update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / pr0 / subst1.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
index 5a67308c4aa025d1681612275a6ec09e78cdac6c..05a93172a34796a6d5becc8c7844b0643bd67a01 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pr0/subst1.ma
@@ -18,70 +18,53 @@ include "basic_1/pr0/subst0.ma".
 
 include "basic_1/subst1/fwd.ma".
 
-theorem pr0_delta1:
+lemma pr0_delta1:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall 
 (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead 
 (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
 \def
  \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (t1: 
 T).(\lambda (t2: T).(\lambda (H0: (pr0 t1 t2)).(\lambda (w: T).(\lambda (H1: 
-(subst1 O u2 t2 w)).(let TMP_5 \def (\lambda (t: T).(let TMP_1 \def (Bind 
-Abbr) in (let TMP_2 \def (THead TMP_1 u1 t1) in (let TMP_3 \def (Bind Abbr) 
-in (let TMP_4 \def (THead TMP_3 u2 t) in (pr0 TMP_2 TMP_4)))))) in (let TMP_6 
-\def (Bind Abbr) in (let TMP_7 \def (pr0_comp u1 u2 H t1 t2 H0 TMP_6) in (let 
-TMP_8 \def (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 
-u2 H t1 t2 H0 t0 H2))) in (subst1_ind O u2 t2 TMP_5 TMP_7 TMP_8 w 
-H1)))))))))))).
+(subst1 O u2 t2 w)).(subst1_ind O u2 t2 (\lambda (t: T).(pr0 (THead (Bind 
+Abbr) u1 t1) (THead (Bind Abbr) u2 t))) (pr0_comp u1 u2 H t1 t2 H0 (Bind 
+Abbr)) (\lambda (t0: T).(\lambda (H2: (subst0 O u2 t2 t0)).(pr0_delta u1 u2 H 
+t1 t2 H0 t0 H2))) w H1)))))))).
 
-theorem pr0_subst1_back:
+lemma pr0_subst1_back:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
 i u2 t1 t2) \to (\forall (u1: T).((pr0 u1 u2) \to (ex2 T (\lambda (t: 
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t2)))))))))
 \def
  \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i u2 t1 t2)).(let TMP_3 \def (\lambda (t: T).(\forall (u1: 
-T).((pr0 u1 u2) \to (let TMP_1 \def (\lambda (t0: T).(subst1 i u1 t1 t0)) in 
-(let TMP_2 \def (\lambda (t0: T).(pr0 t0 t)) in (ex2 T TMP_1 TMP_2)))))) in 
-(let TMP_8 \def (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(let TMP_4 \def 
-(\lambda (t: T).(subst1 i u1 t1 t)) in (let TMP_5 \def (\lambda (t: T).(pr0 t 
-t1)) in (let TMP_6 \def (subst1_refl i u1 t1) in (let TMP_7 \def (pr0_refl 
-t1) in (ex_intro2 T TMP_4 TMP_5 t1 TMP_6 TMP_7))))))) in (let TMP_19 \def 
-(\lambda (t0: T).(\lambda (H0: (subst0 i u2 t1 t0)).(\lambda (u1: T).(\lambda 
-(H1: (pr0 u1 u2)).(let TMP_9 \def (\lambda (t: T).(subst0 i u1 t1 t)) in (let 
-TMP_10 \def (\lambda (t: T).(pr0 t t0)) in (let TMP_11 \def (\lambda (t: 
-T).(subst1 i u1 t1 t)) in (let TMP_12 \def (\lambda (t: T).(pr0 t t0)) in 
-(let TMP_13 \def (ex2 T TMP_11 TMP_12) in (let TMP_17 \def (\lambda (x: 
-T).(\lambda (H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(let TMP_14 
-\def (\lambda (t: T).(subst1 i u1 t1 t)) in (let TMP_15 \def (\lambda (t: 
-T).(pr0 t t0)) in (let TMP_16 \def (subst1_single i u1 t1 x H2) in (ex_intro2 
-T TMP_14 TMP_15 x TMP_16 H3))))))) in (let TMP_18 \def (pr0_subst0_back u2 t1 
-t0 i H0 u1 H1) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_17 TMP_18)))))))))))) in 
-(subst1_ind i u2 t1 TMP_3 TMP_8 TMP_19 t2 H)))))))).
+(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: 
+T).((pr0 u1 u2) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda 
+(t0: T).(pr0 t0 t)))))) (\lambda (u1: T).(\lambda (_: (pr0 u1 u2)).(ex_intro2 
+T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t1)) t1 
+(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 
+i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u1 u2)).(ex2_ind T (\lambda 
+(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) (ex2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0))) (\lambda (x: T).(\lambda 
+(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 x t0)).(ex_intro2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t t0)) x (subst1_single i u1 t1 x 
+H2) H3)))) (pr0_subst0_back u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
 
