update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / pc1 / props.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma b/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma
index c6f910089b2af1c4aa1f10106fd1f2773a0857b5..86e794a79c5771b62b45f6495a8b027dd781216a 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc1/props.ma
@@ -18,113 +18,91 @@ include "basic_1/pc1/defs.ma".
 
 include "basic_1/pr1/pr1.ma".
 
-theorem pc1_pr0_r:
+lemma pc1_pr0_r:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pc1 t1 t2)))
 \def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(let TMP_1 \def 
-(\lambda (t: T).(pr1 t1 t)) in (let TMP_2 \def (\lambda (t: T).(pr1 t2 t)) in 
-(let TMP_3 \def (pr1_pr0 t1 t2 H) in (let TMP_4 \def (pr1_refl t2) in 
-(ex_intro2 T TMP_1 TMP_2 t2 TMP_3 TMP_4))))))).
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t1 t2)).(ex_intro2 T 
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t2 (pr1_pr0 t1 t2 H) 
+(pr1_refl t2)))).
 
-theorem pc1_pr0_x:
+lemma pc1_pr0_x:
  \forall (t1: T).(\forall (t2: T).((pr0 t2 t1) \to (pc1 t1 t2)))
 \def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(let TMP_1 \def 
-(\lambda (t: T).(pr1 t1 t)) in (let TMP_2 \def (\lambda (t: T).(pr1 t2 t)) in 
-(let TMP_3 \def (pr1_refl t1) in (let TMP_4 \def (pr1_pr0 t2 t1 H) in 
-(ex_intro2 T TMP_1 TMP_2 t1 TMP_3 TMP_4))))))).
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr0 t2 t1)).(ex_intro2 T 
+(\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) t1 (pr1_refl t1) 
+(pr1_pr0 t2 t1 H)))).
 
-theorem pc1_refl:
+lemma pc1_refl:
  \forall (t: T).(pc1 t t)
 \def
- \lambda (t: T).(let TMP_1 \def (\lambda (t0: T).(pr1 t t0)) in (let TMP_2 
-\def (\lambda (t0: T).(pr1 t t0)) in (let TMP_3 \def (pr1_refl t) in (let 
-TMP_4 \def (pr1_refl t) in (ex_intro2 T TMP_1 TMP_2 t TMP_3 TMP_4))))).
+ \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:
+lemma pc1_pr0_u:
  \forall (t2: T).(\forall (t1: T).((pr0 t1 t2) \to (\forall (t3: T).((pc1 t2 
 t3) \to (pc1 t1 t3)))))
 \def
  \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pr0 t1 t2)).(\lambda (t3: 
-T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (let TMP_1 \def (\lambda (t: 
-T).(pr1 t2 t)) in (let TMP_2 \def (\lambda (t: T).(pr1 t3 t)) in (let TMP_3 
-\def (pc1 t1 t3) in (let TMP_7 \def (\lambda (x: T).(\lambda (H2: (pr1 t2 
-x)).(\lambda (H3: (pr1 t3 x)).(let TMP_4 \def (\lambda (t: T).(pr1 t1 t)) in 
-(let TMP_5 \def (\lambda (t: T).(pr1 t3 t)) in (let TMP_6 \def (pr1_sing t2 
-t1 H x H2) in (ex_intro2 T TMP_4 TMP_5 x TMP_6 H3))))))) in (ex2_ind T TMP_1 
-TMP_2 TMP_3 TMP_7 H1)))))))))).
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: 
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(ex_intro2 T (\lambda 
+(t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x (pr1_sing t2 t1 H x H2) 
+H3)))) H1)))))).
 
-theorem pc1_s:
+lemma pc1_s:
  \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (pc1 t2 t1)))
 \def
  \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(let H0 \def H in 
-(let TMP_1 \def (\lambda (t: T).(pr1 t1 t)) in (let TMP_2 \def (\lambda (t: 
-T).(pr1 t2 t)) in (let TMP_3 \def (pc1 t2 t1) in (let TMP_6 \def (\lambda (x: 
-T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(let TMP_4 \def 
-(\lambda (t: T).(pr1 t2 t)) in (let TMP_5 \def (\lambda (t: T).(pr1 t1 t)) in 
-(ex_intro2 T TMP_4 TMP_5 x H2 H1)))))) in (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_6 
-H0)))))))).
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t2 
+t1) (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 
+x)).(ex_intro2 T (\lambda (t: T).(pr1 t2 t)) (\lambda (t: T).(pr1 t1 t)) x H2 
+H1)))) H0)))).
 
