C).(\lambda (H3: (csuba g c c2)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2
H2)))))))))) c1 t a H))))).
-theorem csuba_arity_rev:
+axiom csuba_arity_rev:
\forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1
t a) \to (\forall (c2: C).((csuba g c2 c1) \to (arity g c2 t a)))))))
-\def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H:
-(arity g c1 t a)).(arity_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (a0:
-A).(\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 a0)))))) (\lambda (c:
-C).(\lambda (n: nat).(\lambda (c2: C).(\lambda (_: (csuba g c2
-c)).(arity_sort g c2 n))))) (\lambda (c: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr)
-u))).(\lambda (a0: A).(\lambda (H1: (arity g d u a0)).(\lambda (H2: ((\forall
-(c2: C).((csuba g c2 d) \to (arity g c2 u a0))))).(\lambda (c2: C).(\lambda
-(H3: (csuba g c2 c)).(let H4 \def (csuba_getl_abbr_rev g c d u i H0 c2 H3) in
-(or_ind (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u)))
-(\lambda (d2: C).(csuba g d2 d))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1))))) (arity g c2
-(TLRef i) a0) (\lambda (H5: (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2
-(Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d)))).(ex2_ind C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d2 d))
-(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H6: (getl i c2 (CHead x
-(Bind Abbr) u))).(\lambda (H7: (csuba g x d)).(arity_abbr g c2 x u i H6 a0
-(H2 x H7))))) H5)) (\lambda (H5: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2:
-T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abst) u2))))) (\lambda (d2:
-C).(\lambda (_: T).(\lambda (_: A).(csuba g d2 d)))) (\lambda (d2:
-C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2 (asucc g a1))))) (\lambda
-(_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d u a1)))))).(ex4_3_ind C T
-A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2
-(Bind Abst) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g
-d2 d)))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a1: A).(arity g d2 u2
-(asucc g a1))))) (\lambda (_: C).(\lambda (_: T).(\lambda (a1: A).(arity g d
-u a1)))) (arity g c2 (TLRef i) a0) (\lambda (x0: C).(\lambda (x1: T).(\lambda
-(x2: A).(\lambda (H6: (getl i c2 (CHead x0 (Bind Abst) x1))).(\lambda (_:
-(csuba g x0 d)).(\lambda (H8: (arity g x0 x1 (asucc g x2))).(\lambda (H9:
-(arity g d u x2)).(arity_repl g c2 (TLRef i) x2 (arity_abst g c2 x0 x1 i H6
-x2 H8) a0 (arity_mono g d u x2 H9 a0 H1))))))))) H5)) H4)))))))))))) (\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 (_: (arity g d u
-(asucc g a0))).(\lambda (H2: ((\forall (c2: C).((csuba g c2 d) \to (arity g
-c2 u (asucc g a0)))))).(\lambda (c2: C).(\lambda (H3: (csuba g c2 c)).(let H4
-\def (csuba_getl_abst_rev g c d u i H0 c2 H3) in (ex2_ind C (\lambda (d2:
-C).(getl i c2 (CHead d2 (Bind Abst) u))) (\lambda (d2: C).(csuba g d2 d))
-(arity g c2 (TLRef i) a0) (\lambda (x: C).(\lambda (H5: (getl i c2 (CHead x
-(Bind Abst) u))).(\lambda (H6: (csuba g x d)).(arity_abst g c2 x u i H5 a0
-(H2 x H6))))) H4)))))))))))) (\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 (c2: C).((csuba g c2 c) \to (arity g c2 u
-a1))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
-(Bind b) u) t0 a2)).(\lambda (H4: ((\forall (c2: C).((csuba g c2 (CHead c
-(Bind b) u)) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H5: (csuba
-g c2 c)).(arity_bind g b H0 c2 u a1 (H2 c2 H5) t0 a2 (H4 (CHead c2 (Bind b)
-u) (csuba_head g c2 c H5 (Bind b) u)))))))))))))))) (\lambda (c: C).(\lambda
-(u: T).(\lambda (a1: A).(\lambda (_: (arity g c u (asucc g a1))).(\lambda