-theorem pr0_subst1_fwd:
+lemma pr0_subst1_fwd:
  \forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (i: nat).((subst1 
 i u2 t1 t2) \to (\forall (u1: T).((pr0 u2 u1) \to (ex2 T (\lambda (t: 
 T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))
 \def
  \lambda (u2: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (i: nat).(\lambda 
-(H: (subst1 i u2 t1 t2)).(let TMP_3 \def (\lambda (t: T).(\forall (u1: 
-T).((pr0 u2 u1) \to (let TMP_1 \def (\lambda (t0: T).(subst1 i u1 t1 t0)) in 
-(let TMP_2 \def (\lambda (t0: T).(pr0 t t0)) in (ex2 T TMP_1 TMP_2)))))) in 
-(let TMP_8 \def (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(let TMP_4 \def 
-(\lambda (t: T).(subst1 i u1 t1 t)) in (let TMP_5 \def (\lambda (t: T).(pr0 
-t1 t)) in (let TMP_6 \def (subst1_refl i u1 t1) in (let TMP_7 \def (pr0_refl 
-t1) in (ex_intro2 T TMP_4 TMP_5 t1 TMP_6 TMP_7))))))) in (let TMP_19 \def 
-(\lambda (t0: T).(\lambda (H0: (subst0 i u2 t1 t0)).(\lambda (u1: T).(\lambda 
-(H1: (pr0 u2 u1)).(let TMP_9 \def (\lambda (t: T).(subst0 i u1 t1 t)) in (let 
-TMP_10 \def (\lambda (t: T).(pr0 t0 t)) in (let TMP_11 \def (\lambda (t: 
-T).(subst1 i u1 t1 t)) in (let TMP_12 \def (\lambda (t: T).(pr0 t0 t)) in 
-(let TMP_13 \def (ex2 T TMP_11 TMP_12) in (let TMP_17 \def (\lambda (x: 
-T).(\lambda (H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(let TMP_14 
-\def (\lambda (t: T).(subst1 i u1 t1 t)) in (let TMP_15 \def (\lambda (t: 
-T).(pr0 t0 t)) in (let TMP_16 \def (subst1_single i u1 t1 x H2) in (ex_intro2 
-T TMP_14 TMP_15 x TMP_16 H3))))))) in (let TMP_18 \def (pr0_subst0_fwd u2 t1 
-t0 i H0 u1 H1) in (ex2_ind T TMP_9 TMP_10 TMP_13 TMP_17 TMP_18)))))))))))) in 
-(subst1_ind i u2 t1 TMP_3 TMP_8 TMP_19 t2 H)))))))).
+(H: (subst1 i u2 t1 t2)).(subst1_ind i u2 t1 (\lambda (t: T).(\forall (u1: 
+T).((pr0 u2 u1) \to (ex2 T (\lambda (t0: T).(subst1 i u1 t1 t0)) (\lambda 
+(t0: T).(pr0 t t0)))))) (\lambda (u1: T).(\lambda (_: (pr0 u2 u1)).(ex_intro2 
+T (\lambda (t: T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 
+(subst1_refl i u1 t1) (pr0_refl t1)))) (\lambda (t0: T).(\lambda (H0: (subst0 
+i u2 t1 t0)).(\lambda (u1: T).(\lambda (H1: (pr0 u2 u1)).(ex2_ind T (\lambda 
+(t: T).(subst0 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) (ex2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t))) (\lambda (x: T).(\lambda 
+(H2: (subst0 i u1 t1 x)).(\lambda (H3: (pr0 t0 x)).(ex_intro2 T (\lambda (t: 
+T).(subst1 i u1 t1 t)) (\lambda (t: T).(pr0 t0 t)) x (subst1_single i u1 t1 x 
+H2) H3)))) (pr0_subst0_fwd u2 t1 t0 i H0 u1 H1)))))) t2 H))))).
 
