update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_1 / pc3 / left.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma b/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma
index 9b1bbcfc2fd370bfc846609911d85bc80ed375f7..579d42b240863c6909ecd85c7db15818f20b83cb 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/pc3/left.ma
@@ -16,93 +16,83 @@
 
 include "basic_1/pc3/props.ma".
 
-let rec pc3_left_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall (t: T).(P 
-t t))) (f0: (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall 
-(t3: T).((pc3_left c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (f1: (\forall 
-(t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: T).((pc3_left c t1 
-t3) \to ((P t1 t3) \to (P t2 t3)))))))) (t: T) (t0: T) (p: pc3_left c t t0) 
-on p: P t t0 \def match p with [(pc3_left_r t1) \Rightarrow (f t1) | 
-(pc3_left_ur t1 t2 p0 t3 p1) \Rightarrow (let TMP_2 \def ((pc3_left_ind c P f 
-f0 f1) t2 t3 p1) in (f0 t1 t2 p0 t3 p1 TMP_2)) | (pc3_left_ux t1 t2 p0 t3 p1) 
-\Rightarrow (let TMP_1 \def ((pc3_left_ind c P f f0 f1) t1 t3 p1) in (f1 t1 
-t2 p0 t3 p1 TMP_1))].
+implied rec lemma pc3_left_ind (c: C) (P: (T \to (T \to Prop))) (f: (\forall 
+(t: T).(P t t))) (f0: (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to 
+(\forall (t3: T).((pc3_left c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) (f1: 
+(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
+T).((pc3_left c t1 t3) \to ((P t1 t3) \to (P t2 t3)))))))) (t: T) (t0: T) (p: 
+pc3_left c t t0) on p: P t t0 \def match p with [(pc3_left_r t1) \Rightarrow 
+(f t1) | (pc3_left_ur t1 t2 p0 t3 p1) \Rightarrow (f0 t1 t2 p0 t3 p1 
+((pc3_left_ind c P f f0 f1) t2 t3 p1)) | (pc3_left_ux t1 t2 p0 t3 p1) 
+\Rightarrow (f1 t1 t2 p0 t3 p1 ((pc3_left_ind c P f f0 f1) t1 t3 p1))].
 
-theorem pc3_ind_left__pc3_left_pr3:
+fact pc3_ind_left__pc3_left_pr3:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to 
 (pc3_left c t1 t2))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(let TMP_1 \def (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) in 
-(let TMP_2 \def (\lambda (t: T).(pc3_left_r c t)) in (let TMP_3 \def (\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 t0)).(\lambda (t4: 
-T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (pc3_left c t0 t4)).(pc3_left_ur 
-c t3 t0 H0 t4 H2))))))) in (pr3_ind c TMP_1 TMP_2 TMP_3 t1 t2 H))))))).
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t t0))) (\lambda 
+(t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 
+c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: 
+(pc3_left c t0 t4)).(pc3_left_ur c t3 t0 H0 t4 H2))))))) t1 t2 H)))).
 
-theorem pc3_ind_left__pc3_left_trans:
+fact pc3_ind_left__pc3_left_trans:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
 (\forall (t3: T).((pc3_left c t2 t3) \to (pc3_left c t1 t3))))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
-t2)).(let TMP_1 \def (\lambda (t: T).(\lambda (t0: T).(\forall (t3: 
-T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) in (let TMP_2 \def (\lambda 
-(t: T).(\lambda (t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) in (let TMP_4 
-\def (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda 
-(t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: ((\forall (t5: 
-T).((pc3_left c t4 t5) \to (pc3_left c t3 t5))))).(\lambda (t5: T).(\lambda 
-(H3: (pc3_left c t4 t5)).(let TMP_3 \def (H2 t5 H3) in (pc3_left_ur c t0 t3 
-H0 t5 TMP_3)))))))))) in (let TMP_6 \def (\lambda (t0: T).(\lambda (t3: 
-T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 
-t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 
-t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(let TMP_5 \def 
-(H2 t5 H3) in (pc3_left_ux c t0 t3 H0 t5 TMP_5)))))))))) in (pc3_left_ind c 
-TMP_1 TMP_2 TMP_4 TMP_6 t1 t2 H)))))))).
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(\forall (t3: 
+T).((pc3_left c t0 t3) \to (pc3_left c t t3))))) (\lambda (t: T).(\lambda 
+(t3: T).(\lambda (H0: (pc3_left c t t3)).H0))) (\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 
+t4)).(\lambda (H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t3 
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ur c t0 
+t3 H0 t5 (H2 t5 H3)))))))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: 
+(pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda 
+(H2: ((\forall (t5: T).((pc3_left c t4 t5) \to (pc3_left c t0 
+t5))))).(\lambda (t5: T).(\lambda (H3: (pc3_left c t4 t5)).(pc3_left_ux c t0 
+t3 H0 t5 (H2 t5 H3)))))))))) t1 t2 H)))).
 
