3)
-\to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda
-(H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e
-(CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c
-(lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2
-(\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to
-((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9:
-(getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
-(H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2))
-(\lambda (H12: (lt i d)).(let H13 \def (drop_getl_trans_le i d (le_S_n i d
-(le_S (S i) d H12)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C
-(\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda
-(e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear
-e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda
-(x0: C).(\lambda (x1: C).(\lambda (H14: (drop i O c x0)).(\lambda (H15: (drop
-h (minus d i) x0 x1)).(\lambda (H16: (clear x1 (CHead d0 (Bind Abbr)
-u))).(let H17 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0
-x1)) H15 (S (minus d (S i))) (minus_x_Sy d i H12)) in (let H18 \def
-(drop_clear_S x1 x0 h (minus d (S i)) H17 Abbr d0 u H16) in (ex2_ind C
-(\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i))
-u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1)
-(lift h d t2)) (\lambda (x: C).(\lambda (H19: (clear x0 (CHead x (Bind Abbr)
-(lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x
-d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x
-(Bind Abbr) (lift h (minus d (S i)) u)) x0 H14 H19) (lift h d t1) (lift h d
-t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d
-H12 h))))) H18)))))))) H13))) (\lambda (H12: (le d i)).(pr2_delta c d0 u
-(plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H12)
-(lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2)
-(subst0_lift_ge t3 t2 u i h H11 d H12))))))) t (sym_eq T t t2 H8))) t0
-(sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1
-(refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))).
+t2)).(insert_eq C e (\lambda (c0: C).(pr2 c0 t1 t2)) (\lambda (_: C).(pr2 c
+(lift h d t1) (lift h d t2))) (\lambda (y: C).(\lambda (H1: (pr2 y t1
+t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e)
+\to (pr2 c (lift h d t) (lift h d t0)))))) (\lambda (c0: C).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_: (eq C c0
+e)).(pr2_free c (lift h d t3) (lift h d t4) (pr0_lift t3 t4 H2 h d)))))))
+(\lambda (c0: C).(\lambda (d0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda
+(H2: (getl i c0 (CHead d0 (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4:
+T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4: (subst0 i u t4
+t)).(\lambda (H5: (eq C c0 e)).(let H6 \def (eq_ind C c0 (\lambda (c1:
+C).(getl i c1 (CHead d0 (Bind Abbr) u))) H2 e H5) in (lt_le_e i d (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (H7: (lt i d)).(let H8 \def
+(drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H7)) c e h H (CHead d0
+(Bind Abbr) u) H6) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i
+O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1)))
+(\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H9:
+(drop i O c x0)).(\lambda (H10: (drop h (minus d i) x0 x1)).(\lambda (H11:
+(clear x1 (CHead d0 (Bind Abbr) u))).(let H12 \def (eq_ind nat (minus d i)
+(\lambda (n: nat).(drop h n x0 x1)) H10 (S (minus d (S i))) (minus_x_Sy d i
+H7)) in (let H13 \def (drop_clear_S x1 x0 h (minus d (S i)) H12 Abbr d0 u
+H11) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h
+(minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c
+(lift h d t3) (lift h d t)) (\lambda (x: C).(\lambda (H14: (clear x0 (CHead x
+(Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S
+i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead
+x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H9 H14) (lift h d t3) (lift h d
+t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_lt t4 t u i H4 d H7
+h))))) H13)))))))) H8))) (\lambda (H7: (le d i)).(pr2_delta c d0 u (plus i h)
+(drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H6 H7) (lift h d t3)
+(lift h d t4) (pr0_lift t3 t4 H3 h d) (lift h d t) (subst0_lift_ge t4 t u i h
+H4 d H7)))))))))))))))) y t1 t2 H1))) H0)))))))).
(* This file was automatically generated: do not edit *********************)
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
-include "pr0/subst1.ma".
+include "LambdaDelta-1/pr0/subst1.ma".
-include "pr0/fwd.ma".
+include "LambdaDelta-1/pr0/fwd.ma".
-include "csubst1/getl.ma".
+include "LambdaDelta-1/csubst1/getl.ma".
-include "csubst1/fwd.ma".
+include "LambdaDelta-1/csubst1/fwd.ma".
-include "subst1/subst1.ma".
+include "LambdaDelta-1/subst1/subst1.ma".
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
theorem pr2_delta1:
\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c
\def
\lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
(H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2:
-T).(\lambda (H0: (pr2 c t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda
-(c0: C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C
-c0 c) \to ((eq T t t1) \to ((eq T t0 t2) \to (\forall (w1: T).((subst1 i v t1
-w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v
-t2 w2)))))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq
-C c0 c)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c
-(\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (\forall
-(w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2))
-(\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H5: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (\forall
-(w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2))
-(\lambda (w2: T).(subst1 i v t2 w2)))))))) (\lambda (H6: (eq T t3
-t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (\forall (w1: T).((subst1 i
-v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1
-i v t2 w2))))))) (\lambda (H7: (pr0 t1 t2)).(\lambda (w1: T).(\lambda (H8:
-(subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2:
-T).(subst1 i v t2 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
-T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H9: (pr0 w1 x)).(\lambda
-(H10: (subst1 i v t2 x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2))
-(\lambda (w2: T).(subst1 i v t2 w2)) x (pr2_free c w1 x H9) H10))))
-(pr0_subst1 t1 t2 H7 v w1 i H8 v (pr0_refl v)))))) t3 (sym_eq T t3 t2 H6)))
-t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d
-u i0 H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5:
-(eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c (\lambda (c1: C).((eq T
-t0 t1) \to ((eq T t t2) \to ((getl i0 c1 (CHead d (Bind Abbr) u)) \to ((pr0
-t0 t3) \to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to
-(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2
-w2))))))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq
-T t t2) \to ((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to
-((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T
-(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))
-(\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i0 c (CHead d
-(Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i0 u t3 t4) \to (\forall (w1:
-T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda
-(w2: T).(subst1 i v t2 w2))))))))) (\lambda (H9: (getl i0 c (CHead d (Bind
-Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i0 u t3
-t2)).(\lambda (w1: T).(\lambda (H12: (subst1 i v t1 w1)).(ex2_ind T (\lambda
-(w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T (\lambda
-(w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x:
-T).(\lambda (H13: (pr0 w1 x)).(\lambda (H14: (subst1 i v t3 x)).(neq_eq_e i
-i0 (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2
-w2))) (\lambda (H15: (not (eq nat i i0))).(ex2_ind T (\lambda (t4: T).(subst1
-i v t2 t4)) (\lambda (t4: T).(subst1 i0 u x t4)) (ex2 T (\lambda (w2: T).(pr2
-c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda
-(H16: (subst1 i v t2 x0)).(\lambda (H17: (subst1 i0 u x x0)).(ex_intro2 T
-(\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0
-(pr2_delta1 c d u i0 H9 w1 x H13 x0 H17) H16)))) (subst1_confluence_neq t3 t2
-u i0 (subst1_single i0 u t3 t2 H11) x v i H14 (sym_not_eq nat i i0 H15))))
-(\lambda (H15: (eq nat i i0)).(let H16 \def (eq_ind_r nat i0 (\lambda (n:
-nat).(subst0 n u t3 t2)) H11 i H15) in (let H17 \def (eq_ind_r nat i0
-(\lambda (n: nat).(getl n c (CHead d (Bind Abbr) u))) H9 i H15) in (let H18
-\def (eq_ind C (CHead e (Bind Abbr) v) (\lambda (c1: C).(getl i c c1)) H
+T).(\lambda (H0: (pr2 c t1 t2)).(insert_eq C c (\lambda (c0: C).(pr2 c0 t1
+t2)) (\lambda (c0: C).(\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T
+(\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))
+(\lambda (y: C).(\lambda (H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0:
+C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to (\forall (w1:
+T).((subst1 i v t w1) \to (ex2 T (\lambda (w2: T).(pr2 c0 w1 w2)) (\lambda
+(w2: T).(subst1 i v t0 w2))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda
+(t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (H3: (eq C c0 c)).(\lambda (w1:
+T).(\lambda (H4: (subst1 i v t3 w1)).(eq_ind_r C c (\lambda (c1: C).(ex2 T
+(\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))))
+(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2))
+(ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t4 w2)))
+(\lambda (x: T).(\lambda (H5: (pr0 w1 x)).(\lambda (H6: (subst1 i v t4
+x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v
+t4 w2)) x (pr2_free c w1 x H5) H6)))) (pr0_subst1 t3 t4 H2 v w1 i H4 v
+(pr0_refl v))) c0 H3)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i0: nat).(\lambda (H2: (getl i0 c0 (CHead d (Bind Abbr)
+u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda
+(t: T).(\lambda (H4: (subst0 i0 u t4 t)).(\lambda (H5: (eq C c0 c)).(\lambda
+(w1: T).(\lambda (H6: (subst1 i v t3 w1)).(let H7 \def (eq_ind C c0 (\lambda
+(c1: C).(getl i0 c1 (CHead d (Bind Abbr) u))) H2 c H5) in (eq_ind_r C c
+(\lambda (c1: C).(ex2 T (\lambda (w2: T).(pr2 c1 w1 w2)) (\lambda (w2:
+T).(subst1 i v t w2)))) (ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda
+(w2: T).(subst1 i v t4 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda
+(w2: T).(subst1 i v t w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1
+x)).(\lambda (H9: (subst1 i v t4 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2:
+T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t w2))) (\lambda (H10: (not
+(eq nat i i0))).(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0:
+T).(subst1 i0 u x t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
+T).(subst1 i v t w2))) (\lambda (x0: T).(\lambda (H11: (subst1 i v t
+x0)).(\lambda (H12: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c
+w1 w2)) (\lambda (w2: T).(subst1 i v t w2)) x0 (pr2_delta1 c d u i0 H7 w1 x
+H8 x0 H12) H11)))) (subst1_confluence_neq t4 t u i0 (subst1_single i0 u t4 t
+H4) x v i H9 (sym_not_eq nat i i0 H10)))) (\lambda (H10: (eq nat i i0)).(let
+H11 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t4 t)) H4 i H10) in
+(let H12 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind
+Abbr) u))) H7 i H10) in (let H13 \def (eq_ind C (CHead e (Bind Abbr) v)
+(\lambda (c1: C).(getl i c c1)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead
+e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in (let H14 \def (f_equal
+C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with [(CSort _)
+\Rightarrow e | (CHead c1 _ _) \Rightarrow c1])) (CHead e (Bind Abbr) v)
(CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d
-(Bind Abbr) u) H17)) in (let H19 \def (f_equal C C (\lambda (e0: C).(match e0
-in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c1 _ _)
-\Rightarrow c1])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono
-c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H17)) in ((let H20 \def
-(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow v | (CHead _ _ t4) \Rightarrow t4])) (CHead e (Bind
-Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H
-(CHead d (Bind Abbr) u) H17)) in (\lambda (H21: (eq C e d)).(let H22 \def
-(eq_ind_r T u (\lambda (t4: T).(getl i c (CHead d (Bind Abbr) t4))) H18 v
-H20) in (let H23 \def (eq_ind_r T u (\lambda (t4: T).(subst0 i t4 t3 t2)) H16
-v H20) in (let H24 \def (eq_ind_r C d (\lambda (c1: C).(getl i c (CHead c1
-(Bind Abbr) v))) H22 e H21) in (ex2_ind T (\lambda (t4: T).(subst1 i v t2
-t4)) (\lambda (t4: T).(subst1 i v x t4)) (ex2 T (\lambda (w2: T).(pr2 c w1
-w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H25:
-(subst1 i v t2 x0)).(\lambda (H26: (subst1 i v x x0)).(ex_intro2 T (\lambda
-(w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c
-e v i H24 w1 x H13 x0 H26) H25)))) (subst1_confluence_eq t3 t2 v i
-(subst1_single i v t3 t2 H23) x H14))))))) H19)))))))))) (pr0_subst1 t1 t3
-H10 v w1 i H12 v (pr0_refl v)))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0
-t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c)
-(refl_equal T t1) (refl_equal T t2)))))))))).
+(Bind Abbr) u) H12)) in ((let H15 \def (f_equal C T (\lambda (e0: C).(match
+e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _
+t0) \Rightarrow t0])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u)
+(getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H12)) in
+(\lambda (H16: (eq C e d)).(let H17 \def (eq_ind_r T u (\lambda (t0: T).(getl
+i c (CHead d (Bind Abbr) t0))) H13 v H15) in (let H18 \def (eq_ind_r T u
+(\lambda (t0: T).(subst0 i t0 t4 t)) H11 v H15) in (let H19 \def (eq_ind_r C
+d (\lambda (c1: C).(getl i c (CHead c1 (Bind Abbr) v))) H17 e H16) in
+(ex2_ind T (\lambda (t0: T).(subst1 i v t t0)) (\lambda (t0: T).(subst1 i v x
+t0)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t
+w2))) (\lambda (x0: T).(\lambda (H20: (subst1 i v t x0)).(\lambda (H21:
+(subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2:
+T).(subst1 i v t w2)) x0 (pr2_delta1 c e v i H19 w1 x H8 x0 H21) H20))))
+(subst1_confluence_eq t4 t v i (subst1_single i v t4 t H18) x H9)))))))
+H14)))))))))) (pr0_subst1 t3 t4 H3 v w1 i H6 v (pr0_refl v))) c0
+H5))))))))))))))) y t1 t2 H1))) H0)))))))).
theorem pr2_gen_cabbr:
\forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall
(* This file was automatically generated: do not edit *********************)
-include "pr2/defs.ma".
+include "LambdaDelta-1/pr2/defs.ma".
inductive pr3 (c: C): T \to (T \to Prop) \def
| pr3_refl: \forall (t: T).(pr3 c t t)
(* This file was automatically generated: do not edit *********************)
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
-include "pr2/fwd.ma".
+include "LambdaDelta-1/pr2/fwd.ma".
theorem pr3_gen_sort:
\forall (c: C).(\forall (x: T).(\forall (n: nat).((pr3 c (TSort n) x) \to
(* This file was automatically generated: do not edit *********************)
-include "pr3/fwd.ma".
+include "LambdaDelta-1/pr3/fwd.ma".
-include "iso/props.ma".
+include "LambdaDelta-1/iso/props.ma".
-include "tlist/props.ma".
+include "LambdaDelta-1/tlist/props.ma".
theorem pr3_iso_appls_abbr:
\forall (c: C).(\forall (d: C).(\forall (w: T).(\forall (i: nat).((getl i c
(* This file was automatically generated: do not edit *********************)
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
-include "pr1/defs.ma".
+include "LambdaDelta-1/pr1/defs.ma".
theorem pr3_pr1:
\forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (c: C).(pr3 c t1
(* This file was automatically generated: do not edit *********************)
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
-include "pr2/pr2.ma".
+include "LambdaDelta-1/pr2/pr2.ma".
theorem pr3_strip:
\forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr3 c t0 t1) \to (\forall
(* This file was automatically generated: do not edit *********************)
-include "pr3/pr1.ma".
+include "LambdaDelta-1/pr3/pr1.ma".
-include "pr2/props.ma".
+include "LambdaDelta-1/pr2/props.ma".
-include "pr1/props.ma".
+include "LambdaDelta-1/pr1/props.ma".
theorem clear_pr3_trans:
\forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr3 c2 t1 t2) \to
\def
\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c:
C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2
-(CHead c k u2) t1 t2)).(let H1 \def (match H0 in pr2 return (\lambda (c0:
-C).(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0
-(CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1
-t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0
-(CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3
-t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2)
-\to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3
-(CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda
-(t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1
-t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3
-(sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2)
-H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda
-(H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq
-T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t
-t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i
-u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0
-t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2)
-(CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3
-(CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda
-(t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to
-((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i
-(CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda
-(H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2)
-(CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1
-t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr)
-u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k0: K).((getl O
-(CHead c k0 u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t1 t2)))
-(\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind
-Abbr) u))).(let H15 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow
-c1])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c
-(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind
-Abbr) u) H14))) in ((let H16 \def (f_equal C B (\lambda (e: C).(match e in C
-return (\lambda (_: C).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _)
-\Rightarrow (match k0 in K return (\lambda (_: K).B) with [(Bind b0)
-\Rightarrow b0 | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u)
-(CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2
-(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H17
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u | (CHead _ _ t4) \Rightarrow t4])) (CHead d
-(Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr)
-u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in
-(\lambda (H18: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H20 \def (eq_ind
-T u (\lambda (t4: T).(subst0 O t4 t3 t2)) H13 u2 H17) in (eq_ind B Abbr
-(\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t4:
-T).(subst0 O u1 t3 t4)) (\lambda (t4: T).(pr0 t4 t2)) (pr3 (CHead c (Bind
-Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H21: (subst0 O u1 t3 x)).(\lambda
-(H22: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr) u1) x t1 (pr2_delta (CHead c
-(Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H21) t2 (pr3_pr2
-(CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c (Bind Abbr) u1) x t2
-H22)))))) (pr0_subst0_back u2 t3 t2 O H20 u1 H)) b H18))))) H16)) H15))))
-(\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat f) u2) (CHead d (Bind
-Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta
-c d u O (getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c)
-(clear_gen_flat f c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f)
-u2) (CHead d (Bind Abbr) u) H14))) t1 t3 H10 t2 H13) f u1)))) k H12)))
-(\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind
-Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1
-t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr)
-u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k0: K).((getl
-(S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k0
-u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k0
-u1) t1 t2)))) \to (pr3 (CHead c k0 u1) t1 t2)))) (\lambda (b: B).(\lambda
-(H14: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(\lambda
-(_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0
-u t3 t2) \to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b)
-u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0
-c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u) u2
-i0 H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (getl (S i0)
-(CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0
-(CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to
-(pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2
-(pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c
-(CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi)))))
-i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0
-(CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2))
-(refl_equal T t1) (refl_equal T t2)))))))))).
+(CHead c k u2) t1 t2)).(insert_eq C (CHead c k u2) (\lambda (c0: C).(pr2 c0
+t1 t2)) (\lambda (_: C).(pr3 (CHead c k u1) t1 t2)) (\lambda (y: C).(\lambda
+(H1: (pr2 y t1 t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0:
+T).((eq C c0 (CHead c k u2)) \to (pr3 (CHead c k u1) t t0))))) (\lambda (c0:
+C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H2: (pr0 t3 t4)).(\lambda (_:
+(eq C c0 (CHead c k u2))).(pr3_pr2 (CHead c k u1) t3 t4 (pr2_free (CHead c k
+u1) t3 t4 H2))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H2: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t3:
+T).(\lambda (t4: T).(\lambda (H3: (pr0 t3 t4)).(\lambda (t: T).(\lambda (H4:
+(subst0 i u t4 t)).(\lambda (H5: (eq C c0 (CHead c k u2))).(let H6 \def
+(eq_ind C c0 (\lambda (c1: C).(getl i c1 (CHead d (Bind Abbr) u))) H2 (CHead
+c k u2) H5) in (nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d
+(Bind Abbr) u)) \to ((subst0 n u t4 t) \to (pr3 (CHead c k u1) t3 t))))
+(\lambda (H7: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H8:
+(subst0 O u t4 t)).(K_ind (\lambda (k0: K).((getl O (CHead c k0 u2) (CHead d
+(Bind Abbr) u)) \to (pr3 (CHead c k0 u1) t3 t))) (\lambda (b: B).(\lambda
+(H9: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H10 \def
+(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow d | (CHead c1 _ _) \Rightarrow c1])) (CHead d (Bind
+Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2
+(getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H9))) in ((let H11
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
+\Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2)
+(clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b)
+u2) (CHead d (Bind Abbr) u) H9))) in ((let H12 \def (f_equal C T (\lambda (e:
+C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u |
+(CHead _ _ t0) \Rightarrow t0])) (CHead d (Bind Abbr) u) (CHead c (Bind b)
+u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind
+b) u2) (CHead d (Bind Abbr) u) H9))) in (\lambda (H13: (eq B Abbr
+b)).(\lambda (_: (eq C d c)).(let H15 \def (eq_ind T u (\lambda (t0:
+T).(subst0 O t0 t4 t)) H8 u2 H12) in (eq_ind B Abbr (\lambda (b0: B).(pr3
+(CHead c (Bind b0) u1) t3 t)) (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4
+t0)) (\lambda (t0: T).(pr0 t0 t)) (pr3 (CHead c (Bind Abbr) u1) t3 t)
+(\lambda (x: T).(\lambda (H16: (subst0 O u1 t4 x)).(\lambda (H17: (pr0 x
+t)).(pr3_sing (CHead c (Bind Abbr) u1) x t3 (pr2_delta (CHead c (Bind Abbr)
+u1) c u1 O (getl_refl Abbr c u1) t3 t4 H3 x H16) t (pr3_pr2 (CHead c (Bind
+Abbr) u1) x t (pr2_free (CHead c (Bind Abbr) u1) x t H17))))))
+(pr0_subst0_back u2 t4 t O H15 u1 H)) b H13))))) H11)) H10)))) (\lambda (f:
+F).(\lambda (H9: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr)
+u))).(pr3_pr2 (CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u O
+(getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c
+(CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind
+Abbr) u) H9))) t3 t4 H3 t H8) f u1)))) k H7))) (\lambda (i0: nat).(\lambda
+(IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4
+t) \to (pr3 (CHead c k u1) t3 t))))).(\lambda (H7: (getl (S i0) (CHead c k
+u2) (CHead d (Bind Abbr) u))).(\lambda (H8: (subst0 (S i0) u t4 t)).(K_ind
+(\lambda (k0: K).((getl (S i0) (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to
+((((getl i0 (CHead c k0 u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t4 t)
+\to (pr3 (CHead c k0 u1) t3 t)))) \to (pr3 (CHead c k0 u1) t3 t)))) (\lambda
+(b: B).(\lambda (H9: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr)
+u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))
+\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Bind b) u1) t3 t))))).(pr3_pr2
+(CHead c (Bind b) u1) t3 t (pr2_delta (CHead c (Bind b) u1) d u (S i0)
+(getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c
+(CHead d (Bind Abbr) u) u2 i0 H9) u1) t3 t4 H3 t H8))))) (\lambda (f:
+F).(\lambda (H9: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr)
+u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))
+\to ((subst0 i0 u t4 t) \to (pr3 (CHead c (Flat f) u1) t3 t))))).(pr3_pr2
+(CHead c (Flat f) u1) t3 t (pr2_cflat c t3 t (pr2_delta c d u (r (Flat f) i0)
+(getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H9) t3 t4 H3 t H8) f
+u1))))) k H7 IHi))))) i H6 H4))))))))))))) y t1 t2 H1))) H0)))))))).
theorem pr3_pr2_pr2_t:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall
(CHead c k u1) t1 t2))))))))
\def
\lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1
-u2)).(let H0 \def (match H in pr2 return (\lambda (c0: C).(\lambda (t:
-T).(\lambda (t0: T).(\lambda (_: (pr2 c0 t t0)).((eq C c0 c) \to ((eq T t u1)
-\to ((eq T t0 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2
-(CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))))))) with
-[(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2:
-(eq T t1 u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T
-t1 u1) \to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4:
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3
-t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T
-t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
-K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda
-(H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3:
-T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3
-(CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H7: (pr2 (CHead c k u2) t3
-t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H7)))))) t2 (sym_eq T t2 u2 H5))) t1
-(sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u
-i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq
-T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1
-u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1
-t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k:
-K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))))))
-(\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to
-((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t)
-\to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3
-t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t
-u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to
-((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4:
-T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3
-t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9:
-(pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda
-(t0: T).(\lambda (k: K).(\lambda (H11: (pr2 (CHead c k u2) t3 t0)).(let H12
-\def (match H11 in pr2 return (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5:
-T).(\lambda (_: (pr2 c1 t4 t5)).((eq C c1 (CHead c k u2)) \to ((eq T t4 t3)
-\to ((eq T t5 t0) \to (pr3 (CHead c k u1) t3 t0)))))))) with [(pr2_free c1 t4
-t5 H12) \Rightarrow (\lambda (H13: (eq C c1 (CHead c k u2))).(\lambda (H14:
-(eq T t4 t3)).(\lambda (H15: (eq T t5 t0)).(eq_ind C (CHead c k u2) (\lambda
-(_: C).((eq T t4 t3) \to ((eq T t5 t0) \to ((pr0 t4 t5) \to (pr3 (CHead c k
-u1) t3 t0))))) (\lambda (H16: (eq T t4 t3)).(eq_ind T t3 (\lambda (t6:
-T).((eq T t5 t0) \to ((pr0 t6 t5) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda
-(H17: (eq T t5 t0)).(eq_ind T t0 (\lambda (t6: T).((pr0 t3 t6) \to (pr3
-(CHead c k u1) t3 t0))) (\lambda (H18: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1)
-t3 t0 (pr2_free (CHead c k u1) t3 t0 H18))) t5 (sym_eq T t5 t0 H17))) t4
-(sym_eq T t4 t3 H16))) c1 (sym_eq C c1 (CHead c k u2) H13) H14 H15 H12)))) |
-(pr2_delta c1 d0 u0 i0 H12 t4 t5 H13 t6 H14) \Rightarrow (\lambda (H15: (eq C
-c1 (CHead c k u2))).(\lambda (H16: (eq T t4 t3)).(\lambda (H17: (eq T t6
-t0)).(eq_ind C (CHead c k u2) (\lambda (c2: C).((eq T t4 t3) \to ((eq T t6
-t0) \to ((getl i0 c2 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t4 t5) \to ((subst0
-i0 u0 t5 t6) \to (pr3 (CHead c k u1) t3 t0))))))) (\lambda (H18: (eq T t4
-t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t6 t0) \to ((getl i0 (CHead c k u2)
-(CHead d0 (Bind Abbr) u0)) \to ((pr0 t7 t5) \to ((subst0 i0 u0 t5 t6) \to
-(pr3 (CHead c k u1) t3 t0)))))) (\lambda (H19: (eq T t6 t0)).(eq_ind T t0
-(\lambda (t7: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to
-((pr0 t3 t5) \to ((subst0 i0 u0 t5 t7) \to (pr3 (CHead c k u1) t3 t0)))))
-(\lambda (H20: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda
-(H21: (pr0 t3 t5)).(\lambda (H22: (subst0 i0 u0 t5 t0)).(nat_ind (\lambda (n:
-nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
-t0) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda (H23: (getl O (CHead c k u2)
-(CHead d0 (Bind Abbr) u0))).(\lambda (H24: (subst0 O u0 t5 t0)).(K_ind
-(\lambda (k0: K).((clear (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pr3
-(CHead c k0 u1) t3 t0))) (\lambda (b: B).(\lambda (H25: (clear (CHead c (Bind
-b) u2) (CHead d0 (Bind Abbr) u0))).(let H26 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0 |
-(CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b)
-u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H27 \def
-(f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B) with
-[(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
+u2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (t1:
+T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c0 k t0) t1 t2) \to (pr3
+(CHead c0 k t) t1 t2)))))))) (\lambda (c0: C).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (H0: (pr0 t1 t2)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k:
+K).(\lambda (H1: (pr2 (CHead c0 k t2) t0 t3)).(pr3_pr0_pr2_t t1 t2 H0 c0 t0
+t3 k H1))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda
+(i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (t1:
+T).(\lambda (t2: T).(\lambda (H1: (pr0 t1 t2)).(\lambda (t: T).(\lambda (H2:
+(subst0 i u t2 t)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda
+(H3: (pr2 (CHead c0 k t) t0 t3)).(insert_eq C (CHead c0 k t) (\lambda (c1:
+C).(pr2 c1 t0 t3)) (\lambda (_: C).(pr3 (CHead c0 k t1) t0 t3)) (\lambda (y:
+C).(\lambda (H4: (pr2 y t0 t3)).(pr2_ind (\lambda (c1: C).(\lambda (t4:
+T).(\lambda (t5: T).((eq C c1 (CHead c0 k t)) \to (pr3 (CHead c0 k t1) t4
+t5))))) (\lambda (c1: C).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H5: (pr0
+t4 t5)).(\lambda (_: (eq C c1 (CHead c0 k t))).(pr3_pr2 (CHead c0 k t1) t4 t5
+(pr2_free (CHead c0 k t1) t4 t5 H5))))))) (\lambda (c1: C).(\lambda (d0:
+C).(\lambda (u0: T).(\lambda (i0: nat).(\lambda (H5: (getl i0 c1 (CHead d0
+(Bind Abbr) u0))).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H6: (pr0 t4
+t5)).(\lambda (t6: T).(\lambda (H7: (subst0 i0 u0 t5 t6)).(\lambda (H8: (eq C
+c1 (CHead c0 k t))).(let H9 \def (eq_ind C c1 (\lambda (c2: C).(getl i0 c2
+(CHead d0 (Bind Abbr) u0))) H5 (CHead c0 k t) H8) in (nat_ind (\lambda (n:
+nat).((getl n (CHead c0 k t) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t5
+t6) \to (pr3 (CHead c0 k t1) t4 t6)))) (\lambda (H10: (getl O (CHead c0 k t)
+(CHead d0 (Bind Abbr) u0))).(\lambda (H11: (subst0 O u0 t5 t6)).(K_ind
+(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3
+(CHead c0 k0 t1) t4 t6))) (\lambda (b: B).(\lambda (H12: (clear (CHead c0
+(Bind b) t) (CHead d0 (Bind Abbr) u0))).(let H13 \def (f_equal C C (\lambda
+(e: C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d0
+| (CHead c2 _ _) \Rightarrow c2])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind
+b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H14
+\def (f_equal C B (\lambda (e: C).(match e in C return (\lambda (_: C).B)
+with [(CSort _) \Rightarrow Abbr | (CHead _ k0 _) \Rightarrow (match k0 in K
return (\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _)
-\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2)
-(clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H25)) in ((let H28 \def
+\Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c0 (Bind b) t)
+(clear_gen_bind b c0 (CHead d0 (Bind Abbr) u0) t H12)) in ((let H15 \def
(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
[(CSort _) \Rightarrow u0 | (CHead _ _ t7) \Rightarrow t7])) (CHead d0 (Bind
-Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0)
-u2 H25)) in (\lambda (H29: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H31
-\def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t0)) H24 u2 H28) in
-(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t3 t0)) (ex2_ind
+Abbr) u0) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead d0 (Bind Abbr)
+u0) t H12)) in (\lambda (H16: (eq B Abbr b)).(\lambda (_: (eq C d0 c0)).(let
+H18 \def (eq_ind T u0 (\lambda (t7: T).(subst0 O t7 t5 t6)) H11 t H15) in
+(eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c0 (Bind b0) t1) t4 t6)) (ex2_ind
T (\lambda (t7: T).(subst0 O t2 t5 t7)) (\lambda (t7: T).(subst0 (S (plus i
-O)) u t7 t0)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x: T).(\lambda
-(H32: (subst0 O t2 t5 x)).(\lambda (H33: (subst0 (S (plus i O)) u x t0)).(let
-H34 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O
-i))) in (let H35 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n
-u x t0)) H33 (S i) H34) in (ex2_ind T (\lambda (t7: T).(subst0 O u1 t5 t7))
-(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda
-(x0: T).(\lambda (H36: (subst0 O u1 t5 x0)).(\lambda (H37: (pr0 x0
-x)).(pr3_sing (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr)
-u1) c u1 O (getl_refl Abbr c u1) t3 t5 H21 x0 H36) t0 (pr3_pr2 (CHead c (Bind
-Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i)
-(getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1 (clear_bind Abbr c u1)
-(CHead d (Bind Abbr) u) i H8) x0 x H37 t0 H35)))))) (pr0_subst0_back t2 t5 x
-O H32 u1 H9))))))) (subst0_subst0 t5 t0 u2 O H31 t2 u i H10)) b H29)))))
-H27)) H26)))) (\lambda (f: F).(\lambda (H25: (clear (CHead c (Flat f) u2)
-(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c
-t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0 (Bind Abbr) u0) c
-(drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H25)) t3 t5
-H21 t0 H24) f u1)))) k (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0)
-H23)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0
-(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t0) \to (pr3 (CHead c k u1) t3
-t0))))).(\lambda (H23: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr)
-u0))).(\lambda (H24: (subst0 (S i1) u0 t5 t0)).(K_ind (\lambda (k0: K).((getl
-(S i1) (CHead c k0 u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k0 u1) t3
-t0))) (\lambda (b: B).(\lambda (H25: (getl (S i1) (CHead c (Bind b) u2)
-(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Bind b) u1) t3 t0 (pr2_delta
-(CHead c (Bind b) u1) d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind
-Abbr) u0) (getl_gen_S (Bind b) c (CHead d0 (Bind Abbr) u0) u2 i1 H25) u1) t3
-t5 H21 t0 H24)))) (\lambda (f: F).(\lambda (H25: (getl (S i1) (CHead c (Flat
-f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0
-(pr2_cflat c t3 t0 (pr2_delta c d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f) c
-(CHead d0 (Bind Abbr) u0) u2 i1 H25) t3 t5 H21 t0 H24) f u1)))) k H23))))) i0
-H20 H22)))) t6 (sym_eq T t6 t0 H19))) t4 (sym_eq T t4 t3 H18))) c1 (sym_eq C
-c1 (CHead c k u2) H15) H16 H17 H12 H13 H14))))]) in (H12 (refl_equal C (CHead
-c k u2)) (refl_equal T t3) (refl_equal T t0)))))))))) t (sym_eq T t u2 H7)))
-t1 (sym_eq T t1 u1 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0
-(refl_equal C c) (refl_equal T u1) (refl_equal T u2)))))).