 theorem pr0_subst1:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (v1: T).(\forall 
@@ -90,33 +73,21 @@ v2) \to (ex2 T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v2 t2
 w2)))))))))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(\lambda (v1: 
-T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 w1)).(let 
-TMP_3 \def (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to (let TMP_1 \def 
-(\lambda (w2: T).(pr0 t w2)) in (let TMP_2 \def (\lambda (w2: T).(subst1 i v2 
-t2 w2)) in (ex2 T TMP_1 TMP_2)))))) in (let TMP_7 \def (\lambda (v2: 
-T).(\lambda (_: (pr0 v1 v2)).(let TMP_4 \def (\lambda (w2: T).(pr0 t1 w2)) in 
-(let TMP_5 \def (\lambda (w2: T).(subst1 i v2 t2 w2)) in (let TMP_6 \def 
-(subst1_refl i v2 t2) in (ex_intro2 T TMP_4 TMP_5 t2 H TMP_6)))))) in (let 
-TMP_30 \def (\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: 
-T).(\lambda (H2: (pr0 v1 v2)).(let TMP_8 \def (pr0 t0 t2) in (let TMP_9 \def 
-(\lambda (w2: T).(pr0 t0 w2)) in (let TMP_10 \def (\lambda (w2: T).(subst0 i 
-v2 t2 w2)) in (let TMP_11 \def (ex2 T TMP_9 TMP_10) in (let TMP_12 \def 
-(\lambda (w2: T).(pr0 t0 w2)) in (let TMP_13 \def (\lambda (w2: T).(subst1 i 
-v2 t2 w2)) in (let TMP_14 \def (ex2 T TMP_12 TMP_13) in (let TMP_18 \def 
-(\lambda (H3: (pr0 t0 t2)).(let TMP_15 \def (\lambda (w2: T).(pr0 t0 w2)) in 
-(let TMP_16 \def (\lambda (w2: T).(subst1 i v2 t2 w2)) in (let TMP_17 \def 
-(subst1_refl i v2 t2) in (ex_intro2 T TMP_15 TMP_16 t2 H3 TMP_17))))) in (let 
-TMP_28 \def (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: 
-T).(subst0 i v2 t2 w2)))).(let TMP_19 \def (\lambda (w2: T).(pr0 t0 w2)) in 
-(let TMP_20 \def (\lambda (w2: T).(subst0 i v2 t2 w2)) in (let TMP_21 \def 
-(\lambda (w2: T).(pr0 t0 w2)) in (let TMP_22 \def (\lambda (w2: T).(subst1 i 
-v2 t2 w2)) in (let TMP_23 \def (ex2 T TMP_21 TMP_22) in (let TMP_27 \def 
-(\lambda (x: T).(\lambda (H4: (pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 
-x)).(let TMP_24 \def (\lambda (w2: T).(pr0 t0 w2)) in (let TMP_25 \def 
-(\lambda (w2: T).(subst1 i v2 t2 w2)) in (let TMP_26 \def (subst1_single i v2 
-t2 x H5) in (ex_intro2 T TMP_24 TMP_25 x H4 TMP_26))))))) in (ex2_ind T 
-TMP_19 TMP_20 TMP_23 TMP_27 H3)))))))) in (let TMP_29 \def (pr0_subst0 t1 t2 
-H v1 t0 i H1 v2 H2) in (or_ind TMP_8 TMP_11 TMP_14 TMP_18 TMP_28 
-TMP_29))))))))))))))) in (subst1_ind i v1 t1 TMP_3 TMP_7 TMP_30 w1 
-H0)))))))))).
+T).(\lambda (w1: T).(\lambda (i: nat).(\lambda (H0: (subst1 i v1 t1 
+w1)).(subst1_ind i v1 t1 (\lambda (t: T).(\forall (v2: T).((pr0 v1 v2) \to 
+(ex2 T (\lambda (w2: T).(pr0 t w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)))))) 
+(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(ex_intro2 T (\lambda (w2: T).(pr0 
+t1 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H (subst1_refl i v2 t2)))) 
+(\lambda (t0: T).(\lambda (H1: (subst0 i v1 t1 t0)).(\lambda (v2: T).(\lambda 
+(H2: (pr0 v1 v2)).(or_ind (pr0 t0 t2) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
+(\lambda (w2: T).(subst0 i v2 t2 w2))) (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
+(\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (H3: (pr0 t0 t2)).(ex_intro2 
+T (\lambda (w2: T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) t2 H3 
+(subst1_refl i v2 t2))) (\lambda (H3: (ex2 T (\lambda (w2: T).(pr0 t0 w2)) 
+(\lambda (w2: T).(subst0 i v2 t2 w2)))).(ex2_ind T (\lambda (w2: T).(pr0 t0 
+w2)) (\lambda (w2: T).(subst0 i v2 t2 w2)) (ex2 T (\lambda (w2: T).(pr0 t0 
+w2)) (\lambda (w2: T).(subst1 i v2 t2 w2))) (\lambda (x: T).(\lambda (H4: 
+(pr0 t0 x)).(\lambda (H5: (subst0 i v2 t2 x)).(ex_intro2 T (\lambda (w2: 
+T).(pr0 t0 w2)) (\lambda (w2: T).(subst1 i v2 t2 w2)) x H4 (subst1_single i 
+v2 t2 x H5))))) H3)) (pr0_subst0 t1 t2 H v1 t0 i H1 v2 H2)))))) w1 H0))))))).