- partial commit (just the components before computation)

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / star.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/star.ma b/matita/matita/contribs/lambdadelta/ground_2/star.ma
index 1e46a48c6d536e131a1d0bcf9ba7c3de414387a6..1358f8da242cab36a609fa84d86382c3641093e3 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/star.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/star.ma
@@ -20,10 +20,6 @@ include "ground_2/notation.ma".
 
 definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
 
-definition Confluent: ∀A. ∀R: relation A. Prop ≝ λA,R.
-                      ∀a0,a1. R a0 a1 → ∀a2. R a0 a2 →
-                      ∃∃a. R a1 a & R a2 a.
-
 definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
                        ∀a1,a0. R a1 a0 → ∀a2. R a0 a2 → R a1 a2.
 
@@ -39,6 +35,15 @@ definition bi_confluent:  ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
                           ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. R a0 b0 a2 b2 →
                           ∃∃a,b. R a1 b1 a b & R a2 b2 a b.
 
+definition LTC: ∀A:Type[0]. ∀B. (A→relation B) → (A→relation B) ≝
+                λA,B,R,a. TC … (R a).
+
+definition s_r_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
+                      ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 L1 L2 → LTC … R1 L1 T1 T2.
+
+definition s_rs_trans: ∀A,B. relation2 (A→relation B) (relation A) ≝ λA,B,R1,R2.
+                       ∀L2,T1,T2. LTC … R1 L2 T1 T2 → ∀L1. R2 L1 L2 → LTC … R1 L1 T1 T2.
+
 lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
                  ∀a0,a1. TC … R1 a0 a1 → ∀a2. R2 a0 a2 →
                  ∃∃a. R2 a1 a & TC … R1 a2 a.
@@ -47,7 +52,7 @@ lemma TC_strip1: ∀A,R1,R2. confluent2 A R1 R2 →
   elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3/
 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
   elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
-  elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=3/
+  elim (HR12 … Ha1 … Ha0) -HR12 -a /4 width=5/
 ]
 qed.
 
@@ -70,7 +75,7 @@ lemma TC_confluent2: ∀A,R1,R2.
   elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3/
 | #a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
   elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
-  elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=3/
+  elim (TC_strip2 … HR12 … Ha0 … Ha1) -HR12 -a /4 width=5/
 ]
 qed.
 
@@ -82,7 +87,7 @@ lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 →
   elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3/
 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
   elim (HR12 … Ha0 … Ha02) -HR12 -a0 #a0 #Ha0 #Ha02
-  elim (IHa … Ha0) -a /4 width=3/
+  elim (IHa … Ha0) -a /4 width=5/
 ]
 qed.
 
@@ -105,7 +110,7 @@ lemma TC_transitive2: ∀A,R1,R2.
   elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3/
 | #a #a0 #_ #Ha0 #IHa #a2 #Ha02
   elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 -a0 #a0 #Ha0 #Ha02
-  elim (IHa … Ha0) -a /4 width=3/
+  elim (IHa … Ha0) -a /4 width=5/
 ]
 qed.
 
@@ -119,7 +124,7 @@ inductive SN (A) (R,S:relation A): predicate A ≝
 lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
 #A #R #S #a1 #Ha1
 @SN_intro #a2 #HRa12 #HSa12
-elim (HSa12 ?) -HSa12 /2 width=1/
+elim HSa12 -HSa12 /2 width=1/
 qed.
 
 definition NF_sn: ∀A. relation A → relation A → predicate A ≝
@@ -132,7 +137,7 @@ inductive SN_sn (A) (R,S:relation A): predicate A ≝
 lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
 #A #R #S #a2 #Ha2
 @SN_sn_intro #a1 #HRa12 #HSa12
-elim (HSa12 ?) -HSa12 /2 width=1/
+elim HSa12 -HSa12 /2 width=1/
 qed.
 
 lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
@@ -156,3 +161,30 @@ lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R →
   elim (bi_TC_strip … HR … H13 … H10) -a1 -b1 /3 width=7/
 ]
 qed.
+
+lemma bi_TC_decomp_r: ∀A,B. ∀R:bi_relation A B.
+                      ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
+                      R a1 b1 a2 b2 ∨
+                      ∃∃a,b. bi_TC … R a1 b1 a b & R a b a2 b2.
+#A #B #R #a1 #a2 #b1 #b2 * -a2 -b2 /2 width=1/ /3 width=4/
+qed-.
+
+lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B.
+                      ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 →
+                      R a1 b1 a2 b2 ∨
+                      ∃∃a,b. R a1 b1 a b & bi_TC … R a b a2 b2.
+#A #B #R #a1 #a2 #b1 #b2 #H @(bi_TC_ind_dx … a1 b1 H) -a1 -b1
+[ /2 width=1/
+| #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4/
+]
+qed-.
+
+lemma s_r_trans_TC1: ∀A,B,R,S. s_r_trans A B R S → s_rs_trans A B R S.
+#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
+#T #T2 #_ #HT2 #IHT1 #L1 #HL12
+lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
+qed-.
+
+lemma s_r_trans_TC2: ∀A,B,R,S. s_rs_trans A B R S → s_r_trans A B R (TC … S).
+#A #B #R #S #HRS #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /2 width=3/ /3 width=3/
+qed-.