include "ty3/arity.ma".
+include "pc3/nf2.ma".
+
include "nf2/arity.ma".
+definition ty3_nf2_inv_abst_premise:
+ C \to (T \to (T \to Prop))
+\def
+ \lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\forall (d: C).(\forall (wi:
+T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi)) \to (\forall (vs:
+TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi)) (THead (Bind Abst) w
+u)) \to False)))))))).
+
+theorem ty3_nf2_inv_abst_premise_csort:
+ \forall (w: T).(\forall (u: T).(\forall (m: nat).(ty3_nf2_inv_abst_premise
+(CSort m) w u)))
+\def
+ \lambda (w: T).(\lambda (u: T).(\lambda (m: nat).(\lambda (d: C).(\lambda
+(wi: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort m) (CHead d (Bind Abst)
+wi))).(\lambda (vs: TList).(\lambda (_: (pc3 (CSort m) (THeads (Flat Appl) vs
+(lift (S i) O wi)) (THead (Bind Abst) w u))).(getl_gen_sort m i (CHead d
+(Bind Abst) wi) H False))))))))).
+
theorem ty3_nf2_inv_all:
\forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (u: T).((ty3 g c t
u) \to ((nf2 c t) \to (or3 (ex3_2 T T (\lambda (w: T).(\lambda (u0: T).(eq T
x0 x1 (refl_equal T (THeads (Flat Appl) x0 (TLRef x1))) H4 H5)) t H3)))))))
H2)) H1)))))))).
+theorem ty3_nf2_gen__ty3_nf2_inv_abst_aux:
+ \forall (c: C).(\forall (w1: T).(\forall (u1: T).((ty3_nf2_inv_abst_premise
+c w1 u1) \to (\forall (t: T).(\forall (w2: T).(\forall (u2: T).((pc3 c (THead
+(Flat Appl) t (THead (Bind Abst) w2 u2)) (THead (Bind Abst) w1 u1)) \to
+(ty3_nf2_inv_abst_premise c w2 u2))))))))
+\def
+ \lambda (c: C).(\lambda (w1: T).(\lambda (u1: T).(\lambda (H: ((\forall (d:
+C).(\forall (wi: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) wi))
+\to (\forall (vs: TList).((pc3 c (THeads (Flat Appl) vs (lift (S i) O wi))
+(THead (Bind Abst) w1 u1)) \to False)))))))).(\lambda (t: T).(\lambda (w2:
+T).(\lambda (u2: T).(\lambda (H0: (pc3 c (THead (Flat Appl) t (THead (Bind
+Abst) w2 u2)) (THead (Bind Abst) w1 u1))).(\lambda (d: C).(\lambda (wi:
+T).(\lambda (i: nat).(\lambda (H1: (getl i c (CHead d (Bind Abst)
+wi))).(\lambda (vs: TList).(\lambda (H2: (pc3 c (THeads (Flat Appl) vs (lift
+(S i) O wi)) (THead (Bind Abst) w2 u2))).(H d wi i H1 (TCons t vs) (pc3_t
+(THead (Flat Appl) t (THead (Bind Abst) w2 u2)) c (THead (Flat Appl) t
+(THeads (Flat Appl) vs (lift (S i) O wi))) (pc3_thin_dx c (THeads (Flat Appl)
+vs (lift (S i) O wi)) (THead (Bind Abst) w2 u2) H2 t Appl) (THead (Bind Abst)
+w1 u1) H0))))))))))))))).