-theorem pc3_ind_left__pc3_left_sym:
+fact pc3_ind_left__pc3_left_sym:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
 (pc3_left c t2 t1))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
-t2)).(let TMP_1 \def (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) in 
-(let TMP_2 \def (\lambda (t: T).(pc3_left_r c t)) in (let TMP_5 \def (\lambda 
-(t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: 
-T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3_left c t4 t3)).(let 
-TMP_3 \def (pc3_left_r c t0) in (let TMP_4 \def (pc3_left_ux c t0 t3 H0 t0 
-TMP_3) in (pc3_ind_left__pc3_left_trans c t4 t3 H2 t0 TMP_4))))))))) in (let 
-TMP_8 \def (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 
-t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda (H2: 
-(pc3_left c t4 t0)).(let TMP_6 \def (pc3_left_r c t3) in (let TMP_7 \def 
-(pc3_left_ur c t0 t3 H0 t3 TMP_6) in (pc3_ind_left__pc3_left_trans c t4 t0 H2 
-t3 TMP_7))))))))) in (pc3_left_ind c TMP_1 TMP_2 TMP_5 TMP_8 t1 t2 H)))))))).
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3_left c t0 t))) 
+(\lambda (t: T).(pc3_left_r c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 
+t4)).(\lambda (H2: (pc3_left c t4 t3)).(pc3_ind_left__pc3_left_trans c t4 t3 
+H2 t0 (pc3_left_ux c t0 t3 H0 t0 (pc3_left_r c t0))))))))) (\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda 
+(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3_left c t4 
+t0)).(pc3_ind_left__pc3_left_trans c t4 t0 H2 t3 (pc3_left_ur c t0 t3 H0 t3 
+(pc3_left_r c t3))))))))) t1 t2 H)))).
 
-theorem pc3_ind_left__pc3_left_pc3:
+fact pc3_ind_left__pc3_left_pc3:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3 c t1 t2) \to 
 (pc3_left c t1 t2))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3 c t1 
-t2)).(let H0 \def H in (let TMP_1 \def (\lambda (t: T).(pr3 c t1 t)) in (let 
-TMP_2 \def (\lambda (t: T).(pr3 c t2 t)) in (let TMP_3 \def (pc3_left c t1 
-t2) in (let TMP_7 \def (\lambda (x: T).(\lambda (H1: (pr3 c t1 x)).(\lambda 
-(H2: (pr3 c t2 x)).(let TMP_4 \def (pc3_ind_left__pc3_left_pr3 c t1 x H1) in 
-(let TMP_5 \def (pc3_ind_left__pc3_left_pr3 c t2 x H2) in (let TMP_6 \def 
-(pc3_ind_left__pc3_left_sym c t2 x TMP_5) in (pc3_ind_left__pc3_left_trans c 
-t1 x TMP_4 t2 TMP_6))))))) in (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_7 H0))))))))).
+t2)).(let H0 \def H in (ex2_ind T (\lambda (t: T).(pr3 c t1 t)) (\lambda (t: 
+T).(pr3 c t2 t)) (pc3_left c t1 t2) (\lambda (x: T).(\lambda (H1: (pr3 c t1 
+x)).(\lambda (H2: (pr3 c t2 x)).(pc3_ind_left__pc3_left_trans c t1 x 
+(pc3_ind_left__pc3_left_pr3 c t1 x H1) t2 (pc3_ind_left__pc3_left_sym c t2 x 
+(pc3_ind_left__pc3_left_pr3 c t2 x H2)))))) H0))))).
 