+O)) u t7 t6)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda (x: T).(\lambda
+(H19: (subst0 O t2 t5 x)).(\lambda (H20: (subst0 (S (plus i O)) u x t6)).(let
+H21 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O
+i))) in (let H22 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n
+u x t6)) H20 (S i) H21) in (ex2_ind T (\lambda (t7: T).(subst0 O t1 t5 t7))
+(\lambda (t7: T).(pr0 t7 x)) (pr3 (CHead c0 (Bind Abbr) t1) t4 t6) (\lambda
+(x0: T).(\lambda (H23: (subst0 O t1 t5 x0)).(\lambda (H24: (pr0 x0
+x)).(pr3_sing (CHead c0 (Bind Abbr) t1) x0 t4 (pr2_delta (CHead c0 (Bind
+Abbr) t1) c0 t1 O (getl_refl Abbr c0 t1) t4 t5 H6 x0 H23) t6 (pr3_pr2 (CHead
+c0 (Bind Abbr) t1) x0 t6 (pr2_delta (CHead c0 (Bind Abbr) t1) d u (S i)
+(getl_clear_bind Abbr (CHead c0 (Bind Abbr) t1) c0 t1 (clear_bind Abbr c0 t1)
+(CHead d (Bind Abbr) u) i H0) x0 x H24 t6 H22)))))) (pr0_subst0_back t2 t5 x
+O H19 t1 H1))))))) (subst0_subst0 t5 t6 t O H18 t2 u i H2)) b H16))))) H14))
+H13)))) (\lambda (f: F).(\lambda (H12: (clear (CHead c0 (Flat f) t) (CHead d0
+(Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6 (pr2_cflat c0 t4 t6
+(pr2_delta c0 d0 u0 O (getl_intro O c0 (CHead d0 (Bind Abbr) u0) c0
+(drop_refl c0) (clear_gen_flat f c0 (CHead d0 (Bind Abbr) u0) t H12)) t4 t5
+H6 t6 H11) f t1)))) k (getl_gen_O (CHead c0 k t) (CHead d0 (Bind Abbr) u0)
+H10)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c0 k t) (CHead d0
+(Bind Abbr) u0)) \to ((subst0 i1 u0 t5 t6) \to (pr3 (CHead c0 k t1) t4
+t6))))).(\lambda (H10: (getl (S i1) (CHead c0 k t) (CHead d0 (Bind Abbr)
+u0))).(\lambda (H11: (subst0 (S i1) u0 t5 t6)).(K_ind (\lambda (k0: K).((getl
+(S i1) (CHead c0 k0 t) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c0 k0 t1)
+t4 t6))) (\lambda (b: B).(\lambda (H12: (getl (S i1) (CHead c0 (Bind b) t)
+(CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Bind b) t1) t4 t6 (pr2_delta
+(CHead c0 (Bind b) t1) d0 u0 (S i1) (getl_head (Bind b) i1 c0 (CHead d0 (Bind
+Abbr) u0) (getl_gen_S (Bind b) c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t1) t4
+t5 H6 t6 H11)))) (\lambda (f: F).(\lambda (H12: (getl (S i1) (CHead c0 (Flat
+f) t) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c0 (Flat f) t1) t4 t6
+(pr2_cflat c0 t4 t6 (pr2_delta c0 d0 u0 (r (Flat f) i1) (getl_gen_S (Flat f)
+c0 (CHead d0 (Bind Abbr) u0) t i1 H12) t4 t5 H6 t6 H11) f t1)))) k H10)))))
+i0 H9 H7))))))))))))) y t0 t3 H4))) H3))))))))))))))) c u1 u2 H)))).
theorem pr3_pr2_pr3_t:
\forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall
(* This file was automatically generated: do not edit *********************)
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
-include "pr2/subst1.ma".
+include "LambdaDelta-1/pr2/subst1.ma".
theorem pr3_subst1:
\forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c
(* This file was automatically generated: do not edit *********************)
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
-include "wcpr0/getl.ma".
+include "LambdaDelta-1/wcpr0/getl.ma".
theorem pr3_wcpr0_t:
\forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1:
T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda
(H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead
c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3
-(CHead c0 k u1) (let H7 \def (match H4 in pr2 return (\lambda (c: C).(\lambda
-(t: T).(\lambda (t5: T).(\lambda (_: (pr2 c t t5)).((eq C c (CHead c3 k u1))
-\to ((eq T t t3) \to ((eq T t5 t0) \to (pr3 (CHead c0 k u1) t3 t0))))))))
-with [(pr2_free c t5 t6 H7) \Rightarrow (\lambda (H8: (eq C c (CHead c3 k
-u1))).(\lambda (H9: (eq T t5 t3)).(\lambda (H10: (eq T t6 t0)).(eq_ind C
-(CHead c3 k u1) (\lambda (_: C).((eq T t5 t3) \to ((eq T t6 t0) \to ((pr0 t5
-t6) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H11: (eq T t5 t3)).(eq_ind
-T t3 (\lambda (t: T).((eq T t6 t0) \to ((pr0 t t6) \to (pr3 (CHead c0 k u1)
-t3 t0)))) (\lambda (H12: (eq T t6 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3
-t) \to (pr3 (CHead c0 k u1) t3 t0))) (\lambda (H13: (pr0 t3 t0)).(pr3_pr2
-(CHead c0 k u1) t3 t0 (pr2_free (CHead c0 k u1) t3 t0 H13))) t6 (sym_eq T t6
-t0 H12))) t5 (sym_eq T t5 t3 H11))) c (sym_eq C c (CHead c3 k u1) H8) H9 H10
-H7)))) | (pr2_delta c d u i H7 t5 t6 H8 t H9) \Rightarrow (\lambda (H10: (eq
-C c (CHead c3 k u1))).(\lambda (H11: (eq T t5 t3)).(\lambda (H12: (eq T t
-t0)).(eq_ind C (CHead c3 k u1) (\lambda (c4: C).((eq T t5 t3) \to ((eq T t
-t0) \to ((getl i c4 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i
-u t6 t) \to (pr3 (CHead c0 k u1) t3 t0))))))) (\lambda (H13: (eq T t5
-t3)).(eq_ind T t3 (\lambda (t7: T).((eq T t t0) \to ((getl i (CHead c3 k u1)
-(CHead d (Bind Abbr) u)) \to ((pr0 t7 t6) \to ((subst0 i u t6 t) \to (pr3
-(CHead c0 k u1) t3 t0)))))) (\lambda (H14: (eq T t t0)).(eq_ind T t0 (\lambda
-(t7: T).((getl i (CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t3 t6)
-\to ((subst0 i u t6 t7) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H15:
-(getl i (CHead c3 k u1) (CHead d (Bind Abbr) u))).(\lambda (H16: (pr0 t3
-t6)).(\lambda (H17: (subst0 i u t6 t0)).(ex3_2_ind C T (\lambda (e2:
+(CHead c0 k u1) (insert_eq C (CHead c3 k u1) (\lambda (c: C).(pr2 c t3 t0))
+(\lambda (_: C).(pr3 (CHead c0 k u1) t3 t0)) (\lambda (y: C).(\lambda (H7:
+(pr2 y t3 t0)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t5: T).((eq
+C c (CHead c3 k u1)) \to (pr3 (CHead c0 k u1) t t5))))) (\lambda (c:
+C).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H8: (pr0 t5 t6)).(\lambda (_:
+(eq C c (CHead c3 k u1))).(pr3_pr2 (CHead c0 k u1) t5 t6 (pr2_free (CHead c0
+k u1) t5 t6 H8))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (i: nat).(\lambda (H8: (getl i c (CHead d (Bind Abbr)
+u))).(\lambda (t5: T).(\lambda (t6: T).(\lambda (H9: (pr0 t5 t6)).(\lambda
+(t: T).(\lambda (H10: (subst0 i u t6 t)).(\lambda (H11: (eq C c (CHead c3 k
+u1))).(let H12 \def (eq_ind C c (\lambda (c4: C).(getl i c4 (CHead d (Bind
+Abbr) u))) H8 (CHead c3 k u1) H11) in (ex3_2_ind C T (\lambda (e2:
C).(\lambda (u3: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u3))))
(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u3:
-T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x0: C).(\lambda (x1:
-T).(\lambda (H18: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr)
-x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H20: (pr0 x1 u)).(ex2_ind T
-(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t0)) (pr3
-(CHead c0 k u1) t3 t0) (\lambda (x: T).(\lambda (H21: (subst0 i x1 t6
-x)).(\lambda (H22: (pr0 x t0)).(pr3_sing (CHead c0 k u1) x t3 (pr2_delta
-(CHead c0 k u1) x0 x1 i H18 t3 t6 H16 x H21) t0 (pr3_pr2 (CHead c0 k u1) x t0
-(pr2_free (CHead c0 k u1) x t0 H22)))))) (pr0_subst0_back u t6 t0 i H17 x1
-H20))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3
-H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H15))))) t (sym_eq T t t0 H14)))
-t5 (sym_eq T t5 t3 H13))) c (sym_eq C c (CHead c3 k u1) H10) H11 H12 H7 H8
-H9))))]) in (H7 (refl_equal C (CHead c3 k u1)) (refl_equal T t3) (refl_equal
-T t0))) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1
-u2 H2)))))))))))))) c2 c1 H))).
+T).(pr0 u3 u))) (pr3 (CHead c0 k u1) t5 t) (\lambda (x0: C).(\lambda (x1:
+T).(\lambda (H13: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr)
+x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H15: (pr0 x1 u)).(ex2_ind T
+(\lambda (t7: T).(subst0 i x1 t6 t7)) (\lambda (t7: T).(pr0 t7 t)) (pr3
+(CHead c0 k u1) t5 t) (\lambda (x: T).(\lambda (H16: (subst0 i x1 t6
+x)).(\lambda (H17: (pr0 x t)).(pr3_sing (CHead c0 k u1) x t5 (pr2_delta
+(CHead c0 k u1) x0 x1 i H13 t5 t6 H9 x H16) t (pr3_pr2 (CHead c0 k u1) x t
+(pr2_free (CHead c0 k u1) x t H17)))))) (pr0_subst0_back u t6 t i H10 x1
+H15))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3
+H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H12)))))))))))))) y t3 t0 H7)))
+H4) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2
+H2)))))))))))))) c2 c1 H))).
(* *)
(**************************************************************************)
-include "../Base-1/theory.ma".
+include "Base-1/theory.ma".
alias id "and_ind" = "cic:/Coq/Init/Logic/and_ind.con".
alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con".
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
definition r:
K \to (nat \to nat)
(* This file was automatically generated: do not edit *********************)
-include "r/defs.ma".
+include "LambdaDelta-1/r/defs.ma".
-include "s/defs.ma".
+include "LambdaDelta-1/s/defs.ma".
theorem r_S:
\forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r k i))))
+++ /dev/null
-baseuri=cic:/matita/LAMBDA-TYPES/LambdaDelta-1
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
definition s:
K \to (nat \to nat)
(* This file was automatically generated: do not edit *********************)
-include "s/defs.ma".
+include "LambdaDelta-1/s/defs.ma".
theorem s_S:
\forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
(* This file was automatically generated: do not edit *********************)
-include "csubc/arity.ma".
+include "LambdaDelta-1/csubc/arity.ma".
-include "csubc/getl.ma".
+include "LambdaDelta-1/csubc/getl.ma".
-include "csubc/drop1.ma".
+include "LambdaDelta-1/csubc/drop1.ma".
-include "csubc/props.ma".
+include "LambdaDelta-1/csubc/props.ma".
theorem sc3_arity_csubc:
\forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
C).(csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2
(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H9: (getl (trans is i) c2
x0)).(\lambda (H10: (csubc g (CHead x (Bind Abbr) (lift1 (ptrans is i) u))
-x0)).(let H11 \def (match H10 in csubc return (\lambda (c0: C).(\lambda (c3:
-C).(\lambda (_: (csubc ? c0 c3)).((eq C c0 (CHead x (Bind Abbr) (lift1
-(ptrans is i) u))) \to ((eq C c3 x0) \to (sc3 g a0 c2 (lift1 is (TLRef
-i)))))))) with [(csubc_sort n) \Rightarrow (\lambda (H11: (eq C (CSort n)
-(CHead x (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (H12: (eq C (CSort
-n) x0)).((let H13 \def (eq_ind C (CSort n) (\lambda (e: C).(match e in C
-return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _)
-\Rightarrow False])) I (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H11) in
-(False_ind ((eq C (CSort n) x0) \to (sc3 g a0 c2 (lift1 is (TLRef i)))) H13))
-H12))) | (csubc_head c0 c3 H11 k v) \Rightarrow (\lambda (H12: (eq C (CHead
-c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)))).(\lambda (H13: (eq C
-(CHead c3 k v) x0)).((let H14 \def (f_equal C T (\lambda (e: C).(match e in C
-return (\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0)
-\Rightarrow t0])) (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i)
-u)) H12) in ((let H15 \def (f_equal C K (\lambda (e: C).(match e in C return
-(\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k0 _) \Rightarrow
-k0])) (CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H12) in
-((let H16 \def (f_equal C C (\lambda (e: C).(match e in C return (\lambda (_:
-C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4]))
-(CHead c0 k v) (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H12) in (eq_ind
-C x (\lambda (c4: C).((eq K k (Bind Abbr)) \to ((eq T v (lift1 (ptrans is i)
-u)) \to ((eq C (CHead c3 k v) x0) \to ((csubc g c4 c3) \to (sc3 g a0 c2
-(lift1 is (TLRef i)))))))) (\lambda (H17: (eq K k (Bind Abbr))).(eq_ind K
-(Bind Abbr) (\lambda (k0: K).((eq T v (lift1 (ptrans is i) u)) \to ((eq C
-(CHead c3 k0 v) x0) \to ((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef
-i))))))) (\lambda (H18: (eq T v (lift1 (ptrans is i) u))).(eq_ind T (lift1
-(ptrans is i) u) (\lambda (t0: T).((eq C (CHead c3 (Bind Abbr) t0) x0) \to
-((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))) (\lambda (H19: (eq
-C (CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)) x0)).(eq_ind C (CHead c3
-(Bind Abbr) (lift1 (ptrans is i) u)) (\lambda (_: C).((csubc g x c3) \to (sc3
-g a0 c2 (lift1 is (TLRef i))))) (\lambda (_: (csubc g x c3)).(let H21 \def
-(eq_ind_r C x0 (\lambda (c4: C).(getl (trans is i) c2 c4)) H9 (CHead c3 (Bind
-Abbr) (lift1 (ptrans is i) u)) H19) in (let H_y \def (sc3_abbr g a0 TNil) in
+x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind
+Abbr) H10) in (let H11 \def H_x1 in (or_ind (ex2 C (\lambda (c3: C).(eq C x0
+(CHead c3 (Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
+(Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (sc3 g a0 c2 (lift1
+is (TLRef i))) (\lambda (H12: (ex2 C (\lambda (c3: C).(eq C x0 (CHead c3
+(Bind Abbr) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abbr) (lift1
+(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is
+(TLRef i))) (\lambda (x1: C).(\lambda (H13: (eq C x0 (CHead x1 (Bind Abbr)
+(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H15 \def (eq_ind
+C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr)
+(lift1 (ptrans is i) u)) H13) in (let H_y \def (sc3_abbr g a0 TNil) in
(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y
-(trans is i) c3 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O
+(trans is i) x1 (lift1 (ptrans is i) u) c2 (eq_ind T (lift1 is (lift (S i) O
u)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (eq_ind T (lift1 (PConsTail is (S i)
O) u) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H2 d1 (PConsTail is (S i) O)
(drop1_cons_tail c d (S i) O (getl_drop Abbr c d u i H0) is d1 H3) c2 H4)
(lift1 is (lift (S i) O u)) (lift1_cons_tail u (S i) O is)) (lift (S (trans
-is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H21) (lift1 is (TLRef
-i)) (lift1_lref is i))))) x0 H19)) v (sym_eq T v (lift1 (ptrans is i) u)
-H18))) k (sym_eq K k (Bind Abbr) H17))) c0 (sym_eq C c0 x H16))) H15)) H14))
-H13 H11))) | (csubc_abst c0 c3 H11 v a1 H12 w H13) \Rightarrow (\lambda (H14:
-(eq C (CHead c0 (Bind Abst) v) (CHead x (Bind Abbr) (lift1 (ptrans is i)
-u)))).(\lambda (H15: (eq C (CHead c3 (Bind Abbr) w) x0)).((let H16 \def
-(eq_ind C (CHead c0 (Bind Abst) v) (\lambda (e: C).(match e in C return
-(\lambda (_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _)
-\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b)
-\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow
-False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _)
-\Rightarrow False])])) I (CHead x (Bind Abbr) (lift1 (ptrans is i) u)) H14)
-in (False_ind ((eq C (CHead c3 (Bind Abbr) w) x0) \to ((csubc g c0 c3) \to
-((sc3 g (asucc g a1) c0 v) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is
-(TLRef i))))))) H16)) H15 H11 H12 H13)))]) in (H11 (refl_equal C (CHead x
-(Bind Abbr) (lift1 (ptrans is i) u))) (refl_equal C x0)))))) H8))))))
-H5)))))))))))))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda
-(i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abst) u))).(\lambda (a0:
-A).(\lambda (H1: (arity g d u (asucc g a0))).(\lambda (_: ((\forall (d1:
-C).(\forall (is: PList).((drop1 is d1 d) \to (\forall (c2: C).((csubc g d1
-c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (d1: C).(\lambda
-(is: PList).(\lambda (H3: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H4:
-(csubc g d1 c2)).(let H5 \def H0 in (let H_x \def (drop1_getl_trans is c d1
-H3 Abst d u i H5) in (let H6 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1
-(ptrans is i) e2 d)) (\lambda (e2: C).(getl (trans is i) d1 (CHead e2 (Bind
-Abst) (lift1 (ptrans is i) u)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda
-(x: C).(\lambda (H7: (drop1 (ptrans is i) x d)).(\lambda (H8: (getl (trans is
-i) d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)))).(let H_x0 \def
-(csubc_getl_conf g d1 (CHead x (Bind Abst) (lift1 (ptrans is i) u)) (trans is
-i) H8 c2 H4) in (let H9 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans
-is i) c2 e2)) (\lambda (e2: C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans
-is i) u)) e2)) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x0: C).(\lambda
-(H10: (getl (trans is i) c2 x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst)
-(lift1 (ptrans is i) u)) x0)).(let H12 \def (match H11 in csubc return
-(\lambda (c0: C).(\lambda (c3: C).(\lambda (_: (csubc ? c0 c3)).((eq C c0
-(CHead x (Bind Abst) (lift1 (ptrans is i) u))) \to ((eq C c3 x0) \to (sc3 g
-a0 c2 (lift1 is (TLRef i)))))))) with [(csubc_sort n) \Rightarrow (\lambda
-(H12: (eq C (CSort n) (CHead x (Bind Abst) (lift1 (ptrans is i)
-u)))).(\lambda (H13: (eq C (CSort n) x0)).((let H14 \def (eq_ind C (CSort n)
-(\lambda (e: C).(match e in C return (\lambda (_: C).Prop) with [(CSort _)
-\Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x (Bind Abst)
-(lift1 (ptrans is i) u)) H12) in (False_ind ((eq C (CSort n) x0) \to (sc3 g
-a0 c2 (lift1 is (TLRef i)))) H14)) H13))) | (csubc_head c0 c3 H12 k v)
-\Rightarrow (\lambda (H13: (eq C (CHead c0 k v) (CHead x (Bind Abst) (lift1
-(ptrans is i) u)))).(\lambda (H14: (eq C (CHead c3 k v) x0)).((let H15 \def
-(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
-[(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k v)
-(CHead x (Bind Abst) (lift1 (ptrans is i) u)) H13) in ((let H16 \def (f_equal
-C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with [(CSort _)
-\Rightarrow k | (CHead _ k0 _) \Rightarrow k0])) (CHead c0 k v) (CHead x
-(Bind Abst) (lift1 (ptrans is i) u)) H13) in ((let H17 \def (f_equal C C
-(\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow c0 | (CHead c4 _ _) \Rightarrow c4])) (CHead c0 k v) (CHead x
-(Bind Abst) (lift1 (ptrans is i) u)) H13) in (eq_ind C x (\lambda (c4:
-C).((eq K k (Bind Abst)) \to ((eq T v (lift1 (ptrans is i) u)) \to ((eq C
-(CHead c3 k v) x0) \to ((csubc g c4 c3) \to (sc3 g a0 c2 (lift1 is (TLRef
-i)))))))) (\lambda (H18: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda
-(k0: K).((eq T v (lift1 (ptrans is i) u)) \to ((eq C (CHead c3 k0 v) x0) \to
-((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i))))))) (\lambda (H19: (eq
-T v (lift1 (ptrans is i) u))).(eq_ind T (lift1 (ptrans is i) u) (\lambda (t0:
-T).((eq C (CHead c3 (Bind Abst) t0) x0) \to ((csubc g x c3) \to (sc3 g a0 c2
-(lift1 is (TLRef i)))))) (\lambda (H20: (eq C (CHead c3 (Bind Abst) (lift1
-(ptrans is i) u)) x0)).(eq_ind C (CHead c3 (Bind Abst) (lift1 (ptrans is i)
-u)) (\lambda (_: C).((csubc g x c3) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))
-(\lambda (_: (csubc g x c3)).(let H22 \def (eq_ind_r C x0 (\lambda (c4:
-C).(getl (trans is i) c2 c4)) H10 (CHead c3 (Bind Abst) (lift1 (ptrans is i)
-u)) H20) in (let H_y \def (sc3_abst g a0 TNil) in (eq_ind_r T (TLRef (trans
-is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y c2 (trans is i)
-(csubc_arity_conf g d1 c2 H4 (TLRef (trans is i)) a0 (eq_ind T (lift1 is
-(TLRef i)) (\lambda (t0: T).(arity g d1 t0 a0)) (arity_lift1 g a0 c is d1
-(TLRef i) H3 (arity_abst g c d u i H0 a0 H1)) (TLRef (trans is i))
-(lift1_lref is i))) (nf2_lref_abst c2 c3 (lift1 (ptrans is i) u) (trans is i)
-H22) I) (lift1 is (TLRef i)) (lift1_lref is i))))) x0 H20)) v (sym_eq T v
-(lift1 (ptrans is i) u) H19))) k (sym_eq K k (Bind Abst) H18))) c0 (sym_eq C
-c0 x H17))) H16)) H15)) H14 H12))) | (csubc_abst c0 c3 H12 v a1 H13 w H14)
-\Rightarrow (\lambda (H15: (eq C (CHead c0 (Bind Abst) v) (CHead x (Bind
-Abst) (lift1 (ptrans is i) u)))).(\lambda (H16: (eq C (CHead c3 (Bind Abbr)
-w) x0)).((let H17 \def (f_equal C T (\lambda (e: C).(match e in C return
-(\lambda (_: C).T) with [(CSort _) \Rightarrow v | (CHead _ _ t0) \Rightarrow
-t0])) (CHead c0 (Bind Abst) v) (CHead x (Bind Abst) (lift1 (ptrans is i) u))
-H15) in ((let H18 \def (f_equal C C (\lambda (e: C).(match e in C return
-(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c4 _ _)
-\Rightarrow c4])) (CHead c0 (Bind Abst) v) (CHead x (Bind Abst) (lift1
-(ptrans is i) u)) H15) in (eq_ind C x (\lambda (c4: C).((eq T v (lift1
-(ptrans is i) u)) \to ((eq C (CHead c3 (Bind Abbr) w) x0) \to ((csubc g c4
-c3) \to ((sc3 g (asucc g a1) c4 v) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2
-(lift1 is (TLRef i))))))))) (\lambda (H19: (eq T v (lift1 (ptrans is i)
-u))).(eq_ind T (lift1 (ptrans is i) u) (\lambda (t0: T).((eq C (CHead c3
-(Bind Abbr) w) x0) \to ((csubc g x c3) \to ((sc3 g (asucc g a1) x t0) \to
-((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))))) (\lambda (H20:
-(eq C (CHead c3 (Bind Abbr) w) x0)).(eq_ind C (CHead c3 (Bind Abbr) w)
-(\lambda (_: C).((csubc g x c3) \to ((sc3 g (asucc g a1) x (lift1 (ptrans is
-i) u)) \to ((sc3 g a1 c3 w) \to (sc3 g a0 c2 (lift1 is (TLRef i)))))))
-(\lambda (_: (csubc g x c3)).(\lambda (H22: (sc3 g (asucc g a1) x (lift1
-(ptrans is i) u))).(\lambda (H23: (sc3 g a1 c3 w)).(let H24 \def (eq_ind_r C
-x0 (\lambda (c4: C).(getl (trans is i) c2 c4)) H10 (CHead c3 (Bind Abbr) w)
-H20) in (let H_y \def (sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef (trans is
-i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y (trans is i) c3 w c2 (let H_y0
-\def (arity_lift1 g (asucc g a0) d (ptrans is i) x u H7 H1) in (let H_y1 \def
-(sc3_arity_gen g x (lift1 (ptrans is i) u) (asucc g a1) H22) in (sc3_repl g
-a1 c2 (lift (S (trans is i)) O w) (sc3_lift g a1 c3 w H23 c2 (S (trans is i))
-O (getl_drop Abbr c2 c3 w (trans is i) H24)) a0 (asucc_inj g a1 a0
-(arity_mono g x (lift1 (ptrans is i) u) (asucc g a1) H_y1 (asucc g a0)
-H_y0))))) H24) (lift1 is (TLRef i)) (lift1_lref is i))))))) x0 H20)) v
-(sym_eq T v (lift1 (ptrans is i) u) H19))) c0 (sym_eq C c0 x H18))) H17)) H16
-H12 H13 H14)))]) in (H12 (refl_equal C (CHead x (Bind Abst) (lift1 (ptrans is
-i) u))) (refl_equal C x0)))))) H9)))))) H6))))))))))))))))) (\lambda (b:
-B).(\lambda (H0: (not (eq B b Abst))).(\lambda (c: C).(\lambda (u:
-T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda (H2: ((\forall
-(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g
-d1 c2) \to (sc3 g a1 c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2:
-A).(\lambda (_: (arity g (CHead c (Bind b) u) t0 a2)).(\lambda (H4: ((\forall
-(d1: C).(\forall (is: PList).((drop1 is d1 (CHead c (Bind b) u)) \to (\forall
-(c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1:
-C).(\lambda (is: PList).(\lambda (H5: (drop1 is d1 c)).(\lambda (c2:
-C).(\lambda (H6: (csubc g d1 c2)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil)
-in (eq_ind_r T (THead (Bind b) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1:
-T).(sc3 g a2 c2 t1)) (H_y c2 (lift1 is u) (lift1 (Ss is) t0) (H4 (CHead d1
-(Bind b) (lift1 is u)) (Ss is) (drop1_skip_bind b c is d1 u H5) (CHead c2
-(Bind b) (lift1 is u)) (csubc_head g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1
-is H5 c2 H6)) (lift1 is (THead (Bind b) u t0)) (lift1_bind b is u
-t0))))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1:
-A).(\lambda (H0: (arity g c u (asucc g a1))).(\lambda (H1: ((\forall (d1:
-C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1
-c2) \to (sc3 g (asucc g a1) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda
-(a2: A).(\lambda (H2: (arity g (CHead c (Bind Abst) u) t0 a2)).(\lambda (H3:
-((\forall (d1: C).(\forall (is: PList).((drop1 is d1 (CHead c (Bind Abst) u))
-\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a2 c2 (lift1 is
-t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1
-c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(eq_ind_r T (THead (Bind
-Abst) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: T).(land (arity g c2 t1
-(AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall
-(is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1
-is0 t1)))))))))) (conj (arity g c2 (THead (Bind Abst) (lift1 is u) (lift1 (Ss
-is) t0)) (AHead a1 a2)) (\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to
-(\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w
-(lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0))))))))))
-(csubc_arity_conf g d1 c2 H5 (THead (Bind Abst) (lift1 is u) (lift1 (Ss is)
-t0)) (AHead a1 a2) (arity_head g d1 (lift1 is u) a1 (arity_lift1 g (asucc g
-a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2 (arity_lift1 g a2 (CHead c (Bind
-Abst) u) (Ss is) (CHead d1 (Bind Abst) (lift1 is u)) t0 (drop1_skip_bind Abst
-c is d1 u H4) H2))) (\lambda (d: C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d
-w)).(\lambda (is0: PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead
-(Bind Abst) (lift1 is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0)))
-(\lambda (t1: T).(sc3 g a2 d (THead (Flat Appl) w t1))) (let H8 \def
-(sc3_appl g a1 a2 TNil) in (H8 d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let
-H_y \def (sc3_bind g Abbr (\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9))
-a1 a2 TNil) in (H_y d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def
-(csubc_drop1_conf_rev g is0 d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C
-(\lambda (c3: C).(drop1 is0 c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g
-a2 (CHead d (Bind Abbr) w) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x:
-C).(\lambda (H10: (drop1 is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T
-(lift1 (papp (Ss is0) (Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind
-Abbr) w) t1)) (eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g
-a2 (CHead d (Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1
-(papp is0 is) u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x
-u (drop1_trans is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x
-d H11 (lift1 (papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1
-H10 is c H4) x (csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0
-is)) (lift1 (Ss is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is)
-t0))))) H9))) H6)) H6 (lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1)
-is0 d (lift1 is u) (H1 d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst)
-(lift1 is u) (lift1 (Ss is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1
-(Ss is) t0))))))))) (lift1 is (THead (Bind Abst) u t0)) (lift1_bind Abst is u
-t0)))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda
-(_: (arity g c u a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is:
-PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1
-c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity
-g c t0 (AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is:
-PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
-(AHead a1 a2) c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is:
-PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g
-d1 c2)).(let H_y \def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2
-H5) in (let H6 \def H_y0 in (and_ind (arity g c2 (lift1 is t0) (AHead a1 a2))
-(\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0:
-PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0
-(lift1 is t0))))))))) (sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0)))
-(\lambda (_: (arity g c2 (lift1 is t0) (AHead a1 a2))).(\lambda (H8:
-((\forall (d: C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0:
-PList).((drop1 is0 d c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0
-(lift1 is t0))))))))))).(let H_y1 \def (H8 c2 (lift1 is u) H_y PNil) in
-(eq_ind_r T (THead (Flat Appl) (lift1 is u) (lift1 is t0)) (\lambda (t1:
-T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2)) (lift1 is (THead (Flat Appl) u
-t0)) (lift1_flat Appl is u t0))))) H6)))))))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g
-a0))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
+is i)) O (lift1 (ptrans is i) u)) (lift1_free is i u)) H15) (lift1 is (TLRef
+i)) (lift1_lref is i))))))) H12)) (\lambda (H12: (ex5_3 C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abbr) (Bind Abst))))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3
+w)))))).(ex5_3_ind C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq
+K (Bind Abbr) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w)))) (sc3 g a0 c2 (lift1 is
+(TLRef i))) (\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: A).(\lambda (H13:
+(eq K (Bind Abbr) (Bind Abst))).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abbr)
+x2))).(\lambda (_: (csubc g x x1)).(\lambda (_: (sc3 g (asucc g x3) x (lift1
+(ptrans is i) u))).(\lambda (_: (sc3 g x3 x1 x2)).(let H18 \def (eq_ind C x0
+(\lambda (c0: C).(getl (trans is i) c2 c0)) H9 (CHead x1 (Bind Abbr) x2) H14)
+in (let H19 \def (eq_ind K (Bind Abbr) (\lambda (ee: K).(match ee in K return
+(\lambda (_: K).Prop) with [(Bind b) \Rightarrow (match b in B return
+(\lambda (_: B).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False |
+Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abst) H13)
+in (False_ind (sc3 g a0 c2 (lift1 is (TLRef i))) H19))))))))))) H12))
+H11)))))) H8)))))) H5)))))))))))))))) (\lambda (c: C).(\lambda (d:
+C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind
+Abst) u))).(\lambda (a0: A).(\lambda (H1: (arity g d u (asucc g
+a0))).(\lambda (_: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 d)
\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a0) c2 (lift1 is
-u))))))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
-((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2:
-C).((csubc g d1 c2) \to (sc3 g a0 c2 (lift1 is t0))))))))).(\lambda (d1:
-C).(\lambda (is: PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2:
-C).(\lambda (H5: (csubc g d1 c2)).(let H_y \def (sc3_cast g a0 TNil) in
-(eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 is t0)) (\lambda (t1:
-T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1 is H4 c2 H5) (lift1 is t0)
-(H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast) u t0)) (lift1_flat Cast is
-u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1:
-A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall (d1: C).(\forall
-(is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g
-a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1