+
+theorem ty3_nf2_inv_abst:
+ \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (w: T).(\forall (u:
+T).((ty3 g c t (THead (Bind Abst) w u)) \to ((nf2 c t) \to ((nf2 c w) \to
+((ty3_nf2_inv_abst_premise c w u) \to (ex4_2 T T (\lambda (v: T).(\lambda (_:
+T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g
+c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v
+u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w)
+v))))))))))))
+\def
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (w: T).(\lambda (u:
+T).(\lambda (H: (ty3 g c t (THead (Bind Abst) w u))).(\lambda (H0: (nf2 c
+t)).(\lambda (H1: (nf2 c w)).(\lambda (H2: (ty3_nf2_inv_abst_premise c w
+u)).(let H_x \def (ty3_nf2_inv_all g c t (THead (Bind Abst) w u) H H0) in
+(let H3 \def H_x in (or3_ind (ex3_2 T T (\lambda (w0: T).(\lambda (u0: T).(eq
+T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c
+w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))))
+(ex nat (\lambda (n: nat).(eq T t (TSort n)))) (ex3_2 TList nat (\lambda (ws:
+TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl) ws (TLRef i)))))
+(\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws))) (\lambda (_:
+TList).(\lambda (i: nat).(nf2 c (TLRef i))))) (ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v)))) (\lambda (H4: (ex3_2 T T (\lambda (w0: T).(\lambda (u0:
+T).(eq T t (THead (Bind Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2
+c w0))) (\lambda (w0: T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0)
+u0))))).(ex3_2_ind T T (\lambda (w0: T).(\lambda (u0: T).(eq T t (THead (Bind
+Abst) w0 u0)))) (\lambda (w0: T).(\lambda (_: T).(nf2 c w0))) (\lambda (w0:
+T).(\lambda (u0: T).(nf2 (CHead c (Bind Abst) w0) u0))) (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v)))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H5: (eq T t
+(THead (Bind Abst) x0 x1))).(\lambda (H6: (nf2 c x0)).(\lambda (H7: (nf2
+(CHead c (Bind Abst) x0) x1)).(let H8 \def (eq_ind T t (\lambda (t0: T).(ty3
+g c t0 (THead (Bind Abst) w u))) H (THead (Bind Abst) x0 x1) H5) in (eq_ind_r
+T (THead (Bind Abst) x0 x1) (\lambda (t0: T).(ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v))))) (ex_ind T (\lambda (t0: T).(ty3 g c (THead (Bind Abst)
+w u) t0)) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst)
+x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
+(x: T).(\lambda (H9: (ty3 g c (THead (Bind Abst) w u) x)).(ex4_3_ind T T T
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) w
+t2) x)))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c w t0))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst)
+w) u t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead c
+(Bind Abst) w) t2 t1)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda
+(w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
+(Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
+Abst) w) v)))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda
+(_: (pc3 c (THead (Bind Abst) w x2) x)).(\lambda (H11: (ty3 g c w
+x3)).(\lambda (H12: (ty3 g (CHead c (Bind Abst) w) u x2)).(\lambda (_: (ty3 g
+(CHead c (Bind Abst) w) x2 x4)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda
+(_: T).(\lambda (_: T).(pc3 c (THead (Bind Abst) x0 t2) (THead (Bind Abst) w
+u))))) (\lambda (_: T).(\lambda (t0: T).(\lambda (_: T).(ty3 g c x0 t0))))
+(\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst)
+x0) x1 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t1: T).(ty3 g (CHead
+c (Bind Abst) x0) t2 t1)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
+(THead (Bind Abst) x0 x1) (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda
+(w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
+(Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
+Abst) w) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda
+(H14: (pc3 c (THead (Bind Abst) x0 x5) (THead (Bind Abst) w u))).(\lambda (_:
+(ty3 g c x0 x6)).(\lambda (H16: (ty3 g (CHead c (Bind Abst) x0) x1
