(* TRANSITIVE CLOSURE FOR RELATIONS *****************************************)
-definition CTC: ∀A:Type[0]. ∀B. (A→relation B) → (A→relation B) ≝
- λA,B,R,a. TC … (R a).
-
-definition s_r_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
- ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → CTC … R1 L1 T1 T2.
-
-definition s_rs_transitive: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
- ∀L2,T1,T2. CTC … R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → CTC … 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.
+definition CTC (A:Type[0]) (B):
+ (A→relation B) → (A→relation B) ≝
+ λR,a. TC … (R a).
+
+definition s_r_transitive (A) (B):
+ relation2 (A→relation B) (B→relation A) ≝
+ λR1,R2.
+ ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → CTC … R1 L1 T1 T2.
+
+definition s_rs_transitive (A) (B):
+ relation2 (A→relation B) (B→relation A) ≝
+ λR1,R2.
+ ∀L2,T1,T2. CTC … R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → CTC … R1 L1 T1 T2.
+
+lemma TC_strip (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.
#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
[ #a1 #Ha01 #a2 #Ha02
elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
]
qed.
-lemma TC_strip2: ∀A,R1,R2. confluent2 A R1 R2 →
- ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
- ∃∃a. TC … R2 a1 a & R1 a2 a.
+lemma TC_strip2 (A) (R1) (R2):
+ confluent2 A R1 R2 →
+ ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
+ ∃∃a. TC … R2 a1 a & R1 a2 a.
#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
[ #a2 #Ha02 #a1 #Ha01
elim (HR12 … Ha01 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
]
qed.
-lemma TC_confluent2: ∀A,R1,R2.
- confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
+lemma TC_confluent2 (A) (R1) (R2):
+ confluent2 A R1 R2 → confluent2 A (TC … R1) (TC … R2).
#A #R1 #R2 #HR12 #a0 #a1 #H elim H -a1
[ #a1 #Ha01 #a2 #Ha02
elim (TC_strip2 … HR12 … Ha02 … Ha01) -HR12 -a0 /3 width=3 by inj, ex2_intro/
]
qed.
-lemma TC_strap1: ∀A,R1,R2. transitive2 A R1 R2 →
- ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
- ∃∃a. R2 a1 a & TC … R1 a a2.
+lemma TC_strap1 (A) (R1) (R2):
+ transitive2 A R1 R2 →
+ ∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
+ ∃∃a. R2 a1 a & TC … R1 a a2.
#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
[ #a0 #Ha10 #a2 #Ha02
elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
]
qed.
-lemma TC_strap2: ∀A,R1,R2. transitive2 A R1 R2 →
- ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
- ∃∃a. TC … R2 a1 a & R1 a a2.
+lemma TC_strap2 (A) (R1) (R2):
+ transitive2 A R1 R2 →
+ ∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
+ ∃∃a. TC … R2 a1 a & R1 a a2.
#A #R1 #R2 #HR12 #a0 #a2 #H elim H -a2
[ #a2 #Ha02 #a1 #Ha10
elim (HR12 … Ha10 … Ha02) -HR12 -a0 /3 width=3 by inj, ex2_intro/
]
qed.
-lemma TC_transitive2: ∀A,R1,R2.
- transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2).
+lemma TC_transitive2 (A) (R1) (R2):
+ transitive2 A R1 R2 → transitive2 A (TC … R1) (TC … R2).
#A #R1 #R2 #HR12 #a1 #a0 #H elim H -a0
[ #a0 #Ha10 #a2 #Ha02
elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 -a0 /3 width=3 by inj, ex2_intro/
]
qed.
-lemma CTC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (CTC … R) S.
+lemma CTC_lsub_trans (A) (B) (R) (S):
+ lsub_trans A B R S → lsub_trans A B (CTC … R) S.
#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 /3 width=3 by inj/
#T #T2 #_ #HT2 #IHT1 #L1 #HL12
lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3 by step/
qed-.
-lemma s_r_conf1_CTC1: ∀A,B,S,R. s_r_confluent1 A B S R → s_r_confluent1 A B (CTC … S) R.
+lemma s_r_conf1_CTC1 (A) (B) (S) (R):
+ s_r_confluent1 A B S R → s_r_confluent1 A B (CTC … S) R.
#A #B #S #R #HSR #L1 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3 by/
qed-.
-lemma s_r_trans_CTC1: ∀A,B,S,R. s_r_confluent1 A B S R →
- s_r_transitive A B S R → s_rs_transitive A B S R.
+lemma s_r_trans_CTC1 (A) (B) (S) (R):
+ s_r_confluent1 A B S R →
+ s_r_transitive A B S R → s_rs_transitive A B S R.
#A #B #S #R #H1SR #H2SR #L2 #T1 #T2 #H @(TC_ind_dx … T1 H) -T1 /2 width=3 by/
#T1 #T #HT1 #_ #IHT2 #L1 #HL12 lapply (H2SR … HT1 … HL12) -H2SR -HT1
/4 width=5 by s_r_conf1_CTC1, trans_TC/
qed-.
-lemma s_r_trans_CTC2: ∀A,B,S,R. s_rs_transitive A B S R → s_r_transitive A B S (CTC … R).