-a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1
-c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2 (lift1
-is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
+u))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is d1
+c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(let H5 \def H0 in (let
+H_x \def (drop1_getl_trans is c d1 H3 Abst d u i H5) in (let H6 \def H_x in
+(ex2_ind C (\lambda (e2: C).(drop1 (ptrans is i) e2 d)) (\lambda (e2:
+C).(getl (trans is i) d1 (CHead e2 (Bind Abst) (lift1 (ptrans is i) u))))
+(sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x: C).(\lambda (H7: (drop1
+(ptrans is i) x d)).(\lambda (H8: (getl (trans is i) d1 (CHead x (Bind Abst)
+(lift1 (ptrans is i) u)))).(let H_x0 \def (csubc_getl_conf g d1 (CHead x
+(Bind Abst) (lift1 (ptrans is i) u)) (trans is i) H8 c2 H4) in (let H9 \def
+H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans is i) c2 e2)) (\lambda (e2:
+C).(csubc g (CHead x (Bind Abst) (lift1 (ptrans is i) u)) e2)) (sc3 g a0 c2
+(lift1 is (TLRef i))) (\lambda (x0: C).(\lambda (H10: (getl (trans is i) c2
+x0)).(\lambda (H11: (csubc g (CHead x (Bind Abst) (lift1 (ptrans is i) u))
+x0)).(let H_x1 \def (csubc_gen_head_l g x x0 (lift1 (ptrans is i) u) (Bind
+Abst) H11) in (let H12 \def H_x1 in (or_ind (ex2 C (\lambda (c3: C).(eq C x0
+(CHead c3 (Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x
+c3))) (ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K
+(Bind Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_:
+A).(eq C x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_:
+T).(\lambda (_: A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda
+(a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3:
+C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3 w))))) (sc3 g a0 c2 (lift1
+is (TLRef i))) (\lambda (H13: (ex2 C (\lambda (c3: C).(eq C x0 (CHead c3
+(Bind Abst) (lift1 (ptrans is i) u)))) (\lambda (c3: C).(csubc g x
+c3)))).(ex2_ind C (\lambda (c3: C).(eq C x0 (CHead c3 (Bind Abst) (lift1
+(ptrans is i) u)))) (\lambda (c3: C).(csubc g x c3)) (sc3 g a0 c2 (lift1 is
+(TLRef i))) (\lambda (x1: C).(\lambda (H14: (eq C x0 (CHead x1 (Bind Abst)
+(lift1 (ptrans is i) u)))).(\lambda (_: (csubc g x x1)).(let H16 \def (eq_ind
+C x0 (\lambda (c0: C).(getl (trans is i) c2 c0)) H10 (CHead x1 (Bind Abst)
+(lift1 (ptrans is i) u)) H14) in (let H_y \def (sc3_abst g a0 TNil) in
+(eq_ind_r T (TLRef (trans is i)) (\lambda (t0: T).(sc3 g a0 c2 t0)) (H_y c2
+(trans is i) (csubc_arity_conf g d1 c2 H4 (TLRef (trans is i)) a0 (eq_ind T
+(lift1 is (TLRef i)) (\lambda (t0: T).(arity g d1 t0 a0)) (arity_lift1 g a0 c
+is d1 (TLRef i) H3 (arity_abst g c d u i H0 a0 H1)) (TLRef (trans is i))
+(lift1_lref is i))) (nf2_lref_abst c2 x1 (lift1 (ptrans is i) u) (trans is i)
+H16) I) (lift1 is (TLRef i)) (lift1_lref is i))))))) H13)) (\lambda (H13:
+(ex5_3 C T A (\lambda (_: C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind
+Abst) (Bind Abst))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (_: A).(eq C
+x0 (CHead c3 (Bind Abbr) w))))) (\lambda (c3: C).(\lambda (_: T).(\lambda (_:
+A).(csubc g x c3)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g
+(asucc g a1) x (lift1 (ptrans is i) u))))) (\lambda (c3: C).(\lambda (w:
+T).(\lambda (a1: A).(sc3 g a1 c3 w)))))).(ex5_3_ind C T A (\lambda (_:
+C).(\lambda (_: T).(\lambda (_: A).(eq K (Bind Abst) (Bind Abst))))) (\lambda
+(c3: C).(\lambda (w: T).(\lambda (_: A).(eq C x0 (CHead c3 (Bind Abbr) w)))))
+(\lambda (c3: C).(\lambda (_: T).(\lambda (_: A).(csubc g x c3)))) (\lambda
+(_: C).(\lambda (_: T).(\lambda (a1: A).(sc3 g (asucc g a1) x (lift1 (ptrans
+is i) u))))) (\lambda (c3: C).(\lambda (w: T).(\lambda (a1: A).(sc3 g a1 c3
+w)))) (sc3 g a0 c2 (lift1 is (TLRef i))) (\lambda (x1: C).(\lambda (x2:
+T).(\lambda (x3: A).(\lambda (_: (eq K (Bind Abst) (Bind Abst))).(\lambda
+(H15: (eq C x0 (CHead x1 (Bind Abbr) x2))).(\lambda (_: (csubc g x
+x1)).(\lambda (H17: (sc3 g (asucc g x3) x (lift1 (ptrans is i) u))).(\lambda
+(H18: (sc3 g x3 x1 x2)).(let H19 \def (eq_ind C x0 (\lambda (c0: C).(getl
+(trans is i) c2 c0)) H10 (CHead x1 (Bind Abbr) x2) H15) in (let H_y \def
+(sc3_abbr g a0 TNil) in (eq_ind_r T (TLRef (trans is i)) (\lambda (t0:
+T).(sc3 g a0 c2 t0)) (H_y (trans is i) x1 x2 c2 (let H_y0 \def (arity_lift1 g
+(asucc g a0) d (ptrans is i) x u H7 H1) in (let H_y1 \def (sc3_arity_gen g x
+(lift1 (ptrans is i) u) (asucc g x3) H17) in (sc3_repl g x3 c2 (lift (S
+(trans is i)) O x2) (sc3_lift g x3 x1 x2 H18 c2 (S (trans is i)) O (getl_drop
+Abbr c2 x1 x2 (trans is i) H19)) a0 (asucc_inj g x3 a0 (arity_mono g x (lift1
+(ptrans is i) u) (asucc g x3) H_y1 (asucc g a0) H_y0))))) H19) (lift1 is
+(TLRef i)) (lift1_lref is i)))))))))))) H13)) H12)))))) H9))))))
+H6))))))))))))))))) (\lambda (b: B).(\lambda (H0: (not (eq B b
+Abst))).(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity
+g c u a1)).(\lambda (H2: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1
+c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is
+u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
+(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (d1: C).(\forall (is:
+PList).((drop1 is d1 (CHead c (Bind b) u)) \to (\forall (c2: C).((csubc g d1
+c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is:
+PList).(\lambda (H5: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H6: (csubc g
+d1 c2)).(let H_y \def (sc3_bind g b H0 a1 a2 TNil) in (eq_ind_r T (THead
+(Bind b) (lift1 is u) (lift1 (Ss is) t0)) (\lambda (t1: T).(sc3 g a2 c2 t1))
+(H_y c2 (lift1 is u) (lift1 (Ss is) t0) (H4 (CHead d1 (Bind b) (lift1 is u))
+(Ss is) (drop1_skip_bind b c is d1 u H5) (CHead c2 (Bind b) (lift1 is u))
+(csubc_head g d1 c2 H6 (Bind b) (lift1 is u))) (H2 d1 is H5 c2 H6)) (lift1 is
+(THead (Bind b) u t0)) (lift1_bind b is u t0))))))))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c u (asucc g
+a1))).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (asucc g a1) c2 (lift1 is
+u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (H2: (arity g (CHead c
+(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (d1: C).(\forall (is:
+PList).((drop1 is d1 (CHead c (Bind Abst) u)) \to (\forall (c2: C).((csubc g
+d1 c2) \to (sc3 g a2 c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is:
+PList).(\lambda (H4: (drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g
+d1 c2)).(eq_ind_r T (THead (Bind Abst) (lift1 is u) (lift1 (Ss is) t0))
+(\lambda (t1: T).(land (arity g c2 t1 (AHead a1 a2)) (\forall (d: C).(\forall
+(w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g
+a2 d (THead (Flat Appl) w (lift1 is0 t1)))))))))) (conj (arity g c2 (THead
+(Bind Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2)) (\forall (d:
+C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d
+c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (THead (Bind Abst) (lift1
+is u) (lift1 (Ss is) t0)))))))))) (csubc_arity_conf g d1 c2 H5 (THead (Bind
+Abst) (lift1 is u) (lift1 (Ss is) t0)) (AHead a1 a2) (arity_head g d1 (lift1
+is u) a1 (arity_lift1 g (asucc g a1) c is d1 u H4 H0) (lift1 (Ss is) t0) a2
+(arity_lift1 g a2 (CHead c (Bind Abst) u) (Ss is) (CHead d1 (Bind Abst)
+(lift1 is u)) t0 (drop1_skip_bind Abst c is d1 u H4) H2))) (\lambda (d:
+C).(\lambda (w: T).(\lambda (H6: (sc3 g a1 d w)).(\lambda (is0:
+PList).(\lambda (H7: (drop1 is0 d c2)).(eq_ind_r T (THead (Bind Abst) (lift1
+is0 (lift1 is u)) (lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (t1: T).(sc3
+g a2 d (THead (Flat Appl) w t1))) (let H8 \def (sc3_appl g a1 a2 TNil) in (H8
+d w (lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_y \def (sc3_bind g Abbr
+(\lambda (H9: (eq B Abbr Abst)).(not_abbr_abst H9)) a1 a2 TNil) in (H_y d w
+(lift1 (Ss is0) (lift1 (Ss is) t0)) (let H_x \def (csubc_drop1_conf_rev g is0
+d c2 H7 d1 H5) in (let H9 \def H_x in (ex2_ind C (\lambda (c3: C).(drop1 is0
+c3 d1)) (\lambda (c3: C).(csubc g c3 d)) (sc3 g a2 (CHead d (Bind Abbr) w)
+(lift1 (Ss is0) (lift1 (Ss is) t0))) (\lambda (x: C).(\lambda (H10: (drop1
+is0 x d1)).(\lambda (H11: (csubc g x d)).(eq_ind_r T (lift1 (papp (Ss is0)
+(Ss is)) t0) (\lambda (t1: T).(sc3 g a2 (CHead d (Bind Abbr) w) t1))
+(eq_ind_r PList (Ss (papp is0 is)) (\lambda (p: PList).(sc3 g a2 (CHead d
+(Bind Abbr) w) (lift1 p t0))) (H3 (CHead x (Bind Abst) (lift1 (papp is0 is)
+u)) (Ss (papp is0 is)) (drop1_skip_bind Abst c (papp is0 is) x u (drop1_trans
+is0 x d1 H10 is c H4)) (CHead d (Bind Abbr) w) (csubc_abst g x d H11 (lift1
+(papp is0 is) u) a1 (H1 x (papp is0 is) (drop1_trans is0 x d1 H10 is c H4) x
+(csubc_refl g x)) w H6)) (papp (Ss is0) (Ss is)) (papp_ss is0 is)) (lift1 (Ss
+is0) (lift1 (Ss is) t0)) (lift1_lift1 (Ss is0) (Ss is) t0))))) H9))) H6)) H6
+(lift1 is0 (lift1 is u)) (sc3_lift1 g c2 (asucc g a1) is0 d (lift1 is u) (H1
+d1 is H4 c2 H5) H7))) (lift1 is0 (THead (Bind Abst) (lift1 is u) (lift1 (Ss
+is) t0))) (lift1_bind Abst is0 (lift1 is u) (lift1 (Ss is) t0))))))))) (lift1
+is (THead (Bind Abst) u t0)) (lift1_bind Abst is u t0))))))))))))))))
+(\lambda (c: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u
+a1)).(\lambda (H1: ((\forall (d1: C).(\forall (is: PList).((drop1 is d1 c)
+\to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a1 c2 (lift1 is
+u))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0
+(AHead a1 a2))).(\lambda (H3: ((\forall (d1: C).(\forall (is: PList).((drop1
+is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g (AHead a1 a2) c2
+(lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4:
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y
+\def (H1 d1 is H4 c2 H5) in (let H_y0 \def (H3 d1 is H4 c2 H5) in (let H6
+\def H_y0 in (and_ind (arity g c2 (lift1 is t0) (AHead a1 a2)) (\forall (d:
+C).(\forall (w: T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d
+c2) \to (sc3 g a2 d (THead (Flat Appl) w (lift1 is0 (lift1 is t0)))))))))
+(sc3 g a2 c2 (lift1 is (THead (Flat Appl) u t0))) (\lambda (_: (arity g c2
+(lift1 is t0) (AHead a1 a2))).(\lambda (H8: ((\forall (d: C).(\forall (w:
+T).((sc3 g a1 d w) \to (\forall (is0: PList).((drop1 is0 d c2) \to (sc3 g a2
+d (THead (Flat Appl) w (lift1 is0 (lift1 is t0))))))))))).(let H_y1 \def (H8
+c2 (lift1 is u) H_y PNil) in (eq_ind_r T (THead (Flat Appl) (lift1 is u)
+(lift1 is t0)) (\lambda (t1: T).(sc3 g a2 c2 t1)) (H_y1 (drop1_nil c2))
+(lift1 is (THead (Flat Appl) u t0)) (lift1_flat Appl is u t0)))))
+H6)))))))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (a0:
+A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1: ((\forall (d1:
+C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1
+c2) \to (sc3 g (asucc g a0) c2 (lift1 is u))))))))).(\lambda (t0: T).(\lambda
+(_: (arity g c t0 a0)).(\lambda (H3: ((\forall (d1: C).(\forall (is:
+PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g d1 c2) \to (sc3 g a0
+c2 (lift1 is t0))))))))).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H4:
+(drop1 is d1 c)).(\lambda (c2: C).(\lambda (H5: (csubc g d1 c2)).(let H_y
+\def (sc3_cast g a0 TNil) in (eq_ind_r T (THead (Flat Cast) (lift1 is u)
+(lift1 is t0)) (\lambda (t1: T).(sc3 g a0 c2 t1)) (H_y c2 (lift1 is u) (H1 d1
+is H4 c2 H5) (lift1 is t0) (H3 d1 is H4 c2 H5)) (lift1 is (THead (Flat Cast)
+u t0)) (lift1_flat Cast is u t0)))))))))))))))) (\lambda (c: C).(\lambda (t0:
+T).(\lambda (a1: A).(\lambda (_: (arity g c t0 a1)).(\lambda (H1: ((\forall
+(d1: C).(\forall (is: PList).((drop1 is d1 c) \to (\forall (c2: C).((csubc g
+d1 c2) \to (sc3 g a1 c2 (lift1 is t0))))))))).(\lambda (a2: A).(\lambda (H2:
+(leq g a1 a2)).(\lambda (d1: C).(\lambda (is: PList).(\lambda (H3: (drop1 is
+d1 c)).(\lambda (c2: C).(\lambda (H4: (csubc g d1 c2)).(sc3_repl g a1 c2
+(lift1 is t0) (H1 d1 is H3 c2 H4) a2 H2))))))))))))) c1 t a H))))).
theorem sc3_arity:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t
(* This file was automatically generated: do not edit *********************)
-include "sn3/defs.ma".
+include "LambdaDelta-1/sn3/defs.ma".
-include "arity/defs.ma".
+include "LambdaDelta-1/arity/defs.ma".
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
definition sc3:
G \to (A \to (C \to (T \to Prop)))
(* This file was automatically generated: do not edit *********************)
-include "sc3/defs.ma".
+include "LambdaDelta-1/sc3/defs.ma".
-include "sn3/lift1.ma".
+include "LambdaDelta-1/sn3/lift1.ma".
-include "nf2/lift1.ma".
+include "LambdaDelta-1/nf2/lift1.ma".
-include "csuba/arity.ma".
+include "LambdaDelta-1/csuba/arity.ma".
-include "arity/lift1.ma".
+include "LambdaDelta-1/arity/lift1.ma".
-include "arity/aprem.ma".
+include "LambdaDelta-1/arity/aprem.ma".
-include "llt/props.ma".
+include "LambdaDelta-1/llt/props.ma".
-include "drop1/getl.ma".
+include "LambdaDelta-1/drop1/getl.ma".
-include "drop1/props.ma".
+include "LambdaDelta-1/drop1/props.ma".
-include "lift1/props.ma".
+include "LambdaDelta-1/lift1/props.ma".
theorem sc3_arity_gen:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((sc3 g a c
(* This file was automatically generated: do not edit *********************)
-include "pr3/defs.ma".
+include "LambdaDelta-1/pr3/defs.ma".
inductive sn3 (c: C): T \to Prop \def
| sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall
(* This file was automatically generated: do not edit *********************)
-include "sn3/defs.ma".
+include "LambdaDelta-1/sn3/defs.ma".
-include "pr3/props.ma".
+include "LambdaDelta-1/pr3/props.ma".
theorem sn3_gen_bind:
\forall (b: B).(\forall (c: C).(\forall (u: T).(\forall (t: T).((sn3 c
(* This file was automatically generated: do not edit *********************)
-include "sn3/props.ma".
+include "LambdaDelta-1/sn3/props.ma".
-include "drop1/defs.ma".
+include "LambdaDelta-1/drop1/defs.ma".
-include "lift1/fwd.ma".
+include "LambdaDelta-1/lift1/fwd.ma".
theorem sns3_lifts1:
\forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to
(* This file was automatically generated: do not edit *********************)
-include "sn3/defs.ma".
+include "LambdaDelta-1/sn3/defs.ma".
-include "nf2/dec.ma".
+include "LambdaDelta-1/nf2/dec.ma".
-include "nf2/pr3.ma".
+include "LambdaDelta-1/nf2/pr3.ma".
theorem sn3_nf2:
\forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
(* This file was automatically generated: do not edit *********************)
-include "sn3/nf2.ma".
+include "LambdaDelta-1/sn3/nf2.ma".
-include "sn3/fwd.ma".
+include "LambdaDelta-1/sn3/fwd.ma".
-include "nf2/iso.ma".
+include "LambdaDelta-1/nf2/iso.ma".
-include "pr3/iso.ma".
-
-include "iso/props.ma".
+include "LambdaDelta-1/pr3/iso.ma".
theorem sn3_pr3_trans:
\forall (c: C).(\forall (t1: T).((sn3 c t1) \to (\forall (t2: T).((pr3 c t1
(pr3_pr2 (CHead c (Bind b) v1) t1 t2 (pr2_change b H c v2 t1 t2 H4
v1)))))))))) t H0))))))).
+theorem sn3_gen_def:
+ \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
+(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
+\def
+ \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
+(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef
+i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
+(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
+i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
+Abbr c d v i H))))))).
+
+theorem sn3_cdelta:
+ \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
+(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
+C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
+\def
+ \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
+T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
+\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
+C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
+(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
+(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
+(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
+\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
+C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
+v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
+(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
+(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
+c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
+H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
+(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
+(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
+v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
+C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
+v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
+(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
+c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
+(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
+C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
+(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
+(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3
+(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
+(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
+(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
+H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
+t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
+c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
+(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
+in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_:
+(sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
+H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
+(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
+C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
+(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
+(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
+C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
+v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
+(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
+(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
+H0)))))).
+
theorem sn3_cpr3_trans:
\forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr3 c u1 u2) \to (\forall
(k: K).(\forall (t: T).((sn3 (CHead c k u1) t) \to (sn3 (CHead c k u2)
x0 H12 (Bind b) x1 (H3 x1 H18 H13)))) H17)))) H15))) t3 H11))))))) H10))
(\lambda (H10: (pr3 (CHead c (Bind b) t1) t2 (lift (S O) O
t3))).(sn3_gen_lift (CHead c (Bind b) t1) t3 (S O) O (sn3_pr3_trans (CHead c
-(Bind b) t1) t2 (sn3_pr2_intro (CHead c (Bind b) t1) t2 (\lambda (t0:
-T).(\lambda (H11: (((eq T t2 t0) \to (\forall (P: Prop).P)))).(\lambda (H12:
-(pr2 (CHead c (Bind b) t1) t2 t0)).(H3 t0 H11 (pr3_pr2 (CHead c (Bind b) t1)
-t2 t0 H12)))))) (lift (S O) O t3) H10) c (drop_drop (Bind b) O c c (drop_refl
-c) t1))) H9)))) H7)))))))))) t H2)))))) u H)))).
+(Bind b) t1) t2 (sn3_sing (CHead c (Bind b) t1) t2 H3) (lift (S O) O t3) H10)
+c (drop_drop (Bind b) O c c (drop_refl c) t1))) H9)))) H7)))))))))) t
+H2)))))) u H)))).
theorem sn3_beta:
\forall (c: C).(\forall (v: T).(\forall (t: T).((sn3 c (THead (Bind Abbr) v
(sn3 c (lift h i t)) (sns3 c (lifts h i t0)) (sn3_lift d t H3 c h i H) (H0
H4)))) H2)))))) ts)))))).
-theorem sn3_gen_def:
- \forall (c: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i c
-(CHead d (Bind Abbr) v)) \to ((sn3 c (TLRef i)) \to (sn3 d v))))))
-\def
- \lambda (c: C).(\lambda (d: C).(\lambda (v: T).(\lambda (i: nat).(\lambda
-(H: (getl i c (CHead d (Bind Abbr) v))).(\lambda (H0: (sn3 c (TLRef
-i))).(sn3_gen_lift c v (S i) O (sn3_pr3_trans c (TLRef i) H0 (lift (S i) O v)
-(pr3_pr2 c (TLRef i) (lift (S i) O v) (pr2_delta c d v i H (TLRef i) (TLRef
-i) (pr0_refl (TLRef i)) (lift (S i) O v) (subst0_lref v i)))) d (getl_drop
-Abbr c d v i H))))))).
-
-theorem sn3_cdelta:
- \forall (v: T).(\forall (t: T).(\forall (i: nat).(((\forall (w: T).(ex T
-(\lambda (u: T).(subst0 i w t u))))) \to (\forall (c: C).(\forall (d:
-C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to (sn3 d v))))))))
-\def
- \lambda (v: T).(\lambda (t: T).(\lambda (i: nat).(\lambda (H: ((\forall (w:
-T).(ex T (\lambda (u: T).(subst0 i w t u)))))).(let H_x \def (H v) in (let H0
-\def H_x in (ex_ind T (\lambda (u: T).(subst0 i v t u)) (\forall (c:
-C).(\forall (d: C).((getl i c (CHead d (Bind Abbr) v)) \to ((sn3 c t) \to
-(sn3 d v))))) (\lambda (x: T).(\lambda (H1: (subst0 i v t x)).(subst0_ind
-(\lambda (n: nat).(\lambda (t0: T).(\lambda (t1: T).(\lambda (_: T).(\forall
-(c: C).(\forall (d: C).((getl n c (CHead d (Bind Abbr) t0)) \to ((sn3 c t1)
-\to (sn3 d t0))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (c:
-C).(\lambda (d: C).(\lambda (H2: (getl i0 c (CHead d (Bind Abbr)
-v0))).(\lambda (H3: (sn3 c (TLRef i0))).(sn3_gen_def c d v0 i0 H2 H3)))))))
-(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
-nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
-(sn3 d v0))))))).(\lambda (t0: T).(\lambda (k: K).(\lambda (c: C).(\lambda
-(d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr) v0))).(\lambda (H5: (sn3
-c (THead k u1 t0))).(let H_y \def (sn3_gen_head k c u1 t0 H5) in (H3 c d H4
-H_y)))))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2: T).(\lambda
-(t1: T).(\lambda (i0: nat).(\lambda (H2: (subst0 (s k i0) v0 t1 t2)).(\lambda
-(H3: ((\forall (c: C).(\forall (d: C).((getl (s k i0) c (CHead d (Bind Abbr)
-v0)) \to ((sn3 c t1) \to (sn3 d v0))))))).(\lambda (u: T).(\lambda (c:
-C).(\lambda (d: C).(\lambda (H4: (getl i0 c (CHead d (Bind Abbr)
-v0))).(\lambda (H5: (sn3 c (THead k u t1))).(K_ind (\lambda (k0: K).((subst0
-(s k0 i0) v0 t1 t2) \to (((\forall (c0: C).(\forall (d0: C).((getl (s k0 i0)
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0)))))) \to ((sn3
-c (THead k0 u t1)) \to (sn3 d v0))))) (\lambda (b: B).(\lambda (_: (subst0 (s
-(Bind b) i0) v0 t1 t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0:
-C).((getl (s (Bind b) i0) c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to
-(sn3 d0 v0))))))).(\lambda (H8: (sn3 c (THead (Bind b) u t1))).(let H_x0 \def
-(sn3_gen_bind b c u t1 H8) in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3
-(CHead c (Bind b) u) t1) (sn3 d v0) (\lambda (_: (sn3 c u)).(\lambda (H11:
-(sn3 (CHead c (Bind b) u) t1)).(H7 (CHead c (Bind b) u) d (getl_clear_bind b
-(CHead c (Bind b) u) c u (clear_bind b c u) (CHead d (Bind Abbr) v0) i0 H4)
-H11))) H9))))))) (\lambda (f: F).(\lambda (_: (subst0 (s (Flat f) i0) v0 t1
-t2)).(\lambda (H7: ((\forall (c0: C).(\forall (d0: C).((getl (s (Flat f) i0)
-c0 (CHead d0 (Bind Abbr) v0)) \to ((sn3 c0 t1) \to (sn3 d0 v0))))))).(\lambda
-(H8: (sn3 c (THead (Flat f) u t1))).(let H_x0 \def (sn3_gen_flat f c u t1 H8)
-in (let H9 \def H_x0 in (and_ind (sn3 c u) (sn3 c t1) (sn3 d v0) (\lambda (_:
-(sn3 c u)).(\lambda (H11: (sn3 c t1)).(H7 c d H4 H11))) H9))))))) k H2 H3
-H5))))))))))))) (\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda
-(i0: nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (H3: ((\forall (c:
-C).(\forall (d: C).((getl i0 c (CHead d (Bind Abbr) v0)) \to ((sn3 c u1) \to
-(sn3 d v0))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: ((\forall (c: C).(\forall (d:
-C).((getl (s k i0) c (CHead d (Bind Abbr) v0)) \to ((sn3 c t1) \to (sn3 d
-v0))))))).(\lambda (c: C).(\lambda (d: C).(\lambda (H6: (getl i0 c (CHead d
-(Bind Abbr) v0))).(\lambda (H7: (sn3 c (THead k u1 t1))).(let H_y \def
-(sn3_gen_head k c u1 t1 H7) in (H3 c d H6 H_y))))))))))))))))) i v t x H1)))
-H0)))))).
-
(* This file was automatically generated: do not edit *********************)
-include "theory.ma".
+include "LambdaDelta-1/theory.ma".
definition cbk:
C \to nat
| tys3_cons: \forall (t: T).(\forall (u: T).((ty3 g c t u) \to (\forall (ts:
TList).((tys3 g c ts u) \to (tys3 g c (TCons t ts) u))))).
-theorem tys3_inv_coq:
- \forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (u: T).(\forall
-(P: ((G \to (C \to (TList \to (T \to Prop)))))).((((tys3 g c ts u) \to
-(\forall (u0: T).(\forall (u1: T).((eq TList TNil ts) \to ((eq T u0 u) \to
-((ty3 g c u0 u1) \to (P g c ts u)))))))) \to ((((tys3 g c ts u) \to (\forall
-(t: T).(\forall (u0: T).(\forall (ts0: TList).((eq TList (TCons t ts0) ts)
-\to ((eq T u0 u) \to ((ty3 g c t u0) \to ((tys3 g c ts0 u0) \to (P g c ts
-u)))))))))) \to ((tys3 g c ts u) \to (P g c ts u))))))))
-\def
- \lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (u: T).(\lambda
-(P: ((G \to (C \to (TList \to (T \to Prop)))))).(\lambda (H: (((tys3 g c ts
-u) \to (\forall (u0: T).(\forall (u1: T).((eq TList TNil ts) \to ((eq T u0 u)
-\to ((ty3 g c u0 u1) \to (P g c ts u))))))))).(\lambda (H0: (((tys3 g c ts u)
-\to (\forall (t: T).(\forall (u0: T).(\forall (ts0: TList).((eq TList (TCons
-t ts0) ts) \to ((eq T u0 u) \to ((ty3 g c t u0) \to ((tys3 g c ts0 u0) \to (P
-g c ts u))))))))))).(\lambda (H1: (tys3 g c ts u)).(let H2 \def (match H1 in
-tys3 return (\lambda (t: TList).(\lambda (t0: T).(\lambda (_: (tys3 ? ? t
-t0)).((eq TList t ts) \to ((eq T t0 u) \to (P g c ts u)))))) with [(tys3_nil
-u0 u1 H2) \Rightarrow (\lambda (H3: (eq TList TNil ts)).(\lambda (H4: (eq T
-u0 u)).(H H1 u0 u1 H3 H4 H2))) | (tys3_cons t u0 H2 ts0 H3) \Rightarrow
-(\lambda (H4: (eq TList (TCons t ts0) ts)).(\lambda (H5: (eq T u0 u)).(H0 H1
-t u0 ts0 H4 H5 H2 H3)))]) in (H2 (refl_equal TList ts) (refl_equal T
-u)))))))))).
-
theorem tys3_gen_nil:
\forall (g: G).(\forall (c: C).(\forall (u: T).((tys3 g c TNil u) \to (ex T
(\lambda (u0: T).(ty3 g c u u0))))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (tys3 g c TNil
-u)).(tys3_inv_coq g c TNil u (\lambda (g0: G).(\lambda (c0: C).(\lambda (_:
-TList).(\lambda (t0: T).(ex T (\lambda (u0: T).(ty3 g0 c0 t0 u0)))))))
-(\lambda (_: (tys3 g c TNil u)).(\lambda (u0: T).(\lambda (u1: T).(\lambda
-(_: (eq TList TNil TNil)).(\lambda (H2: (eq T u0 u)).(\lambda (H3: (ty3 g c
-u0 u1)).(let H4 \def (eq_ind T u0 (\lambda (t: T).(ty3 g c t u1)) H3 u H2) in
-(ex_intro T (\lambda (u2: T).(ty3 g c u u2)) u1 H4)))))))) (\lambda (_: (tys3
-g c TNil u)).(\lambda (t: T).(\lambda (u0: T).(\lambda (ts0: TList).(\lambda
-(H2: (eq TList (TCons t ts0) TNil)).(\lambda (H3: (eq T u0 u)).(\lambda (H1:
-(ty3 g c t u0)).(\lambda (H4: (tys3 g c ts0 u0)).(let H5 \def (eq_ind T u0
-(\lambda (t0: T).(tys3 g c ts0 t0)) H4 u H3) in (let H6 \def (eq_ind T u0
-(\lambda (t0: T).(ty3 g c t t0)) H1 u H3) in (let H7 \def (eq_ind TList
-(TCons t ts0) (\lambda (ee: TList).(match ee in TList return (\lambda (_:
-TList).Prop) with [TNil \Rightarrow False | (TCons _ _) \Rightarrow True])) I
-TNil H2) in (False_ind (ex T (\lambda (u1: T).(ty3 g c u u1))) H7))))))))))))
-H)))).
+u)).(insert_eq TList TNil (\lambda (t: TList).(tys3 g c t u)) (\lambda (_:
+TList).(ex T (\lambda (u0: T).(ty3 g c u u0)))) (\lambda (y: TList).(\lambda
+(H0: (tys3 g c y u)).(tys3_ind g c (\lambda (t: TList).(\lambda (t0: T).((eq
+TList t TNil) \to (ex T (\lambda (u0: T).(ty3 g c t0 u0)))))) (\lambda (u0:
+T).(\lambda (u1: T).(\lambda (H1: (ty3 g c u0 u1)).(\lambda (_: (eq TList
+TNil TNil)).(ex_intro T (\lambda (u2: T).(ty3 g c u0 u2)) u1 H1))))) (\lambda
+(t: T).(\lambda (u0: T).(\lambda (_: (ty3 g c t u0)).(\lambda (ts:
+TList).(\lambda (_: (tys3 g c ts u0)).(\lambda (_: (((eq TList ts TNil) \to
+(ex T (\lambda (u1: T).(ty3 g c u0 u1)))))).(\lambda (H4: (eq TList (TCons t
+ts) TNil)).(let H5 \def (eq_ind TList (TCons t ts) (\lambda (ee:
+TList).(match ee in TList return (\lambda (_: TList).Prop) with [TNil
+\Rightarrow False | (TCons _ _) \Rightarrow True])) I TNil H4) in (False_ind
+(ex T (\lambda (u1: T).(ty3 g c u0 u1))) H5))))))))) y u H0))) H)))).
theorem tys3_gen_cons:
\forall (g: G).(\forall (c: C).(\forall (ts: TList).(\forall (t: T).(\forall
u)))))))
\def
\lambda (g: G).(\lambda (c: C).(\lambda (ts: TList).(\lambda (t: T).(\lambda
-(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(tys3_inv_coq g c (TCons t ts)
-u (\lambda (g0: G).(\lambda (c0: C).(\lambda (_: TList).(\lambda (t1:
-T).(land (ty3 g0 c0 t t1) (tys3 g0 c0 ts t1)))))) (\lambda (_: (tys3 g c
-(TCons t ts) u)).(\lambda (u0: T).(\lambda (u1: T).(\lambda (H1: (eq TList
-TNil (TCons t ts))).(\lambda (H2: (eq T u0 u)).(\lambda (H3: (ty3 g c u0
-u1)).(let H4 \def (eq_ind T u0 (\lambda (t0: T).(ty3 g c t0 u1)) H3 u H2) in
-(let H5 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList
+(u: T).(\lambda (H: (tys3 g c (TCons t ts) u)).(insert_eq TList (TCons t ts)
+(\lambda (t0: TList).(tys3 g c t0 u)) (\lambda (_: TList).(land (ty3 g c t u)
+(tys3 g c ts u))) (\lambda (y: TList).(\lambda (H0: (tys3 g c y u)).(tys3_ind
+g c (\lambda (t0: TList).(\lambda (t1: T).((eq TList t0 (TCons t ts)) \to
+(land (ty3 g c t t1) (tys3 g c ts t1))))) (\lambda (u0: T).(\lambda (u1:
+T).(\lambda (_: (ty3 g c u0 u1)).(\lambda (H2: (eq TList TNil (TCons t
+ts))).(let H3 \def (eq_ind TList TNil (\lambda (ee: TList).(match ee in TList
return (\lambda (_: TList).Prop) with [TNil \Rightarrow True | (TCons _ _)
-\Rightarrow False])) I (TCons t ts) H1) in (False_ind (land (ty3 g c t u)
-(tys3 g c ts u)) H5))))))))) (\lambda (_: (tys3 g c (TCons t ts) u)).(\lambda
-(t0: T).(\lambda (u0: T).(\lambda (ts0: TList).(\lambda (H2: (eq TList (TCons
-t0 ts0) (TCons t ts))).(\lambda (H3: (eq T u0 u)).(\lambda (H1: (ty3 g c t0
-u0)).(\lambda (H4: (tys3 g c ts0 u0)).(let H5 \def (eq_ind T u0 (\lambda (t1:
-T).(tys3 g c ts0 t1)) H4 u H3) in (let H6 \def (eq_ind T u0 (\lambda (t1:
-T).(ty3 g c t0 t1)) H1 u H3) in (let H7 \def (f_equal TList T (\lambda (e:
-TList).(match e in TList return (\lambda (_: TList).T) with [TNil \Rightarrow
-t0 | (TCons t1 _) \Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H2) in ((let
-H8 \def (f_equal TList TList (\lambda (e: TList).(match e in TList return
-(\lambda (_: TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1)
-\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H2) in (\lambda (H9: (eq T t0
-t)).(let H10 \def (eq_ind TList ts0 (\lambda (t1: TList).(tys3 g c t1 u)) H5
-ts H8) in (let H11 \def (eq_ind T t0 (\lambda (t1: T).(ty3 g c t1 u)) H6 t
-H9) in (conj (ty3 g c t u) (tys3 g c ts u) H11 H10))))) H7))))))))))))
-H)))))).