+x5)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x0) x5 x7)).(and_ind (pc3 c x0
+w) (\forall (b: B).(\forall (u0: T).(pc3 (CHead c (Bind b) u0) x5 u))) (ex4_2
+T T (\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead
+(Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
+(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda
+(H18: (pc3 c x0 w)).(\lambda (H19: ((\forall (b: B).(\forall (u0: T).(pc3
+(CHead c (Bind b) u0) x5 u))))).(let H_y \def (pc3_nf2 c x0 w H18 H6 H1) in
+(let H20 \def (eq_ind T x0 (\lambda (t0: T).(ty3 g (CHead c (Bind Abst) t0)
+x1 x5)) H16 w H_y) in (let H21 \def (eq_ind T x0 (\lambda (t0: T).(nf2 (CHead
+c (Bind Abst) t0) x1)) H7 w H_y) in (eq_ind_r T w (\lambda (t0: T).(ex4_2 T T
+(\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) t0 x1) (THead (Bind
+Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) (ex4_2_intro T T
+(\lambda (v: T).(\lambda (_: T).(eq T (THead (Bind Abst) w x1) (THead (Bind
+Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))) x1 x3 (refl_equal T
+(THead (Bind Abst) w x1)) H11 (ty3_conv g (CHead c (Bind Abst) w) u x2 H12 x1
+x5 H20 (H19 Abst w)) H21) x0 H_y)))))) (pc3_gen_abst c x0 w x5 u H14)))))))))
+(ty3_gen_bind g Abst c x0 x1 (THead (Bind Abst) w u) H8)))))))))
+(ty3_gen_bind g Abst c w u x H9)))) (ty3_correct g c (THead (Bind Abst) x0
+x1) (THead (Bind Abst) w u) H8)) t H5))))))) H4)) (\lambda (H4: (ex nat
+(\lambda (n: nat).(eq T t (TSort n))))).(ex_ind nat (\lambda (n: nat).(eq T t
+(TSort n))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T t (THead (Bind
+Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))) (\lambda (x:
+nat).(\lambda (H5: (eq T t (TSort x))).(let H6 \def (eq_ind T t (\lambda (t0:
+T).(ty3 g c t0 (THead (Bind Abst) w u))) H (TSort x) H5) in (eq_ind_r T
+(TSort x) (\lambda (t0: T).(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T
+t0 (THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))))
+(pc3_gen_sort_abst c w u (next g x) (ty3_gen_sort g c (THead (Bind Abst) w u)
+x H6) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TSort x) (THead (Bind
+Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v))))) t H5)))) H4)) (\lambda
+(H4: (ex3_2 TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads
+(Flat Appl) ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c
+ws))) (\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))))).(ex3_2_ind
+TList nat (\lambda (ws: TList).(\lambda (i: nat).(eq T t (THeads (Flat Appl)
+ws (TLRef i))))) (\lambda (ws: TList).(\lambda (_: nat).(nfs2 c ws)))
+(\lambda (_: TList).(\lambda (i: nat).(nf2 c (TLRef i)))) (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T t (THead (Bind Abst) w v)))) (\lambda (_:
+T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g
+(CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c
+(Bind Abst) w) v)))) (\lambda (x0: TList).(\lambda (x1: nat).(\lambda (H5:
+(eq T t (THeads (Flat Appl) x0 (TLRef x1)))).(\lambda (_: (nfs2 c
+x0)).(\lambda (H7: (nf2 c (TLRef x1))).(let H8 \def (eq_ind T t (\lambda (t0:
+T).(ty3 g c t0 (THead (Bind Abst) w u))) H (THeads (Flat Appl) x0 (TLRef x1))
+H5) in (eq_ind_r T (THeads (Flat Appl) x0 (TLRef x1)) (\lambda (t0: T).(ex4_2
+T T (\lambda (v: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) w v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) w) v u))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) w) v))))) (let H9 \def H2 in ((let H10 \def H8
+in (unintro T u (\lambda (t0: T).((ty3 g c (THeads (Flat Appl) x0 (TLRef x1))
+(THead (Bind Abst) w t0)) \to ((ty3_nf2_inv_abst_premise c w t0) \to (ex4_2 T
+T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) x0 (TLRef x1))
+(THead (Bind Abst) w v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c w w0)))
+(\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) w) v t0)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) w) v)))))))
+(unintro T w (\lambda (t0: T).(\forall (x: T).((ty3 g c (THeads (Flat Appl)