-theorem pc1_head_1:
+lemma pc1_head_1:
  \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t: T).(\forall 
 (k: K).(pc1 (THead k u1 t) (THead k u2 t))))))
 \def
  \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t: 
-T).(\lambda (k: K).(let H0 \def H in (let TMP_1 \def (\lambda (t0: T).(pr1 u1 
-t0)) in (let TMP_2 \def (\lambda (t0: T).(pr1 u2 t0)) in (let TMP_3 \def 
-(THead k u1 t) in (let TMP_4 \def (THead k u2 t) in (let TMP_5 \def (pc1 
-TMP_3 TMP_4) in (let TMP_13 \def (\lambda (x: T).(\lambda (H1: (pr1 u1 
-x)).(\lambda (H2: (pr1 u2 x)).(let TMP_7 \def (\lambda (t0: T).(let TMP_6 
-\def (THead k u1 t) in (pr1 TMP_6 t0))) in (let TMP_9 \def (\lambda (t0: 
-T).(let TMP_8 \def (THead k u2 t) in (pr1 TMP_8 t0))) in (let TMP_10 \def 
-(THead k x t) in (let TMP_11 \def (pr1_head_1 u1 x H1 t k) in (let TMP_12 
-\def (pr1_head_1 u2 x H2 t k) in (ex_intro2 T TMP_7 TMP_9 TMP_10 TMP_11 
-TMP_12))))))))) in (ex2_ind T TMP_1 TMP_2 TMP_5 TMP_13 H0)))))))))))).
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t0: T).(pr1 u1 t0)) 
+(\lambda (t0: T).(pr1 u2 t0)) (pc1 (THead k u1 t) (THead k u2 t)) (\lambda 
+(x: T).(\lambda (H1: (pr1 u1 x)).(\lambda (H2: (pr1 u2 x)).(ex_intro2 T 
+(\lambda (t0: T).(pr1 (THead k u1 t) t0)) (\lambda (t0: T).(pr1 (THead k u2 
+t) t0)) (THead k x t) (pr1_head_1 u1 x H1 t k) (pr1_head_1 u2 x H2 t k))))) 
+H0)))))).
 
-theorem pc1_head_2:
+lemma pc1_head_2:
  \forall (t1: T).(\forall (t2: T).((pc1 t1 t2) \to (\forall (u: T).(\forall 
 (k: K).(pc1 (THead k u t1) (THead k u t2))))))
 \def
  \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc1 t1 t2)).(\lambda (u: 
-T).(\lambda (k: K).(let H0 \def H in (let TMP_1 \def (\lambda (t: T).(pr1 t1 
-t)) in (let TMP_2 \def (\lambda (t: T).(pr1 t2 t)) in (let TMP_3 \def (THead 
-k u t1) in (let TMP_4 \def (THead k u t2) in (let TMP_5 \def (pc1 TMP_3 
-TMP_4) in (let TMP_13 \def (\lambda (x: T).(\lambda (H1: (pr1 t1 x)).(\lambda 
-(H2: (pr1 t2 x)).(let TMP_7 \def (\lambda (t: T).(let TMP_6 \def (THead k u 
-t1) in (pr1 TMP_6 t))) in (let TMP_9 \def (\lambda (t: T).(let TMP_8 \def 
-(THead k u t2) in (pr1 TMP_8 t))) in (let TMP_10 \def (THead k u x) in (let 
-TMP_11 \def (pr1_head_2 t1 x H1 u k) in (let TMP_12 \def (pr1_head_2 t2 x H2 
-u k) in (ex_intro2 T TMP_7 TMP_9 TMP_10 TMP_11 TMP_12))))))))) in (ex2_ind T 
-TMP_1 TMP_2 TMP_5 TMP_13 H0)))))))))))).
+T).(\lambda (k: K).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr1 t1 t)) 
+(\lambda (t: T).(pr1 t2 t)) (pc1 (THead k u t1) (THead k u t2)) (\lambda (x: 
+T).(\lambda (H1: (pr1 t1 x)).(\lambda (H2: (pr1 t2 x)).(ex_intro2 T (\lambda 
+(t: T).(pr1 (THead k u t1) t)) (\lambda (t: T).(pr1 (THead k u t2) t)) (THead 
+k u x) (pr1_head_2 t1 x H1 u k) (pr1_head_2 t2 x H2 u k))))) H0)))))).
 
