set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/iso/props".
-include "iso/defs.ma".
+include "iso/fwd.ma".
+
+theorem iso_refl:
+ \forall (t: T).(iso t t)
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(iso t0 t0)) (\lambda (n:
+nat).(iso_sort n n)) (\lambda (n: nat).(iso_lref n n)) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (_: (iso t0 t0)).(\lambda (t1: T).(\lambda (_:
+(iso t1 t1)).(iso_head t0 t0 t1 t1 k)))))) t).
theorem iso_trans:
\forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2
\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda
(t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3)))))
(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso
-(TSort n2) t3)).(let H1 \def (match H0 in iso return (\lambda (t: T).(\lambda
-(t0: T).(\lambda (_: (iso t t0)).((eq T t (TSort n2)) \to ((eq T t0 t3) \to
-(iso (TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H1: (eq
-T (TSort n0) (TSort n2))).(\lambda (H2: (eq T (TSort n3) t3)).((let H3 \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 n2) H1) in (eq_ind nat n2 (\lambda (_:
-nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H4: (eq T
-(TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t))
-(iso_sort n1 n3) t3 H4)) n0 (sym_eq nat n0 n2 H3))) H2))) | (iso_lref i1 i2)
-\Rightarrow (\lambda (H1: (eq T (TLRef i1) (TSort n2))).(\lambda (H2: (eq T
-(TLRef i2) t3)).((let H3 \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 n2) H1) in
-(False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H3)) H2))) |
-(iso_head k v1 v2 t0 t4) \Rightarrow (\lambda (H1: (eq T (THead k v1 t0)
-(TSort n2))).(\lambda (H2: (eq T (THead k v2 t4) t3)).((let H3 \def (eq_ind T
-(THead k v1 t0) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop)
-with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _
-_) \Rightarrow True])) I (TSort n2) H1) in (False_ind ((eq T (THead k v2 t4)
-t3) \to (iso (TSort n1) t3)) H3)) H2)))]) in (H1 (refl_equal T (TSort n2))
-(refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3:
-T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 in iso return
-(\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t t0)).((eq T t (TLRef
-i2)) \to ((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2)
-\Rightarrow (\lambda (H1: (eq T (TSort n1) (TLRef i2))).(\lambda (H2: (eq T
-(TSort n2) t3)).((let H3 \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 (TLRef i2) H1) in
-(False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H3)) H2))) |
-(iso_lref i0 i3) \Rightarrow (\lambda (H1: (eq T (TLRef i0) (TLRef
-i2))).(\lambda (H2: (eq T (TLRef i3) t3)).((let H3 \def (f_equal T nat
-(\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _)
-\Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0]))
-(TLRef i0) (TLRef i2) H1) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef
-i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H4: (eq T (TLRef i3) t3)).(eq_ind
-T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H4)) i0
-(sym_eq nat i0 i2 H3))) H2))) | (iso_head k v1 v2 t0 t4) \Rightarrow (\lambda
-(H1: (eq T (THead k v1 t0) (TLRef i2))).(\lambda (H2: (eq T (THead k v2 t4)
-t3)).((let H3 \def (eq_ind T (THead k v1 t0) (\lambda (e: T).(match e in T
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
-\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H1) in
-(False_ind ((eq T (THead k v2 t4) t3) \to (iso (TLRef i1) t3)) H3)) H2)))])
-in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k:
-K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4:
-T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def
-(match H0 in iso return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: (iso t
-t0)).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3)
-t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort n1)
-(THead k v2 t4))).(\lambda (H2: (eq T (TSort n2) t5)).((let H3 \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 v2 t4) H1) in (False_ind ((eq T (TSort n2)
-t5) \to (iso (THead k v1 t3) t5)) H3)) H2))) | (iso_lref i1 i2) \Rightarrow
-(\lambda (H1: (eq T (TLRef i1) (THead k v2 t4))).(\lambda (H2: (eq T (TLRef
-i2) t5)).((let H3 \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 v2 t4) H1)
-in (False_ind ((eq T (TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H3)) H2)))
-| (iso_head k0 v0 v3 t0 t6) \Rightarrow (\lambda (H1: (eq T (THead k0 v0 t0)
-(THead k v2 t4))).(\lambda (H2: (eq T (THead k0 v3 t6) t5)).((let H3 \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 v2 t4) H1) in ((let H4 \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 v2 t4) H1) in ((let H5 \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 v2 t4) H1) in (eq_ind K k
-(\lambda (k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to ((eq T (THead k1 v3 t6)
-t5) \to (iso (THead k v1 t3) t5))))) (\lambda (H6: (eq T v0 v2)).(eq_ind T v2
-(\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k v3 t6) t5) \to (iso (THead
-k v1 t3) t5)))) (\lambda (H7: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).((eq
-T (THead k v3 t6) t5) \to (iso (THead k v1 t3) t5))) (\lambda (H8: (eq T
-(THead k v3 t6) t5)).(eq_ind T (THead k v3 t6) (\lambda (t: T).(iso (THead k
-v1 t3) t)) (iso_head k v1 v3 t3 t6) t5 H8)) t0 (sym_eq T t0 t4 H7))) v0
-(sym_eq T v0 v2 H6))) k0 (sym_eq K k0 k H5))) H4)) H3)) H2)))]) in (H1
-(refl_equal T (THead k v2 t4)) (refl_equal T t5)))))))))) t1 t2 H))).
+(TSort n2) t3)).(let H_x \def (iso_gen_sort t3 n2 H0) in (let H1 \def H_x in
+(ex_ind nat (\lambda (n3: nat).(eq T t3 (TSort n3))) (iso (TSort n1) t3)
+(\lambda (x: nat).(\lambda (H2: (eq T t3 (TSort x))).(eq_ind_r T (TSort x)
+(\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 x) t3 H2))) H1)))))))
+(\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso
+(TLRef i2) t3)).(let H_x \def (iso_gen_lref t3 i2 H0) in (let H1 \def H_x in
+(ex_ind nat (\lambda (n2: nat).(eq T t3 (TLRef n2))) (iso (TLRef i1) t3)
+(\lambda (x: nat).(\lambda (H2: (eq T t3 (TLRef x))).(eq_ind_r T (TLRef x)
+(\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 x) t3 H2))) H1)))))))
+(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(k: K).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H_x \def
+(iso_gen_head k v2 t4 t5 H0) in (let H1 \def H_x in (ex_2_ind T T (\lambda
+(v3: T).(\lambda (t6: T).(eq T t5 (THead k v3 t6)))) (iso (THead k v1 t3) t5)
+(\lambda (x0: T).(\lambda (x1: T).(\lambda (H2: (eq T t5 (THead k x0
+x1))).(eq_ind_r T (THead k x0 x1) (\lambda (t: T).(iso (THead k v1 t3) t))
+(iso_head v1 x0 t3 x1 k) t5 H2)))) H1)))))))))) t1 t2 H))).
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