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14
15 include "ground/relocation/pr_after_eq.ma".
16
17 (* RELATIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
18
19 (* Main constructions *******************************************************)
20
21 (*** after_trans1 *)
22 corec theorem pr_after_trans_sn:
23               ∀f0,f3,f4. f0 ⊚ f3 ≘ f4 →
24               ∀f1,f2. f1 ⊚ f2 ≘ f0 →
25               ∀f. f2 ⊚ f3 ≘ f → f1 ⊚ f ≘ f4.
26 #f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
27 [ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
28   cases (pr_after_inv_push … Hg0 … H0) -g0
29   #f1 #f2 #Hf0 #H1 #H2
30   cases (pr_after_inv_push_bi … Hg … H2 H3) -g2 -g3
31   #f #Hf #H /3 width=7 by pr_after_refl/
32 | #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
33   cases (pr_after_inv_push … Hg0 … H0) -g0
34   #f1 #f2 #Hf0 #H1 #H2
35   cases (pr_after_inv_push_next … Hg … H2 H3) -g2 -g3
36   #f #Hf #H /3 width=7 by pr_after_push/
37 | #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
38   cases (pr_after_inv_next … Hg0 … H0) -g0 *
39   [ #f1 #f2 #Hf0 #H1 #H2
40     cases (pr_after_inv_next_sn … Hg … H2) -g2
41     #f #Hf #H /3 width=7 by pr_after_push/
42   | #f1 #Hf0 #H1 /3 width=6 by pr_after_next/
43   ]
44 ]
45 qed-.
46
47 (*** after_trans2 *)
48 corec theorem pr_after_trans_dx:
49               ∀f1,f0,f4. f1 ⊚ f0 ≘ f4 →
50               ∀f2, f3. f2 ⊚ f3 ≘ f0 →
51               ∀f. f1 ⊚ f2 ≘ f → f ⊚ f3 ≘ f4.
52 #f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
53 [ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
54   cases (pr_after_inv_push … Hg0 … H0) -g0
55   #f2 #f3 #Hf0 #H2 #H3
56   cases (pr_after_inv_push_bi … Hg … H1 H2) -g1 -g2
57   #f #Hf #H /3 width=7 by pr_after_refl/
58 | #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
59   cases (pr_after_inv_next … Hg0 … H0) -g0 *
60   [ #f2 #f3 #Hf0 #H2 #H3
61     cases (pr_after_inv_push_bi … Hg … H1 H2) -g1 -g2
62     #f #Hf #H /3 width=7 by pr_after_push/
63   | #f2 #Hf0 #H2
64     cases (pr_after_inv_push_next … Hg … H1 H2) -g1 -g2
65     #f #Hf #H /3 width=6 by pr_after_next/
66   ]
67 | #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
68   cases (pr_after_inv_next_sn … Hg … H1) -g1
69   #f #Hg #H /3 width=6 by pr_after_next/
70 ]
71 qed-.
72
73 (* Main inversions **********************************************************)
74
75 (*** after_mono *)
76 corec theorem pr_after_mono:
77               ∀f1,f2,x,y. f1 ⊚ f2 ≘ x → f1 ⊚ f2 ≘ y → x ≡ y.
78 #f1 #f2 #x #y * -f1 -f2 -x
79 #f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
80 [ cases (pr_after_inv_push_bi … Hy … H1 H2) -g1 -g2 /3 width=8 by pr_eq_push/
81 | cases (pr_after_inv_push_next … Hy … H1 H2) -g1 -g2 /3 width=8 by pr_eq_next/
82 | cases (pr_after_inv_next_sn … Hy … H1) -g1 /3 width=8 by pr_eq_next/
83 ]
84 qed-.
85
86 (*** after_mono_eq *)
87 lemma pr_after_mono_eq:
88       ∀f1,f2,f. f1 ⊚ f2 ≘ f → ∀g1,g2,g. g1 ⊚ g2 ≘ g →
89       f1 ≡ g1 → f2 ≡ g2 → f ≡ g.
90 /4 width=4 by pr_after_mono, pr_after_eq_repl_back_dx, pr_after_eq_repl_back_sn/ qed-.