+\Rightarrow False])) I (TCons t ts) H2) in (False_ind (land (ty3 g c t u0)
+(tys3 g c ts u0)) H3)))))) (\lambda (t0: T).(\lambda (u0: T).(\lambda (H1:
+(ty3 g c t0 u0)).(\lambda (ts0: TList).(\lambda (H2: (tys3 g c ts0
+u0)).(\lambda (H3: (((eq TList ts0 (TCons t ts)) \to (land (ty3 g c t u0)
+(tys3 g c ts u0))))).(\lambda (H4: (eq TList (TCons t0 ts0) (TCons t
+ts))).(let H5 \def (f_equal TList T (\lambda (e: TList).(match e in TList
+return (\lambda (_: TList).T) with [TNil \Rightarrow t0 | (TCons t1 _)
+\Rightarrow t1])) (TCons t0 ts0) (TCons t ts) H4) in ((let H6 \def (f_equal
+TList TList (\lambda (e: TList).(match e in TList return (\lambda (_:
+TList).TList) with [TNil \Rightarrow ts0 | (TCons _ t1) \Rightarrow t1]))
+(TCons t0 ts0) (TCons t ts) H4) in (\lambda (H7: (eq T t0 t)).(let H8 \def
+(eq_ind TList ts0 (\lambda (t1: TList).((eq TList t1 (TCons t ts)) \to (land
+(ty3 g c t u0) (tys3 g c ts u0)))) H3 ts H6) in (let H9 \def (eq_ind TList
+ts0 (\lambda (t1: TList).(tys3 g c t1 u0)) H2 ts H6) in (let H10 \def (eq_ind
+T t0 (\lambda (t1: T).(ty3 g c t1 u0)) H1 t H7) in (conj (ty3 g c t u0) (tys3
+g c ts u0) H10 H9)))))) H5))))))))) y u H0))) H)))))).
theorem ty3_getl_subst0:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
(* This file was automatically generated: do not edit *********************)
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
-include "lift/props.ma".
+include "LambdaDelta-1/lift/props.ma".
theorem dnf_dec2:
\forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v:
(* This file was automatically generated: do not edit *********************)
-include "lift/defs.ma".
+include "LambdaDelta-1/lift/defs.ma".
inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
| subst0_lref: \forall (v: T).(\forall (i: nat).(subst0 i v (TLRef i) (lift
(* This file was automatically generated: do not edit *********************)
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
-include "lift/props.ma".
-
-theorem subst0_inv_coq:
- \forall (i: nat).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).(\forall
-(P: ((nat \to (T \to (T \to (T \to Prop)))))).((((subst0 i v t1 t2) \to
-(\forall (v0: T).(\forall (i0: nat).((eq nat i0 i) \to ((eq T v0 v) \to ((eq
-T (TLRef i0) t1) \to ((eq T (lift (S i0) O v0) t2) \to (P i v t1 t2)))))))))
-\to ((((subst0 i v t1 t2) \to (\forall (v0: T).(\forall (u2: T).(\forall (u1:
-T).(\forall (i0: nat).(\forall (t: T).(\forall (k: K).((eq nat i0 i) \to ((eq
-T v0 v) \to ((eq T (THead k u1 t) t1) \to ((eq T (THead k u2 t) t2) \to
-((subst0 i0 v0 u1 u2) \to (P i v t1 t2)))))))))))))) \to ((((subst0 i v t1
-t2) \to (\forall (k: K).(\forall (v0: T).(\forall (t0: T).(\forall (t3:
-T).(\forall (i0: nat).(\forall (u: T).((eq nat i0 i) \to ((eq T v0 v) \to
-((eq T (THead k u t3) t1) \to ((eq T (THead k u t0) t2) \to ((subst0 (s k i0)
-v0 t3 t0) \to (P i v t1 t2)))))))))))))) \to ((((subst0 i v t1 t2) \to
-(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).(\forall (i0:
-nat).(\forall (k: K).(\forall (t0: T).(\forall (t3: T).((eq nat i0 i) \to
-((eq T v0 v) \to ((eq T (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2)
-\to ((subst0 i0 v0 u1 u2) \to ((subst0 (s k i0) v0 t0 t3) \to (P i v t1
-t2)))))))))))))))) \to ((subst0 i v t1 t2) \to (P i v t1 t2))))))))))
-\def
- \lambda (i: nat).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(P: ((nat \to (T \to (T \to (T \to Prop)))))).(\lambda (H: (((subst0 i v t1
-t2) \to (\forall (v0: T).(\forall (i0: nat).((eq nat i0 i) \to ((eq T v0 v)
-\to ((eq T (TLRef i0) t1) \to ((eq T (lift (S i0) O v0) t2) \to (P i v t1
-t2)))))))))).(\lambda (H0: (((subst0 i v t1 t2) \to (\forall (v0: T).(\forall
-(u2: T).(\forall (u1: T).(\forall (i0: nat).(\forall (t: T).(\forall (k:
-K).((eq nat i0 i) \to ((eq T v0 v) \to ((eq T (THead k u1 t) t1) \to ((eq T
-(THead k u2 t) t2) \to ((subst0 i0 v0 u1 u2) \to (P i v t1
-t2))))))))))))))).(\lambda (H1: (((subst0 i v t1 t2) \to (\forall (k:
-K).(\forall (v0: T).(\forall (t0: T).(\forall (t3: T).(\forall (i0:
-nat).(\forall (u: T).((eq nat i0 i) \to ((eq T v0 v) \to ((eq T (THead k u
-t3) t1) \to ((eq T (THead k u t0) t2) \to ((subst0 (s k i0) v0 t3 t0) \to (P
-i v t1 t2))))))))))))))).(\lambda (H2: (((subst0 i v t1 t2) \to (\forall (v0:
-T).(\forall (u1: T).(\forall (u2: T).(\forall (i0: nat).(\forall (k:
-K).(\forall (t0: T).(\forall (t3: T).((eq nat i0 i) \to ((eq T v0 v) \to ((eq
-T (THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) \to ((subst0 i0 v0 u1
-u2) \to ((subst0 (s k i0) v0 t0 t3) \to (P i v t1
-t2))))))))))))))))).(\lambda (H3: (subst0 i v t1 t2)).(let H4 \def (match H3
-in subst0 return (\lambda (n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda
-(t3: T).(\lambda (_: (subst0 n t t0 t3)).((eq nat n i) \to ((eq T t v) \to
-((eq T t0 t1) \to ((eq T t3 t2) \to (P i v t1 t2)))))))))) with [(subst0_lref
-v0 i0) \Rightarrow (\lambda (H4: (eq nat i0 i)).(\lambda (H5: (eq T v0
-v)).(\lambda (H6: (eq T (TLRef i0) t1)).(\lambda (H7: (eq T (lift (S i0) O
-v0) t2)).(H H3 v0 i0 H4 H5 H6 H7))))) | (subst0_fst v0 u2 u1 i0 H4 t k)
-\Rightarrow (\lambda (H5: (eq nat i0 i)).(\lambda (H6: (eq T v0 v)).(\lambda
-(H7: (eq T (THead k u1 t) t1)).(\lambda (H8: (eq T (THead k u2 t) t2)).(H0 H3
-v0 u2 u1 i0 t k H5 H6 H7 H8 H4))))) | (subst0_snd k v0 t0 t3 i0 H4 u)
-\Rightarrow (\lambda (H5: (eq nat i0 i)).(\lambda (H6: (eq T v0 v)).(\lambda
-(H7: (eq T (THead k u t3) t1)).(\lambda (H8: (eq T (THead k u t0) t2)).(H1 H3
-k v0 t0 t3 i0 u H5 H6 H7 H8 H4))))) | (subst0_both v0 u1 u2 i0 H4 k t0 t3 H5)
-\Rightarrow (\lambda (H6: (eq nat i0 i)).(\lambda (H7: (eq T v0 v)).(\lambda
-(H8: (eq T (THead k u1 t0) t1)).(\lambda (H9: (eq T (THead k u2 t3) t2)).(H2
-H3 v0 u1 u2 i0 k t0 t3 H6 H7 H8 H9 H4 H5)))))]) in (H4 (refl_equal nat i)
-(refl_equal T v) (refl_equal T t1) (refl_equal T t2)))))))))))).
+include "LambdaDelta-1/lift/props.ma".
theorem subst0_gen_sort:
\forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
i v (TSort n) x) \to (\forall (P: Prop).P)))))
\def
\lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(subst0_inv_coq i v (TSort
-n) x (\lambda (_: nat).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).P))))
-(\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: T).(\lambda (i0:
-nat).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3:
-(eq T (TLRef i0) (TSort n))).(\lambda (H4: (eq T (lift (S i0) O v0) x)).(let
-H5 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (lift (S n0) O v0) x)) H4 i
-H1) in (let H6 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (TLRef n0) (TSort
-n))) H3 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(eq T (lift (S i)
-O t) x)) H5 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v
-(TSort n) t)) H0 (lift (S i) O v) H7) in (let H9 \def (eq_ind_r T x (\lambda
-(t: T).(subst0 i v (TSort n) t)) H (lift (S i) O v) H7) in (let H10 \def
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_:
+(H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(insert_eq T (TSort n)
+(\lambda (t: T).(subst0 i v t x)) (\lambda (_: T).P) (\lambda (y: T).(\lambda
+(H0: (subst0 i v y x)).(subst0_ind (\lambda (_: nat).(\lambda (_: T).(\lambda
+(t0: T).(\lambda (_: T).((eq T t0 (TSort n)) \to P))))) (\lambda (_:
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TSort n))).(let H2 \def
+(eq_ind T (TLRef i0) (\lambda (ee: T).(match ee in T return (\lambda (_:
T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
-(THead _ _ _) \Rightarrow False])) I (TSort n) H6) in (False_ind P
-H10)))))))))))))) (\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0:
-T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (t:
-T).(\lambda (k: K).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
-v)).(\lambda (H3: (eq T (THead k u1 t) (TSort n))).(\lambda (H4: (eq T (THead
-k u2 t) x)).(\lambda (H5: (subst0 i0 v0 u1 u2)).(let H6 \def (eq_ind nat i0
-(\lambda (n0: nat).(subst0 n0 v0 u1 u2)) H5 i H1) in (let H7 \def (eq_ind T
-v0 (\lambda (t0: T).(subst0 i t0 u1 u2)) H6 v H2) in (let H8 \def (eq_ind_r T
-x (\lambda (t0: T).(subst0 i v (TSort n) t0)) H0 (THead k u2 t) H4) in (let
-H9 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v (TSort n) t0)) H (THead k
-u2 t) H4) in (let H10 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match
-ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False |
-(TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n)
-H3) in (False_ind P H10)))))))))))))))))) (\lambda (H0: (subst0 i v (TSort n)
-x)).(\lambda (k: K).(\lambda (v0: T).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (i0: nat).(\lambda (u: T).(\lambda (H1: (eq nat i0 i)).(\lambda
-(H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t3) (TSort n))).(\lambda
-(H4: (eq T (THead k u t0) x)).(\lambda (H5: (subst0 (s k i0) v0 t3 t0)).(let
-H6 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t3 t0)) H5 i
-H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) t t3 t0)) H6
-v H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TSort n) t))
-H0 (THead k u t0) H4) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i
-v (TSort n) t)) H (THead k u t0) H4) in (let H10 \def (eq_ind T (THead k u
-t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TSort n) H3) in (False_ind P H10))))))))))))))))))
-(\lambda (H0: (subst0 i v (TSort n) x)).(\lambda (v0: T).(\lambda (u1:
-T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (k: K).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0
-v)).(\lambda (H4: (eq T (THead k u1 t0) (TSort n))).(\lambda (H5: (eq T
-(THead k u2 t3) x)).(\lambda (H1: (subst0 i0 v0 u1 u2)).(\lambda (H6: (subst0
-(s k i0) v0 t0 t3)).(let H7 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s
-k n0) v0 t0 t3)) H6 i H2) in (let H8 \def (eq_ind nat i0 (\lambda (n0:
-nat).(subst0 n0 v0 u1 u2)) H1 i H2) in (let H9 \def (eq_ind T v0 (\lambda (t:
-T).(subst0 (s k i) t t0 t3)) H7 v H3) in (let H10 \def (eq_ind T v0 (\lambda
-(t: T).(subst0 i t u1 u2)) H8 v H3) in (let H11 \def (eq_ind_r T x (\lambda
-(t: T).(subst0 i v (TSort n) t)) H0 (THead k u2 t3) H5) in (let H12 \def
-(eq_ind_r T x (\lambda (t: T).(subst0 i v (TSort n) t)) H (THead k u2 t3) H5)
-in (let H13 \def (eq_ind T (THead k u1 t0) (\lambda (ee: T).(match ee in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H4) in
-(False_ind P H13)))))))))))))))))))))) H)))))).
+(THead _ _ _) \Rightarrow False])) I (TSort n) H1) in (False_ind P H2)))))
+(\lambda (v0: T).(\lambda (u2: T).(\lambda (u1: T).(\lambda (i0:
+nat).(\lambda (_: (subst0 i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n))
+\to P))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T (THead k u1 t)
+(TSort n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef
+_) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in
+(False_ind P H4))))))))))) (\lambda (k: K).(\lambda (v0: T).(\lambda (t2:
+T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0 (s k i0) v0 t1
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (u: T).(\lambda
+(H3: (eq T (THead k u t1) (TSort n))).(let H4 \def (eq_ind T (THead k u t1)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TSort n) H3) in (False_ind P H4))))))))))) (\lambda (v0:
+T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0
+i0 v0 u1 u2)).(\lambda (_: (((eq T u1 (TSort n)) \to P))).(\lambda (k:
+K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (_: (subst0 (s k i0) v0 t1
+t2)).(\lambda (_: (((eq T t1 (TSort n)) \to P))).(\lambda (H5: (eq T (THead k
+u1 t1) (TSort n))).(let H6 \def (eq_ind T (THead k u1 t1) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(TSort n) H5) in (False_ind P H6)))))))))))))) i v y x H0))) H)))))).
theorem subst0_gen_lref:
\forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0
i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
\def
\lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda
-(H: (subst0 i v (TLRef n) x)).(subst0_inv_coq i v (TLRef n) x (\lambda (n0:
-nat).(\lambda (t: T).(\lambda (_: T).(\lambda (t1: T).(land (eq nat n n0) (eq
-T t1 (lift (S n) O t))))))) (\lambda (H0: (subst0 i v (TLRef n) x)).(\lambda
-(v0: T).(\lambda (i0: nat).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T
-v0 v)).(\lambda (H3: (eq T (TLRef i0) (TLRef n))).(\lambda (H4: (eq T (lift
-(S i0) O v0) x)).(let H5 \def (eq_ind nat i0 (\lambda (n0: nat).(eq T (lift
-(S n0) O v0) x)) H4 i H1) in (let H6 \def (eq_ind nat i0 (\lambda (n0:
-nat).(eq T (TLRef n0) (TLRef n))) H3 i H1) in (let H7 \def (eq_ind T v0
-(\lambda (t: T).(eq T (lift (S i) O t) x)) H5 v H2) in (let H8 \def (eq_ind_r
-T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H0 (lift (S i) O v) H7) in (let
-H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H (lift (S i)
-O v) H7) in (eq_ind T (lift (S i) O v) (\lambda (t: T).(land (eq nat n i) (eq
-T t (lift (S n) O v)))) (let H10 \def (f_equal T nat (\lambda (e: T).(match e
-in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | (TLRef n0)
-\Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H6) in
-(let H11 \def (eq_ind_r nat n (\lambda (n0: nat).(subst0 i v (TLRef n0) (lift
-(S i) O v))) H8 i H10) in (let H12 \def (eq_ind_r nat n (\lambda (n0:
-nat).(subst0 i v (TLRef n0) (lift (S i) O v))) H9 i H10) in (eq_ind nat i
-(\lambda (n0: nat).(land (eq nat n0 i) (eq T (lift (S i) O v) (lift (S n0) O
-v)))) (conj (eq nat i i) (eq T (lift (S i) O v) (lift (S i) O v)) (refl_equal
-nat i) (refl_equal T (lift (S i) O v))) n H10)))) x H7))))))))))))) (\lambda
-(H0: (subst0 i v (TLRef n) x)).(\lambda (v0: T).(\lambda (u2: T).(\lambda
-(u1: T).(\lambda (i0: nat).(\lambda (t: T).(\lambda (k: K).(\lambda (H1: (eq
-nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t)
-(TLRef n))).(\lambda (H4: (eq T (THead k u2 t) x)).(\lambda (H5: (subst0 i0
-v0 u1 u2)).(let H6 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 n0 v0 u1
-u2)) H5 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t0: T).(subst0 i t0 u1
-u2)) H6 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v
-(TLRef n) t0)) H0 (THead k u2 t) H4) in (let H9 \def (eq_ind_r T x (\lambda
-(t0: T).(subst0 i v (TLRef n) t0)) H (THead k u2 t) H4) in (eq_ind T (THead k
-u2 t) (\lambda (t0: T).(land (eq nat n i) (eq T t0 (lift (S n) O v)))) (let
-H10 \def (eq_ind T (THead k u1 t) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in
-(False_ind (land (eq nat n i) (eq T (THead k u2 t) (lift (S n) O v))) H10)) x
-H4))))))))))))))))) (\lambda (H0: (subst0 i v (TLRef n) x)).(\lambda (k:
-K).(\lambda (v0: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (i0:
-nat).(\lambda (u: T).(\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0
-v)).(\lambda (H3: (eq T (THead k u t3) (TLRef n))).(\lambda (H4: (eq T (THead
-k u t0) x)).(\lambda (H5: (subst0 (s k i0) v0 t3 t0)).(let H6 \def (eq_ind
-nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t3 t0)) H5 i H1) in (let H7
-\def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i) t t3 t0)) H6 v H2) in (let
-H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n) t)) H0 (THead k u
-t0) H4) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (TLRef n)
-t)) H (THead k u t0) H4) in (eq_ind T (THead k u t0) (\lambda (t: T).(land
-(eq nat n i) (eq T t (lift (S n) O v)))) (let H10 \def (eq_ind T (THead k u
-t3) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort
-_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
-\Rightarrow True])) I (TLRef n) H3) in (False_ind (land (eq nat n i) (eq T
-(THead k u t0) (lift (S n) O v))) H10)) x H4))))))))))))))))) (\lambda (H0:
-(subst0 i v (TLRef n) x)).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2:
-T).(\lambda (i0: nat).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3:
-T).(\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq
-T (THead k u1 t0) (TLRef n))).(\lambda (H5: (eq T (THead k u2 t3)
-x)).(\lambda (H1: (subst0 i0 v0 u1 u2)).(\lambda (H6: (subst0 (s k i0) v0 t0
-t3)).(let H7 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 (s k n0) v0 t0
-t3)) H6 i H2) in (let H8 \def (eq_ind nat i0 (\lambda (n0: nat).(subst0 n0 v0
-u1 u2)) H1 i H2) in (let H9 \def (eq_ind T v0 (\lambda (t: T).(subst0 (s k i)
-t t0 t3)) H7 v H3) in (let H10 \def (eq_ind T v0 (\lambda (t: T).(subst0 i t
-u1 u2)) H8 v H3) in (let H11 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v
-(TLRef n) t)) H0 (THead k u2 t3) H5) in (let H12 \def (eq_ind_r T x (\lambda
-(t: T).(subst0 i v (TLRef n) t)) H (THead k u2 t3) H5) in (eq_ind T (THead k
-u2 t3) (\lambda (t: T).(land (eq nat n i) (eq T t (lift (S n) O v)))) (let
-H13 \def (eq_ind T (THead k u1 t0) (\lambda (ee: T).(match ee in T return
-(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in
-(False_ind (land (eq nat n i) (eq T (THead k u2 t3) (lift (S n) O v))) H13))
-x H5))))))))))))))))))))) H))))).
+(H: (subst0 i v (TLRef n) x)).(insert_eq T (TLRef n) (\lambda (t: T).(subst0
+i v t x)) (\lambda (_: T).(land (eq nat n i) (eq T x (lift (S n) O v))))
+(\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda (n0:
+nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef n))
+\to (land (eq nat n n0) (eq T t1 (lift (S n) O t)))))))) (\lambda (v0:
+T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef i0) (TLRef n))).(let H2 \def
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with
+[(TSort _) \Rightarrow i0 | (TLRef n0) \Rightarrow n0 | (THead _ _ _)
+\Rightarrow i0])) (TLRef i0) (TLRef n) H1) in (eq_ind_r nat n (\lambda (n0:
+nat).(land (eq nat n n0) (eq T (lift (S n0) O v0) (lift (S n) O v0)))) (conj
+(eq nat n n) (eq T (lift (S n) O v0) (lift (S n) O v0)) (refl_equal nat n)
+(refl_equal T (lift (S n) O v0))) i0 H2))))) (\lambda (v0: T).(\lambda (u2:
+T).(\lambda (u1: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1
+u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2
+(lift (S n) O v0)))))).(\lambda (t: T).(\lambda (k: K).(\lambda (H3: (eq T
+(THead k u1 t) (TLRef n))).(let H4 \def (eq_ind T (THead k u1 t) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u2
+t) (lift (S n) O v0))) H4))))))))))) (\lambda (k: K).(\lambda (v0:
+T).(\lambda (t2: T).(\lambda (t1: T).(\lambda (i0: nat).(\lambda (_: (subst0
+(s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef n)) \to (land (eq nat n (s
+k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda (u: T).(\lambda (H3: (eq T
+(THead k u t1) (TLRef n))).(let H4 \def (eq_ind T (THead k u t1) (\lambda
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H3) in (False_ind (land (eq nat n i0) (eq T (THead k u
+t2) (lift (S n) O v0))) H4))))))))))) (\lambda (v0: T).(\lambda (u1:
+T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (_: (subst0 i0 v0 u1
+u2)).(\lambda (_: (((eq T u1 (TLRef n)) \to (land (eq nat n i0) (eq T u2
+(lift (S n) O v0)))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2:
+T).(\lambda (_: (subst0 (s k i0) v0 t1 t2)).(\lambda (_: (((eq T t1 (TLRef
+n)) \to (land (eq nat n (s k i0)) (eq T t2 (lift (S n) O v0)))))).(\lambda
+(H5: (eq T (THead k u1 t1) (TLRef n))).(let H6 \def (eq_ind T (THead k u1 t1)
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow
+True])) I (TLRef n) H5) in (False_ind (land (eq nat n i0) (eq T (THead k u2
+t2) (lift (S n) O v0))) H6)))))))))))))) i v y x H0))) H))))).
theorem subst0_gen_head:
\forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall
\def
\lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda
(x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1)
-x)).(subst0_inv_coq i v (THead k u1 t1) x (\lambda (n: nat).(\lambda (t:
-T).(\lambda (_: T).(\lambda (t2: T).(or3 (ex2 T (\lambda (u2: T).(eq T t2
-(THead k u2 t1))) (\lambda (u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3:
-T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3)))
-(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead k u2 t3))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 n t u1 u2))) (\lambda (_:
-T).(\lambda (t3: T).(subst0 (s k n) t t1 t3))))))))) (\lambda (H0: (subst0 i
-v (THead k u1 t1) x)).(\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq
-nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (TLRef i0) (THead k
-u1 t1))).(\lambda (H4: (eq T (lift (S i0) O v0) x)).(let H5 \def (eq_ind nat
-i0 (\lambda (n: nat).(eq T (lift (S n) O v0) x)) H4 i H1) in (let H6 \def
-(eq_ind nat i0 (\lambda (n: nat).(eq T (TLRef n) (THead k u1 t1))) H3 i H1)
-in (let H7 \def (eq_ind T v0 (\lambda (t: T).(eq T (lift (S i) O t) x)) H5 v
-H2) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1)
-t)) H0 (lift (S i) O v) H7) in (let H9 \def (eq_ind_r T x (\lambda (t:
-T).(subst0 i v (THead k u1 t1) t)) H (lift (S i) O v) H7) in (eq_ind T (lift
-(S i) O v) (\lambda (t: T).(or3 (ex2 T (\lambda (u2: T).(eq T t (THead k u2
-t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T t
-(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T
-(\lambda (u2: T).(\lambda (t2: T).(eq T t (THead k u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i) v t1 t2)))))) (let H10 \def (eq_ind T (TLRef i) (\lambda
-(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _)
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow
-False])) I (THead k u1 t1) H6) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq
-T (lift (S i) O v) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2)))
-(ex2 T (\lambda (t2: T).(eq T (lift (S i) O v) (THead k u1 t2))) (\lambda
-(t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
-T).(eq T (lift (S i) O v) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
-T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
-t2))))) H10)) x H7))))))))))))) (\lambda (H0: (subst0 i v (THead k u1 t1)
-x)).(\lambda (v0: T).(\lambda (u2: T).(\lambda (u0: T).(\lambda (i0:
-nat).(\lambda (t: T).(\lambda (k0: K).(\lambda (H1: (eq nat i0 i)).(\lambda
-(H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1
-t1))).(\lambda (H4: (eq T (THead k0 u2 t) x)).(\lambda (H5: (subst0 i0 v0 u0
-u2)).(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(subst0 n v0 u0 u2)) H5 i
-H1) in (let H7 \def (eq_ind T v0 (\lambda (t0: T).(subst0 i t0 u0 u2)) H6 v
-H2) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).(subst0 i v (THead k u1
-t1) t0)) H0 (THead k0 u2 t) H4) in (let H9 \def (eq_ind_r T x (\lambda (t0:
-T).(subst0 i v (THead k u1 t1) t0)) H (THead k0 u2 t) H4) in (eq_ind T (THead
-k0 u2 t) (\lambda (t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T t0 (THead k u3
-t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t0
-(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T
-(\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i) v t1 t2)))))) (let H10 \def (f_equal T K (\lambda (e:
-T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k0 |
-(TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t)
-(THead k u1 t1) H3) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _)
-\Rightarrow u0 | (THead _ t0 _) \Rightarrow t0])) (THead k0 u0 t) (THead k u1
-t1) H3) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e in T return
-(\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t |
-(THead _ _ t0) \Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in
-(\lambda (H13: (eq T u0 u1)).(\lambda (H14: (eq K k0 k)).(let H15 \def
-(eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u2 t)))
-H9 k H14) in (let H16 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k
-u1 t1) (THead k1 u2 t))) H8 k H14) in (eq_ind_r K k (\lambda (k1: K).(or3
-(ex2 T (\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3:
-T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t) (THead
-k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda
-(u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2)))) (\lambda
-(u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i) v t1 t2)))))) (let H17 \def (eq_ind T t (\lambda (t0:
-T).(subst0 i v (THead k u1 t1) (THead k u2 t0))) H15 t1 H12) in (let H18 \def
-(eq_ind T t (\lambda (t0: T).(subst0 i v (THead k u1 t1) (THead k u2 t0)))
-H16 t1 H12) in (eq_ind_r T t1 (\lambda (t0: T).(or3 (ex2 T (\lambda (u3:
-T).(eq T (THead k u2 t0) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1
-u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t0) (THead k u1 t2)))
-(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i) v t1 t2)))))) (let H19 \def (eq_ind T u0 (\lambda (t0:
-T).(subst0 i v t0 u2)) H7 u1 H13) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq
-T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3)))
-(ex2 T (\lambda (t2: T).(eq T (THead k u2 t1) (THead k u1 t2))) (\lambda (t2:
-T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2:
-T).(eq T (THead k u2 t1) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_:
-T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
-t2)))) (ex_intro2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1)))
-(\lambda (u3: T).(subst0 i v u1 u3)) u2 (refl_equal T (THead k u2 t1)) H19)))
-t H12))) k0 H14)))))) H11)) H10)) x H4))))))))))))))))) (\lambda (H0: (subst0
-i v (THead k u1 t1) x)).(\lambda (k0: K).(\lambda (v0: T).(\lambda (t0:
-T).(\lambda (t3: T).(\lambda (i0: nat).(\lambda (u: T).(\lambda (H1: (eq nat
-i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u t3) (THead
-k u1 t1))).(\lambda (H4: (eq T (THead k0 u t0) x)).(\lambda (H5: (subst0 (s
-k0 i0) v0 t3 t0)).(let H6 \def (eq_ind nat i0 (\lambda (n: nat).(subst0 (s k0
-n) v0 t3 t0)) H5 i H1) in (let H7 \def (eq_ind T v0 (\lambda (t: T).(subst0
-(s k0 i) t t3 t0)) H6 v H2) in (let H8 \def (eq_ind_r T x (\lambda (t:
-T).(subst0 i v (THead k u1 t1) t)) H0 (THead k0 u t0) H4) in (let H9 \def
-(eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) t)) H (THead k0 u
-t0) H4) in (eq_ind T (THead k0 u t0) (\lambda (t: T).(or3 (ex2 T (\lambda
-(u2: T).(eq T t (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2
-T (\lambda (t2: T).(eq T t (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i)
-v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t (THead k u2
-t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (let H10 \def (f_equal T K
-(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _)
-\Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _) \Rightarrow k1]))
-(THead k0 u t3) (THead k u1 t1) H3) in ((let H11 \def (f_equal T T (\lambda
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u
-| (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k0 u t3)
-(THead k u1 t1) H3) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e
-in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t3 | (TLRef _)
-\Rightarrow t3 | (THead _ _ t) \Rightarrow t])) (THead k0 u t3) (THead k u1
-t1) H3) in (\lambda (H13: (eq T u u1)).(\lambda (H14: (eq K k0 k)).(let H15
-\def (eq_ind T u (\lambda (t: T).(subst0 i v (THead k u1 t1) (THead k0 t
-t0))) H9 u1 H13) in (let H16 \def (eq_ind T u (\lambda (t: T).(subst0 i v
-(THead k u1 t1) (THead k0 t t0))) H8 u1 H13) in (eq_ind_r T u1 (\lambda (t:
-T).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k0 t t0) (THead k u2 t1)))
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k0
-t t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k0 t t0) (THead k u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (let H17 \def (eq_ind T t3
-(\lambda (t: T).(subst0 (s k0 i) v t t0)) H7 t1 H12) in (let H18 \def (eq_ind
-K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u1 t0))) H15 k
-H14) in (let H19 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1
-t1) (THead k1 u1 t0))) H16 k H14) in (let H20 \def (eq_ind K k0 (\lambda (k1:
-K).(subst0 (s k1 i) v t1 t0)) H17 k H14) in (eq_ind_r K k (\lambda (k1:
-K).(or3 (ex2 T (\lambda (u2: T).(eq T (THead k1 u1 t0) (THead k u2 t1)))
-(\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k1
-u1 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T
-T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead k1 u1 t0) (THead k u2 t2))))
-(\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (or3_intro1 (ex2 T (\lambda
-(u2: T).(eq T (THead k u1 t0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v
-u1 u2))) (ex2 T (\lambda (t2: T).(eq T (THead k u1 t0) (THead k u1 t2)))
+x)).(insert_eq T (THead k u1 t1) (\lambda (t: T).(subst0 i v t x)) (\lambda
+(_: T).(or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2)))
(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2:
-T).(\lambda (t2: T).(eq T (THead k u1 t0) (THead k u2 t2)))) (\lambda (u2:
-T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (t2: T).(eq T (THead k
-u1 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2)) t0
-(refl_equal T (THead k u1 t0)) H20)) k0 H14))))) u H13)))))) H11)) H10)) x
-H4))))))))))))))))) (\lambda (H0: (subst0 i v (THead k u1 t1) x)).(\lambda
-(v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (i0: nat).(\lambda (k0:
-K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H2: (eq nat i0 i)).(\lambda
-(H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k u1
-t1))).(\lambda (H5: (eq T (THead k0 u2 t3) x)).(\lambda (H1: (subst0 i0 v0 u0
-u2)).(\lambda (H6: (subst0 (s k0 i0) v0 t0 t3)).(let H7 \def (eq_ind nat i0
-(\lambda (n: nat).(subst0 (s k0 n) v0 t0 t3)) H6 i H2) in (let H8 \def
-(eq_ind nat i0 (\lambda (n: nat).(subst0 n v0 u0 u2)) H1 i H2) in (let H9
-\def (eq_ind T v0 (\lambda (t: T).(subst0 (s k0 i) t t0 t3)) H7 v H3) in (let
-H10 \def (eq_ind T v0 (\lambda (t: T).(subst0 i t u0 u2)) H8 v H3) in (let
-H11 \def (eq_ind_r T x (\lambda (t: T).(subst0 i v (THead k u1 t1) t)) H0
-(THead k0 u2 t3) H5) in (let H12 \def (eq_ind_r T x (\lambda (t: T).(subst0 i
-v (THead k u1 t1) t)) H (THead k0 u2 t3) H5) in (eq_ind T (THead k0 u2 t3)
-(\lambda (t: T).(or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1)))
-(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t (THead
-k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda
-(u3: T).(\lambda (t2: T).(eq T t (THead k u3 t2)))) (\lambda (u3: T).(\lambda
-(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i)
-v t1 t2)))))) (let H13 \def (f_equal T K (\lambda (e: T).(match e in T return
-(\lambda (_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0
-| (THead k1 _ _) \Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H4) in
-((let H14 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_:
-T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t
-_) \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H4) in ((let H15 \def
+T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1
+t2)))))) (\lambda (y: T).(\lambda (H0: (subst0 i v y x)).(subst0_ind (\lambda
+(n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq T t0 (THead k
+u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 t1))) (\lambda
+(u2: T).(subst0 n t u1 u2))) (ex2 T (\lambda (t3: T).(eq T t2 (THead k u1
+t3))) (\lambda (t3: T).(subst0 (s k n) t t1 t3))) (ex3_2 T T (\lambda (u2:
+T).(\lambda (t3: T).(eq T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 n t u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k n) t t1
+t3)))))))))) (\lambda (v0: T).(\lambda (i0: nat).(\lambda (H1: (eq T (TLRef
+i0) (THead k u1 t1))).(let H2 \def (eq_ind T (TLRef i0) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead k u1 t1) H1) in (False_ind (or3 (ex2 T (\lambda (u2: T).(eq T (lift (S
+i0) O v0) (THead k u2 t1))) (\lambda (u2: T).(subst0 i0 v0 u1 u2))) (ex2 T
+(\lambda (t2: T).(eq T (lift (S i0) O v0) (THead k u1 t2))) (\lambda (t2:
+T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2:
+T).(eq T (lift (S i0) O v0) (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_:
+T).(subst0 i0 v0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0)
+v0 t1 t2))))) H2))))) (\lambda (v0: T).(\lambda (u2: T).(\lambda (u0:
+T).(\lambda (i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq
+T u0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T
+u2 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (t: T).(\lambda (k0:
+K).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H4 \def
+(f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with
+[(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
+\Rightarrow k1])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H5 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t0 _)
+\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in ((let H6 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t0)
+\Rightarrow t0])) (THead k0 u0 t) (THead k u1 t1) H3) in (\lambda (H7: (eq T
+u0 u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T
+(\lambda (u3: T).(eq T (THead k1 u2 t) (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1 u2 t)
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t) (THead k u3 t2))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_:
+T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (eq_ind_r T t1 (\lambda
+(t0: T).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t0) (THead k u3 t1)))
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead
+k u2 t0) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t0) (THead k
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda
+(_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))))) (let H9 \def (eq_ind
+T u0 (\lambda (t0: T).((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda
+(u3: T).(eq T u2 (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3)))
+(ex2 T (\lambda (t2: T).(eq T u2 (THead k u1 t2))) (\lambda (t2: T).(subst0
+(s k i0) v0 t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T u2
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))))))) H2 u1 H7)
+in (let H10 \def (eq_ind T u0 (\lambda (t0: T).(subst0 i0 v0 t0 u2)) H1 u1
+H7) in (or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3
+t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T
+(THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1
+t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1)
+(THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u1 u3)))
+(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t1 t2)))) (ex_intro2 T
+(\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3:
+T).(subst0 i0 v0 u1 u3)) u2 (refl_equal T (THead k u2 t1)) H10)))) t H6) k0
+H8)))) H5)) H4))))))))))) (\lambda (k0: K).(\lambda (v0: T).(\lambda (t2:
+T).(\lambda (t0: T).(\lambda (i0: nat).(\lambda (H1: (subst0 (s k0 i0) v0 t0
+t2)).(\lambda (H2: (((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2:
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
+t3)))))))).(\lambda (u: T).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1
+t1))).(let H4 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda
+(_: T).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead
+k1 _ _) \Rightarrow k1])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H5
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T)
+with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _)
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in ((let H6 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
+\Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H3) in (\lambda (H7: (eq T u
+u1)).(\lambda (H8: (eq K k0 k)).(eq_ind_r T u1 (\lambda (t: T).(or3 (ex2 T
+(\lambda (u2: T).(eq T (THead k0 t t2) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k0 t t2)
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k0 t t2) (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (let H9 \def (eq_ind T t0
+(\lambda (t: T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2:
+T).(eq T t2 (THead k u2 t1))) (\lambda (u2: T).(subst0 (s k0 i0) v0 u1 u2)))
+(ex2 T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0
+(s k (s k0 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq
+T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k0 i0) v0
+u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
+t3))))))) H2 t1 H6) in (let H10 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s
+k0 i0) v0 t t2)) H1 t1 H6) in (let H11 \def (eq_ind K k0 (\lambda (k1:
+K).((eq T t1 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u2: T).(eq T t2
+(THead k u2 t1))) (\lambda (u2: T).(subst0 (s k1 i0) v0 u1 u2))) (ex2 T
+(\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s
+k1 i0)) v0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2
+(THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1
+u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1
+t3))))))) H9 k H8) in (let H12 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s
+k1 i0) v0 t1 t2)) H10 k H8) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T
+(\lambda (u2: T).(eq T (THead k1 u1 t2) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k1 u1 t2)
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k1 u1 t2) (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))))) (or3_intro1 (ex2 T
+(\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2:
+T).(subst0 i0 v0 u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2)
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
+(\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3))))
+(\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u1 u2))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3)))) (ex_intro2 T (\lambda (t3:
+T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0)
+v0 t1 t3)) t2 (refl_equal T (THead k u1 t2)) H12)) k0 H8))))) u H7)))) H5))
+H4))))))))))) (\lambda (v0: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda
+(i0: nat).(\lambda (H1: (subst0 i0 v0 u0 u2)).(\lambda (H2: (((eq T u0 (THead
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1)))
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t2: T).(eq T u2
+(THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t1 t2))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t2: T).(eq T u2 (THead k u3 t2)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t2:
+T).(subst0 (s k i0) v0 t1 t2)))))))).(\lambda (k0: K).(\lambda (t0:
+T).(\lambda (t2: T).(\lambda (H3: (subst0 (s k0 i0) v0 t0 t2)).(\lambda (H4:
+(((eq T t0 (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0))
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1
+t3)))))))).(\lambda (H5: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H6
+\def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K)
+with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k1 _ _)
+\Rightarrow k1])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H7 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _)
+\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in ((let H8 \def
(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
[(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t)
-\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H4) in (\lambda (H16: (eq T
-u0 u1)).(\lambda (H17: (eq K k0 k)).(let H18 \def (eq_ind T t0 (\lambda (t:
-T).(subst0 (s k0 i) v t t3)) H9 t1 H15) in (let H19 \def (eq_ind K k0
-(\lambda (k1: K).(subst0 i v (THead k u1 t1) (THead k1 u2 t3))) H12 k H17) in
-(let H20 \def (eq_ind K k0 (\lambda (k1: K).(subst0 i v (THead k u1 t1)
-(THead k1 u2 t3))) H11 k H17) in (let H21 \def (eq_ind K k0 (\lambda (k1:
-K).(subst0 (s k1 i) v t1 t3)) H18 k H17) in (eq_ind_r K k (\lambda (k1:
-K).(or3 (ex2 T (\lambda (u3: T).(eq T (THead k1 u2 t3) (THead k u3 t1)))
-(\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k1
-u2 t3) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T
-T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k1 u2 t3) (THead k u3 t2))))
-(\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_:
-T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) (let H22 \def (eq_ind T u0
-(\lambda (t: T).(subst0 i v t u2)) H10 u1 H16) in (or3_intro2 (ex2 T (\lambda
-(u3: T).(eq T (THead k u2 t3) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v
-u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t3) (THead k u1 t2)))
-(\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3:
-T).(\lambda (t2: T).(eq T (THead k u2 t3) (THead k u3 t2)))) (\lambda (u3:
-T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2:
-T).(subst0 (s k i) v t1 t2)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda
-(t2: T).(eq T (THead k u2 t3) (THead k u3 t2)))) (\lambda (u3: T).(\lambda
-(_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i)
-v t1 t2))) u2 t3 (refl_equal T (THead k u2 t3)) H22 H21))) k0 H17))))))))
-H14)) H13)) x H5))))))))))))))))))))) H))))))).