+lemma s_r_trans_CTC2 (A) (B) (S) (R):
+ s_rs_transitive A B S R → s_r_transitive A B S (CTC … R).
#A #B #S #R #HSR #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /3 width=3 by inj/
qed-.
-lemma s_r_to_s_rs_trans: ∀A,B,S,R. s_r_transitive A B (CTC … S) R →
- s_rs_transitive A B S R.
+lemma s_r_to_s_rs_trans (A) (B) (S) (R):
+ s_r_transitive A B (CTC … S) R → s_rs_transitive A B S R.
#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
elim (TC_idem … (S L1) … T1 T2)
#_ #H @H @HSR //
qed-.
-lemma s_rs_to_s_r_trans: ∀A,B,S,R. s_rs_transitive A B S R →
- s_r_transitive A B (CTC … S) R.
+lemma s_rs_to_s_r_trans (A) (B) (S) (R):
+ s_rs_transitive A B S R → s_r_transitive A B (CTC … S) R.
#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
elim (TC_idem … (S L1) … T1 T2)
#H #_ @H @HSR //
qed-.
-lemma s_rs_trans_TC1: ∀A,B,S,R. s_rs_transitive A B S R →
- s_rs_transitive A B (CTC … S) R.
+lemma s_rs_trans_TC1 (A) (B) (S) (R):
+ s_rs_transitive A B S R → s_rs_transitive A B (CTC … S) R.
#A #B #S #R #HSR #L2 #T1 #T2 #HL2 #L1 #HT1
elim (TC_idem … (S L1) … T1 T2)
elim (TC_idem … (S L2) … T1 T2)
(* NOTE: Normal form and strong normalization *******************************)
-lemma SN_to_NF: ∀A,R,S. NF_dec A R S →
- ∀a1. SN A R S a1 →
- ∃∃a2. star … R a1 a2 & NF A R S a2.
+lemma SN_to_NF (A) (R) (S):
+ NF_dec A R S →
+ ∀a1. SN A R S a1 →
+ ∃∃a2. star … R a1 a2 & NF A R S a2.
#A #R #S #HRS #a1 #H elim H -a1
#a1 #_ #IHa1 elim (HRS a1) -HRS /2 width=3 by srefl, ex2_intro/
* #a0 #Ha10 #Ha01 elim (IHa1 … Ha10 Ha01) -IHa1 -Ha01 /3 width=3 by star_compl, ex2_intro/
(* NOTE: Relations with unboxed pairs ***************************************)
-lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
- ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
- ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
+lemma bi_TC_strip (A) (B) (R):
+ bi_confluent A B R →
+ ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
+ ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
#A #B #R #HR #a0 #a1 #b0 #b1 #H01 #a2 #b2 #H elim H -a2 -b2
[ #a2 #b2 #H02
elim (HR … H01 … H02) -HR -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/
]
qed.
-lemma bi_TC_confluent: ∀A,B,R. bi_confluent A B R →
- bi_confluent A B (bi_TC … R).
+lemma bi_TC_confluent (A) (B) (R):
+ bi_confluent A B R → bi_confluent A B (bi_TC … R).
#A #B #R #HR #a0 #a1 #b0 #b1 #H elim H -a1 -b1
[ #a1 #b1 #H01 #a2 #b2 #H02
elim (bi_TC_strip … HR … H01 … H02) -a0 -b0 /3 width=4 by ex2_2_intro, bi_inj/
]
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.
+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 by ex2_2_intro, or_intror/
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.
+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 by or_introl/
| #a1 #a #b1 #b #Hab1 #Hab2 #_ /3 width=4 by ex2_2_intro, or_intror/ (* * auto fails without #_ *)
(* NOTE: Relations with unboxed triples *************************************)
-definition tri_star: ∀A,B,C,R. tri_relation A B C ≝
- λA,B,C,R. tri_RC A B C (tri_TC … R).
+definition tri_star (A) (B) (C) (R):
+ tri_relation A B C ≝
+ tri_RC A B C (tri_TC … R).
-lemma tri_star_tri_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_star … R).
+lemma tri_star_tri_reflexive (A) (B) (C) (R):
+ tri_reflexive A B C (tri_star … R).
/2 width=1 by/ qed.
-lemma tri_TC_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
- tri_TC A B C R a1 b1 c1 a2 b2 c2 →
- tri_star A B C R a1 b1 c1 a2 b2 c2.
+lemma tri_TC_to_tri_star (A) (B) (C) (R):
+ ∀a1,b1,c1,a2,b2,c2.
+ tri_TC A B C R a1 b1 c1 a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
/2 width=1 by or_introl/ qed.
-lemma tri_R_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
- R a1 b1 c1 a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
+lemma tri_R_to_tri_star (A) (B) (C) (R):
+ ∀a1,b1,c1,a2,b2,c2.
+ R a1 b1 c1 a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
/3 width=1 by tri_TC_to_tri_star, tri_inj/ qed.
-lemma tri_star_strap1: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
- tri_star A B C R a1 b1 c1 a b c →
- R a b c a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
+lemma tri_star_strap1 (A) (B) (C) (R):
+ ∀a1,a,a2,b1,b,b2,c1,c,c2.