 theorem pc1_t:
  \forall (t2: T).(\forall (t1: T).((pc1 t1 t2) \to (\forall (t3: T).((pc1 t2 
 t3) \to (pc1 t1 t3)))))
 \def
  \lambda (t2: T).(\lambda (t1: T).(\lambda (H: (pc1 t1 t2)).(\lambda (t3: 
-T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (let TMP_1 \def (\lambda (t: 
-T).(pr1 t2 t)) in (let TMP_2 \def (\lambda (t: T).(pr1 t3 t)) in (let TMP_3 
-\def (pc1 t1 t3) in (let TMP_17 \def (\lambda (x: T).(\lambda (H2: (pr1 t2 
-x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in (let TMP_4 \def (\lambda (t: 
-T).(pr1 t1 t)) in (let TMP_5 \def (\lambda (t: T).(pr1 t2 t)) in (let TMP_6 
-\def (pc1 t1 t3) in (let TMP_16 \def (\lambda (x0: T).(\lambda (H5: (pr1 t1 
-x0)).(\lambda (H6: (pr1 t2 x0)).(let TMP_7 \def (\lambda (t: T).(pr1 x0 t)) 
-in (let TMP_8 \def (\lambda (t: T).(pr1 x t)) in (let TMP_9 \def (pc1 t1 t3) 
-in (let TMP_14 \def (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: 
-(pr1 x x1)).(let TMP_10 \def (\lambda (t: T).(pr1 t1 t)) in (let TMP_11 \def 
-(\lambda (t: T).(pr1 t3 t)) in (let TMP_12 \def (pr1_t x0 t1 H5 x1 H7) in 
-(let TMP_13 \def (pr1_t x t3 H3 x1 H8) in (ex_intro2 T TMP_10 TMP_11 x1 
-TMP_12 TMP_13)))))))) in (let TMP_15 \def (pr1_confluence t2 x0 H6 x H2) in 
-(ex2_ind T TMP_7 TMP_8 TMP_9 TMP_14 TMP_15))))))))) in (ex2_ind T TMP_4 TMP_5 
-TMP_6 TMP_16 H4))))))))) in (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_17 H1)))))))))).
+T).(\lambda (H0: (pc1 t2 t3)).(let H1 \def H0 in (ex2_ind T (\lambda (t: 
+T).(pr1 t2 t)) (\lambda (t: T).(pr1 t3 t)) (pc1 t1 t3) (\lambda (x: 
+T).(\lambda (H2: (pr1 t2 x)).(\lambda (H3: (pr1 t3 x)).(let H4 \def H in 
+(ex2_ind T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t2 t)) (pc1 t1 
+t3) (\lambda (x0: T).(\lambda (H5: (pr1 t1 x0)).(\lambda (H6: (pr1 t2 
+x0)).(ex2_ind T (\lambda (t: T).(pr1 x0 t)) (\lambda (t: T).(pr1 x t)) (pc1 
+t1 t3) (\lambda (x1: T).(\lambda (H7: (pr1 x0 x1)).(\lambda (H8: (pr1 x 
+x1)).(ex_intro2 T (\lambda (t: T).(pr1 t1 t)) (\lambda (t: T).(pr1 t3 t)) x1 
+(pr1_t x0 t1 H5 x1 H7) (pr1_t x t3 H3 x1 H8))))) (pr1_confluence t2 x0 H6 x 
+H2))))) H4))))) H1)))))).
 
-theorem pc1_pr0_u2:
+lemma pc1_pr0_u2:
  \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pc1 t0 
 t2) \to (pc1 t1 t2)))))
 \def
  \lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr0 t0 t1)).(\lambda (t2: 
-T).(\lambda (H0: (pc1 t0 t2)).(let TMP_1 \def (pc1_pr0_x t1 t0 H) in (pc1_t 
-t0 t1 TMP_1 t2 H0)))))).
+T).(\lambda (H0: (pc1 t0 t2)).(pc1_t t0 t1 (pc1_pr0_x t1 t0 H) t2 H0))))).
 
 theorem pc1_head:
  \forall (u1: T).(\forall (u2: T).((pc1 u1 u2) \to (\forall (t1: T).(\forall 
@@ -132,9 +110,7 @@ theorem pc1_head:
 t2))))))))
 \def
  \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pc1 u1 u2)).(\lambda (t1: 
-T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(let TMP_1 
-\def (THead k u2 t1) in (let TMP_2 \def (THead k u1 t1) in (let TMP_3 \def 
-(pc1_head_1 u1 u2 H t1 k) in (let TMP_4 \def (THead k u2 t2) in (let TMP_5 
-\def (pc1_head_2 t1 t2 H0 u2 k) in (pc1_t TMP_1 TMP_2 TMP_3 TMP_4 
-TMP_5)))))))))))).
+T).(\lambda (t2: T).(\lambda (H0: (pc1 t1 t2)).(\lambda (k: K).(pc1_t (THead 
+k u2 t1) (THead k u1 t1) (pc1_head_1 u1 u2 H t1 k) (THead k u2 t2) 
+(pc1_head_2 t1 t2 H0 u2 k)))))))).