-theorem pc3_ind_left__pc3_pc3_left:
+fact pc3_ind_left__pc3_pc3_left:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pc3_left c t1 t2) \to 
 (pc3 c t1 t2))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pc3_left c t1 
-t2)).(let TMP_1 \def (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) in (let 
-TMP_2 \def (\lambda (t: T).(pc3_refl c t)) in (let TMP_4 \def (\lambda (t0: 
+t2)).(pc3_left_ind c (\lambda (t: T).(\lambda (t0: T).(pc3 c t t0))) (\lambda 
+(t: T).(pc3_refl c t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c 
+t0 t3)).(\lambda (t4: T).(\lambda (_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 
+c t3 t4)).(pc3_t t3 c t0 (pc3_pr2_r c t0 t3 H0) t4 H2))))))) (\lambda (t0: 
 T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda (t4: T).(\lambda 
-(_: (pc3_left c t3 t4)).(\lambda (H2: (pc3 c t3 t4)).(let TMP_3 \def 
-(pc3_pr2_r c t0 t3 H0) in (pc3_t t3 c t0 TMP_3 t4 H2)))))))) in (let TMP_6 
-\def (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t0 t3)).(\lambda 
-(t4: T).(\lambda (_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(let 
-TMP_5 \def (pc3_pr2_x c t3 t0 H0) in (pc3_t t0 c t3 TMP_5 t4 H2)))))))) in 
-(pc3_left_ind c TMP_1 TMP_2 TMP_4 TMP_6 t1 t2 H)))))))).
+(_: (pc3_left c t0 t4)).(\lambda (H2: (pc3 c t0 t4)).(pc3_t t0 c t3 
+(pc3_pr2_x c t3 t0 H0) t4 H2))))))) t1 t2 H)))).
 
-theorem pc3_ind_left:
+lemma pc3_ind_left:
  \forall (c: C).(\forall (P: ((T \to (T \to Prop)))).(((\forall (t: T).(P t 
 t))) \to (((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (t3: 
 T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 t3)))))))) \to (((\forall (t1: 
@@ -115,14 +105,13 @@ t t0) \to (P t t0))))))))
 t2) \to (\forall (t3: T).((pc3 c t2 t3) \to ((P t2 t3) \to (P t1 
 t3))))))))).(\lambda (H1: ((\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) 
 \to (\forall (t3: T).((pc3 c t1 t3) \to ((P t1 t3) \to (P t2 
-t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t t0)).(let 
-TMP_1 \def (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) in (let TMP_3 \def 
-(\lambda (t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: 
-T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(let TMP_2 \def 
-(pc3_ind_left__pc3_pc3_left c t2 t3 H4) in (H0 t1 t2 H3 t3 TMP_2 H5)))))))) 
-in (let TMP_5 \def (\lambda (t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 
-t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left c t1 t3)).(\lambda (H5: (P t1 
-t3)).(let TMP_4 \def (pc3_ind_left__pc3_pc3_left c t1 t3 H4) in (H1 t1 t2 H3 
-t3 TMP_4 H5)))))))) in (let TMP_6 \def (pc3_ind_left__pc3_left_pc3 c t t0 H2) 
-in (pc3_left_ind c TMP_1 H TMP_3 TMP_5 t t0 TMP_6)))))))))))).
+t3))))))))).(\lambda (t: T).(\lambda (t0: T).(\lambda (H2: (pc3 c t 
+t0)).(pc3_left_ind c (\lambda (t1: T).(\lambda (t2: T).(P t1 t2))) H (\lambda 
+(t1: T).(\lambda (t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: 
+T).(\lambda (H4: (pc3_left c t2 t3)).(\lambda (H5: (P t2 t3)).(H0 t1 t2 H3 t3 
+(pc3_ind_left__pc3_pc3_left c t2 t3 H4) H5))))))) (\lambda (t1: T).(\lambda 
+(t2: T).(\lambda (H3: (pr2 c t1 t2)).(\lambda (t3: T).(\lambda (H4: (pc3_left 
+c t1 t3)).(\lambda (H5: (P t1 t3)).(H1 t1 t2 H3 t3 
+(pc3_ind_left__pc3_pc3_left c t1 t3 H4) H5))))))) t t0 
+(pc3_ind_left__pc3_left_pc3 c t t0 H2))))))))).