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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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15 include "ground_2/relocation/rtmap_isid.ma".
16 include "ground_2/relocation/rtmap_isdiv.ma".
18 (* RELOCATION MAP ***********************************************************)
20 coinductive sle: relation rtmap ≝
21 | sle_push: ∀f1,f2,g1,g2. sle f1 f2 → ⫯f1 = g1 → ⫯f2 = g2 → sle g1 g2
22 | sle_next: ∀f1,f2,g1,g2. sle f1 f2 → ↑f1 = g1 → ↑f2 = g2 → sle g1 g2
23 | sle_weak: ∀f1,f2,g1,g2. sle f1 f2 → ⫯f1 = g1 → ↑f2 = g2 → sle g1 g2
26 interpretation "inclusion (rtmap)"
27 'subseteq f1 f2 = (sle f1 f2).
29 (* Basic properties *********************************************************)
31 axiom sle_eq_repl_back1: ∀f2. eq_repl_back … (λf1. f1 ⊆ f2).
33 lemma sle_eq_repl_fwd1: ∀f2. eq_repl_fwd … (λf1. f1 ⊆ f2).
34 #f2 @eq_repl_sym /2 width=3 by sle_eq_repl_back1/
37 axiom sle_eq_repl_back2: ∀f1. eq_repl_back … (λf2. f1 ⊆ f2).
39 lemma sle_eq_repl_fwd2: ∀f1. eq_repl_fwd … (λf2. f1 ⊆ f2).
40 #f1 @eq_repl_sym /2 width=3 by sle_eq_repl_back2/
43 corec lemma sle_refl: ∀f. f ⊆ f.
44 #f cases (pn_split f) * #g #H
45 [ @(sle_push … H H) | @(sle_next … H H) ] -H //
48 lemma sle_refl_eq: ∀f1,f2. f1 ≡ f2 → f1 ⊆ f2.
49 /2 width=3 by sle_eq_repl_back2/ qed.
51 (* Basic inversion lemmas ***************************************************)
53 lemma sle_inv_xp: ∀g1,g2. g1 ⊆ g2 → ∀f2. ⫯f2 = g2 →
54 ∃∃f1. f1 ⊆ f2 & ⫯f1 = g1.
56 #f1 #f2 #g1 #g2 #H #H1 #H2 #x2 #Hx2 destruct
57 [ lapply (injective_push … Hx2) -Hx2 /2 width=3 by ex2_intro/ ]
58 elim (discr_push_next … Hx2)
61 lemma sle_inv_nx: ∀g1,g2. g1 ⊆ g2 → ∀f1. ↑f1 = g1 →
62 ∃∃f2. f1 ⊆ f2 & ↑f2 = g2.
64 #f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #Hx1 destruct
65 [2: lapply (injective_next … Hx1) -Hx1 /2 width=3 by ex2_intro/ ]
66 elim (discr_next_push … Hx1)
69 lemma sle_inv_pn: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 → f1 ⊆ f2.
71 #f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #x2 #Hx1 #Hx2 destruct
72 [ elim (discr_next_push … Hx2)
73 | elim (discr_push_next … Hx1)
74 | lapply (injective_push … Hx1) -Hx1
75 lapply (injective_next … Hx2) -Hx2 //
79 (* Advanced inversion lemmas ************************************************)
81 lemma sle_inv_pp: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 → f1 ⊆ f2.
82 #g1 #g2 #H #f1 #f2 #H1 #H2 elim (sle_inv_xp … H … H2) -g2
83 #x1 #H #Hx1 destruct lapply (injective_push … Hx1) -Hx1 //
86 lemma sle_inv_nn: ∀g1,g2. g1 ⊆ g2 → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 → f1 ⊆ f2.
87 #g1 #g2 #H #f1 #f2 #H1 #H2 elim (sle_inv_nx … H … H1) -g1
88 #x2 #H #Hx2 destruct lapply (injective_next … Hx2) -Hx2 //
91 lemma sle_inv_px: ∀g1,g2. g1 ⊆ g2 → ∀f1. ⫯f1 = g1 →
92 (∃∃f2. f1 ⊆ f2 & ⫯f2 = g2) ∨ ∃∃f2. f1 ⊆ f2 & ↑f2 = g2.
93 #g1 #g2 elim (pn_split g2) * #f2 #H2 #H #f1 #H1
94 [ lapply (sle_inv_pp … H … H1 H2) | lapply (sle_inv_pn … H … H1 H2) ] -H -H1
95 /3 width=3 by ex2_intro, or_introl, or_intror/
98 lemma sle_inv_xn: ∀g1,g2. g1 ⊆ g2 → ∀f2. ↑f2 = g2 →
99 (∃∃f1. f1 ⊆ f2 & ⫯f1 = g1) ∨ ∃∃f1. f1 ⊆ f2 & ↑f1 = g1.