+x0 (TLRef x1)) (THead (Bind Abst) t0 x)) \to ((ty3_nf2_inv_abst_premise c t0
+x) \to (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl)
+x0 (TLRef x1)) (THead (Bind Abst) t0 v)))) (\lambda (_: T).(\lambda (w0:
+T).(ty3 g c t0 w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind
+Abst) t0) v x))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst)
+t0) v)))))))) (TList_ind (\lambda (t0: TList).(\forall (x: T).(\forall (x2:
+T).((ty3 g c (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x x2)) \to
+((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
+(_: T).(eq T (THeads (Flat Appl) t0 (TLRef x1)) (THead (Bind Abst) x v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) x) v))))))))) (\lambda (x: T).(\lambda (x2:
+T).(\lambda (H11: (ty3 g c (TLRef x1) (THead (Bind Abst) x x2))).(\lambda
+(H12: (ty3_nf2_inv_abst_premise c x x2)).(or_ind (ex3_3 C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind
+Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda
+(_: T).(pc3 c (lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0)))))
+(\lambda (e: C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0))))) (ex4_2
+T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind Abst) x
+v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (H13: (ex3_3 C
+T T (\lambda (_: C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O
+t0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda
+(_: T).(getl x1 c (CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T (\lambda (_:
+C).(\lambda (_: T).(\lambda (t0: T).(pc3 c (lift (S x1) O t0) (THead (Bind
+Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_: T).(getl x1 c
+(CHead e (Bind Abbr) u0))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (t0:
+T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef
+x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda
+(x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c (lift (S x1) O
+x5) (THead (Bind Abst) x x2))).(\lambda (H15: (getl x1 c (CHead x3 (Bind
+Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(nf2_gen_lref c x3 x4 x1 H15 H7
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind
+Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))))))))) H13)) (\lambda
+(H13: (ex3_3 C T T (\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c
+(lift (S x1) O u0) (THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (_: T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e:
+C).(\lambda (u0: T).(\lambda (t0: T).(ty3 g e u0 t0)))))).(ex3_3_ind C T T
+(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(pc3 c (lift (S x1) O u0)
+(THead (Bind Abst) x x2))))) (\lambda (e: C).(\lambda (u0: T).(\lambda (_:
+T).(getl x1 c (CHead e (Bind Abst) u0))))) (\lambda (e: C).(\lambda (u0:
+T).(\lambda (t0: T).(ty3 g e u0 t0)))) (ex4_2 T T (\lambda (v: T).(\lambda
+(_: T).(eq T (TLRef x1) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
+Abst) x) v)))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda
+(H14: (pc3 c (lift (S x1) O x4) (THead (Bind Abst) x x2))).(\lambda (H15:
+(getl x1 c (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let
+H_x0 \def (H12 x3 x4 x1 H15 TNil H14) in (let H17 \def H_x0 in (False_ind
+(ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (TLRef x1) (THead (Bind
+Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v:
+T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v:
+T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v)))) H17))))))))) H13))
+(ty3_gen_lref g c (THead (Bind Abst) x x2) x1 H11)))))) (\lambda (t0:
+T).(\lambda (t1: TList).(\lambda (H11: ((\forall (x: T).(\forall (x2:
+T).((ty3 g c (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2)) \to
+((ty3_nf2_inv_abst_premise c x x2) \to (ex4_2 T T (\lambda (v: T).(\lambda
+(_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) x) v)))))))))).(\lambda (x: T).(\lambda (x2:
+T).(\lambda (H12: (ty3 g c (THead (Flat Appl) t0 (THeads (Flat Appl) t1