+\Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H5) in (\lambda (H9: (eq T
+u0 u1)).(\lambda (H10: (eq K k0 k)).(let H11 \def (eq_ind T t0 (\lambda (t:
+T).((eq T t (THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead
+k u3 t1))) (\lambda (u3: T).(subst0 (s k0 i0) v0 u1 u3))) (ex2 T (\lambda
+(t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k0 i0))
+v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3
+t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 (s k0 i0) v0 u1 u3)))
+(\lambda (_: T).(\lambda (t3: T).(subst0 (s k (s k0 i0)) v0 t1 t3))))))) H4
+t1 H8) in (let H12 \def (eq_ind T t0 (\lambda (t: T).(subst0 (s k0 i0) v0 t
+t2)) H3 t1 H8) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq T t1
+(THead k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 t1)))
+(\lambda (u3: T).(subst0 (s k1 i0) v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T
+t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3)))
+(ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3))))
+(\lambda (u3: T).(\lambda (_: T).(subst0 (s k1 i0) v0 u1 u3))) (\lambda (_:
+T).(\lambda (t3: T).(subst0 (s k (s k1 i0)) v0 t1 t3))))))) H11 k H10) in
+(let H14 \def (eq_ind K k0 (\lambda (k1: K).(subst0 (s k1 i0) v0 t1 t2)) H12
+k H10) in (eq_ind_r K k (\lambda (k1: K).(or3 (ex2 T (\lambda (u3: T).(eq T
+(THead k1 u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0 v0 u1 u3)))
+(ex2 T (\lambda (t3: T).(eq T (THead k1 u2 t2) (THead k u1 t3))) (\lambda
+(t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead k1 u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k
+i0) v0 t1 t3)))))) (let H15 \def (eq_ind T u0 (\lambda (t: T).((eq T t (THead
+k u1 t1)) \to (or3 (ex2 T (\lambda (u3: T).(eq T u2 (THead k u3 t1)))
+(\lambda (u3: T).(subst0 i0 v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T u2
+(THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T
+(\lambda (u3: T).(\lambda (t3: T).(eq T u2 (THead k u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i0) v0 t1 t3))))))) H2 u1 H9) in (let H16 \def (eq_ind T u0
+(\lambda (t: T).(subst0 i0 v0 t u2)) H1 u1 H9) in (or3_intro2 (ex2 T (\lambda
+(u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i0
+v0 u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3)))
+(\lambda (t3: T).(subst0 (s k i0) v0 t1 t3))) (ex3_2 T T (\lambda (u3:
+T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3:
+T).(\lambda (_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3:
+T).(subst0 (s k i0) v0 t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda
+(t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda
+(_: T).(subst0 i0 v0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k
+i0) v0 t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H16 H14)))) k0
+H10)))))))) H7)) H6)))))))))))))) i v y x H0))) H))))))).
theorem subst0_gen_lift_lt:
\forall (u: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).(\forall
(* This file was automatically generated: do not edit *********************)
-include "subst0/fwd.ma".
-
-include "lift/props.ma".
+include "LambdaDelta-1/subst0/fwd.ma".
theorem subst0_refl:
\forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to
(* This file was automatically generated: do not edit *********************)
-include "subst0/props.ma".
+include "LambdaDelta-1/subst0/props.ma".
theorem subst0_subst0:
\forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst0
(* This file was automatically generated: do not edit *********************)
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
-include "lift/props.ma".
+include "LambdaDelta-1/lift/props.ma".
-include "lift/tlt.ma".
+include "LambdaDelta-1/lift/tlt.ma".
theorem subst0_weight_le:
\forall (u: T).(\forall (t: T).(\forall (z: T).(\forall (d: nat).((subst0 d
(* This file was automatically generated: do not edit *********************)
-include "subst0/defs.ma".
+include "LambdaDelta-1/subst0/defs.ma".
inductive subst1 (i: nat) (v: T) (t1: T): T \to Prop \def
| subst1_refl: subst1 i v t1 t1
(* This file was automatically generated: do not edit *********************)
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
-include "subst0/props.ma".
+include "LambdaDelta-1/subst0/props.ma".
theorem subst1_gen_sort:
\forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst1
(* This file was automatically generated: do not edit *********************)
-include "subst1/defs.ma".
+include "LambdaDelta-1/subst1/defs.ma".
-include "subst0/props.ma".
+include "LambdaDelta-1/subst0/props.ma".
theorem subst1_head:
\forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1
(* This file was automatically generated: do not edit *********************)
-include "subst1/fwd.ma".
+include "LambdaDelta-1/subst1/fwd.ma".
-include "subst0/subst0.ma".
+include "LambdaDelta-1/subst0/subst0.ma".
theorem subst1_subst1:
\forall (t1: T).(\forall (t2: T).(\forall (u2: T).(\forall (j: nat).((subst1
(* This file was automatically generated: do not edit *********************)
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
-include "getl/defs.ma".
+include "LambdaDelta-1/getl/defs.ma".
inductive tau0 (g: G): C \to (T \to (T \to Prop)) \def
| tau0_sort: \forall (c: C).(\forall (n: nat).(tau0 g c (TSort n) (TSort
(* This file was automatically generated: do not edit *********************)
-include "tau0/defs.ma".
+include "LambdaDelta-1/tau0/defs.ma".
-include "getl/drop.ma".
+include "LambdaDelta-1/getl/drop.ma".
theorem tau0_lift:
\forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((tau0 g e
(* This file was automatically generated: do not edit *********************)
-include "tau1/props.ma".
+include "LambdaDelta-1/tau1/props.ma".
-include "cnt/props.ma".
+include "LambdaDelta-1/cnt/props.ma".
theorem tau1_cnt:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau0 g c
(* This file was automatically generated: do not edit *********************)
-include "tau0/defs.ma".
+include "LambdaDelta-1/tau0/defs.ma".
inductive tau1 (g: G) (c: C) (t1: T): T \to Prop \def
| tau1_tau0: \forall (t2: T).((tau0 g c t1 t2) \to (tau1 g c t1 t2))
(* This file was automatically generated: do not edit *********************)
-include "tau1/defs.ma".
+include "LambdaDelta-1/tau1/defs.ma".
-include "tau0/props.ma".
+include "LambdaDelta-1/tau0/props.ma".
theorem tau1_trans:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t: T).((tau1 g c
(* This file was automatically generated: do not edit *********************)
-include "subst0/tlt.ma".
+include "LambdaDelta-1/subst0/tlt.ma".
-include "tau1/cnt.ma".
+include "LambdaDelta-1/tau1/cnt.ma".
-include "ex0/props.ma".
+include "LambdaDelta-1/ex0/props.ma".
-include "wcpr0/fwd.ma".
+include "LambdaDelta-1/wcpr0/fwd.ma".
-include "pr3/wcpr0.ma".
+include "LambdaDelta-1/pr3/wcpr0.ma".
-include "ex2/props.ma".
+include "LambdaDelta-1/ex2/props.ma".
-include "ex1/props.ma".
+include "LambdaDelta-1/ex1/props.ma".
-include "ty3/tau0.ma".
+include "LambdaDelta-1/ty3/tau0.ma".
-include "ty3/nf2.ma".
+include "LambdaDelta-1/ty3/nf2.ma".
-include "ty3/dec.ma".
+include "LambdaDelta-1/ty3/dec.ma".
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
inductive TList: Set \def
| TNil: TList
(* This file was automatically generated: do not edit *********************)
-include "tlist/defs.ma".
+include "LambdaDelta-1/tlist/defs.ma".
theorem tslt_wf__q_ind:
\forall (P: ((TList \to Prop))).(((\forall (n: nat).((\lambda (P0: ((TList
(* This file was automatically generated: do not edit *********************)
-include "T/defs.ma".
+include "LambdaDelta-1/T/defs.ma".
definition wadd:
((nat \to nat)) \to (nat \to (nat \to nat))
(* This file was automatically generated: do not edit *********************)
-include "tlt/defs.ma".
+include "LambdaDelta-1/tlt/defs.ma".
theorem wadd_le:
\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall (n:
(* This file was automatically generated: do not edit *********************)
-include "ty3/pr3_props.ma".
+include "LambdaDelta-1/ty3/pr3_props.ma".
-include "arity/pr3.ma".
+include "LambdaDelta-1/arity/pr3.ma".
-include "asucc/fwd.ma".
+include "LambdaDelta-1/asucc/fwd.ma".
theorem ty3_arity:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
(* This file was automatically generated: do not edit *********************)
-include "ty3/arity.ma".
+include "LambdaDelta-1/ty3/arity.ma".
-include "ty3/fwd.ma".
-
-include "sc3/arity.ma".
+include "LambdaDelta-1/sc3/arity.ma".
theorem ty3_predicative:
\forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u:
(* This file was automatically generated: do not edit *********************)
-include "ty3/pr3_props.ma".
+include "LambdaDelta-1/pc3/dec.ma".
-include "pc3/dec.ma".
+include "LambdaDelta-1/getl/flt.ma".
-include "getl/flt.ma".
-
-include "getl/dec.ma".
+include "LambdaDelta-1/getl/dec.ma".
theorem ty3_inference:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2:
(* This file was automatically generated: do not edit *********************)
-include "G/defs.ma".
+include "LambdaDelta-1/G/defs.ma".
-include "pc3/defs.ma".
+include "LambdaDelta-1/pc3/defs.ma".
inductive ty3 (g: G): C \to (T \to (T \to Prop)) \def
| ty3_conv: \forall (c: C).(\forall (t2: T).(\forall (t: T).((ty3 g c t2 t)
(* This file was automatically generated: do not edit *********************)
-include "ty3/props.ma".
+include "LambdaDelta-1/ty3/props.ma".
-include "pc3/fsubst0.ma".
+include "LambdaDelta-1/pc3/fsubst0.ma".
-include "csubst0/props.ma".
+include "LambdaDelta-1/csubst0/props.ma".
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
theorem ty3_fsubst0:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
(* This file was automatically generated: do not edit *********************)
-include "ty3/defs.ma".
+include "LambdaDelta-1/ty3/defs.ma".
-include "pc3/props.ma".
+include "LambdaDelta-1/pc3/props.ma".
theorem ty3_gen_sort:
\forall (g: G).(\forall (c: C).(\forall (x: T).(\forall (n: nat).((ty3 g c
(* This file was automatically generated: do not edit *********************)
-include "ty3/arity.ma".
+include "LambdaDelta-1/ty3/arity.ma".
-include "pc3/nf2.ma".
+include "LambdaDelta-1/pc3/nf2.ma".
-include "nf2/arity.ma".
+include "LambdaDelta-1/nf2/arity.ma".
definition ty3_nf2_inv_abst_premise:
C \to (T \to (T \to Prop))
(* This file was automatically generated: do not edit *********************)
-include "csubt/ty3.ma".
+include "LambdaDelta-1/csubt/ty3.ma".
-include "ty3/subst1.ma".
+include "LambdaDelta-1/ty3/subst1.ma".
-include "ty3/fsubst0.ma".
+include "LambdaDelta-1/ty3/fsubst0.ma".
-include "pc3/pc1.ma".
+include "LambdaDelta-1/pc3/pc1.ma".
-include "pc3/wcpr0.ma".
+include "LambdaDelta-1/pc3/wcpr0.ma".
-include "pc1/props.ma".
+include "LambdaDelta-1/pc1/props.ma".
theorem ty3_sred_wcpr0_pr0:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1
(* This file was automatically generated: do not edit *********************)
-include "ty3/pr3.ma".
+include "LambdaDelta-1/ty3/pr3.ma".
theorem ty3_cred_pr2:
\forall (g: G).(\forall (c: C).(\forall (v1: T).(\forall (v2: T).((pr2 c v1
(* This file was automatically generated: do not edit *********************)
-include "ty3/fwd.ma".
+include "LambdaDelta-1/ty3/fwd.ma".
-include "pc3/fwd.ma".
+include "LambdaDelta-1/pc3/fwd.ma".
theorem ty3_lift:
\forall (g: G).(\forall (e: C).(\forall (t1: T).(\forall (t2: T).((ty3 g e
(* This file was automatically generated: do not edit *********************)
-include "ty3/props.ma".
+include "LambdaDelta-1/ty3/props.ma".
-include "pc3/subst1.ma".
+include "LambdaDelta-1/pc3/subst1.ma".
-include "pc3/fwd.ma".
-
-include "csubst1/getl.ma".
-
-include "csubst1/fwd.ma".
-
-include "getl/getl.ma".
+include "LambdaDelta-1/getl/getl.ma".
theorem ty3_gen_cabbr:
\forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c
(* This file was automatically generated: do not edit *********************)
-include "ty3/pr3_props.ma".
+include "LambdaDelta-1/ty3/pr3_props.ma".
-include "tau0/defs.ma".
+include "LambdaDelta-1/tau0/defs.ma".
theorem ty3_tau0:
\forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u
(* This file was automatically generated: do not edit *********************)
-include "pr0/defs.ma".
+include "LambdaDelta-1/pr0/defs.ma".
-include "C/defs.ma".
+include "LambdaDelta-1/C/defs.ma".
inductive wcpr0: C \to (C \to Prop) \def
| wcpr0_refl: \forall (c: C).(wcpr0 c c)
(* This file was automatically generated: do not edit *********************)
-include "wcpr0/defs.ma".
+include "LambdaDelta-1/wcpr0/defs.ma".
theorem wcpr0_gen_sort:
\forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort
n))))
\def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let H0
-\def (match H in wcpr0 return (\lambda (c: C).(\lambda (c0: C).(\lambda (_:
-(wcpr0 c c0)).((eq C c (CSort n)) \to ((eq C c0 x) \to (eq C x (CSort
-n))))))) with [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CSort
-n))).(\lambda (H1: (eq C c x)).(eq_ind C (CSort n) (\lambda (c0: C).((eq C c0
-x) \to (eq C x (CSort n)))) (\lambda (H2: (eq C (CSort n) x)).(eq_ind C
-(CSort n) (\lambda (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x
-H2)) c (sym_eq C c (CSort n) H0) H1))) | (wcpr0_comp c1 c2 H0 u1 u2 H1 k)
-\Rightarrow (\lambda (H2: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H3: (eq
-C (CHead c2 k u2) x)).((let H4 \def (eq_ind C (CHead c1 k u1) (\lambda (e:
-C).(match e in C return (\lambda (_: C).Prop) with [(CSort _) \Rightarrow
-False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in (False_ind ((eq
-C (CHead c2 k u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (eq C x (CSort
-n))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C
-x))))).
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n)
+x)).(insert_eq C (CSort n) (\lambda (c: C).(wcpr0 c x)) (\lambda (c: C).(eq C
+x c)) (\lambda (y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c:
+C).(\lambda (c0: C).((eq C c (CSort n)) \to (eq C c0 c)))) (\lambda (c:
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 c0))
+(refl_equal C (CSort n)) c H2)))) (\lambda (c1: C).(\lambda (c2: C).(\lambda
+(_: (wcpr0 c1 c2)).(\lambda (_: (((eq C c1 (CSort n)) \to (eq C c2
+c1)))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda
+(k: K).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(let H5 \def (eq_ind C
+(CHead c1 k u1) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
+(CSort n) H4) in (False_ind (eq C (CHead c2 k u2) (CHead c1 k u1))
+H5))))))))))) y x H0))) H))).
theorem wcpr0_gen_head:
\forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0
T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
\def
\lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda
-(H: (wcpr0 (CHead c1 k u1) x)).(let H0 \def (match H in wcpr0 return (\lambda
-(c: C).(\lambda (c0: C).(\lambda (_: (wcpr0 c c0)).((eq C c (CHead c1 k u1))
-\to ((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2:
+(H: (wcpr0 (CHead c1 k u1) x)).(insert_eq C (CHead c1 k u1) (\lambda (c:
+C).(wcpr0 c x)) (\lambda (c: C).(or (eq C x c) (ex3_2 C T (\lambda (c2:
C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
-T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))) with
-[(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CHead c1 k u1))).(\lambda
-(H1: (eq C c x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).((eq C c0 x) \to
-(or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq
-C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2)))
-(\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))) (\lambda (H2: (eq C (CHead
-c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).(or (eq C c0 (CHead
-c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C c0 (CHead c2 k
-u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_:
-C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C (CHead c1 k u1) (CHead
-c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C (CHead c1 k u1)
-(CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda
-(_: C).(\lambda (u2: T).(pr0 u1 u2)))) (refl_equal C (CHead c1 k u1))) x H2))
-c (sym_eq C c (CHead c1 k u1) H0) H1))) | (wcpr0_comp c0 c2 H0 u0 u2 H1 k0)
-\Rightarrow (\lambda (H2: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda
-(H3: (eq C (CHead c2 k0 u2) x)).((let H4 \def (f_equal C T (\lambda (e:
-C).(match e in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 |
-(CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let
-H5 \def (f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K)
-with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0
-k0 u0) (CHead c1 k u1) H2) in ((let H6 \def (f_equal C C (\lambda (e:
-C).(match e in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
-(CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in
-(eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead
-c2 k0 u2) x) \to ((wcpr0 c c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k
-u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3))))
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (\lambda
+(y: C).(\lambda (H0: (wcpr0 y x)).(wcpr0_ind (\lambda (c: C).(\lambda (c0:
+C).((eq C c (CHead c1 k u1)) \to (or (eq C c0 c) (ex3_2 C T (\lambda (c2:
+C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))
+(\lambda (c: C).(\lambda (H1: (eq C c (CHead c1 k u1))).(let H2 \def (f_equal
+C C (\lambda (e: C).e) c (CHead c1 k u1) H1) in (eq_ind_r C (CHead c1 k u1)
+(\lambda (c0: C).(or (eq C c0 c0) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
+T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1
+c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C
+(CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2:
+T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_:
+T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))
+(refl_equal C (CHead c1 k u1))) c H2)))) (\lambda (c0: C).(\lambda (c2:
+C).(\lambda (H1: (wcpr0 c0 c2)).(\lambda (H2: (((eq C c0 (CHead c1 k u1)) \to
+(or (eq C c2 c0) (ex3_2 C T (\lambda (c3: C).(\lambda (u2: T).(eq C c2 (CHead
+c3 k u2)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_:
+C).(\lambda (u2: T).(pr0 u1 u2)))))))).(\lambda (u0: T).(\lambda (u2:
+T).(\lambda (H3: (pr0 u0 u2)).(\lambda (k0: K).(\lambda (H4: (eq C (CHead c0
+k0 u0) (CHead c1 k u1))).(let H5 \def (f_equal C C (\lambda (e: C).(match e
+in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _)
+\Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in ((let H6 \def
+(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
+[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
+(CHead c1 k u1) H4) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e in
+C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H4) in (\lambda (H8: (eq K
+k0 k)).(\lambda (H9: (eq C c0 c1)).(eq_ind_r K k (\lambda (k1: K).(or (eq C
+(CHead c2 k1 u2) (CHead c0 k1 u0)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3:
+T).(eq C (CHead c2 k1 u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_:
+T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))) (let H10
+\def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H3 u1 H7) in (eq_ind_r T u1
+(\lambda (t: T).(or (eq C (CHead c2 k u2) (CHead c0 k t)) (ex3_2 C T (\lambda
+(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
+(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u1 u3)))))) (let H11 \def (eq_ind C c0 (\lambda (c: C).((eq C c (CHead c1 k
+u1)) \to (or (eq C c2 c) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C
+c2 (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))) H2 c1 H9) in (let H12 \def
+(eq_ind C c0 (\lambda (c: C).(wcpr0 c c2)) H1 c1 H9) in (eq_ind_r C c1
+(\lambda (c: C).(or (eq C (CHead c2 k u2) (CHead c k u1)) (ex3_2 C T (\lambda
+(c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda
+(c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u1 u3)))))) (or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T
+(\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3))))
(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u1 u3))))))))))) (\lambda (H7: (eq K k0 k)).(eq_ind K k (\lambda
-(k1: K).((eq T u0 u1) \to ((eq C (CHead c2 k1 u2) x) \to ((wcpr0 c1 c2) \to
-((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3:
-C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_:
-T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))))))
-(\lambda (H8: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k
-u2) x) \to ((wcpr0 c1 c2) \to ((pr0 t u2) \to (or (eq C x (CHead c1 k u1))
-(ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3))))
-(\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u1 u3))))))))) (\lambda (H9: (eq C (CHead c2 k u2) x)).(eq_ind C
-(CHead c2 k u2) (\lambda (c: C).((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (or (eq C
-c (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c
-(CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 u1 u3)))))))) (\lambda (H10: (wcpr0 c1
-c2)).(\lambda (H11: (pr0 u1 u2)).(or_intror (eq C (CHead c2 k u2) (CHead c1 k
-u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2)
-(CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3:
-C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3:
-C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1
-u3))) c2 u2 (refl_equal C (CHead c2 k u2)) H10 H11)))) x H9)) u0 (sym_eq T u0
-u1 H8))) k0 (sym_eq K k0 k H7))) c0 (sym_eq C c0 c1 H6))) H5)) H4)) H3 H0
-H1)))]) in (H0 (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))).
+(u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq
+C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0
+c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C
+(CHead c2 k u2)) H12 H10)) c0 H9))) u0 H7)) k0 H8)))) H6)) H5))))))))))) y x
+H0))) H))))).
(* This file was automatically generated: do not edit *********************)
-include "wcpr0/defs.ma".
+include "LambdaDelta-1/wcpr0/defs.ma".
-include "getl/props.ma".
+include "LambdaDelta-1/getl/props.ma".
theorem wcpr0_drop:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
-c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2:
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop h O c4 (CHead e2 k u2)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c0 k u1) (CHead
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u1) (CHead
e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
-c3 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+c4 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c0 k u1)
-(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c0 k u1) (CHead
-e1 k0 u0) H3) in eq return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c
-(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O
-(CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))))) with [refl_equal
-\Rightarrow (\lambda (H4: (eq C (CHead c0 k u1) (CHead e1 k0 u0))).(let H5
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k
-u1) (CHead e1 k0 u0) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match
-e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1
-_) \Rightarrow k1])) (CHead c0 k u1) (CHead e1 k0 u0) H4) in ((let H7 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u1)
-(CHead e1 k0 u0) H4) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T
-u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k
-u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))))) (\lambda (H8: (eq K k
-k0)).(eq_ind K k0 (\lambda (k1: K).((eq T u1 u0) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop O O (CHead c3 k1 u2) (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3)))))) (\lambda (H9: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C
-T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0
-u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
-C).(\lambda (u3: T).(pr0 u0 u3))))) (let H10 \def (eq_ind T u1 (\lambda (t:
-T).(pr0 t u2)) H2 u0 H9) in (let H11 \def (eq_ind C c0 (\lambda (c: C).(wcpr0
-c c3)) H0 e1 H7) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop
-O O (CHead c3 k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) c3 u2
-(drop_refl (CHead c3 k0 u2)) H11 H10))) u1 (sym_eq T u1 u0 H9))) k (sym_eq K
-k k0 H8))) c0 (sym_eq C c0 e1 H7))) H6)) H5)))]) in (H4 (refl_equal C (CHead
-e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c0 k0 u1) (CHead e1 k1
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u1)
+(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _)
+\Rightarrow c])) (CHead c3 k u1) (CHead e1 k0 u0) (drop_gen_refl (CHead c3 k
+u1) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k1 _) \Rightarrow k1])) (CHead c3 k u1) (CHead e1 k0 u0)
+(drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in ((let H6 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u1)
+(CHead e1 k0 u0) (drop_gen_refl (CHead c3 k u1) (CHead e1 k0 u0) H3)) in
+(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c3 e1)).(eq_ind K k (\lambda
+(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k
+u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind T u1 (\lambda (t:
+T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2)
+(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind C c3 (\lambda (c: C).(ex3_2 C
+T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(drop O O (CHead c4 k u2) (CHead e2 k u3)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u1 u3))) c4 u2 (drop_refl (CHead c4 k u2)) H0 H2) e1 H8) u0 H6) k0 H7))))
+H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
+C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u1) (CHead e1 k1
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 k0
u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) \to (\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c0 k0 u1) (CHead
+C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u1) (CHead
e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
-(CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+(CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b:
B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c0 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u2) (CHead e2
+(k0: K).((drop n O (CHead c3 (Bind b) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Bind b) u2) (CHead e2
k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Bind b) u1) (CHead
-e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c3
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda
-(u3: T).(drop (S n) O (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c3 (CHead
-x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u1) (CHead
+e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1
+k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
+O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
+n O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0
x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
-c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+c4 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop
-(Bind b) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 n e1 u0 k0
-(drop_gen_drop (Bind b) c0 (CHead e1 k0 u0) u1 n H4)))))))))) (\lambda (f:
+(Bind b) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f:
F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c0 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u2) (CHead e2
+(k0: K).((drop n O (CHead c3 (Flat f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c4 (Flat f) u2) (CHead e2
k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
C).(\lambda (u4: T).(pr0 u3 u4)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Flat f) u1) (CHead
-e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c3
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda
-(u3: T).(drop (S n) O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c3
-(CHead x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0
-x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
-c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop
-(Flat f) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 (S n) e1 u0 k0
-(drop_gen_drop (Flat f) c0 (CHead e1 k0 u0) u1 n H4)))))))))) k) h))))))))))
-c1 c2 H))).