100 #g1 #g2 elim (pn_split g1) * #f1 #H1 #H #f2 #H2
101 [ lapply (sle_inv_pn … H … H1 H2) | lapply (sle_inv_nn … H … H1 H2) ] -H -H2
102 /3 width=3 by ex2_intro, or_introl, or_intror/
105 (* Main properties **********************************************************)
107 corec theorem sle_trans: Transitive … sle.
109 #f1 #f #g1 #g #Hf #H1 #H #g2 #H0
110 [ cases (sle_inv_px … H0 … H) * |*: cases (sle_inv_nx … H0 … H) ] -g
111 /3 width=5 by sle_push, sle_next, sle_weak/
114 (* Properties with iteraded push ********************************************)
116 lemma sle_pushs: ∀f1,f2. f1 ⊆ f2 → ∀i. ⫯*[i] f1 ⊆ ⫯*[i] f2.
117 #f1 #f2 #Hf12 #i elim i -i /2 width=5 by sle_push/
120 (* Properties with tail *****************************************************)
122 lemma sle_px_tl: ∀g1,g2. g1 ⊆ g2 → ∀f1. ⫯f1 = g1 → f1 ⊆ ⫱g2.
123 #g1 #g2 #H #f1 #H1 elim (sle_inv_px … H … H1) -H -H1 * //
126 lemma sle_xn_tl: ∀g1,g2. g1 ⊆ g2 → ∀f2. ↑f2 = g2 → ⫱g1 ⊆ f2.
127 #g1 #g2 #H #f2 #H2 elim (sle_inv_xn … H … H2) -H -H2 * //
130 lemma sle_tl: ∀f1,f2. f1 ⊆ f2 → ⫱f1 ⊆ ⫱f2.
131 #f1 elim (pn_split f1) * #g1 #H1 #f2 #H
132 [ lapply (sle_px_tl … H … H1) -H //
133 | elim (sle_inv_nx … H … H1) -H //
137 (* Inversion lemmas with tail ***********************************************)
139 lemma sle_inv_tl_sn: ∀f1,f2. ⫱f1 ⊆ f2 → f1 ⊆ ↑f2.
140 #f1 elim (pn_split f1) * #g1 #H destruct
141 /2 width=5 by sle_next, sle_weak/
144 lemma sle_inv_tl_dx: ∀f1,f2. f1 ⊆ ⫱f2 → ⫯f1 ⊆ f2.
145 #f1 #f2 elim (pn_split f2) * #g2 #H destruct
146 /2 width=5 by sle_push, sle_weak/
149 (* Properties with iteraded tail ********************************************)
151 lemma sle_tls: ∀f1,f2. f1 ⊆ f2 → ∀i. ⫱*[i] f1 ⊆ ⫱*[i] f2.
152 #f1 #f2 #Hf12 #i elim i -i /2 width=5 by sle_tl/
155 (* Properties with isid *****************************************************)
157 corec lemma sle_isid_sn: ∀f1. 𝐈❪f1❫ → ∀f2. f1 ⊆ f2.
159 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) *
160 /3 width=5 by sle_weak, sle_push/
163 (* Inversion lemmas with isid ***********************************************)
165 corec lemma sle_inv_isid_dx: ∀f1,f2. f1 ⊆ f2 → 𝐈❪f2❫ → 𝐈❪f1❫.
167 #f1 #f2 #g1 #g2 #Hf * * #H
168 [2,3: elim (isid_inv_next … H) // ]
169 lapply (isid_inv_push … H ??) -H
170 /3 width=3 by isid_push/
173 (* Properties with isdiv ****************************************************)
175 corec lemma sle_isdiv_dx: ∀f2. 𝛀❪f2❫ → ∀f1. f1 ⊆ f2.
177 #f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) *
178 /3 width=5 by sle_weak, sle_next/
181 (* Inversion lemmas with isdiv **********************************************)
183 corec lemma sle_inv_isdiv_sn: ∀f1,f2. f1 ⊆ f2 → 𝛀❪f1❫ → 𝛀❪f2❫.
185 #f1 #f2 #g1 #g2 #Hf * * #H
186 [1,3: elim (isdiv_inv_push … H) // ]
187 lapply (isdiv_inv_next … H ??) -H
188 /3 width=3 by isdiv_next/