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7 (* ||T|| The HELM team. *)
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15 include "ground/generated/insert_eq_1.ma".
16 include "ground/lib/functions.ma".
18 (* LABELLED TRANSITIVE CLOSURE FOR RELATIONS ********************************)
20 inductive ltc (A:Type[0]) (f) (B) (R:relation3 A B B): relation3 A B B ≝
21 | ltc_rc : ∀a,b1,b2. R a b1 b2 → ltc … a b1 b2
22 | ltc_trans: ∀a1,a2,b1,b,b2. ltc … a1 b1 b → ltc … a2 b b2 → ltc … (f a1 a2) b1 b2
25 (* Basic constructions ******************************************************)
27 lemma ltc_sn (A) (f) (B) (R):
29 ∀a2,b2. ltc A f B R a2 b b2 → ltc … f … R (f a1 a2) b1 b2.
30 /3 width=3 by ltc_rc, ltc_trans/ qed.
32 lemma ltc_dx (A) (f) (B) (R):
33 ∀a1,b1,b. ltc A f B R a1 b1 b →
34 ∀a2,b2. R a2 b b2 → ltc … f … R (f a1 a2) b1 b2.
35 /3 width=3 by ltc_rc, ltc_trans/ qed.
37 (* Basic eliminations *******************************************************)
39 lemma ltc_ind_sn (A) (f) (B) (R) (Q:relation2 A B) (b2):
41 (∀a,b1. R a b1 b2 → Q a b1) →
42 (∀a1,a2,b1,b. R a1 b1 b → ltc … f … R a2 b b2 → Q a2 b → Q (f a1 a2) b1) →
43 ∀a,b1. ltc … f … R a b1 b2 → Q a b1.
44 #A #f #B #R #Q #b2 #Hf #IH1 #IH2 #a #b1 @(insert_eq_1 … b2)
45 #b0 #H elim H -a -b1 -b0 /2 width=2 by/
46 #a1 #a2 #b1 #b #b0 #H #Hb2 #_
47 generalize in match Hb2; generalize in match a2; -Hb2 -a2
48 elim H -a1 -b1 -b /4 width=4 by ltc_trans/
51 lemma ltc_ind_dx (A) (f) (B) (R) (Q:A→predicate B) (b1):
53 (∀a,b2. R a b1 b2 → Q a b2) →
54 (∀a1,a2,b,b2. ltc … f … R a1 b1 b → Q a1 b → R a2 b b2 → Q (f a1 a2) b2) →
55 ∀a,b2. ltc … f … R a b1 b2 → Q a b2.
56 #A #f #B #R #Q #b1 #Hf #IH1 #IH2 #a #b2 @(insert_eq_1 … b1)
57 #b0 #H elim H -a -b0 -b2 /2 width=2 by/
58 #a1 #a2 #b0 #b #b2 #Hb0 #H #IHb0 #_
59 generalize in match IHb0; generalize in match Hb0; generalize in match a1; -IHb0 -Hb0 -a1
60 elim H -a2 -b -b2 /4 width=4 by ltc_trans/
63 (* Advanced eliminations (with reflexivity) *********************************)
65 lemma ltc_ind_sn_refl (A) (i) (f) (B) (R) (Q:relation2 A B) (b2):
66 associative … f → right_identity … f i → reflexive B (R i) →
68 (∀a1,a2,b1,b. R a1 b1 b → ltc … f … R a2 b b2 → Q a2 b → Q (f a1 a2) b1) →
69 ∀a,b1. ltc … f … R a b1 b2 → Q a b1.
70 #A #i #f #B #R #Q #b2 #H1f #H2f #HR #IH1 #IH2 #a #b1 #H
71 @(ltc_ind_sn … R … H1f … IH2 … H) -a -b1 -H1f #a #b1 #Hb12
72 >(H2f a) -H2f /3 width=4 by ltc_rc/
75 lemma ltc_ind_dx_refl (A) (i) (f) (B) (R) (Q:A→predicate B) (b1):
76 associative … f → left_identity … f i → reflexive B (R i) →
78 (∀a1,a2,b,b2. ltc … f … R a1 b1 b → Q a1 b → R a2 b b2 → Q (f a1 a2) b2) →
79 ∀a,b2. ltc … f … R a b1 b2 → Q a b2.
80 #A #i #f #B #R #Q #b1 #H1f #H2f #HR #IH1 #IH2 #a #b2 #H
81 @(ltc_ind_dx … R … H1f … IH2 … H) -a -b2 -H1f #a #b2 #Hb12
82 >(H2f a) -H2f /3 width=4 by ltc_rc/
85 (* Constructions with lsub **************************************************)
87 lemma ltc_lsub_trans (A) (f) (B) (C) (R) (S):
89 (∀n. lsub_trans B C (λL. R L n) S) →
90 ∀n. lsub_trans B C (λL. ltc A f … (R L) n) S.
91 #A #f #B #C #R #S #Hf #HRS #n #L2 #T1 #T2 #H
92 @(ltc_ind_dx … Hf ???? H) -n -T2
93 /3 width=5 by ltc_dx, ltc_rc/