-(H1: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u (asucc g
-a1)))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c
-(Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (c2: C).((csuba g c2 (CHead c
-(Bind Abst) u)) \to (arity g c2 t0 a2))))).(\lambda (c2: C).(\lambda (H4:
-(csuba g c2 c)).(arity_head g c2 u a1 (H1 c2 H4) t0 a2 (H3 (CHead c2 (Bind
-Abst) u) (csuba_head g c2 c H4 (Bind Abst) u)))))))))))))) (\lambda (c:
-C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c u a1)).(\lambda
-(H1: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u a1))))).(\lambda
-(t0: T).(\lambda (a2: A).(\lambda (_: (arity g c t0 (AHead a1 a2))).(\lambda
-(H3: ((\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 (AHead a1
-a2)))))).(\lambda (c2: C).(\lambda (H4: (csuba g c2 c)).(arity_appl g c2 u a1
-(H1 c2 H4) t0 a2 (H3 c2 H4))))))))))))) (\lambda (c: C).(\lambda (u:
-T).(\lambda (a0: A).(\lambda (_: (arity g c u (asucc g a0))).(\lambda (H1:
-((\forall (c2: C).((csuba g c2 c) \to (arity g c2 u (asucc g
-a0)))))).(\lambda (t0: T).(\lambda (_: (arity g c t0 a0)).(\lambda (H3:
-((\forall (c2: C).((csuba g c2 c) \to (arity g c2 t0 a0))))).(\lambda (c2:
-C).(\lambda (H4: (csuba g c2 c)).(arity_cast g c2 u a0 (H1 c2 H4) t0 (H3 c2
-H4)))))))))))) (\lambda (c: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_:
-(arity g c t0 a1)).(\lambda (H1: ((\forall (c2: C).((csuba g c2 c) \to (arity
-g c2 t0 a1))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (c2:
-C).(\lambda (H3: (csuba g c2 c)).(arity_repl g c2 t0 a1 (H1 c2 H3) a2
-H2)))))))))) c1 t a H))))).
+.
theorem arity_appls_appl:
\forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (a1: A).((arity g c
(refl_equal T (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2
t2)))) (refl_equal T (THead (Bind b) v t))))))))) vs)))))))).
+theorem iso_gen_sort:
+ \forall (u2: T).(\forall (n1: nat).((iso (TSort n1) u2) \to (ex nat (\lambda
+(n2: nat).(eq T u2 (TSort n2))))))
+\def
+ \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TSort n1) u2)).(let H0
+\def (match H in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_:
+(iso t t0)).((eq T t (TSort n1)) \to ((eq T t0 u2) \to (ex nat (\lambda (n2:
+nat).(eq T u2 (TSort n2))))))))) with [(iso_sort n0 n2) \Rightarrow (\lambda
+(H0: (eq T (TSort n0) (TSort n1))).(\lambda (H1: (eq T (TSort n2) u2)).((let
+H2 \def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_:
+T).nat) with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _
+_) \Rightarrow n0])) (TSort n0) (TSort n1) H0) in (eq_ind nat n1 (\lambda (_:
+nat).((eq T (TSort n2) u2) \to (ex nat (\lambda (n3: nat).(eq T u2 (TSort
+n3)))))) (\lambda (H3: (eq T (TSort n2) u2)).(eq_ind T (TSort n2) (\lambda
+(t: T).(ex nat (\lambda (n3: nat).(eq T t (TSort n3))))) (ex_intro nat
+(\lambda (n3: nat).(eq T (TSort n2) (TSort n3))) n2 (refl_equal T (TSort
+n2))) u2 H3)) n0 (sym_eq nat n0 n1 H2))) H1))) | (iso_lref i1 i2) \Rightarrow
+(\lambda (H0: (eq T (TLRef i1) (TSort n1))).(\lambda (H1: (eq T (TLRef i2)
+u2)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n1) H0) in
+(False_ind ((eq T (TLRef i2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
+(TSort n2))))) H2)) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow (\lambda
+(H0: (eq T (THead k v1 t1) (TSort n1))).(\lambda (H1: (eq T (THead k v2 t2)
+u2)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n1) H0) in
+(False_ind ((eq T (THead k v2 t2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
+(TSort n2))))) H2)) H1)))]) in (H0 (refl_equal T (TSort n1)) (refl_equal T
+u2))))).