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u1) (CHead
+e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead
+e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
+(S n) O c4 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
+e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(drop (S n) O (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
+(S n) O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0
+u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
+(CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1
+(drop_drop (Flat f) n c4 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k)
+h)))))))))) c1 c2 H))).
theorem wcpr0_drop_back:
\forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h:
u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead
e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
-c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
-(e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2:
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((drop h O c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda
+(e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u2) (CHead
+(e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c4 k u2) (CHead
e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead
-c0 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+c3 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u2)
-(CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c3 k u2) (CHead
-e1 k0 u0) H3) in eq return (\lambda (c: C).(\lambda (_: (eq ? ? c)).((eq C c
-(CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O
-(CHead c0 k u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))))) with [refl_equal
-\Rightarrow (\lambda (H4: (eq C (CHead c3 k u2) (CHead e1 k0 u0))).(let H5
-\def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
-with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k
-u2) (CHead e1 k0 u0) H4) in ((let H6 \def (f_equal C K (\lambda (e: C).(match
-e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k1
-_) \Rightarrow k1])) (CHead c3 k u2) (CHead e1 k0 u0) H4) in ((let H7 \def
-(f_equal C C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with
-[(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u2)
-(CHead e1 k0 u0) H4) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T
-u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c0 k
-u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
-(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))))) (\lambda (H8: (eq K k
-k0)).(eq_ind K k0 (\lambda (k1: K).((eq T u2 u0) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u3: T).(drop O O (CHead c0 k1 u1) (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0)))))) (\lambda (H9: (eq T u2 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C
-T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0
-u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
-C).(\lambda (u3: T).(pr0 u3 u0))))) (let H10 \def (eq_ind T u2 (\lambda (t:
-T).(pr0 u1 t)) H2 u0 H9) in (let H11 \def (eq_ind C c3 (\lambda (c: C).(wcpr0
-c0 c)) H0 e1 H7) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop
-O O (CHead c0 k0 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) c0 u1
-(drop_refl (CHead c0 k0 u1)) H11 H10))) u2 (sym_eq T u2 u0 H9))) k (sym_eq K
-k k0 H8))) c3 (sym_eq C c3 e1 H7))) H6)) H5)))]) in (H4 (refl_equal C (CHead
-e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c3 k0 u2) (CHead e1 k1
-u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 k0
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c4 k u2)
+(CHead e1 k0 u0))).(let H4 \def (f_equal C C (\lambda (e: C).(match e in C
+return (\lambda (_: C).C) with [(CSort _) \Rightarrow c4 | (CHead c _ _)
+\Rightarrow c])) (CHead c4 k u2) (CHead e1 k0 u0) (drop_gen_refl (CHead c4 k
+u2) (CHead e1 k0 u0) H3)) in ((let H5 \def (f_equal C K (\lambda (e:
+C).(match e in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k |
+(CHead _ k1 _) \Rightarrow k1])) (CHead c4 k u2) (CHead e1 k0 u0)
+(drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in ((let H6 \def
+(f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c4 k u2)
+(CHead e1 k0 u0) (drop_gen_refl (CHead c4 k u2) (CHead e1 k0 u0) H3)) in
+(\lambda (H7: (eq K k k0)).(\lambda (H8: (eq C c4 e1)).(eq_ind K k (\lambda
+(k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k
+u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind T u2 (\lambda (t:
+T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1)
+(CHead e2 k u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
+(_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind C c4 (\lambda (c: C).(ex3_2 C
+T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k
+u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_:
+C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2:
+C).(\lambda (u3: T).(drop O O (CHead c3 k u1) (CHead e2 k u3)))) (\lambda
+(e2: C).(\lambda (_: T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0
+u3 u2))) c3 u1 (drop_refl (CHead c3 k u1)) H0 H2) e1 H8) u0 H6) k0 H7))))
+H5)) H4)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1:
+C).(\forall (u3: T).(\forall (k1: K).((drop n O (CHead c4 k0 u2) (CHead e1 k1
+u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0
u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) \to (\forall (e1:
-C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u2) (CHead
+C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c4 k0 u2) (CHead
e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O
-(CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+(CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b:
B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c3 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 (Bind b) u1) (CHead e2
+(k0: K).((drop n O (CHead c4 (Bind b) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u1) (CHead e2
k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u2) (CHead
-e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c0
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda
-(u3: T).(drop (S n) O (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c0 (CHead
-x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Bind b) u2) (CHead
+e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c4 (CHead e1
+k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n
+O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
+(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
+n O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1
u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
-c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
+c3 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop
-(Bind b) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 n e1 u0 k0
-(drop_gen_drop (Bind b) c3 (CHead e1 k0 u0) u2 n H4)))))))))) (\lambda (f:
+(Bind b) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f:
F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall
-(k0: K).((drop n O (CHead c3 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
-(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 (Flat f) u1) (CHead e2
+(k0: K).((drop n O (CHead c4 (Flat f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T
+(\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u1) (CHead e2
k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
C).(\lambda (u4: T).(pr0 u4 u3)))))))))).(\lambda (e1: C).(\lambda (u0:
-T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u2) (CHead
-e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c0
-(CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda
-(_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda
-(u3: T).(drop (S n) O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda
-(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c0
-(CHead x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1
-u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead
-c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0
-e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop
-(Flat f) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 (S n) e1 u0 k0
-(drop_gen_drop (Flat f) c3 (CHead e1 k0 u0) u2 n H4)))))))))) k) h))))))))))
-c2 c1 H))).
+T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c4 (Flat f) u2) (CHead
+e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c4 (CHead
+e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop
+(S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
+e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2:
+C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
+(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
+(u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (drop
+(S n) O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0
+x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O
+(CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_:
+T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1
+(drop_drop (Flat f) n c3 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k)
+h)))))))))) c2 c1 H))).
theorem wcpr0_getl:
\forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h:
u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_:
C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
-c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(getl h c4 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1
u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 k u1) (CHead e1
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u1) (CHead e1
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c4 k
u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
(\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c0 k u1)
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c0 k1 u1) (CHead e1
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u1)
+(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u1) (CHead e1
+k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 k1
u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) (\lambda (b: B).(\lambda
-(H4: (clear (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C
+(H4: (clear (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C
C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c0
-(Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def
+\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3
+(Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def
(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
-(CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in ((let
+(CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in ((let
H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
-u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in
-(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c0)).(eq_ind_r K
+u0) (CHead c3 (Bind b) u1) (clear_gen_bind b c3 (CHead e1 k0 u0) u1 H4)) in
+(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K
(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
-O (CHead c3 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
+O (CHead c4 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r
T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
-(CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
+(CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
(_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3)))))
-(eq_ind_r C c0 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
-T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2:
+(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl O (CHead c4 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1
-u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3
+u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4
(Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 c0 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c3 u2
-(getl_refl b c3 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let H5
-\def (H1 O e1 u0 k0 (getl_intro O c0 (CHead e1 k0 u0) c0 (drop_refl c0)
-(clear_gen_flat f c0 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda
-(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2:
+T).(wcpr0 c3 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c4 u2
+(getl_refl b c4 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
+F).(\lambda (H4: (clear (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5
+\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3)
+(clear_gen_flat f c3 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda
+(e2: C).(\lambda (u3: T).(getl O c4 (CHead e2 k0 u3)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
-u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f)
+u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c4 (Flat f)
u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2)))
(\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
+(x1: T).(\lambda (H6: (getl O c4 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1
x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
-(u3: T).(getl O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2:
+(u3: T).(getl O (CHead c4 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0
-u3))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k
-(getl_gen_O (CHead c0 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
+u3))) x0 x1 (getl_flat c4 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))) k
+(getl_gen_O (CHead c3 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
-K).((getl n (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl n (CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda
+K).((getl n (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl n (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
u3 u4))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
-(S n) (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda
+(S n) (CHead c3 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl (S n) (CHead c4 k0 u2) (CHead e2 k1 u4)))) (\lambda
(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0
u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 (Bind b) u1)
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u1)
(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
-(CHead c3 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
+(CHead c4 (Bind b) u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
-u0 k0 (getl_gen_S (Bind b) c0 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda
+(getl (S n) (CHead c3 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
+u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(getl n c4 (CHead e2 k0 u3)))) (\lambda
(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
+u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
(Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7:
+C).(\lambda (x1: T).(\lambda (H6: (getl n c4 (CHead x0 k0 x1))).(\lambda (H7:
(wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u2) (CHead e2 k0 u3))))
+C).(\lambda (u3: T).(getl (S n) (CHead c4 (Bind b) u2) (CHead e2 k0 u3))))
(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u2)
+(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c4 (CHead x0 k0 x1) H6 u2)
H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 (Flat
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat
f) u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
-T).(getl n (CHead c3 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2:
+T).(getl n (CHead c4 (Flat f) u2) (CHead e2 k0 u4)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3
u4)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
-e1 u0 k0 (getl_gen_S (Flat f) c0 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda
+(getl (S n) (CHead c3 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
+e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c4 (CHead e2 k0 u3)))) (\lambda
(e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
+u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c4
(Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1
e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda
+C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c4 (CHead x0 k0 x1))).(\lambda
(H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u2) (CHead e2 k0 u3))))
+C).(\lambda (u3: T).(getl (S n) (CHead c4 (Flat f) u2) (CHead e2 k0 u3))))
(\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u2)
+(u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c4 (CHead x0 k0 x1) H6 u2)
H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))).
theorem wcpr0_getl_back:
u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2
k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_:
C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl
-u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0
-c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
-T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
+u1)))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (wcpr0 c3
+c4)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1:
+T).(\forall (k: K).((getl h c4 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda
(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2
u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1
u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u2) (CHead e1
-k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c0 k
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 k u2) (CHead e1
+k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k
u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
(\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1:
-C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u2)
-(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c3 k1 u2) (CHead e1
-k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 k1
+C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c4 k u2)
+(CHead e1 k0 u0))).(K_ind (\lambda (k1: K).((clear (CHead c4 k1 u2) (CHead e1
+k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1
u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) (\lambda (b: B).(\lambda
-(H4: (clear (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C
+(H4: (clear (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C
C (\lambda (e: C).(match e in C return (\lambda (_: C).C) with [(CSort _)
-\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3
-(Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def
+\Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c4
+(Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def
(f_equal C K (\lambda (e: C).(match e in C return (\lambda (_: C).K) with
[(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead e1 k0 u0)
-(CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in ((let
+(CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in ((let
H7 \def (f_equal C T (\lambda (e: C).(match e in C return (\lambda (_: C).T)
with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0
-u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in
-(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K
+u0) (CHead c4 (Bind b) u2) (clear_gen_bind b c4 (CHead e1 k0 u0) u2 H4)) in
+(\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c4)).(eq_ind_r K
(Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl
-O (CHead c0 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
+O (CHead c3 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_:
T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r
T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O
-(CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
+(CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda
(_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t)))))
-(eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
-T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2:
+(eq_ind_r C c4 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3:
+T).(getl O (CHead c3 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0
+u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3
(Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_:
-T).(wcpr0 e2 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c0 u1
-(getl_refl b c0 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
-F).(\lambda (H4: (clear (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let H5
-\def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3)
-(clear_gen_flat f c3 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda
-(e2: C).(\lambda (u3: T).(getl O c0 (CHead e2 k0 u3)))) (\lambda (e2:
+T).(wcpr0 e2 c4))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c3 u1
+(getl_refl b c3 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5)))) (\lambda (f:
+F).(\lambda (H4: (clear (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5
+\def (H1 O e1 u0 k0 (getl_intro O c4 (CHead e1 k0 u0) c4 (drop_refl c4)
+(clear_gen_flat f c4 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda
+(e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Flat f)
+u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f)
u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1)))
(\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda
-(x1: T).(\lambda (H6: (getl O c0 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
+(x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0
e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda
-(u3: T).(getl O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2:
+(u3: T).(getl O (CHead c3 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3
-u0))) x0 x1 (getl_flat c0 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k
-(getl_gen_O (CHead c3 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
+u0))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))) k
+(getl_gen_O (CHead c4 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0:
K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k1:
-K).((getl n (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl n (CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda
+K).((getl n (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl n (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
u4 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl
-(S n) (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
-C).(\lambda (u4: T).(getl (S n) (CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda
+(S n) (CHead c4 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2:
+C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u1) (CHead e2 k1 u4)))) (\lambda
(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0
u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall
-(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Bind b) u2)
+(e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Bind b) u2)
(CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n
-(CHead c0 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
+(CHead c3 (Bind b) u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_:
T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
-u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl n c0 (CHead e2 k0 u3)))) (\lambda
+(getl (S n) (CHead c4 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1
+u0 k0 (getl_gen_S (Bind b) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda
(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0
+u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
(Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl n c0 (CHead x0 k0 x1))).(\lambda (H7:
+C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7:
(wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c0 (Bind b) u1) (CHead e2 k0 u3))))
+C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u1) (CHead e2 k0 u3))))
(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c0 (CHead x0 k0 x1) H6 u1)
+(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u1)
H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_:
-((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 (Flat
+((\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c4 (Flat
f) u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4:
-T).(getl n (CHead c0 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2:
+T).(getl n (CHead c3 (Flat f) u1) (CHead e2 k0 u4)))) (\lambda (e2:
C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4
u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4:
-(getl (S n) (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
-e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
-(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c0 (CHead e2 k0 u3)))) (\lambda
+(getl (S n) (CHead c4 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n)
+e1 u0 k0 (getl_gen_S (Flat f) c4 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T
+(\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda
(e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0
-u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0
+u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3
(Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2
e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0:
-C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c0 (CHead x0 k0 x1))).(\lambda
+C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda
(H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2:
-C).(\lambda (u3: T).(getl (S n) (CHead c0 (Flat f) u1) (CHead e2 k0 u3))))
+C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u1) (CHead e2 k0 u3))))
(\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda
-(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c0 (CHead x0 k0 x1) H6 u1)
+(u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u1)
H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))).
+H=@
+
+MATITAOPTIONS=-onepass
+
DIR=$(shell basename $$PWD)
-MMAS = $(shell find -name "*.mma")
+MMAS = $(shell find Base-2 -name "*.mma")
MAS = $(MMAS:%.mma=%.ma)
%.ma: %.mma
- ../../matitac.opt $< 2> /dev/null
- ../../matitac.opt -dump $@ $< 2> /dev/null
- $(MAKE) depend.opt
+ $(H)../../matitac.opt $(MATITAOPTIONS) $< 2> /dev/null
+ $(H)../../matitac.opt $(MATITAOPTIONS) -dump $@ $< 2> /dev/null
+ $(H)$(MAKE) depend.opt
$(DIR) all: $(MAS)
- ../../matitac
+ $(H)../../matitac $(MATITAOPTIONS) 2> /dev/null
$(DIR).opt opt all.opt: $(MAS)
- ../../matitac.opt
+ $(H)../../matitac.opt $(MATITAOPTIONS) 2> /dev/null
clean:
- ../../matitaclean
- rm -f $(MAS)
- $(MAKE) depend
+ $(H)../../matitaclean
+ $(H)rm -f $(MAS)
+ $(H)$(MAKE) depend
clean.opt:
- ../../matitaclean.opt
- rm -f $(MAS)
- $(MAKE) depend.opt
+ $(H)../../matitaclean.opt
+ $(H)rm -f $(MAS)
+ $(H)$(MAKE) depend.opt
depend:
- ../../matitadep
- cat Base-2/depends_mma >> depends
+ $(H)../../matitadep
+ $(H)cat Base-2/depends >> depends
depend.opt:
- ../../matitadep.opt
- cat Base-2/depends_mma >> depends
+ $(H)../../matitadep.opt
+ $(H)cat Base-2/depends >> depends
include Base-2/.depend
(* *)
(**************************************************************************)
-
-
(* LIFT RELATION
- Usage: invoke with positive polarity
*)
-include "datatypes/Term.ma".
+include "Unified-Sub/datatypes/Term.ma".
inductive Lift (l: Nat): Nat \to Term \to Term \to Prop \def
| lift_sort : \forall i,h.
(* *)
(**************************************************************************)
-
-
-include "Lift/inv.ma".
+include "Unified-Sub/Lift/inv.ma".
(* Functional properties ****************************************************)
(* *)
(**************************************************************************)
-
-
-include "Lift/defs.ma".
+include "Unified-Sub/Lift/defs.ma".
(* Inversion properties *****************************************************)
(* *)
(**************************************************************************)
-
-
-include "Lift/fun.ma".
+include "Unified-Sub/Lift/fun.ma".
(* NOTE: this holds because of nplus_comm_1 *)
theorem lift_comp: \forall l1,i1,t1,t2. Lift l1 i1 t1 t2 \to
+++ /dev/null
-DIR=$(shell basename $$PWD)
-
-$(DIR) all:
- ../../../matitac
-$(DIR).opt opt all.opt:
- ../../../matitac.opt
-clean:
- ../../../matitaclean
-clean.opt:
- ../../../matitaclean.opt
-depend:
- ../matitadep
-depend.opt:
- ../matitadep.opt
(* *)
(**************************************************************************)
-
-
(* FLAT CONTEXTS
- Naming policy:
- contexts: c d
*)
