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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/syntax/lenv.ma".
16 include "apps_2/models/model_push.ma".
17 include "apps_2/notation/models/inwbrackets_4.ma".
19 (* LOCAL ENVIRONMENT INTERPRETATION ****************************************)
21 inductive li (M) (gv): relation2 lenv (evaluation M) ≝
22 | li_atom: ∀lv. li M gv (⋆) lv
23 | li_abbr: ∀lv,d,L,V. li M gv L lv → ⟦V⟧[gv, lv] ≗ d → li M gv (L.ⓓV) (⫯[d]lv)
24 | li_abst: ∀lv,d,L,W. li M gv L lv → li M gv (L.ⓛW) (⫯[d]lv)
25 | li_unit: ∀lv,d,I,L. li M gv L lv → li M gv (L.ⓤ{I}) (⫯[d]lv)
28 interpretation "local environment interpretation (model)"
29 'InWBrackets M gv L lv = (li M gv L lv).
31 (* Basic inversion lemmas ***************************************************)
33 fact li_inv_abbr_aux (M) (gv): ∀v,Y. v ϵ ⟦Y⟧{M}[gv] → ∀L,V. Y = L.ⓓV →
34 ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⟦V⟧{M}[gv, lv] ≗ d & ⫯{M}[d]lv = v.
36 [ #lv #K #W #H destruct
37 | #lv #d #L #V #HL #HV #K #W #H destruct /2 width=5 by ex3_2_intro/
38 | #lv #d #L #V #_ #K #W #H destruct
39 | #lv #d #I #L #_ #K #W #H destruct
43 lemma li_inv_abbr (M) (gv): ∀v,L,V. v ϵ ⟦L.ⓓV⟧{M}[gv] →
44 ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⟦V⟧{M}[gv, lv] ≗ d & ⫯{M}[d]lv = v.
45 /2 width=3 by li_inv_abbr_aux/ qed-.
47 fact li_inv_abst_aux (M) (gv): ∀v,Y. v ϵ ⟦Y⟧{M}[gv] → ∀L,W. Y = L.ⓛW →
48 ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⫯{M}[d]lv = v.
50 [ #lv #K #U #H destruct
51 | #lv #d #L #V #_ #_ #K #U #H destruct
52 | #lv #d #L #V #HL #K #U #H destruct /2 width=4 by ex2_2_intro/
53 | #lv #d #I #L #_ #K #U #H destruct
57 lemma li_inv_abst (M) (gv): ∀v,L,W. v ϵ ⟦L.ⓛW⟧{M}[gv] →
58 ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⫯{M}[d]lv = v.
59 /2 width=4 by li_inv_abst_aux/ qed-.
61 fact li_inv_unit_aux (M) (gv): ∀v,Y. v ϵ ⟦Y⟧{M}[gv] → ∀I,L. Y = L.ⓤ{I} →
62 ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⫯{M}[d]lv = v.
64 [ #lv #J #K #H destruct
65 | #lv #d #L #V #_ #_ #J #K #H destruct
66 | #lv #d #L #V #_ #J #K #H destruct
67 | #lv #d #I #L #HL #J #K #H destruct /2 width=4 by ex2_2_intro/
71 lemma li_inv_unit (M) (gv): ∀v,I,L. v ϵ ⟦L.ⓤ{I}⟧{M}[gv] →
72 ∃∃lv,d. lv ϵ ⟦L⟧{M}[gv] & ⫯{M}[d]lv = v.
73 /2 width=4 by li_inv_unit_aux/ qed-.