+
+theorem iso_gen_lref:
+ \forall (u2: T).(\forall (n1: nat).((iso (TLRef n1) u2) \to (ex nat (\lambda
+(n2: nat).(eq T u2 (TLRef n2))))))
+\def
+ \lambda (u2: T).(\lambda (n1: nat).(\lambda (H: (iso (TLRef n1) u2)).(let H0
+\def (match H in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_:
+(iso t t0)).((eq T t (TLRef n1)) \to ((eq T t0 u2) \to (ex nat (\lambda (n2:
+nat).(eq T u2 (TLRef n2))))))))) with [(iso_sort n0 n2) \Rightarrow (\lambda
+(H0: (eq T (TSort n0) (TLRef n1))).(\lambda (H1: (eq T (TSort n2) u2)).((let
+H2 \def (eq_ind T (TSort n0) (\lambda (e: T).(match e in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (TLRef n1) H0) in (False_ind ((eq T
+(TSort n2) u2) \to (ex nat (\lambda (n3: nat).(eq T u2 (TLRef n3))))) H2))
+H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TLRef
+n1))).(\lambda (H1: (eq T (TLRef i2) u2)).((let H2 \def (f_equal T nat
+(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
+\Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i1]))
+(TLRef i1) (TLRef n1) H0) in (eq_ind nat n1 (\lambda (_: nat).((eq T (TLRef
+i2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2 (TLRef n2)))))) (\lambda (H3:
+(eq T (TLRef i2) u2)).(eq_ind T (TLRef i2) (\lambda (t: T).(ex nat (\lambda
+(n2: nat).(eq T t (TLRef n2))))) (ex_intro nat (\lambda (n2: nat).(eq T
+(TLRef i2) (TLRef n2))) i2 (refl_equal T (TLRef i2))) u2 H3)) i1 (sym_eq nat
+i1 n1 H2))) H1))) | (iso_head v1 v2 t1 t2 k) \Rightarrow (\lambda (H0: (eq T
+(THead k v1 t1) (TLRef n1))).(\lambda (H1: (eq T (THead k v2 t2) u2)).((let
+H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e in T return
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n1) H0) in
+(False_ind ((eq T (THead k v2 t2) u2) \to (ex nat (\lambda (n2: nat).(eq T u2
+(TLRef n2))))) H2)) H1)))]) in (H0 (refl_equal T (TLRef n1)) (refl_equal T
+u2))))).