-include "datatypes/Term.ma".
+include "Unified-Sub/datatypes/Term.ma".
inductive Context: Type \def
| leaf: Context
(* *)
(**************************************************************************)
-
-
(* POLARIZED TERMS
- Naming policy:
- natural numbers : sorts h k, local references i j, lengths l o
- terms : t u
*)
-include "preamble4.ma".
+include "Unified-Sub/preamble.ma".
inductive Bind: Type \def
| abbr: Bind
+++ /dev/null
-preamble4.ma NLE/nplus.ma NLE/props.ma NPlus/monoid.ma datatypes/Bool.ma logic/equality.ma
-datatypes/Context.ma datatypes/Term.ma
-datatypes/Term.ma preamble4.ma
-Lift/props.ma Lift/fun.ma
-Lift/inv.ma Lift/defs.ma
-Lift/defs.ma datatypes/Term.ma
-Lift/fun.ma Lift/inv.ma
-NLE/nplus.ma
-NLE/props.ma
-NPlus/monoid.ma
-datatypes/Bool.ma
-logic/equality.ma
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+(* Project started Tue Aug 22, 2006 ***************************************)
+
+(* PREAMBLE
+*)
+
+include "logic/equality.ma".
+include "datatypes/Bool.ma".
+include "NPlus/monoid.ma".
+include "NLE/props.ma".
+include "NLE/nplus.ma".
+
+axiom f_equal_3: \forall (A,B,C,D:Set).
+ \forall (f:A \to B \to C \to D).
+ \forall (x1,x2:A).
+ \forall (y1,y2:B).
+ \forall (z1,z2:C).
+ x1 = x2 \to y1 = y2 \to z1 = z2 \to
+ f x1 y1 z1 = f x2 y2 z2.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-(* Project started Tue Aug 22, 2006 ***************************************)
-
-
-
-(* PREAMBLE
-*)
-
-include "logic/equality.ma".
-include "datatypes/Bool.ma".
-include "NPlus/monoid.ma".
-include "NLE/props.ma".
-include "NLE/nplus.ma".
-
-axiom f_equal_3: \forall (A,B,C,D:Set).
- \forall (f:A \to B \to C \to D).
- \forall (x1,x2:A).
- \forall (y1,y2:B).
- \forall (z1,z2:C).
- x1 = x2 \to y1 = y2 \to z1 = z2 \to
- f x1 y1 z1 = f x2 y2 z2.
+++ /dev/null
-baseuri=cic:/matita/LAMBDA-TYPES/Unified-Sub
-include_paths= ../../RELATIONAL/
+Unified-Sub/preamble.ma NLE/nplus.ma NLE/props.ma NPlus/monoid.ma datatypes/Bool.ma logic/equality.ma
+Unified-Sub/datatypes/Term.ma Unified-Sub/preamble.ma
+Unified-Sub/datatypes/Context.ma Unified-Sub/datatypes/Term.ma
+Unified-Sub/Lift/inv.ma Unified-Sub/Lift/defs.ma
+Unified-Sub/Lift/defs.ma Unified-Sub/datatypes/Term.ma
+Unified-Sub/Lift/props.ma Unified-Sub/Lift/fun.ma
+Unified-Sub/Lift/fun.ma Unified-Sub/Lift/inv.ma
+LambdaDelta-2/preamble.ma Base-2/preamble.ma LambdaDelta-1/definitions.ma
+LambdaDelta-1/theory.ma LambdaDelta-1/ex0/props.ma LambdaDelta-1/ex1/props.ma LambdaDelta-1/ex2/props.ma LambdaDelta-1/pr3/wcpr0.ma LambdaDelta-1/subst0/tlt.ma LambdaDelta-1/tau1/cnt.ma LambdaDelta-1/ty3/dec.ma LambdaDelta-1/ty3/nf2.ma LambdaDelta-1/ty3/tau0.ma LambdaDelta-1/wcpr0/fwd.ma
+LambdaDelta-1/spare.ma LambdaDelta-1/theory.ma
+LambdaDelta-1/preamble.ma Base-1/theory.ma
+LambdaDelta-1/definitions.ma LambdaDelta-1/aprem/defs.ma LambdaDelta-1/cimp/defs.ma LambdaDelta-1/clen/defs.ma LambdaDelta-1/cnt/defs.ma LambdaDelta-1/csuba/defs.ma LambdaDelta-1/csubc/defs.ma LambdaDelta-1/csubst1/defs.ma LambdaDelta-1/csubt/defs.ma LambdaDelta-1/ex0/defs.ma LambdaDelta-1/ex1/defs.ma LambdaDelta-1/ex2/defs.ma LambdaDelta-1/flt/defs.ma LambdaDelta-1/fsubst0/defs.ma LambdaDelta-1/iso/defs.ma LambdaDelta-1/llt/defs.ma LambdaDelta-1/next_plus/defs.ma LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pc1/defs.ma LambdaDelta-1/subst1/defs.ma LambdaDelta-1/tau1/defs.ma LambdaDelta-1/tlt/defs.ma LambdaDelta-1/wcpr0/defs.ma
+LambdaDelta-1/ex0/defs.ma LambdaDelta-1/A/defs.ma LambdaDelta-1/G/defs.ma
+LambdaDelta-1/ex0/props.ma LambdaDelta-1/aplus/props.ma LambdaDelta-1/ex0/defs.ma LambdaDelta-1/leq/defs.ma
+LambdaDelta-1/ex2/defs.ma LambdaDelta-1/C/defs.ma
+LambdaDelta-1/ex2/props.ma LambdaDelta-1/arity/fwd.ma LambdaDelta-1/ex2/defs.ma LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pr2/fwd.ma
+LambdaDelta-1/tlist/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/tlist/props.ma LambdaDelta-1/tlist/defs.ma
+LambdaDelta-1/csubt/defs.ma LambdaDelta-1/ty3/defs.ma
+LambdaDelta-1/csubt/fwd.ma LambdaDelta-1/csubt/defs.ma
+LambdaDelta-1/csubt/props.ma LambdaDelta-1/csubt/defs.ma
+LambdaDelta-1/csubt/clear.ma LambdaDelta-1/clear/fwd.ma LambdaDelta-1/csubt/defs.ma
+LambdaDelta-1/csubt/drop.ma LambdaDelta-1/csubt/defs.ma LambdaDelta-1/drop/fwd.ma
+LambdaDelta-1/csubt/getl.ma LambdaDelta-1/csubt/clear.ma LambdaDelta-1/csubt/drop.ma LambdaDelta-1/csubt/fwd.ma LambdaDelta-1/getl/clear.ma
+LambdaDelta-1/csubt/pc3.ma LambdaDelta-1/csubt/getl.ma LambdaDelta-1/pc3/left.ma
+LambdaDelta-1/csubt/ty3.ma LambdaDelta-1/csubt/pc3.ma LambdaDelta-1/csubt/props.ma
+LambdaDelta-1/csubc/defs.ma LambdaDelta-1/sc3/defs.ma
+LambdaDelta-1/csubc/props.ma LambdaDelta-1/csubc/defs.ma LambdaDelta-1/sc3/props.ma
+LambdaDelta-1/csubc/csuba.ma LambdaDelta-1/csubc/defs.ma LambdaDelta-1/sc3/props.ma
+LambdaDelta-1/csubc/drop.ma LambdaDelta-1/csubc/fwd.ma LambdaDelta-1/sc3/props.ma
+LambdaDelta-1/csubc/drop1.ma LambdaDelta-1/csubc/drop.ma
+LambdaDelta-1/csubc/clear.ma LambdaDelta-1/csubc/fwd.ma
+LambdaDelta-1/csubc/getl.ma LambdaDelta-1/csubc/clear.ma LambdaDelta-1/csubc/drop.ma
+LambdaDelta-1/csubc/arity.ma LambdaDelta-1/csubc/csuba.ma
+LambdaDelta-1/csubc/fwd.ma LambdaDelta-1/csubc/defs.ma
+LambdaDelta-1/ty3/fsubst0.ma LambdaDelta-1/csubst0/props.ma LambdaDelta-1/getl/getl.ma LambdaDelta-1/pc3/fsubst0.ma LambdaDelta-1/ty3/props.ma
+LambdaDelta-1/ty3/defs.ma LambdaDelta-1/G/defs.ma LambdaDelta-1/pc3/defs.ma
+LambdaDelta-1/ty3/fwd.ma LambdaDelta-1/pc3/props.ma LambdaDelta-1/ty3/defs.ma
+LambdaDelta-1/ty3/props.ma LambdaDelta-1/pc3/fwd.ma LambdaDelta-1/ty3/fwd.ma
+LambdaDelta-1/ty3/subst1.ma LambdaDelta-1/getl/getl.ma LambdaDelta-1/pc3/subst1.ma LambdaDelta-1/ty3/props.ma
+LambdaDelta-1/ty3/pr3_props.ma LambdaDelta-1/ty3/pr3.ma
+LambdaDelta-1/ty3/pr3.ma LambdaDelta-1/csubt/ty3.ma LambdaDelta-1/pc1/props.ma LambdaDelta-1/pc3/pc1.ma LambdaDelta-1/pc3/wcpr0.ma LambdaDelta-1/ty3/fsubst0.ma LambdaDelta-1/ty3/subst1.ma
+LambdaDelta-1/ty3/tau0.ma LambdaDelta-1/tau0/defs.ma LambdaDelta-1/ty3/pr3_props.ma
+LambdaDelta-1/ty3/arity.ma LambdaDelta-1/arity/pr3.ma LambdaDelta-1/asucc/fwd.ma LambdaDelta-1/ty3/pr3_props.ma
+LambdaDelta-1/ty3/dec.ma LambdaDelta-1/getl/dec.ma LambdaDelta-1/getl/flt.ma LambdaDelta-1/pc3/dec.ma
+LambdaDelta-1/ty3/arity_props.ma LambdaDelta-1/sc3/arity.ma LambdaDelta-1/ty3/arity.ma
+LambdaDelta-1/ty3/nf2.ma LambdaDelta-1/nf2/arity.ma LambdaDelta-1/pc3/nf2.ma LambdaDelta-1/ty3/arity.ma
+LambdaDelta-1/pc3/defs.ma LambdaDelta-1/pr3/defs.ma
+LambdaDelta-1/pc3/props.ma LambdaDelta-1/pc3/defs.ma LambdaDelta-1/pr3/pr3.ma
+LambdaDelta-1/pc3/fsubst0.ma LambdaDelta-1/csubst0/getl.ma LambdaDelta-1/fsubst0/defs.ma LambdaDelta-1/pc3/left.ma
+LambdaDelta-1/pc3/pc1.ma LambdaDelta-1/pc1/defs.ma LambdaDelta-1/pc3/defs.ma LambdaDelta-1/pr3/pr1.ma
+LambdaDelta-1/pc3/wcpr0.ma LambdaDelta-1/pc3/props.ma LambdaDelta-1/wcpr0/getl.ma
+LambdaDelta-1/pc3/left.ma LambdaDelta-1/pc3/props.ma
+LambdaDelta-1/pc3/fwd.ma LambdaDelta-1/pc3/props.ma LambdaDelta-1/pr3/fwd.ma
+LambdaDelta-1/pc3/nf2.ma LambdaDelta-1/nf2/pr3.ma LambdaDelta-1/pc3/defs.ma
+LambdaDelta-1/pc3/subst1.ma LambdaDelta-1/pc3/props.ma LambdaDelta-1/pr3/subst1.ma
+LambdaDelta-1/pc3/dec.ma LambdaDelta-1/nf2/fwd.ma LambdaDelta-1/ty3/arity_props.ma
+LambdaDelta-1/pc1/defs.ma LambdaDelta-1/pr1/defs.ma
+LambdaDelta-1/pc1/props.ma LambdaDelta-1/pc1/defs.ma LambdaDelta-1/pr1/pr1.ma
+LambdaDelta-1/ex1/defs.ma LambdaDelta-1/C/defs.ma
+LambdaDelta-1/ex1/props.ma LambdaDelta-1/arity/defs.ma LambdaDelta-1/ex1/defs.ma LambdaDelta-1/leq/props.ma LambdaDelta-1/nf2/pr3.ma LambdaDelta-1/nf2/props.ma LambdaDelta-1/pc3/fwd.ma LambdaDelta-1/ty3/fwd.ma
+LambdaDelta-1/sc3/defs.ma LambdaDelta-1/arity/defs.ma LambdaDelta-1/drop1/defs.ma LambdaDelta-1/sn3/defs.ma
+LambdaDelta-1/sc3/props.ma LambdaDelta-1/arity/aprem.ma LambdaDelta-1/arity/lift1.ma LambdaDelta-1/csuba/arity.ma LambdaDelta-1/drop1/getl.ma LambdaDelta-1/drop1/props.ma LambdaDelta-1/lift1/props.ma LambdaDelta-1/llt/props.ma LambdaDelta-1/nf2/lift1.ma LambdaDelta-1/sc3/defs.ma LambdaDelta-1/sn3/lift1.ma
+LambdaDelta-1/sc3/arity.ma LambdaDelta-1/csubc/arity.ma LambdaDelta-1/csubc/drop1.ma LambdaDelta-1/csubc/getl.ma LambdaDelta-1/csubc/props.ma
+LambdaDelta-1/sn3/defs.ma LambdaDelta-1/pr3/defs.ma
+LambdaDelta-1/sn3/nf2.ma LambdaDelta-1/nf2/dec.ma LambdaDelta-1/nf2/pr3.ma LambdaDelta-1/sn3/defs.ma
+LambdaDelta-1/sn3/props.ma LambdaDelta-1/nf2/iso.ma LambdaDelta-1/pr3/iso.ma LambdaDelta-1/sn3/fwd.ma LambdaDelta-1/sn3/nf2.ma
+LambdaDelta-1/sn3/lift1.ma LambdaDelta-1/drop1/defs.ma LambdaDelta-1/lift1/fwd.ma LambdaDelta-1/sn3/props.ma
+LambdaDelta-1/sn3/fwd.ma LambdaDelta-1/pr3/props.ma LambdaDelta-1/sn3/defs.ma
+LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pr2/defs.ma
+LambdaDelta-1/nf2/fwd.ma LambdaDelta-1/T/props.ma LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pr2/clen.ma LambdaDelta-1/subst0/dec.ma
+LambdaDelta-1/nf2/props.ma LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pr2/fwd.ma
+LambdaDelta-1/nf2/arity.ma LambdaDelta-1/arity/subst0.ma LambdaDelta-1/nf2/fwd.ma
+LambdaDelta-1/nf2/pr3.ma LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pr3/pr3.ma
+LambdaDelta-1/nf2/lift1.ma LambdaDelta-1/drop1/defs.ma LambdaDelta-1/nf2/props.ma
+LambdaDelta-1/nf2/iso.ma LambdaDelta-1/iso/props.ma LambdaDelta-1/nf2/pr3.ma LambdaDelta-1/pr3/fwd.ma
+LambdaDelta-1/nf2/dec.ma LambdaDelta-1/C/props.ma LambdaDelta-1/nf2/defs.ma LambdaDelta-1/pr0/dec.ma LambdaDelta-1/pr2/clen.ma LambdaDelta-1/pr2/fwd.ma
+LambdaDelta-1/csuba/defs.ma LambdaDelta-1/arity/defs.ma
+LambdaDelta-1/csuba/fwd.ma LambdaDelta-1/csuba/defs.ma
+LambdaDelta-1/csuba/props.ma LambdaDelta-1/csuba/defs.ma
+LambdaDelta-1/csuba/arity.ma LambdaDelta-1/T/props.ma LambdaDelta-1/arity/props.ma LambdaDelta-1/csuba/getl.ma LambdaDelta-1/csuba/props.ma
+LambdaDelta-1/csuba/clear.ma LambdaDelta-1/clear/fwd.ma LambdaDelta-1/csuba/defs.ma
+LambdaDelta-1/csuba/drop.ma LambdaDelta-1/csuba/fwd.ma LambdaDelta-1/drop/fwd.ma
+LambdaDelta-1/csuba/getl.ma LambdaDelta-1/csuba/clear.ma LambdaDelta-1/csuba/drop.ma LambdaDelta-1/getl/clear.ma
+LambdaDelta-1/wcpr0/getl.ma LambdaDelta-1/getl/props.ma LambdaDelta-1/wcpr0/defs.ma
+LambdaDelta-1/wcpr0/fwd.ma LambdaDelta-1/wcpr0/defs.ma
+LambdaDelta-1/wcpr0/defs.ma LambdaDelta-1/C/defs.ma LambdaDelta-1/pr0/defs.ma
+LambdaDelta-1/pr3/defs.ma LambdaDelta-1/pr2/defs.ma
+LambdaDelta-1/pr3/subst1.ma LambdaDelta-1/pr2/subst1.ma LambdaDelta-1/pr3/defs.ma
+LambdaDelta-1/pr3/props.ma LambdaDelta-1/pr1/props.ma LambdaDelta-1/pr2/props.ma LambdaDelta-1/pr3/pr1.ma
+LambdaDelta-1/pr3/pr1.ma LambdaDelta-1/pr1/defs.ma LambdaDelta-1/pr3/defs.ma
+LambdaDelta-1/pr3/wcpr0.ma LambdaDelta-1/pr3/props.ma LambdaDelta-1/wcpr0/getl.ma
+LambdaDelta-1/pr3/fwd.ma LambdaDelta-1/pr2/fwd.ma LambdaDelta-1/pr3/props.ma
+LambdaDelta-1/pr3/pr3.ma LambdaDelta-1/pr2/pr2.ma LambdaDelta-1/pr3/props.ma
+LambdaDelta-1/pr3/iso.ma LambdaDelta-1/iso/props.ma LambdaDelta-1/pr3/fwd.ma LambdaDelta-1/tlist/props.ma
+LambdaDelta-1/pr2/defs.ma LambdaDelta-1/getl/defs.ma LambdaDelta-1/pr0/defs.ma
+LambdaDelta-1/pr2/props.ma LambdaDelta-1/getl/clear.ma LambdaDelta-1/getl/drop.ma LambdaDelta-1/pr0/props.ma LambdaDelta-1/pr2/defs.ma
+LambdaDelta-1/pr2/fwd.ma LambdaDelta-1/getl/clear.ma LambdaDelta-1/getl/drop.ma LambdaDelta-1/pr0/fwd.ma LambdaDelta-1/pr2/defs.ma
+LambdaDelta-1/pr2/subst1.ma LambdaDelta-1/csubst1/fwd.ma LambdaDelta-1/csubst1/getl.ma LambdaDelta-1/getl/drop.ma LambdaDelta-1/pr0/fwd.ma LambdaDelta-1/pr0/subst1.ma LambdaDelta-1/pr2/defs.ma LambdaDelta-1/subst1/subst1.ma
+LambdaDelta-1/pr2/clen.ma LambdaDelta-1/clen/getl.ma LambdaDelta-1/pr2/props.ma
+LambdaDelta-1/pr2/pr2.ma LambdaDelta-1/getl/props.ma LambdaDelta-1/pr0/pr0.ma LambdaDelta-1/pr2/defs.ma
+LambdaDelta-1/pr1/defs.ma LambdaDelta-1/pr0/defs.ma
+LambdaDelta-1/pr1/props.ma LambdaDelta-1/T/props.ma LambdaDelta-1/pr0/subst1.ma LambdaDelta-1/pr1/defs.ma LambdaDelta-1/subst1/props.ma
+LambdaDelta-1/pr1/pr1.ma LambdaDelta-1/pr0/pr0.ma LambdaDelta-1/pr1/props.ma
+LambdaDelta-1/pr0/subst1.ma LambdaDelta-1/pr0/props.ma LambdaDelta-1/subst1/defs.ma
+LambdaDelta-1/pr0/defs.ma LambdaDelta-1/subst0/defs.ma
+LambdaDelta-1/pr0/fwd.ma LambdaDelta-1/pr0/props.ma
+LambdaDelta-1/pr0/props.ma LambdaDelta-1/pr0/defs.ma LambdaDelta-1/subst0/subst0.ma
+LambdaDelta-1/pr0/pr0.ma LambdaDelta-1/lift/tlt.ma LambdaDelta-1/pr0/fwd.ma
+LambdaDelta-1/pr0/dec.ma LambdaDelta-1/T/dec.ma LambdaDelta-1/T/props.ma LambdaDelta-1/pr0/fwd.ma LambdaDelta-1/subst0/dec.ma
+LambdaDelta-1/tau1/defs.ma LambdaDelta-1/tau0/defs.ma
+LambdaDelta-1/tau1/props.ma LambdaDelta-1/tau0/props.ma LambdaDelta-1/tau1/defs.ma
+LambdaDelta-1/tau1/cnt.ma LambdaDelta-1/cnt/props.ma LambdaDelta-1/tau1/props.ma
+LambdaDelta-1/arity/defs.ma LambdaDelta-1/getl/defs.ma LambdaDelta-1/leq/defs.ma
+LambdaDelta-1/arity/fwd.ma LambdaDelta-1/arity/defs.ma LambdaDelta-1/getl/drop.ma LambdaDelta-1/leq/asucc.ma LambdaDelta-1/leq/fwd.ma
+LambdaDelta-1/arity/subst0.ma LambdaDelta-1/arity/props.ma LambdaDelta-1/csubst0/getl.ma LambdaDelta-1/csubst0/props.ma LambdaDelta-1/fsubst0/fwd.ma LambdaDelta-1/getl/getl.ma LambdaDelta-1/subst0/dec.ma LambdaDelta-1/subst0/fwd.ma
+LambdaDelta-1/arity/props.ma LambdaDelta-1/arity/fwd.ma
+LambdaDelta-1/arity/pr3.ma LambdaDelta-1/arity/subst0.ma LambdaDelta-1/csuba/arity.ma LambdaDelta-1/pr0/fwd.ma LambdaDelta-1/pr1/defs.ma LambdaDelta-1/pr3/defs.ma LambdaDelta-1/wcpr0/getl.ma
+LambdaDelta-1/arity/lift1.ma LambdaDelta-1/arity/props.ma LambdaDelta-1/drop1/defs.ma
+LambdaDelta-1/arity/cimp.ma LambdaDelta-1/arity/defs.ma LambdaDelta-1/cimp/props.ma
+LambdaDelta-1/arity/aprem.ma LambdaDelta-1/aprem/props.ma LambdaDelta-1/arity/cimp.ma LambdaDelta-1/arity/props.ma
+LambdaDelta-1/aprem/defs.ma LambdaDelta-1/A/defs.ma
+LambdaDelta-1/aprem/props.ma LambdaDelta-1/aprem/defs.ma LambdaDelta-1/leq/defs.ma
+LambdaDelta-1/llt/defs.ma LambdaDelta-1/A/defs.ma
+LambdaDelta-1/llt/props.ma LambdaDelta-1/leq/defs.ma LambdaDelta-1/llt/defs.ma
+LambdaDelta-1/leq/defs.ma LambdaDelta-1/aplus/defs.ma
+LambdaDelta-1/leq/asucc.ma LambdaDelta-1/leq/props.ma
+LambdaDelta-1/leq/props.ma LambdaDelta-1/aplus/props.ma LambdaDelta-1/leq/defs.ma
+LambdaDelta-1/leq/fwd.ma LambdaDelta-1/leq/defs.ma
+LambdaDelta-1/aplus/defs.ma LambdaDelta-1/asucc/defs.ma
+LambdaDelta-1/aplus/props.ma LambdaDelta-1/aplus/defs.ma LambdaDelta-1/next_plus/props.ma
+LambdaDelta-1/asucc/defs.ma LambdaDelta-1/A/defs.ma LambdaDelta-1/G/defs.ma
+LambdaDelta-1/asucc/fwd.ma LambdaDelta-1/asucc/defs.ma
+LambdaDelta-1/A/defs.ma LambdaDelta-1/preamble.ma
+LambdaDelta-1/fsubst0/defs.ma LambdaDelta-1/csubst0/defs.ma
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+LambdaDelta-1/csubst1/fwd.ma LambdaDelta-1/csubst0/fwd.ma LambdaDelta-1/csubst1/defs.ma LambdaDelta-1/subst1/props.ma
+LambdaDelta-1/csubst1/getl.ma LambdaDelta-1/csubst0/getl.ma LambdaDelta-1/csubst0/props.ma LambdaDelta-1/csubst1/props.ma LambdaDelta-1/drop/props.ma LambdaDelta-1/subst1/props.ma
+LambdaDelta-1/csubst0/defs.ma LambdaDelta-1/C/defs.ma LambdaDelta-1/subst0/defs.ma
+LambdaDelta-1/csubst0/fwd.ma LambdaDelta-1/csubst0/defs.ma
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+LambdaDelta-1/subst0/fwd.ma LambdaDelta-1/lift/props.ma LambdaDelta-1/subst0/defs.ma
+LambdaDelta-1/subst0/subst0.ma LambdaDelta-1/subst0/props.ma
+LambdaDelta-1/subst0/props.ma LambdaDelta-1/subst0/fwd.ma
+LambdaDelta-1/subst0/tlt.ma LambdaDelta-1/lift/props.ma LambdaDelta-1/lift/tlt.ma LambdaDelta-1/subst0/defs.ma
+LambdaDelta-1/subst0/dec.ma LambdaDelta-1/lift/props.ma LambdaDelta-1/subst0/defs.ma
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+LambdaDelta-1/tau0/props.ma LambdaDelta-1/getl/drop.ma LambdaDelta-1/tau0/defs.ma
+LambdaDelta-1/subst1/subst1.ma LambdaDelta-1/subst0/subst0.ma LambdaDelta-1/subst1/fwd.ma
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+LambdaDelta-1/subst1/props.ma LambdaDelta-1/subst0/props.ma LambdaDelta-1/subst1/defs.ma
+LambdaDelta-1/cimp/defs.ma LambdaDelta-1/getl/defs.ma
+LambdaDelta-1/cimp/props.ma LambdaDelta-1/cimp/defs.ma LambdaDelta-1/getl/getl.ma
+LambdaDelta-1/getl/defs.ma LambdaDelta-1/clear/defs.ma LambdaDelta-1/drop/defs.ma
+LambdaDelta-1/getl/props.ma LambdaDelta-1/clear/props.ma LambdaDelta-1/drop/props.ma LambdaDelta-1/getl/fwd.ma
+LambdaDelta-1/getl/fwd.ma LambdaDelta-1/clear/fwd.ma LambdaDelta-1/drop/fwd.ma LambdaDelta-1/getl/defs.ma
+LambdaDelta-1/getl/drop.ma LambdaDelta-1/clear/drop.ma LambdaDelta-1/getl/props.ma
+LambdaDelta-1/getl/clear.ma LambdaDelta-1/clear/drop.ma LambdaDelta-1/getl/props.ma
+LambdaDelta-1/getl/getl.ma LambdaDelta-1/getl/clear.ma LambdaDelta-1/getl/drop.ma
+LambdaDelta-1/getl/dec.ma LambdaDelta-1/getl/props.ma
+LambdaDelta-1/getl/flt.ma LambdaDelta-1/clear/props.ma LambdaDelta-1/flt/props.ma LambdaDelta-1/getl/fwd.ma
+LambdaDelta-1/clear/defs.ma LambdaDelta-1/C/defs.ma
+LambdaDelta-1/clear/fwd.ma LambdaDelta-1/clear/defs.ma
+LambdaDelta-1/clear/props.ma LambdaDelta-1/clear/fwd.ma
+LambdaDelta-1/clear/drop.ma LambdaDelta-1/clear/fwd.ma LambdaDelta-1/drop/fwd.ma
+LambdaDelta-1/drop1/defs.ma LambdaDelta-1/drop/defs.ma LambdaDelta-1/lift1/defs.ma
+LambdaDelta-1/drop1/props.ma LambdaDelta-1/drop/props.ma LambdaDelta-1/drop1/defs.ma LambdaDelta-1/getl/defs.ma
+LambdaDelta-1/drop1/getl.ma LambdaDelta-1/drop1/defs.ma LambdaDelta-1/getl/drop.ma
+LambdaDelta-1/drop/defs.ma LambdaDelta-1/C/defs.ma LambdaDelta-1/lift/defs.ma LambdaDelta-1/r/defs.ma
+LambdaDelta-1/drop/fwd.ma LambdaDelta-1/drop/defs.ma
+LambdaDelta-1/drop/props.ma LambdaDelta-1/drop/fwd.ma LambdaDelta-1/lift/props.ma LambdaDelta-1/r/props.ma
+LambdaDelta-1/lift1/defs.ma LambdaDelta-1/lift/defs.ma
+LambdaDelta-1/lift1/fwd.ma LambdaDelta-1/lift/fwd.ma LambdaDelta-1/lift1/defs.ma
+LambdaDelta-1/lift1/props.ma LambdaDelta-1/drop1/defs.ma LambdaDelta-1/lift/props.ma
+LambdaDelta-1/cnt/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/cnt/props.ma LambdaDelta-1/cnt/defs.ma LambdaDelta-1/lift/fwd.ma
+LambdaDelta-1/lift/defs.ma LambdaDelta-1/s/defs.ma LambdaDelta-1/tlist/defs.ma
+LambdaDelta-1/lift/fwd.ma LambdaDelta-1/lift/defs.ma
+LambdaDelta-1/lift/props.ma LambdaDelta-1/lift/fwd.ma LambdaDelta-1/s/props.ma
+LambdaDelta-1/lift/tlt.ma LambdaDelta-1/lift/fwd.ma LambdaDelta-1/tlt/props.ma
+LambdaDelta-1/iso/props.ma LambdaDelta-1/iso/fwd.ma
+LambdaDelta-1/iso/fwd.ma LambdaDelta-1/iso/defs.ma LambdaDelta-1/tlist/defs.ma
+LambdaDelta-1/iso/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/G/defs.ma LambdaDelta-1/preamble.ma
+LambdaDelta-1/flt/props.ma LambdaDelta-1/C/props.ma LambdaDelta-1/flt/defs.ma
+LambdaDelta-1/flt/defs.ma LambdaDelta-1/C/defs.ma
+LambdaDelta-1/tlt/props.ma LambdaDelta-1/tlt/defs.ma
+LambdaDelta-1/tlt/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/T/props.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/T/dec.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/T/defs.ma LambdaDelta-1/preamble.ma
+LambdaDelta-1/s/props.ma LambdaDelta-1/s/defs.ma
+LambdaDelta-1/s/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/C/props.ma LambdaDelta-1/C/defs.ma LambdaDelta-1/T/props.ma
+LambdaDelta-1/C/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/r/props.ma LambdaDelta-1/r/defs.ma LambdaDelta-1/s/defs.ma
+LambdaDelta-1/r/defs.ma LambdaDelta-1/T/defs.ma
+LambdaDelta-1/next_plus/props.ma LambdaDelta-1/next_plus/defs.ma
+LambdaDelta-1/next_plus/defs.ma LambdaDelta-1/G/defs.ma
+LambdaDelta-1/clen/getl.ma LambdaDelta-1/clen/defs.ma LambdaDelta-1/getl/props.ma
+LambdaDelta-1/clen/defs.ma LambdaDelta-1/C/defs.ma LambdaDelta-1/s/defs.ma
Base-1/theory.ma Base-1/blt/props.ma Base-1/ext/arith.ma Base-1/ext/tactics.ma Base-1/plist/props.ma Base-1/types/props.ma
Base-1/definitions.ma Base-1/blt/defs.ma Base-1/plist/defs.ma Base-1/types/defs.ma
Base-1/preamble.ma coq.ma
Base-2/ext/tactics.ma Base-2/preamble.ma
Base-2/blt/defs.ma Base-2/preamble.ma
Base-2/blt/props.ma Base-2/blt/defs.ma
+NLE/nplus.ma
+NLE/props.ma
+NPlus/monoid.ma
coq.ma
+datatypes/Bool.ma
+logic/equality.ma
Base-2/theory.mma Base-2/preamble.ma Base-1/theory.ma
Base-2/ext/tactics.mma Base-2/preamble.ma Base-1/ext/tactics.ma
Base-2/ext/arith.mma Base-2/preamble.ma Base-1/ext/arith.ma
+++ /dev/null
-\e[0;32mInfo: \e[0mexecution of blt/defs.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/blt ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./blt/defs.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of blt/defs.mma completed in 1''.
-\e[0;32mInfo: \e[0mexecution of blt/props.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/blt ...''
-\e[0;32mInfo: \e[0mbaseuri cic:/matita/LAMBDA-TYPES/Base-2/blt/props is not empty
-\e[0;32mInfo: \e[0mcleaning baseuri cic:/matita/LAMBDA-TYPES/Base-2/blt/props
-\e[0;32mInfo: \e[0mRemoving: cic:/matita/LAMBDA-TYPES/Base-2/blt/props/*
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./blt/defs.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/blt/defs" ...''
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_blt
-Pre Nodes : 257
-nat
-nat
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: nested application
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((lt y n)\to (eq bool (blt y n) true)))) (\lambda y:nat.(\lambda H:(lt y O). let H0 \def (le_ind (S y) (\lambda n:nat.((eq nat n O)\to (eq bool (blt y O) true))) (\lambda H0:(eq nat (S y) O). let H1 \def (eq_ind nat (S y) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind (eq bool (blt y O) true) H1)) (\lambda m:nat.(\lambda H0:(le (S y) m).(\lambda _:((eq nat m O)\to (eq bool (blt y O) true)).(\lambda H1:(eq nat (S m) O). let H2 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in let DEFINED \def (False_ind ((le (S y) m)\to (eq bool (blt y O) true)) H2) in (DEFINED H0))))) O H) in (H0 (refl_equal nat O)))) (\lambda n:nat.(\lambda H:(\forall y:nat.((lt y n)\to (eq bool (blt y n) true))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((lt n0 (S n))\to (eq bool (blt n0 (S n)) true))) (\lambda _:(lt O (S n)).(refl_equal bool true)) (\lambda n0:nat.(\lambda _:((lt n0 (S n))\to (eq bool
-match n0 return (\lambda n1:nat.bool) with
- [ O => true
- | (S (m:nat)) => (blt m n)
-] true)).(\lambda H1:(lt (S n0) (S n)).(H n0 (le_S_n (S n0) n H1))))) y)))) x))
-Post Nodes: 272
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_bge
-Pre Nodes : 255
-nat
-nat
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: nested application
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((le n y)\to (eq bool (blt y n) false)))) (\lambda y:nat.(\lambda _:(le O y).(refl_equal bool false))) (\lambda n:nat.(\lambda H:(\forall y:nat.((le n y)\to (eq bool (blt y n) false))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((le (S n) n0)\to (eq bool (blt n0 (S n)) false))) (\lambda H0:(le (S n) O). let H1 \def (le_ind (S n) (\lambda n0:nat.((eq nat n0 O)\to (eq bool (blt O (S n)) false))) (\lambda H1:(eq nat (S n) O). let H2 \def (eq_ind nat (S n) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in (False_ind (eq bool (blt O (S n)) false) H2)) (\lambda m:nat.(\lambda H1:(le (S n) m).(\lambda _:((eq nat m O)\to (eq bool (blt O (S n)) false)).(\lambda H2:(eq nat (S m) O). let H3 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H2) in let DEFINED \def (False_ind ((le (S n) m)\to (eq bool (blt O (S n)) false)) H3) in (DEFINED H1))))) O H0) in (H1 (refl_equal nat O))) (\lambda n0:nat.(\lambda _:((le (S n) n0)\to (eq bool (blt n0 (S n)) false)).(\lambda H1:(le (S n) (S n0)).(H n0 (le_S_n n n0 H1))))) y)))) x))
-Post Nodes: 272
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: blt_lt
-Pre Nodes : 221
-bool
-bool
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((eq bool (blt y n) true)\to (lt y n)))) (\lambda y:nat.(\lambda H:(eq bool (blt y O) true). let H0 \def (eq_ind bool (blt y O) (\lambda b:bool.((eq bool b true)\to (lt y O))) (\lambda H0:(eq bool (blt y O) true). let H1 \def (eq_ind bool (blt y O) (\lambda e:bool.
-match e return (\lambda _:bool.Prop) with
- [ true => False
- | false => True
-]) I true H0) in (False_ind (lt y O) H1)) true H) in (H0 (refl_equal bool true)))) (\lambda n:nat.(\lambda H:(\forall y:nat.((eq bool (blt y n) true)\to (lt y n))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((eq bool (blt n0 (S n)) true)\to (lt n0 (S n)))) (\lambda _:(eq bool true true).(le_S_n (S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda n0:nat.(\lambda _:((eq bool
-match n0 return (\lambda n1:nat.bool) with
- [ O => true
- | (S (m:nat)) => (blt m n)
-] true)\to (lt n0 (S n))).(\lambda H1:(eq bool (blt n0 n) true).(lt_le_S (S n0) (S n) (lt_n_S n0 n (H n0 H1)))))) y)))) x))
-Post Nodes: 219
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: bge_le
-Pre Nodes : 222
-bool
-bool
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((eq bool (blt y n) false)\to (le n y)))) (\lambda y:nat.(\lambda _:(eq bool (blt y O) false).(le_O_n y))) (\lambda n:nat.(\lambda H:(\forall y:nat.((eq bool (blt y n) false)\to (le n y))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((eq bool (blt n0 (S n)) false)\to (le (S n) n0))) (\lambda H0:(eq bool (blt O (S n)) false). let H1 \def (eq_ind bool (blt O (S n)) (\lambda b:bool.((eq bool b false)\to (le (S n) O))) (\lambda H1:(eq bool (blt O (S n)) false). let H2 \def (eq_ind bool (blt O (S n)) (\lambda e:bool.
-match e return (\lambda _:bool.Prop) with
- [ true => True
- | false => False
-]) I false H1) in (False_ind (le (S n) O) H2)) false H0) in (H1 (refl_equal bool false))) (\lambda n0:nat.(\lambda _:((eq bool (blt n0 (S n)) false)\to (le (S n) n0)).(\lambda H1:(eq bool (blt (S n0) (S n)) false).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 H1))))))) y)))) x))
-Post Nodes: 220
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./blt/props.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of blt/props.mma completed in 7''.
-\e[0;32mInfo: \e[0mexecution of ext/arith.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/ext ...''
-\e[0;32mInfo: \e[0mbaseuri cic:/matita/LAMBDA-TYPES/Base-2/ext/arith is not empty
-\e[0;32mInfo: \e[0mcleaning baseuri cic:/matita/LAMBDA-TYPES/Base-2/ext/arith
-\e[0;32mInfo: \e[0mRemoving: cic:/matita/LAMBDA-TYPES/Base-2/ext/arith/*
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: nat_dec
-Pre Nodes : 474
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda n1:nat.(nat_ind (\lambda n:nat.(\forall n2:nat.(or (eq nat n n2) ((eq nat n n2)\to (\forall P:Prop.P))))) (\lambda n2:nat.(nat_ind (\lambda n:nat.(or (eq nat O n) ((eq nat O n)\to (\forall P:Prop.P)))) (or_introl (eq nat O O) ((eq nat O O)\to (\forall P:Prop.P)) (refl_equal nat O)) (\lambda n3:nat.(\lambda _:(or (eq nat O n3) ((eq nat O n3)\to (\forall P:Prop.P))).(or_intror (eq nat O (S n3)) ((eq nat O (S n3))\to (\forall P:Prop.P)) (\lambda H0:(eq nat O (S n3)).(\lambda P:Prop. let H1 \def (eq_ind nat O (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with
- [ O => True
- | (S (_:nat)) => False
-]) I (S n3) H0) in (False_ind P H1)))))) n2)) (\lambda n2:nat.(\lambda H:(\forall n2:nat.(or (eq nat n2 n2) ((eq nat n2 n2)\to (\forall P:Prop.P)))).(\lambda n3:nat.(nat_ind (\lambda n0:nat.(or (eq nat (S n2) n0) ((eq nat (S n2) n0)\to (\forall P:Prop.P)))) (or_intror (eq nat (S n2) O) ((eq nat (S n2) O)\to (\forall P:Prop.P)) (\lambda H0:(eq nat (S n2) O).(\lambda P:Prop. let H1 \def (eq_ind nat (S n2) (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind P H1)))) (\lambda n4:nat.(\lambda H0:(or (eq nat (S n2) n4) ((eq nat (S n2) n4)\to (\forall P:Prop.P))). let DEFINED \def (H n4) in (or_ind (eq nat n2 n4) ((eq nat n2 n4)\to (\forall P:Prop.P)) (or (eq nat (S n2) (S n4)) ((eq nat (S n2) (S n4))\to (\forall P:Prop.P))) (\lambda H1:(eq nat n2 n4).(eq_ind nat n2 (\lambda n3:nat.(or (eq nat (S n2) (S n3)) ((eq nat (S n2) (S n3))\to (\forall P:Prop.P)))) (or_introl (eq nat (S n2) (S n2)) ((eq nat (S n2) (S n2))\to (\forall P:Prop.P)) (refl_equal nat (S n2))) n4 H1)) (\lambda H1:((eq nat n2 n4)\to (\forall P:Prop.P)).(or_intror (eq nat (S n2) (S n4)) ((eq nat (S n2) (S n4))\to (\forall P:Prop.P)) (\lambda H2:(eq nat (S n2) (S n4)).(\lambda P:Prop. let H3 \def (f_equal nat nat (\lambda e:nat.