+(TLRef x1))) (THead (Bind Abst) x x2))).(\lambda (H13:
+(ty3_nf2_inv_abst_premise c x x2)).(ex3_2_ind T T (\lambda (u0: T).(\lambda
+(t2: T).(pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) u0 t2)) (THead (Bind
+Abst) x x2)))) (\lambda (u0: T).(\lambda (t2: T).(ty3 g c (THeads (Flat Appl)
+t1 (TLRef x1)) (THead (Bind Abst) u0 t2)))) (\lambda (u0: T).(\lambda (_:
+T).(ty3 g c t0 u0))) (ex4_2 T T (\lambda (v: T).(\lambda (_: T).(eq T (THead
+(Flat Appl) t0 (THeads (Flat Appl) t1 (TLRef x1))) (THead (Bind Abst) x v))))
+(\lambda (_: T).(\lambda (w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda
+(_: T).(ty3 g (CHead c (Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_:
+T).(nf2 (CHead c (Bind Abst) x) v)))) (\lambda (x3: T).(\lambda (x4:
+T).(\lambda (H14: (pc3 c (THead (Flat Appl) t0 (THead (Bind Abst) x3 x4))
+(THead (Bind Abst) x x2))).(\lambda (H15: (ty3 g c (THeads (Flat Appl) t1
+(TLRef x1)) (THead (Bind Abst) x3 x4))).(\lambda (_: (ty3 g c t0 x3)).(let
+H_y \def (ty3_nf2_gen__ty3_nf2_inv_abst_aux c x x2 H13 t0 x3 x4 H14) in (let
+H_x0 \def (H11 x3 x4 H15 H_y) in (let H17 \def H_x0 in (ex4_2_ind T T
+(\lambda (v: T).(\lambda (_: T).(eq T (THeads (Flat Appl) t1 (TLRef x1))
+(THead (Bind Abst) x3 v)))) (\lambda (_: T).(\lambda (w0: T).(ty3 g c x3
+w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x3) v x4)))
+(\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x3) v))) (ex4_2 T T
+(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat
+Appl) t1 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
+Abst) x) v)))) (\lambda (x5: T).(\lambda (x6: T).(\lambda (H18: (eq T (THeads
+(Flat Appl) t1 (TLRef x1)) (THead (Bind Abst) x3 x5))).(\lambda (_: (ty3 g c
+x3 x6)).(\lambda (_: (ty3 g (CHead c (Bind Abst) x3) x5 x4)).(\lambda (_:
+(nf2 (CHead c (Bind Abst) x3) x5)).(TList_ind (\lambda (t2: TList).((eq T
+(THeads (Flat Appl) t2 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T
+(\lambda (v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat
+Appl) t2 (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda
+(w0: T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c
+(Bind Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind
+Abst) x) v)))))) (\lambda (H22: (eq T (THeads (Flat Appl) TNil (TLRef x1))
+(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (TLRef x1) (\lambda (ee:
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
+(THead (Bind Abst) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda (v:
+T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) TNil
+(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
+g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v
+x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x) v))))
+H23))) (\lambda (t2: T).(\lambda (t3: TList).(\lambda (_: (((eq T (THeads
+(Flat Appl) t3 (TLRef x1)) (THead (Bind Abst) x3 x5)) \to (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) t3
+(TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0: T).(ty3
+g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind Abst) x) v
+x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x)
+v))))))).(\lambda (H22: (eq T (THeads (Flat Appl) (TCons t2 t3) (TLRef x1))
+(THead (Bind Abst) x3 x5))).(let H23 \def (eq_ind T (THead (Flat Appl) t2
+(THeads (Flat Appl) t3 (TLRef x1))) (\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) x3 x5) H22) in (False_ind (ex4_2 T T (\lambda
+(v: T).(\lambda (_: T).(eq T (THead (Flat Appl) t0 (THeads (Flat Appl) (TCons
+t2 t3) (TLRef x1))) (THead (Bind Abst) x v)))) (\lambda (_: T).(\lambda (w0:
+T).(ty3 g c x w0))) (\lambda (v: T).(\lambda (_: T).(ty3 g (CHead c (Bind
+Abst) x) v x2))) (\lambda (v: T).(\lambda (_: T).(nf2 (CHead c (Bind Abst) x)
+v)))) H23)))))) t1 H18))))))) H17))))))))) (ty3_gen_appl g c t0 (THeads (Flat
+Appl) t1 (TLRef x1)) (THead (Bind Abst) x x2) H12))))))))) x0)) H10)) H9)) t
+H5))))))) H4)) H3))))))))))).
+
projects / helm.git / blobdiff