+
+theorem iso_gen_head:
+ \forall (k: K).(\forall (v1: T).(\forall (t1: T).(\forall (u2: T).((iso
+(THead k v1 t1) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))))))
+\def
+ \lambda (k: K).(\lambda (v1: T).(\lambda (t1: T).(\lambda (u2: T).(\lambda
+(H: (iso (THead k v1 t1) u2)).(let H0 \def (match H in iso return (\lambda
+(t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (THead k v1 t1))
+\to ((eq T t0 u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq
+T (TSort n1) (THead k v1 t1))).(\lambda (H1: (eq T (TSort n2) u2)).((let H2
+\def (eq_ind T (TSort n1) (\lambda (e: T).(match e in T return (\lambda (_:
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False |
+(THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H0) in (False_ind ((eq T
+(TSort n2) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0:
+(eq T (TLRef i1) (THead k v1 t1))).(\lambda (H1: (eq T (TLRef i2) u2)).((let
+H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (THead k v1 t1) H0) in (False_ind ((eq T
+(TLRef i2) u2) \to (ex_2 T T (\lambda (v2: T).(\lambda (t2: T).(eq T u2
+(THead k v2 t2)))))) H2)) H1))) | (iso_head v0 v2 t0 t2 k0) \Rightarrow
+(\lambda (H0: (eq T (THead k0 v0 t0) (THead k v1 t1))).(\lambda (H1: (eq T
+(THead k0 v2 t2) u2)).((let H2 \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 v0 t0) (THead k v1
+t1) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e in T return
+(\lambda (_: T).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0
+| (THead _ t _) \Rightarrow t])) (THead k0 v0 t0) (THead k v1 t1) H0) in
+((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 v0 t0) (THead k v1 t1) H0) in (eq_ind K k
+(\lambda (k1: K).((eq T v0 v1) \to ((eq T t0 t1) \to ((eq T (THead k1 v2 t2)
+u2) \to (ex_2 T T (\lambda (v3: T).(\lambda (t3: T).(eq T u2 (THead k v3
+t3))))))))) (\lambda (H5: (eq T v0 v1)).(eq_ind T v1 (\lambda (_: T).((eq T
+t0 t1) \to ((eq T (THead k v2 t2) u2) \to (ex_2 T T (\lambda (v3: T).(\lambda
+(t3: T).(eq T u2 (THead k v3 t3)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T
+t1 (\lambda (_: T).((eq T (THead k v2 t2) u2) \to (ex_2 T T (\lambda (v3:
+T).(\lambda (t3: T).(eq T u2 (THead k v3 t3))))))) (\lambda (H7: (eq T (THead
+k v2 t2) u2)).(eq_ind T (THead k v2 t2) (\lambda (t: T).(ex_2 T T (\lambda
+(v3: T).(\lambda (t3: T).(eq T t (THead k v3 t3)))))) (ex_2_intro T T
+(\lambda (v3: T).(\lambda (t3: T).(eq T (THead k v2 t2) (THead k v3 t3)))) v2
+t2 (refl_equal T (THead k v2 t2))) u2 H7)) t0 (sym_eq T t0 t1 H6))) v0
+(sym_eq T v0 v1 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H0
+(refl_equal T (THead k v1 t1)) (refl_equal T u2))))))).
+
set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/iso/props".
-include "iso/defs.ma".
+include "iso/fwd.ma".
theorem iso_trans:
\forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
include "lift/props.ma".
+theorem dnf_dec2:
+ \forall (t: T).(\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w t (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t (lift (S
+O) d v))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))))) (ex T (\lambda
+(v: T).(eq T t0 (lift (S O) d v))))))) (\lambda (n: nat).(\lambda (d:
+nat).(or_intror (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (TSort n)
+(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (TSort n) (lift (S O) d
+v)))) (ex_intro T (\lambda (v: T).(eq T (TSort n) (lift (S O) d v))) (TSort
+n) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0)) (refl_equal T
+(TSort n)) (lift (S O) d (TSort n)) (lift_sort n (S O) d)))))) (\lambda (n:
+nat).(\lambda (d: nat).(lt_eq_gt_e n d (or (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(TLRef n) (lift (S O) d v))))) (\lambda (H: (lt n d)).(or_intror (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T
+(\lambda (v: T).(eq T (TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v:
+T).(eq T (TLRef n) (lift (S O) d v))) (TLRef n) (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d