-match e return (\lambda _:nat.nat) with
- [ O => n2
- | (S (n3:nat)) => n3
-]) (S n2) (S n4) H2) in let H4 \def (eq_ind_r nat n4 (\lambda n3:nat.((eq nat n2 n3)\to (\forall P0:Prop.P0))) H1 n2 H3) in (H4 (refl_equal nat n2) P))))) DEFINED))) n3)))) n1))
-Post Nodes: 476
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: simpl_plus_r
-Pre Nodes : 85
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda n:nat.(\lambda m:nat.(\lambda p:nat.(\lambda H:(eq nat (plus m n) (plus p n)). let DEFINED \def (plus_comm n m) in (plus_reg_l n m p (eq_ind_r nat (plus m n) (\lambda n0:nat.(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda n0:nat.(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_comm n p)) (plus m n) H) (plus n m) DEFINED))))))
-Post Nodes: 87
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_plus_r
-Pre Nodes : 33
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda m:nat.(\lambda n:nat. let DEFINED \def (plus_comm m n) in (eq_ind_r nat (plus n m) (\lambda n0:nat.(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) DEFINED)))
-Post Nodes: 35
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: plus_permute_2_in_3
-Pre Nodes : 117
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda z:nat. let DEFINED \def (plus_assoc_reverse x z y) in let DEFINED0 \def (plus_comm y z) in let DEFINED1 \def (plus_assoc_reverse x y z) in (eq_ind_r nat (plus x (plus y z)) (\lambda n:nat.(eq nat n (plus (plus x z) y))) (eq_ind_r nat (plus z y) (\lambda n:nat.(eq nat (plus x n) (plus (plus x z) y))) (eq_ind nat (plus (plus x z) y) (\lambda n:nat.(eq nat n (plus (plus x z) y))) (refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) DEFINED) (plus y z) DEFINED0) (plus (plus x y) z) DEFINED1))))
-Post Nodes: 123
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: plus_permute_2_in_3_assoc
-Pre Nodes : 86
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda n:nat.(\lambda h:nat.(\lambda k:nat. let DEFINED \def (plus_assoc n k h) in let DEFINED0 \def (plus_permute_2_in_3 n h k) in (eq_ind_r nat (plus (plus n k) h) (\lambda n0:nat.(eq nat n0 (plus n (plus k h)))) (eq_ind_r nat (plus (plus n k) h) (\lambda n0:nat.(eq nat (plus (plus n k) h) n0)) (refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) DEFINED) (plus (plus n h) k) DEFINED0))))
-Post Nodes: 90
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: plus_O
-Pre Nodes : 191
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((eq nat (plus n y) O)\to (and (eq nat n O) (eq nat y O))))) (\lambda y:nat.(\lambda H:(eq nat (plus O y) O).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) H))) (\lambda n:nat.(\lambda _:(\forall y:nat.((eq nat (plus n y) O)\to (and (eq nat n O) (eq nat y O)))).(\lambda y:nat.(\lambda H0:(eq nat (plus (S n) y) O). let H1 \def (eq_ind nat (plus (S n) y) (\lambda n0:nat.((eq nat n0 O)\to (and (eq nat (S n) O) (eq nat y O)))) (\lambda H1:(eq nat (plus (S n) y) O). let H2 \def (eq_ind nat (plus (S n) y) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in (False_ind (and (eq nat (S n) O) (eq nat y O)) H2)) O H0) in (H1 (refl_equal nat O)))))) x))
-Post Nodes: 189
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_Sx_SO
-Pre Nodes : 24
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat. let DEFINED \def (minus_n_O x) in (eq_ind nat x (\lambda n:nat.(eq nat n x)) (refl_equal nat x) (minus x O) DEFINED))
-Post Nodes: 26
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: eq_nat_dec
-Pre Nodes : 303
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda i:nat.(nat_ind (\lambda n:nat.(\forall j:nat.(or (not (eq nat n j)) (eq nat n j)))) (\lambda j:nat.(nat_ind (\lambda n:nat.(or (not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) (refl_equal nat O)) (\lambda n:nat.(\lambda _:(or (not (eq nat O n)) (eq nat O n)).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) (\lambda n:nat.(\lambda H:(\forall j:nat.(or (not (eq nat n j)) (eq nat n j))).(\lambda j:nat.(nat_ind (\lambda n0:nat.(or (not (eq nat (S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda n0:nat.(\lambda _:(or (not (eq nat (S n) n0)) (eq nat (S n) n0)). let DEFINED \def (H n0) in (or_ind (not (eq nat n n0)) (eq nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda H1:(not (eq nat n n0)).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (not_eq_S n n0 H1))) (\lambda H1:(eq nat n n0).(or_intror (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) DEFINED))) j)))) i))
-Post Nodes: 305
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: neq_eq_e
-Pre Nodes : 47
-Optimized : (\lambda i:nat.(\lambda j:nat.(\lambda P:Prop.(\lambda H:((not (eq nat i j))\to P).(\lambda H0:((eq nat i j)\to P). let o \def (eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o))))))
-Post Nodes: 47
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_false
-Pre Nodes : 382
-nat
-nat
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: nested application
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-nat
-nat
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: nested application
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda m:nat.(nat_ind (\lambda n:nat.(\forall n0:nat.(\forall P:Prop.((le n n0)\to ((le (S n0) n)\to P))))) (\lambda n:nat.(\lambda P:Prop.(\lambda _:(le O n).(\lambda H0:(le (S n) O). let H1 \def (le_ind (S n) (\lambda n0:nat.((eq nat n0 O)\to P)) (\lambda H1:(eq nat (S n) O). let H2 \def (eq_ind nat (S n) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in (False_ind P H2)) (\lambda m0:nat.(\lambda H1:(le (S n) m0).(\lambda _:((eq nat m0 O)\to P).(\lambda H2:(eq nat (S m0) O). let H3 \def (eq_ind nat (S m0) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H2) in let DEFINED \def (False_ind ((le (S n) m0)\to P) H3) in (DEFINED H1))))) O H0) in (H1 (refl_equal nat O)))))) (\lambda n:nat.(\lambda H:(\forall n0:nat.(\forall P:Prop.((le n n0)\to ((le (S n0) n)\to P)))).(\lambda n0:nat.(nat_ind (\lambda n1:nat.(\forall P:Prop.((le (S n) n1)\to ((le (S n1) (S n))\to P)))) (\lambda P:Prop.(\lambda H0:(le (S n) O).(\lambda _:(le (S O) (S n)). let H2 \def (le_ind (S n) (\lambda n1:nat.((eq nat n1 O)\to P)) (\lambda H2:(eq nat (S n) O). let H3 \def (eq_ind nat (S n) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H2) in (False_ind P H3)) (\lambda m0:nat.(\lambda H2:(le (S n) m0).(\lambda _:((eq nat m0 O)\to P).(\lambda H3:(eq nat (S m0) O). let H4 \def (eq_ind nat (S m0) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H3) in let DEFINED \def (False_ind ((le (S n) m0)\to P) H4) in (DEFINED H2))))) O H0) in (H2 (refl_equal nat O))))) (\lambda n1:nat.(\lambda _:(\forall P:Prop.((le (S n) n1)\to ((le (S n1) (S n))\to P))).(\lambda P:Prop.(\lambda H1:(le (S n) (S n1)).(\lambda H2:(le (S (S n1)) (S n)). let DEFINED \def (le_S_n (S n1) n H2) in (H n1 P (le_S_n n n1 H1) DEFINED)))))) n0)))) m))
-Post Nodes: 400
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_Sx_x
-Pre Nodes : 20
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda H:(le (S x) x).(\lambda P:Prop. let H0 \def le_Sn_n in let DEFINED \def (H0 x H) in (False_ind P DEFINED))))
-Post Nodes: 22
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_le
-Pre Nodes : 88
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.(le (minus n y) n))) (\lambda _:nat.(le_n O)) (\lambda n:nat.(\lambda H:(\forall y:nat.(le (minus n y) n)).(\lambda y:nat.(nat_ind (\lambda n0:nat.(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda n0:nat.(\lambda _:(le
-match n0 return (\lambda n1:nat.nat) with
- [ O => (S n)
- | (S (l:nat)) => (minus n l)
-] (S n)).(le_S (minus n n0) n (H n0)))) y)))) x))
-Post Nodes: 88
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_plus_minus_sym
-Pre Nodes : 45
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda n:nat.(\lambda m:nat.(\lambda H:(le n m). let DEFINED \def (plus_comm (minus m n) n) in (eq_ind_r nat (plus n (minus m n)) (\lambda n0:nat.(eq nat m n0)) (le_plus_minus n m H) (plus (minus m n) n) DEFINED))))
-Post Nodes: 47
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_minus_minus
-Pre Nodes : 84
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(le x y).(\lambda z:nat.(\lambda H0:(le y z). let DEFINED \def (le_plus_minus_r x z (le_trans x y z H H0)) in let DEFINED0 \def (le_plus_minus_r x y H) in (plus_le_reg_l x (minus y x) (minus z x) (eq_ind_r nat y (\lambda n:nat.(le n (plus x (minus z x)))) (eq_ind_r nat z (\lambda n:nat.(le y n)) H0 (plus x (minus z x)) DEFINED) (plus x (minus y x)) DEFINED0)))))))
-Post Nodes: 88
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_minus_plus
-Pre Nodes : 505
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-nat
-nat
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: nested application
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-Optimized : (\lambda z:nat.(nat_ind (\lambda n:nat.(\forall x:nat.((le n x)\to (\forall y:nat.(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) (\lambda x:nat.(\lambda H:(le O x). let H0 \def (le_ind O (\lambda n:nat.((eq nat n x)\to (\forall y:nat.(eq nat (minus (plus x y) O) (plus (minus x O) y))))) (\lambda H0:(eq nat O x).(eq_ind nat O (\lambda n:nat.(\forall y:nat.(eq nat (minus (plus n y) O) (plus (minus n O) y)))) (\lambda y:nat.(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) (minus_n_O (plus O y)))) x H0)) (\lambda m:nat.(\lambda H0:(le O m).(\lambda _:((eq nat m x)\to (\forall y:nat.(eq nat (minus (plus x y) O) (plus (minus x O) y)))).(\lambda H1:(eq nat (S m) x). let DEFINED \def (eq_ind nat (S m) (\lambda n:nat.((le O m)\to (\forall y:nat.(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda _:(le O m).(\lambda y:nat.(refl_equal nat (plus (minus (S m) O) y)))) x H1) in (DEFINED H0))))) x H) in (H0 (refl_equal nat x)))) (\lambda z0:nat.(\lambda H:(\forall x:nat.((le z0 x)\to (\forall y:nat.(eq nat (minus (plus x y) z0) (plus (minus x z0) y))))).(\lambda x:nat.(nat_ind (\lambda n:nat.((le (S z0) n)\to (\forall y:nat.(eq nat (minus (plus n y) (S z0)) (plus (minus n (S z0)) y))))) (\lambda H0:(le (S z0) O).(\lambda y:nat. let H1 \def (le_ind (S z0) (\lambda n:nat.((eq nat n O)\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)))) (\lambda H1:(eq nat (S z0) O). let H2 \def (eq_ind nat (S z0) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in (False_ind (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2)) (\lambda m:nat.(\lambda H1:(le (S z0) m).(\lambda _:((eq nat m O)\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))).(\lambda H2:(eq nat (S m) O). let H3 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H2) in let DEFINED \def (False_ind ((le (S z0) m)\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3) in (DEFINED H1))))) O H0) in (H1 (refl_equal nat O)))) (\lambda n:nat.(\lambda _:((le (S z0) n)\to (\forall y:nat.(eq nat (minus (plus n y) (S z0)) (plus (minus n (S z0)) y)))).(\lambda H1:(le (S z0) (S n)).(\lambda y:nat.(H n (le_S_n z0 n H1) y))))) x)))) z))
-Post Nodes: 560
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_minus
-Pre Nodes : 51
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda z:nat.(\lambda y:nat.(\lambda H:(le (plus x y) z). let DEFINED \def (minus_plus_r x y) in (eq_ind nat (minus (plus x y) y) (\lambda n:nat.(le n (minus z y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x DEFINED)))))
-Post Nodes: 53
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_trans_plus_r
-Pre Nodes : 27
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda z:nat.(\lambda H:(le (plus x y) z).(le_trans y (plus x y) z (le_plus_r x y) H)))))
-Post Nodes: 27
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_gen_S
-Pre Nodes : 217
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-Optimized : (\lambda m:nat.(\lambda x:nat.(\lambda H:(le (S m) x). let H0 \def (le_ind (S m) (\lambda n:nat.((eq nat n x)\to (ex2 nat (\lambda n0:nat.(eq nat x (S n0))) (\lambda n0:nat.(le m n0))))) (\lambda H0:(eq nat (S m) x).(eq_ind nat (S m) (\lambda n:nat.(ex2 nat (\lambda n0:nat.(eq nat n (S n0))) (\lambda n0:nat.(le m n0)))) (ex_intro2 nat (\lambda n:nat.(eq nat (S m) (S n))) (\lambda n:nat.(le m n)) m (refl_equal nat (S m)) (le_n m)) x H0)) (\lambda m0:nat.(\lambda H0:(le (S m) m0).(\lambda _:((eq nat m0 x)\to (ex2 nat (\lambda n0:nat.(eq nat x (S n0))) (\lambda n0:nat.(le m n0)))).(\lambda H1:(eq nat (S m0) x). let DEFINED \def (eq_ind nat (S m0) (\lambda n:nat.((le (S m) m0)\to (ex2 nat (\lambda n0:nat.(eq nat n (S n0))) (\lambda n0:nat.(le m n0))))) (\lambda H2:(le (S m) m0).(ex_intro2 nat (\lambda n:nat.(eq nat (S m0) (S n))) (\lambda n:nat.(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) x H1) in (DEFINED H0))))) x H) in (H0 (refl_equal nat x)))))
-Post Nodes: 243
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_x_plus_x_Sy
-Pre Nodes : 64
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat. let DEFINED \def (plus_comm x (S y)) in (eq_ind_r nat (plus (S y) x) (\lambda n:nat.(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) (le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) DEFINED)))
-Post Nodes: 66
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: simpl_lt_plus_r
-Pre Nodes : 75
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda p:nat.(\lambda n:nat.(\lambda m:nat.(\lambda H:(lt (plus n p) (plus m p)). let DEFINED \def (plus_comm n p) in let H0 \def (eq_ind nat (plus n p) (\lambda n0:nat.(lt n0 (plus m p))) H (plus p n) DEFINED) in let DEFINED0 \def (plus_comm m p) in let H1 \def (eq_ind nat (plus m p) (\lambda n0:nat.(lt (plus p n) n0)) H0 (plus p m) DEFINED0) in (plus_lt_reg_l n m p H1)))))
-Post Nodes: 79
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_x_Sy
-Pre Nodes : 330
-nat
-nat
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: nested application
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((lt y n)\to (eq nat (minus n y) (S (minus n (S y))))))) (\lambda y:nat.(\lambda H:(lt y O). let H0 \def (le_ind (S y) (\lambda n:nat.((eq nat n O)\to (eq nat (minus O y) (S (minus O (S y)))))) (\lambda H0:(eq nat (S y) O). let H1 \def (eq_ind nat (S y) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind (eq nat (minus O y) (S (minus O (S y)))) H1)) (\lambda m:nat.(\lambda H0:(le (S y) m).(\lambda _:((eq nat m O)\to (eq nat (minus O y) (S (minus O (S y))))).(\lambda H1:(eq nat (S m) O). let H2 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in let DEFINED \def (False_ind ((le (S y) m)\to (eq nat (minus O y) (S (minus O (S y))))) H2) in (DEFINED H0))))) O H) in (H0 (refl_equal nat O)))) (\lambda n:nat.(\lambda H:(\forall y:nat.((lt y n)\to (eq nat (minus n y) (S (minus n (S y)))))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((lt n0 (S n))\to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda _:(lt O (S n)). let DEFINED \def (minus_n_O n) in (eq_ind nat n (\lambda n0:nat.(eq nat (S n) (S n0))) (refl_equal nat (S n)) (minus n O) DEFINED)) (\lambda n0:nat.(\lambda _:((lt n0 (S n))\to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))).(\lambda H1:(lt (S n0) (S n)). let H2 \def (le_S_n (S n0) n H1) in (H n0 H2)))) y)))) x))
-Post Nodes: 354
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_plus_minus
-Pre Nodes : 16
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(lt x y).(le_plus_minus (S x) y H))))
-Post Nodes: 16
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_plus_minus_r
-Pre Nodes : 53
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(lt x y). let DEFINED \def (plus_comm (minus y (S x)) x) in (eq_ind_r nat (plus x (minus y (S x))) (\lambda n:nat.(eq nat y (S n))) (lt_plus_minus x y H) (plus (minus y (S x)) x) DEFINED))))
-Post Nodes: 55
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_x_SO
-Pre Nodes : 56
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda H:(lt O x). let DEFINED \def (minus_n_O x) in let DEFINED0 \def (minus_x_Sy x O H) in (eq_ind nat (minus x O) (\lambda n:nat.(eq nat x n)) (eq_ind nat x (\lambda n:nat.(eq nat x n)) (refl_equal nat x) (minus x O) DEFINED) (S (minus x (S O))) DEFINED0)))
-Post Nodes: 60
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_x_pred_y
-Pre Nodes : 175
-nat
-nat
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: nested application
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-Optimized : (\lambda y:nat.(nat_ind (\lambda n:nat.(\forall x:nat.((lt x n)\to (le x (pred n))))) (\lambda x:nat.(\lambda H:(lt x O). let H0 \def (le_ind (S x) (\lambda n:nat.((eq nat n O)\to (le x O))) (\lambda H0:(eq nat (S x) O). let H1 \def (eq_ind nat (S x) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind (le x O) H1)) (\lambda m:nat.(\lambda H0:(le (S x) m).(\lambda _:((eq nat m O)\to (le x O)).(\lambda H1:(eq nat (S m) O). let H2 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in let DEFINED \def (False_ind ((le (S x) m)\to (le x O)) H2) in (DEFINED H0))))) O H) in (H0 (refl_equal nat O)))) (\lambda n:nat.(\lambda _:(\forall x:nat.((lt x n)\to (le x (pred n)))).(\lambda x:nat.(\lambda H0:(lt x (S n)).(le_S_n x n H0))))) y))
-Post Nodes: 186
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_le_minus
-Pre Nodes : 44
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(lt x y). let DEFINED \def (plus_comm x (S O)) in (le_minus x y (S O) (eq_ind_r nat (plus (S O) x) (\lambda n:nat.(le n y)) H (plus x (S O)) DEFINED)))))
-Post Nodes: 46
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_le_e
-Pre Nodes : 39
-Optimized : (\lambda n:nat.(\lambda d:nat.(\lambda P:Prop.(\lambda H:((lt n d)\to P).(\lambda H0:((le d n)\to P). let H1 \def (le_or_lt d n) in (or_ind (le d n) (lt n d) P H0 H H1))))))
-Post Nodes: 39
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_eq_e
-Pre Nodes : 45
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda P:Prop.(\lambda H:((lt x y)\to P).(\lambda H0:((eq nat x y)\to P).(\lambda H1:(le x y). let DEFINED \def (le_lt_or_eq x y H1) in (or_ind (lt x y) (eq nat x y) P H H0 DEFINED)))))))
-Post Nodes: 47
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_eq_gt_e
-Pre Nodes : 60
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda P:Prop.(\lambda H:((lt x y)\to P).(\lambda H0:((eq nat x y)\to P).(\lambda H1:((lt y x)\to P).(lt_le_e x y P H (\lambda H2:(le y x).(lt_eq_e y x P H1 (\lambda H3:(eq nat y x).(H0 (sym_eq nat y x H3))) H2)))))))))
-Post Nodes: 60
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_gen_xS
-Pre Nodes : 190
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall n0:nat.((lt n (S n0))\to (or (eq nat n O) (ex2 nat (\lambda m:nat.(eq nat n (S m))) (\lambda m:nat.(lt m n0))))))) (\lambda n:nat.(\lambda _:(lt O (S n)).(or_introl (eq nat O O) (ex2 nat (\lambda m:nat.(eq nat O (S m))) (\lambda m:nat.(lt m n))) (refl_equal nat O)))) (\lambda n:nat.(\lambda _:(\forall n0:nat.((lt n (S n0))\to (or (eq nat n O) (ex2 nat (\lambda m:nat.(eq nat n (S m))) (\lambda m:nat.(lt m n0)))))).(\lambda n0:nat.(\lambda H0:(lt (S n) (S n0)).(or_intror (eq nat (S n) O) (ex2 nat (\lambda m:nat.(eq nat (S n) (S m))) (\lambda m:nat.(lt m n0))) (ex_intro2 nat (\lambda m:nat.(eq nat (S n) (S m))) (\lambda m:nat.(lt m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x))
-Post Nodes: 190
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_lt_false
-Pre Nodes : 25
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(le x y).(\lambda H0:(lt y x).(\lambda P:Prop. let DEFINED \def (le_not_lt x y H H0) in (False_ind P DEFINED))))))
-Post Nodes: 27
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: lt_neq
-Pre Nodes : 33
-Optimized : (\lambda x:nat.(\lambda y:nat.(\lambda H:(lt x y).(\lambda H0:(eq nat x y). let H1 \def (eq_ind nat x (\lambda n:nat.(lt n y)) H y H0) in (lt_irrefl y H1)))))
-Post Nodes: 33
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: arith0
-Pre Nodes : 185
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda h2:nat.(\lambda d2:nat.(\lambda n:nat.(\lambda H:(le (plus d2 h2) n).(\lambda h3:nat. let DEFINED \def (plus_comm h2 d2) in let DEFINED0 \def (plus_assoc h2 d2 h3) in let DEFINED1 \def (minus_plus h2 (plus d2 h3)) in (eq_ind nat (minus (plus h2 (plus d2 h3)) h2) (\lambda n0:nat.(le n0 (minus (plus n h3) h2))) (le_minus_minus h2 (plus h2 (plus d2 h3)) (le_plus_l h2 (plus d2 h3)) (plus n h3) (eq_ind_r nat (plus (plus h2 d2) h3) (\lambda n0:nat.(le n0 (plus n h3))) (eq_ind_r nat (plus d2 h2) (\lambda n0:nat.(le (plus n0 h3) (plus n h3))) (le_S_n (plus (plus d2 h2) h3) (plus n h3) (lt_le_S (plus (plus d2 h2) h3) (S (plus n h3)) (le_lt_n_Sm (plus (plus d2 h2) h3) (plus n h3) (plus_le_compat (plus d2 h2) n h3 h3 H (le_n h3))))) (plus h2 d2) DEFINED) (plus h2 (plus d2 h3)) DEFINED0)) (plus d2 h3) DEFINED1))))))
-Post Nodes: 191
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: O_minus
-Pre Nodes : 211
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((le n y)\to (eq nat (minus n y) O)))) (\lambda y:nat.(\lambda _:(le O y).(refl_equal nat O))) (\lambda x0:nat.(\lambda H:(\forall y:nat.((le x0 y)\to (eq nat (minus x0 y) O))).(\lambda y:nat.(nat_ind (\lambda n:nat.((le (S x0) n)\to (eq nat
-match n return (\lambda n1:nat.nat) with
- [ O => (S x0)
- | (S (l:nat)) => (minus x0 l)
-] O))) (\lambda H0:(le (S x0) O). let DEFINED \def (le_gen_S x0 O H0) in (ex2_ind nat (\lambda n:nat.(eq nat O (S n))) (\lambda n:nat.(le x0 n)) (eq nat (S x0) O) (\lambda x1:nat.(\lambda H1:(eq nat O (S x1)).(\lambda _:(le x0 x1). let H3 \def (eq_ind nat O (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with
- [ O => True
- | (S (_:nat)) => False
-]) I (S x1) H1) in (False_ind (eq nat (S x0) O) H3)))) DEFINED)) (\lambda n:nat.(\lambda _:((le (S x0) n)\to (eq nat
-match n return (\lambda n1:nat.nat) with
- [ O => (S x0)
- | (S (l:nat)) => (minus x0 l)
-] O)).(\lambda H1:(le (S x0) (S n)).(H n (le_S_n x0 n H1))))) y)))) x))
-Post Nodes: 213
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_minus
-Pre Nodes : 592
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda z:nat.(nat_ind (\lambda n:nat.(\forall x:nat.(\forall y:nat.((le n x)\to ((le n y)\to ((eq nat (minus x n) (minus y n))\to (eq nat x y))))))) (\lambda x:nat.(\lambda y:nat.(\lambda _:(le O x).(\lambda _:(le O y).(\lambda H1:(eq nat (minus x O) (minus y O)). let DEFINED \def (minus_n_O x) in let H2 \def (eq_ind_r nat (minus x O) (\lambda n:nat.(eq nat n (minus y O))) H1 x DEFINED) in let DEFINED0 \def (minus_n_O y) in let H3 \def (eq_ind_r nat (minus y O) (\lambda n:nat.(eq nat x n)) H2 y DEFINED0) in H3))))) (\lambda z0:nat.(\lambda IH:(\forall x:nat.(\forall y:nat.((le z0 x)\to ((le z0 y)\to ((eq nat (minus x z0) (minus y z0))\to (eq nat x y)))))).(\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((le (S z0) n)\to ((le (S z0) y)\to ((eq nat (minus n (S z0)) (minus y (S z0)))\to (eq nat n y)))))) (\lambda y:nat.(\lambda H:(le (S z0) O).(\lambda _:(le (S z0) y).(\lambda _:(eq nat (minus O (S z0)) (minus y (S z0))). let DEFINED \def (le_gen_S z0 O H) in (ex2_ind nat (\lambda n:nat.(eq nat O (S n))) (\lambda n:nat.(le z0 n)) (eq nat O y) (\lambda x0:nat.(\lambda H2:(eq nat O (S x0)).(\lambda _:(le z0 x0). let H4 \def (eq_ind nat O (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with
- [ O => True
- | (S (_:nat)) => False
-]) I (S x0) H2) in (False_ind (eq nat O y) H4)))) DEFINED))))) (\lambda x0:nat.(\lambda _:(\forall y:nat.((le (S z0) x0)\to ((le (S z0) y)\to ((eq nat (minus x0 (S z0)) (minus y (S z0)))\to (eq nat x0 y))))).(\lambda y:nat.(nat_ind (\lambda n:nat.((le (S z0) (S x0))\to ((le (S z0) n)\to ((eq nat (minus (S x0) (S z0)) (minus n (S z0)))\to (eq nat (S x0) n))))) (\lambda _:(le (S z0) (S x0)).(\lambda H0:(le (S z0) O).(\lambda _:(eq nat (minus (S x0) (S z0)) (minus O (S z0))). let DEFINED \def (le_gen_S z0 O H0) in (ex2_ind nat (\lambda n:nat.(eq nat O (S n))) (\lambda n:nat.(le z0 n)) (eq nat (S x0) O) (\lambda x1:nat.(\lambda H2:(eq nat O (S x1)).(\lambda _:(le z0 x1). let H4 \def (eq_ind nat O (\lambda ee:nat.
-match ee return (\lambda _:nat.Prop) with
- [ O => True
- | (S (_:nat)) => False
-]) I (S x1) H2) in (False_ind (eq nat (S x0) O) H4)))) DEFINED)))) (\lambda y0:nat.(\lambda _:((le (S z0) (S x0))\to ((le (S z0) y0)\to ((eq nat (minus (S x0) (S z0)) (minus y0 (S z0)))\to (eq nat (S x0) y0)))).(\lambda H:(le (S z0) (S x0)).(\lambda H0:(le (S z0) (S y0)).(\lambda H1:(eq nat (minus (S x0) (S z0)) (minus (S y0) (S z0))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z))
-Post Nodes: 600
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: plus_plus
-Pre Nodes : 1222
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-\e[0;33mWarn: \e[0mOptimizer: swap 1
-Optimized : (\lambda z:nat.(nat_ind (\lambda n:nat.(\forall x1:nat.(\forall x2:nat.(\forall y1:nat.(\forall y2:nat.((le x1 n)\to ((le x2 n)\to ((eq nat (plus (minus n x1) y1) (plus (minus n x2) y2))\to (eq nat (plus x1 y2) (plus x2 y1)))))))))) (\lambda x1:nat.(\lambda x2:nat.(\lambda y1:nat.(\lambda y2:nat.(\lambda H:(le x1 O).(\lambda H0:(le x2 O).(\lambda H1:(eq nat y1 y2). let H_y \def (le_n_O_eq x2 H0) in let H_y0 \def (le_n_O_eq x1 H) in (eq_ind nat y1 (\lambda n:nat.(eq nat (plus x1 n) (plus x2 y1))) (eq_ind nat O (\lambda n:nat.(eq nat (plus x1 y1) (plus n y1))) (eq_ind nat O (\lambda n:nat.(eq nat (plus n y1) (plus O y1))) (refl_equal nat (plus O y1)) x1 H_y0) x2 H_y) y2 H1)))))))) (\lambda z0:nat.(\lambda IH:(\forall x1:nat.(\forall x2:nat.(\forall y1:nat.(\forall y2:nat.((le x1 z0)\to ((le x2 z0)\to ((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2))\to (eq nat (plus x1 y2) (plus x2 y1))))))))).(\lambda x1:nat.(nat_ind (\lambda n:nat.(\forall x2:nat.(\forall y1:nat.(\forall y2:nat.((le n (S z0))\to ((le x2 (S z0))\to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S z0) x2) y2))\to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda x2:nat.(nat_ind (\lambda n:nat.(\forall y1:nat.(\forall y2:nat.((le O (S z0))\to ((le n (S z0))\to ((eq nat (plus (minus (S z0) O) y1) (plus (minus (S z0) n) y2))\to (eq nat (plus O y2) (plus n y1)))))))) (\lambda y1:nat.(\lambda y2:nat.(\lambda _:(le O (S z0)).(\lambda _:(le O (S z0)).(\lambda H1:(eq nat (S (plus z0 y1)) (S (plus z0 y2))). let H_y \def (IH O O) in let DEFINED \def (minus_n_O z0) in let H2 \def (eq_ind_r nat (minus z0 O) (\lambda n:nat.(\forall y3:nat.(\forall y4:nat.((le O z0)\to ((le O z0)\to ((eq nat (plus n y3) (plus n y4))\to (eq nat y4 y3))))))) H_y z0 DEFINED) in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (H2 (plus z0 y2) (plus z0 y1) (le_O_n z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) (sym_eq nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) H1)))))))))) (\lambda x3:nat.(\lambda _:(\forall y1:nat.(\forall y2:nat.((le O (S z0))\to ((le x3 (S z0))\to ((eq nat (S (plus z0 y1)) (plus
-match x3 return (\lambda n:nat.nat) with
- [ O => (S z0)
- | (S (l:nat)) => (minus z0 l)
-] y2))\to (eq nat y2 (plus x3 y1))))))).(\lambda y1:nat.(\lambda y2:nat.(\lambda _:(le O (S z0)).(\lambda H0:(le (S x3) (S z0)).(\lambda H1:(eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2)). let H_y \def (IH O x3 (S y1)) in let DEFINED \def (minus_n_O z0) in let H2 \def (eq_ind_r nat (minus z0 O) (\lambda n:nat.(\forall y3:nat.((le O z0)\to ((le x3 z0)\to ((eq nat (plus n (S y1)) (plus (minus z0 x3) y3))\to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 DEFINED) in let DEFINED0 \def (plus_n_Sm z0 y1) in let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda n:nat.(\forall y3:nat.((le O z0)\to ((le x3 z0)\to ((eq nat n (plus (minus z0 x3) y3))\to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) DEFINED0) in let DEFINED1 \def (plus_n_Sm x3 y1) in let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda n:nat.(\forall y3:nat.((le O z0)\to ((le x3 z0)\to ((eq nat (S (plus z0 y1)) (plus (minus z0 x3) y3))\to (eq nat y3 n)))))) H3 (S (plus x3 y1)) DEFINED1) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))) x2)) (\lambda x2:nat.(\lambda _:(\forall x3:nat.(\forall y1:nat.(\forall y2:nat.((le x2 (S z0))\to ((le x3 (S z0))\to ((eq nat (plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2))\to (eq nat (plus x2 y2) (plus x3 y1)))))))).(\lambda x3:nat.(nat_ind (\lambda n:nat.(\forall y1:nat.(\forall y2:nat.((le (S x2) (S z0))\to ((le n (S z0))\to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2))\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda y1:nat.(\lambda y2:nat.(\lambda H:(le (S x2) (S z0)).(\lambda _:(le O (S z0)).(\lambda H1:(eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))). let H_y \def (IH x2 O y1 (S y2)) in let DEFINED \def (minus_n_O z0) in let H2 \def (eq_ind_r nat (minus z0 O) (\lambda n:nat.((le x2 z0)\to ((le O z0)\to ((eq nat (plus (minus z0 x2) y1) (plus n (S y2)))\to (eq nat (plus x2 (S y2)) y1))))) H_y z0 DEFINED) in let DEFINED0 \def (plus_n_Sm z0 y2) in let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda n:nat.((le x2 z0)\to ((le O z0)\to ((eq nat (plus (minus z0 x2) y1) n)\to (eq nat (plus x2 (S y2)) y1))))) H2 (S (plus z0 y2)) DEFINED0) in let DEFINED1 \def (plus_n_Sm x2 y2) in let H4 \def (eq_ind_r nat (plus x2 (S y2)) (\lambda n:nat.((le x2 z0)\to ((le O z0)\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))\to (eq nat n y1))))) H3 (S (plus x2 y2)) DEFINED1) in (H4 (le_S_n x2 z0 H) (le_O_n z0) H1)))))) (\lambda x4:nat.(\lambda _:(\forall y1:nat.(\forall y2:nat.((le (S x2) (S z0))\to ((le x4 (S z0))\to ((eq nat (plus (minus z0 x2) y1) (plus
-match x4 return (\lambda n:nat.nat) with
- [ O => (S z0)
- | (S (l:nat)) => (minus z0 l)
-] y2))\to (eq nat (S (plus x2 y2)) (plus x4 y1))))))).(\lambda y1:nat.(\lambda y2:nat.(\lambda H:(le (S x2) (S z0)).(\lambda H0:(le (S x4) (S z0)).(\lambda H1:(eq nat (plus (minus z0 x2) y1) (plus (minus z0 x4) y2)).(f_equal nat nat S (plus x2 y2) (plus x4 y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) x1)))) z))
-Post Nodes: 1236
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: le_S_minus
-Pre Nodes : 41
-\e[0;33mWarn: \e[0mOptimizer: remove 1
-Optimized : (\lambda d:nat.(\lambda h:nat.(\lambda n:nat.(\lambda H:(le (plus d h) n).(le_S d (minus n h) (le_minus d n h H))))))
-Post Nodes: 27
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./ext/arith.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of ext/arith.mma completed in 44''.
-\e[0;32mInfo: \e[0mexecution of ext/tactics.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/ext ...''
-\e[0;32mInfo: \e[0mbaseuri cic:/matita/LAMBDA-TYPES/Base-2/ext/tactics is not empty
-\e[0;32mInfo: \e[0mcleaning baseuri cic:/matita/LAMBDA-TYPES/Base-2/ext/tactics
-\e[0;32mInfo: \e[0mRemoving: cic:/matita/LAMBDA-TYPES/Base-2/ext/tactics/*
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: insert_eq
-Pre Nodes : 36
-Optimized : (\lambda S:Set.(\lambda x:S.(\lambda P:(S\to Prop).(\lambda G:Prop.(\lambda H:(\forall y:S.((P y)\to ((eq S y x)\to G))).(\lambda H0:(P x).(H x H0 (refl_equal S x))))))))
-Post Nodes: 36
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: unintro
-Pre Nodes : 17
-Optimized : (\lambda A:Set.(\lambda a:A.(\lambda P:(A\to Prop).(\lambda H:(\forall x:A.(P x)).(H a)))))
-Post Nodes: 17
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: xinduction
-Pre Nodes : 27
-Optimized : (\lambda A:Set.(\lambda t:A.(\lambda P:(A\to Prop).(\lambda H:(\forall x:A.((eq A t x)\to (P x))).(H t (refl_equal A t))))))
-Post Nodes: 27
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./ext/tactics.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of ext/tactics.mma completed in 4''.
-\e[0;32mInfo: \e[0mexecution of pippo.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/pip ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: minus_x_Sy
-Pre Nodes : 330
-nat
-nat
-nat
-nat
-\e[0;33mWarn: \e[0mOptimizer: remove 3
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: nested application
-\e[0;33mWarn: \e[0mOptimizer: anticipate 2
-\e[0;33mWarn: \e[0mOptimizer: swap 2
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda x:nat.(nat_ind (\lambda n:nat.(\forall y:nat.((lt y n)\to (eq nat (minus n y) (S (minus n (S y))))))) (\lambda y:nat.(\lambda H:(lt y O). let H0 \def (le_ind (S y) (\lambda n:nat.((eq nat n O)\to (eq nat (minus O y) (S (minus O (S y)))))) (\lambda H0:(eq nat (S y) O). let H1 \def (eq_ind nat (S y) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H0) in (False_ind (eq nat (minus O y) (S (minus O (S y)))) H1)) (\lambda m:nat.(\lambda H0:(le (S y) m).(\lambda _:((eq nat m O)\to (eq nat (minus O y) (S (minus O (S y))))).(\lambda H1:(eq nat (S m) O). let H2 \def (eq_ind nat (S m) (\lambda e:nat.
-match e return (\lambda _:nat.Prop) with
- [ O => False
- | (S (_:nat)) => True
-]) I O H1) in let DEFINED \def (False_ind ((le (S y) m)\to (eq nat (minus O y) (S (minus O (S y))))) H2) in (DEFINED H0))))) O H) in (H0 (refl_equal nat O)))) (\lambda n:nat.(\lambda H:(\forall y:nat.((lt y n)\to (eq nat (minus n y) (S (minus n (S y)))))).(\lambda y:nat.(nat_ind (\lambda n0:nat.((lt n0 (S n))\to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda _:(lt O (S n)). let DEFINED \def (minus_n_O n) in (eq_ind nat n (\lambda n0:nat.(eq nat (S n) (S n0))) (refl_equal nat (S n)) (minus n O) DEFINED)) (\lambda n0:nat.(\lambda _:((lt n0 (S n))\to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))).(\lambda H1:(lt (S n0) (S n)). let H2 \def (le_S_n (S n0) n H1) in (H n0 H2)))) y)))) x))
-Post Nodes: 354
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./pippo.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of pippo.mma completed in 2''.
-\e[0;32mInfo: \e[0mexecution of plist/defs.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/pli ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./plist/defs.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of plist/defs.mma completed in 1''.
-\e[0;32mInfo: \e[0mexecution of plist/props.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/pli ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./plist/defs.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/plist/def ...''
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: papp_ss
-Pre Nodes : 121
-\e[0;33mWarn: \e[0mOptimizer: anticipate 3
-\e[0;33mWarn: \e[0mOptimizer: swap 3
-Optimized : (\lambda is1:PList.(PList_ind (\lambda p:PList.(\forall is2:PList.(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda is2:PList.(refl_equal PList (Ss is2))) (\lambda n:nat.(\lambda n0:nat.(\lambda p:PList.(\lambda H:(\forall is2:PList.(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2)))).(\lambda is2:PList. let DEFINED \def (H is2) in (eq_ind_r PList (Ss (papp p is2)) (\lambda p0:PList.(eq PList (PCons n (S n0) p0) (PCons n (S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p is2)))) (papp (Ss p) (Ss is2)) DEFINED)))))) is1))
-Post Nodes: 123
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./plist/props.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of plist/props.mma completed in 2''.
-\e[0;32mInfo: \e[0mexecution of types/defs.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/typ ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./preamble.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/preamble" ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./types/defs.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of types/defs.mma completed in 0''.
-\e[0;32mInfo: \e[0mexecution of types/props.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/typ ...''
-\e[0;32mInfo: \e[0mbaseuri cic:/matita/LAMBDA-TYPES/Base-2/types/props is not empty
-\e[0;32mInfo: \e[0mcleaning baseuri cic:/matita/LAMBDA-TYPES/Base-2/types/props
-\e[0;32mInfo: \e[0mRemoving: cic:/matita/LAMBDA-TYPES/Base-2/types/props/*
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./types/defs.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/types/def ...''
-\e[0;34mDebug: \e[0mExecuting: ``inline procedural "cic:/matita/LAMBDA-TYPES/Base-1 ...''
-BEGIN: ex2_sym
-Pre Nodes : 77
-Optimized : (\lambda A:Set.(\lambda P:(A\to Prop).(\lambda Q:(A\to Prop).(\lambda H:(ex2 A (\lambda x:A.(P x)) (\lambda x:A.(Q x))).(ex2_ind A (\lambda x:A.(P x)) (\lambda x:A.(Q x)) (ex2 A (\lambda x:A.(Q x)) (\lambda x:A.(P x))) (\lambda x:A.(\lambda H0:(P x).(\lambda H1:(Q x).(ex_intro2 A (\lambda x0:A.(Q x0)) (\lambda x0:A.(P x0)) x H1 H0)))) H)))))
-Post Nodes: 77
-\e[0;34mDebug: \e[0mProcedural: level 2 transformation
-\e[0;34mDebug: \e[0mProcedural: grafite rendering
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./types/props.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of types/props.mma completed in 2''.
-\e[0;32mInfo: \e[0mexecution of theory.mma started:
-\e[0;34mDebug: \e[0mExecuting: ``set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-2/the ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./ext/tactics.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/ext/tacti ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./ext/arith.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/ext/arith ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./types/props.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/types/pro ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./blt/props.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/blt/props ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./plist/props.ma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;34mDebug: \e[0mExecuting: ``include "cic:/matita/LAMBDA-TYPES/Base-2/plist/pro ...''
-\e[0;34mDebug: \e[0mIncluding /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/./theory.mma with path: /home/fguidi/svn/software/matita/contribs/LAMBDA-TYPES/Base-2/.
-\e[0;32mInfo: \e[0mexecution of theory.mma completed in 1''.
baseuri=cic:/matita/LAMBDA-TYPES
-include_paths= ../../legacy
+include_paths= ../../legacy ../RELATIONAL
projects / helm.git / commitdiff