+ tri_star A B C R a1 b1 c1 a b c →
+ R a b c a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
[ /3 width=5 by tri_TC_to_tri_star, tri_step/
| * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/
]
qed.
-lemma tri_star_strap2: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. R a1 b1 c1 a b c →
- tri_star A B C R a b c a2 b2 c2 →
- tri_star A B C R a1 b1 c1 a2 b2 c2.
+lemma tri_star_strap2 (A) (B) (C) (R):
+ ∀a1,a,a2,b1,b,b2,c1,c,c2.
+ R a1 b1 c1 a b c → tri_star A B C R a b c a2 b2 c2 →
+ tri_star A B C R a1 b1 c1 a2 b2 c2.
#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
[ /3 width=5 by tri_TC_to_tri_star, tri_TC_strap/
| * #H1 #H2 #H3 destruct /2 width=1 by tri_R_to_tri_star/
]
qed.
-lemma tri_star_to_tri_TC_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
- tri_star A B C R a1 b1 c1 a b c →
- tri_TC A B C R a b c a2 b2 c2 →
- tri_TC A B C R a1 b1 c1 a2 b2 c2.
+lemma tri_star_to_tri_TC_to_tri_TC (A) (B) (C) (R):
+ ∀a1,a,a2,b1,b,b2,c1,c,c2.
+ tri_star A B C R a1 b1 c1 a b c →
+ tri_TC A B C R a b c a2 b2 c2 → tri_TC A B C R a1 b1 c1 a2 b2 c2.
#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
[ /2 width=5 by tri_TC_transitive/
| * #H1 #H2 #H3 destruct /2 width=1 by/
]
qed.
-lemma tri_TC_to_tri_star_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
- tri_TC A B C R a1 b1 c1 a b c →
- tri_star A B C R a b c a2 b2 c2 →
- tri_TC A B C R a1 b1 c1 a2 b2 c2.
+lemma tri_TC_to_tri_star_to_tri_TC (A) (B) (C) (R):
+ ∀a1,a,a2,b1,b,b2,c1,c,c2.
+ tri_TC A B C R a1 b1 c1 a b c →
+ tri_star A B C R a b c a2 b2 c2 → tri_TC A B C R a1 b1 c1 a2 b2 c2.
#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
[ /2 width=5 by tri_TC_transitive/
| * #H1 #H2 #H3 destruct /2 width=1 by/
]
qed.
-lemma tri_tansitive_tri_star: ∀A,B,C,R. tri_transitive A B C (tri_star … R).
+lemma tri_tansitive_tri_star (A) (B) (C) (R):
+ tri_transitive A B C (tri_star … R).
#A #B #C #R #a1 #a #b1 #b #c1 #c #H #a2 #b2 #c2 *
[ /3 width=5 by tri_star_to_tri_TC_to_tri_TC, tri_TC_to_tri_star/
| * #H1 #H2 #H3 destruct /2 width=1 by/
]
qed.
-lemma tri_star_ind: ∀A,B,C,R,a1,b1,c1. ∀P:relation3 A B C. P a1 b1 c1 →
- (∀a,a2,b,b2,c,c2. tri_star … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) →
- ∀a2,b2,c2. tri_star … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2.
-#A #B #C #R #a1 #b1 #c1 #P #H #IH #a2 #b2 #c2 *
+lemma tri_star_ind (A) (B) (C) (R):
+ ∀a1,b1,c1. ∀Q:relation3 A B C. Q a1 b1 c1 →
+ (∀a,a2,b,b2,c,c2. tri_star … R a1 b1 c1 a b c → R a b c a2 b2 c2 → Q a b c → Q a2 b2 c2) →
+ ∀a2,b2,c2. tri_star … R a1 b1 c1 a2 b2 c2 → Q a2 b2 c2.
+#A #B #C #R #a1 #b1 #c1 #Q #H #IH #a2 #b2 #c2 *
[ #H12 elim H12 -a2 -b2 -c2 /3 width=6 by tri_TC_to_tri_star/
| * #H1 #H2 #H3 destruct //
]
qed-.
-lemma tri_star_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 →
- (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_star … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
- ∀a1,b1,c1. tri_star … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
-#A #B #C #R #a2 #b2 #c2 #P #H #IH #a1 #b1 #c1 *
+lemma tri_star_ind_dx (A) (B) (C) (R):
+ ∀a2,b2,c2. ∀Q:relation3 A B C. Q a2 b2 c2 →
+ (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_star … R a b c a2 b2 c2 → Q a b c → Q a1 b1 c1) →
+ ∀a1,b1,c1. tri_star … R a1 b1 c1 a2 b2 c2 → Q a1 b1 c1.
+#A #B #C #R #a2 #b2 #c2 #Q #H #IH #a1 #b1 #c1 *
[ #H12 @(tri_TC_ind_dx … a1 b1 c1 H12) -a1 -b1 -c1 /3 width=6 by tri_TC_to_tri_star/
| * #H1 #H2 #H3 destruct //
]
projects / helm.git / blobdiff