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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.tcs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 include "ground_2/relocation/rtmap_isid.ma".
17 (* RELOCATION MAP ***********************************************************)
19 coinductive sle: relation rtmap ≝
20 | sle_push: ∀f1,f2,g1,g2. sle f1 f2 → ↑f1 = g1 → ↑f2 = g2 → sle g1 g2
21 | sle_next: ∀f1,f2,g1,g2. sle f1 f2 → ⫯f1 = g1 → ⫯f2 = g2 → sle g1 g2
22 | sle_weak: ∀f1,f2,g1,g2. sle f1 f2 → ↑f1 = g1 → ⫯f2 = g2 → sle g1 g2
25 interpretation "inclusion (rtmap)"
26 'subseteq t1 t2 = (sle t1 t2).
28 (* Basic properties *********************************************************)
30 corec lemma sle_refl: ∀f. f ⊆ f.
31 #f cases (pn_split f) * #g #H
32 [ @(sle_push … H H) | @(sle_next … H H) ] -H //
35 (* Basic inversion lemmas ***************************************************)
37 lemma sle_inv_xp: ∀g1,g2. g1 ⊆ g2 → ∀f2. ↑f2 = g2 →
38 ∃∃f1. f1 ⊆ f2 & ↑f1 = g1.
40 #f1 #f2 #g1 #g2 #H #H1 #H2 #x2 #Hx2 destruct
41 [ lapply (injective_push … Hx2) -Hx2 /2 width=3 by ex2_intro/ ]
42 elim (discr_push_next … Hx2)
45 lemma sle_inv_nx: ∀g1,g2. g1 ⊆ g2 → ∀f1. ⫯f1 = g1 →
46 ∃∃f2. f1 ⊆ f2 & ⫯f2 = g2.
48 #f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #Hx1 destruct
49 [2: lapply (injective_next … Hx1) -Hx1 /2 width=3 by ex2_intro/ ]
50 elim (discr_next_push … Hx1)
53 lemma sle_inv_pn: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 → f1 ⊆ f2.
55 #f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #x2 #Hx1 #Hx2 destruct
56 [ elim (discr_next_push … Hx2)
57 | elim (discr_push_next … Hx1)
58 | lapply (injective_push … Hx1) -Hx1
59 lapply (injective_next … Hx2) -Hx2 //
63 (* Advanced inversion lemmas ************************************************)
65 lemma sle_inv_pp: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 → f1 ⊆ f2.
66 #g1 #g2 #H #f1 #f2 #H1 #H2 elim (sle_inv_xp … H … H2) -g2
67 #x1 #H #Hx1 destruct lapply (injective_push … Hx1) -Hx1 //
70 lemma sle_inv_nn: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 → f1 ⊆ f2.
71 #g1 #g2 #H #f1 #f2 #H1 #H2 elim (sle_inv_nx … H … H1) -g1
72 #x2 #H #Hx2 destruct lapply (injective_next … Hx2) -Hx2 //
75 lemma sle_inv_px: ∀g1,g2. g1 ⊆ g2 → ∀f1. ↑f1 = g1 →
76 (∃∃f2. f1 ⊆ f2 & ↑f2 = g2) ∨ ∃∃f2. f1 ⊆ f2 & ⫯f2 = g2.
77 #g1 #g2 elim (pn_split g2) * #f2 #H2 #H #f1 #H1
78 [ lapply (sle_inv_pp … H … H1 H2) | lapply (sle_inv_pn … H … H1 H2) ] -H -H1
79 /3 width=3 by ex2_intro, or_introl, or_intror/
82 lemma sle_inv_xn: ∀g1,g2. g1 ⊆ g2 → ∀f2. ⫯f2 = g2 →
83 (∃∃f1. f1 ⊆ f2 & ↑f1 = g1) ∨ ∃∃f1. f1 ⊆ f2 & ⫯f1 = g1.
84 #g1 #g2 elim (pn_split g1) * #f1 #H1 #H #f2 #H2
85 [ lapply (sle_inv_pn … H … H1 H2) | lapply (sle_inv_nn … H … H1 H2) ] -H -H2
86 /3 width=3 by ex2_intro, or_introl, or_intror/
89 (* Main properties **********************************************************)
91 corec theorem sle_trans: Transitive … sle.
93 #f1 #f #g1 #g #Hf #H1 #H #g2 #H0
94 [ cases (sle_inv_px … H0 … H) * |*: cases (sle_inv_nx … H0 … H) ] -g
95 /3 width=5 by sle_push, sle_next, sle_weak/
98 (* Properties with iteraded push ********************************************)
100 lemma sle_pushs: ∀f1,f2. f1 ⊆ f2 → ∀i. ↑*[i] f1 ⊆ ↑*[i] f2.
101 #f1 #f2 #Hf12 #i elim i -i /2 width=5 by sle_push/
104 (* Properties with tail *****************************************************)
106 lemma sle_px_tl: ∀g1,g2. g1 ⊆ g2 → ∀f1. ↑f1 = g1 → f1 ⊆ ⫱g2.
107 #g1 #g2 #H #f1 #H1 elim (sle_inv_px … H … H1) -H -H1 * //
110 lemma sle_xn_tl: ∀g1,g2. g1 ⊆ g2 → ∀f2. ⫯f2 = g2 → ⫱g1 ⊆ f2.
111 #g1 #g2 #H #f2 #H2 elim (sle_inv_xn … H … H2) -H -H2 * //
114 lemma sle_tl: ∀f1,f2. f1 ⊆ f2 → ⫱f1 ⊆ ⫱f2.
115 #f1 elim (pn_split f1) * #g1 #H1 #f2 #H
116 [ lapply (sle_px_tl … H … H1) -H //
117 | elim (sle_inv_nx … H … H1) -H //
121 (* Inversion lemmas with tail ***********************************************)
123 lemma sle_inv_tl_sn: ∀f1,f2. ⫱f1 ⊆ f2 → f1 ⊆ ⫯f2.
124 #f1 elim (pn_split f1) * #g1 #H destruct
125 /2 width=5 by sle_next, sle_weak/
128 lemma sle_inv_tl_dx: ∀f1,f2. f1 ⊆ ⫱f2 → ↑f1 ⊆ f2.
129 #f1 #f2 elim (pn_split f2) * #g2 #H destruct
130 /2 width=5 by sle_push, sle_weak/
133 (* Properties with isid *****************************************************)
135 corec lemma sle_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⊆ f2.
137 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) *
138 /3 width=5 by sle_weak, sle_push/
141 (* Inversion lemmas with isid ***********************************************)
143 corec lemma sle_inv_isid_dx: ∀f1,f2. f1 ⊆ f2 → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
145 #f1 #f2 #g1 #g2 #Hf * * #H
146 [2,3: elim (isid_inv_next … H) // ]
147 lapply (isid_inv_push … H ??) -H
148 /3 width=3 by isid_push/