+(TLRef n)) (lift_lref_lt n (S O) d H))))) (\lambda (H: (eq nat n d)).(eq_ind
+nat n (\lambda (n0: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 n0
+w (TLRef n) (lift (S O) n0 v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift
+(S O) n0 v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v: T).(subst0 n w
+(TLRef n) (lift (S O) n v))))) (ex T (\lambda (v: T).(eq T (TLRef n) (lift (S
+O) n v)))) (\lambda (w: T).(ex_intro T (\lambda (v: T).(subst0 n w (TLRef n)
+(lift (S O) n v))) (lift n O w) (eq_ind_r T (lift (plus (S O) n) O w)
+(\lambda (t0: T).(subst0 n w (TLRef n) t0)) (subst0_lref w n) (lift (S O) n
+(lift n O w)) (lift_free w n (S O) O n (le_n (plus O n)) (le_O_n n)))))) d
+H)) (\lambda (H: (lt d n)).(or_intror (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (TLRef n) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(TLRef n) (lift (S O) d v)))) (ex_intro T (\lambda (v: T).(eq T (TLRef n)
+(lift (S O) d v))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq T (TLRef n) t0)) (refl_equal T (TLRef n)) (lift (S O) d (TLRef (pred
+n))) (lift_lref_gt d n H)))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
+(H: ((\forall (d: nat).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
+t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v)))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w t1 (lift (S O) d v))))) (ex T (\lambda
+(v: T).(eq T t1 (lift (S O) d v)))))))).(\lambda (d: nat).(let H_x \def (H d)
+in (let H1 \def H_x in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0
+d w t0 (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1)
+(lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O)
+d v))))) (\lambda (H2: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 d w t0
+(lift (S O) d v))))))).(let H_x0 \def (H0 (s k d)) in (let H3 \def H_x0 in
+(or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w t1 (lift (S
+O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))))
+(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift
+(S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d
+v))))) (\lambda (H4: ((\forall (w: T).(ex T (\lambda (v: T).(subst0 (s k d) w
+t1 (lift (S O) (s k d) v))))))).(or_introl (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
+T (THead k t0 t1) (lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H4 w)
+in (let H5 \def H_x1 in (ex_ind T (\lambda (v: T).(subst0 (s k d) w t1 (lift
+(S O) (s k d) v))) (ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S
+O) d v)))) (\lambda (x: T).(\lambda (H6: (subst0 (s k d) w t1 (lift (S O) (s
+k d) x))).(let H_x2 \def (H2 w) in (let H7 \def H_x2 in (ex_ind T (\lambda
+(v: T).(subst0 d w t0 (lift (S O) d v))) (ex T (\lambda (v: T).(subst0 d w
+(THead k t0 t1) (lift (S O) d v)))) (\lambda (x0: T).(\lambda (H8: (subst0 d
+w t0 (lift (S O) d x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0
+t1) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
+(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 t1) t2))
+(subst0_both w t0 (lift (S O) d x0) d H8 k t1 (lift (S O) (s k d) x) H6)
+(lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H7))))) H5))))))
+(\lambda (H4: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
+v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x: T).(\lambda (H5: (eq T t1 (lift (S O) (s k d) x))).(eq_ind_r T
+(lift (S O) (s k d) x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
+(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
+T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex
+T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 (lift (S O) (s k d) x))
+(lift (S O) d v)))) (\lambda (w: T).(let H_x1 \def (H2 w) in (let H6 \def
+H_x1 in (ex_ind T (\lambda (v: T).(subst0 d w t0 (lift (S O) d v))) (ex T
+(\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d) x)) (lift (S O) d
+v)))) (\lambda (x0: T).(\lambda (H7: (subst0 d w t0 (lift (S O) d
+x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k t0 (lift (S O) (s k d)
+x)) (lift (S O) d v))) (THead k x0 x) (eq_ind_r T (THead k (lift (S O) d x0)
+(lift (S O) (s k d) x)) (\lambda (t2: T).(subst0 d w (THead k t0 (lift (S O)
+(s k d) x)) t2)) (subst0_fst w (lift (S O) d x0) t0 d H7 (lift (S O) (s k d)
+x) k) (lift (S O) d (THead k x0 x)) (lift_head k x0 x (S O) d))))) H6))))) t1
+H5))) H4)) H3)))) (\lambda (H2: (ex T (\lambda (v: T).(eq T t0 (lift (S O) d
+v))))).(ex_ind T (\lambda (v: T).(eq T t0 (lift (S O) d v))) (or (\forall (w:
+T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O) d v))))) (ex
+T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v))))) (\lambda (x:
+T).(\lambda (H3: (eq T t0 (lift (S O) d x))).(let H_x0 \def (H0 (s k d)) in
+(let H4 \def H_x0 in (or_ind (\forall (w: T).(ex T (\lambda (v: T).(subst0 (s
+k d) w t1 (lift (S O) (s k d) v))))) (ex T (\lambda (v: T).(eq T t1 (lift (S
+O) (s k d) v)))) (or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
+k t0 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1)
+(lift (S O) d v))))) (\lambda (H5: ((\forall (w: T).(ex T (\lambda (v:
+T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))))))).(eq_ind_r T (lift (S O)
+d x) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w
+(THead k t2 t1) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T (THead k t2
+t1) (lift (S O) d v)))))) (or_introl (\forall (w: T).(ex T (\lambda (v:
+T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v))))) (ex T
+(\lambda (v: T).(eq T (THead k (lift (S O) d x) t1) (lift (S O) d v))))
+(\lambda (w: T).(let H_x1 \def (H5 w) in (let H6 \def H_x1 in (ex_ind T
+(\lambda (v: T).(subst0 (s k d) w t1 (lift (S O) (s k d) v))) (ex T (\lambda
+(v: T).(subst0 d w (THead k (lift (S O) d x) t1) (lift (S O) d v)))) (\lambda
+(x0: T).(\lambda (H7: (subst0 (s k d) w t1 (lift (S O) (s k d)
+x0))).(ex_intro T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x) t1)
+(lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d x) (lift
+(S O) (s k d) x0)) (\lambda (t2: T).(subst0 d w (THead k (lift (S O) d x) t1)
+t2)) (subst0_snd k w (lift (S O) (s k d) x0) t1 d H7 (lift (S O) d x)) (lift
+(S O) d (THead k x x0)) (lift_head k x x0 (S O) d))))) H6))))) t0 H3))
+(\lambda (H5: (ex T (\lambda (v: T).(eq T t1 (lift (S O) (s k d)
+v))))).(ex_ind T (\lambda (v: T).(eq T t1 (lift (S O) (s k d) v))) (or
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k t0 t1) (lift (S O)
+d v))))) (ex T (\lambda (v: T).(eq T (THead k t0 t1) (lift (S O) d v)))))
+(\lambda (x0: T).(\lambda (H6: (eq T t1 (lift (S O) (s k d) x0))).(eq_ind_r T
+(lift (S O) (s k d) x0) (\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda
+(v: T).(subst0 d w (THead k t0 t2) (lift (S O) d v))))) (ex T (\lambda (v:
+T).(eq T (THead k t0 t2) (lift (S O) d v)))))) (eq_ind_r T (lift (S O) d x)
+(\lambda (t2: T).(or (\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead
+k t2 (lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq
+T (THead k t2 (lift (S O) (s k d) x0)) (lift (S O) d v)))))) (or_intror
+(\forall (w: T).(ex T (\lambda (v: T).(subst0 d w (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0)) (lift (S O) d v))))) (ex T (\lambda (v: T).(eq T
+(THead k (lift (S O) d x) (lift (S O) (s k d) x0)) (lift (S O) d v))))
+(ex_intro T (\lambda (v: T).(eq T (THead k (lift (S O) d x) (lift (S O) (s k
+d) x0)) (lift (S O) d v))) (THead k x x0) (eq_ind_r T (THead k (lift (S O) d
+x) (lift (S O) (s k d) x0)) (\lambda (t2: T).(eq T (THead k (lift (S O) d x)
+(lift (S O) (s k d) x0)) t2)) (refl_equal T (THead k (lift (S O) d x) (lift
+(S O) (s k d) x0))) (lift (S O) d (THead k x x0)) (lift_head k x x0 (S O)
+d)))) t0 H3) t1 H6))) H5)) H4))))) H2)) H1))))))))) t).
+
theorem dnf_dec:
\forall (w: T).(\forall (t: T).(\forall (d: nat).(ex T (\lambda (v: T).(or
(subst0 d w t (lift (S O) d v)) (eq T t (lift (S O) d v)))))))
projects / helm.git / commitdiff