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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "ground/notation/relations/ist_1.ma".
16 include "ground/relocation/rtmap_at.ma".
18 (* RELOCATION MAP ***********************************************************)
20 definition istot: predicate rtmap ≝ λf. ∀i. ∃j. @❪i,f❫ ≘ j.
22 interpretation "test for totality (rtmap)"
25 (* Basic inversion lemmas ***************************************************)
27 lemma istot_inv_push: ∀g. 𝐓❪g❫ → ∀f. ⫯f = g → 𝐓❪f❫.
28 #g #Hg #f #H #i elim (Hg (↑i)) -Hg
29 #j #Hg elim (at_inv_npx … Hg … H) -Hg -H /2 width=3 by ex_intro/
32 lemma istot_inv_next: ∀g. 𝐓❪g❫ → ∀f. ↑f = g → 𝐓❪f❫.
33 #g #Hg #f #H #i elim (Hg i) -Hg
34 #j #Hg elim (at_inv_xnx … Hg … H) -Hg -H /2 width=2 by ex_intro/
37 (* Properties on tl *********************************************************)
39 lemma istot_tl: ∀f. 𝐓❪f❫ → 𝐓❪⫱f❫.
40 #f cases (pn_split f) *
41 #g * -f /2 width=3 by istot_inv_next, istot_inv_push/
44 (* Properties on tls ********************************************************)
46 lemma istot_tls: ∀n,f. 𝐓❪f❫ → 𝐓❪⫱*[n]f❫.
47 #n @(nat_ind_succ … n) -n //
49 /3 width=1 by istot_tl/
52 (* Main forward lemmas on at ************************************************)
54 corec theorem at_ext: ∀f1,f2. 𝐓❪f1❫ → 𝐓❪f2❫ →
55 (∀i,i1,i2. @❪i,f1❫ ≘ i1 → @❪i,f2❫ ≘ i2 → i1 = i2) →
57 #f1 cases (pn_split f1) * #g1 #H1
58 #f2 cases (pn_split f2) * #g2 #H2
60 [ @(eq_push … H1 H2) @at_ext -at_ext /2 width=3 by istot_inv_push/ -Hf1 -Hf2
61 #i #i1 #i2 #Hg1 #Hg2 lapply (Hi (↑i) (↑i1) (↑i2) ??) /2 width=7 by at_push/
62 | cases (Hf2 (𝟏)) -Hf1 -Hf2 -at_ext
63 #j2 #Hf2 cases (at_increasing_strict … Hf2 … H2) -H2
64 lapply (Hi (𝟏) (𝟏) j2 … Hf2) /2 width=2 by at_refl/ -Hi -Hf2 -H1
65 #H2 #H cases (plt_ge_false … H) -H //
66 | cases (Hf1 (𝟏)) -Hf1 -Hf2 -at_ext
67 #j1 #Hf1 cases (at_increasing_strict … Hf1 … H1) -H1
68 lapply (Hi (𝟏) j1 (𝟏) Hf1 ?) /2 width=2 by at_refl/ -Hi -Hf1 -H2
69 #H1 #H cases (plt_ge_false … H) -H //
70 | @(eq_next … H1 H2) @at_ext -at_ext /2 width=3 by istot_inv_next/ -Hf1 -Hf2
71 #i #i1 #i2 #Hg1 #Hg2 lapply (Hi i (↑i1) (↑i2) ??) /2 width=5 by at_next/
75 (* Advanced properties on at ************************************************)
77 lemma at_dec: ∀f,i1,i2. 𝐓❪f❫ → Decidable (@❪i1,f❫ ≘ i2).
78 #f #i1 #i2 #Hf lapply (Hf i1) -Hf *
79 #j2 #Hf elim (eq_pnat_dec i2 j2)
80 [ #H destruct /2 width=1 by or_introl/
81 | /4 width=6 by at_mono, or_intror/
85 lemma is_at_dec: ∀f,i2. 𝐓❪f❫ → Decidable (∃i1. @❪i1,f❫ ≘ i2).
87 lapply (dec_plt (λi1.@❪i1,f❫ ≘ i2) … (↑i2)) [| * ]
88 [ /2 width=1 by at_dec/
89 | * /3 width=2 by ex_intro, or_introl/
90 | #H @or_intror * #i1 #Hi12
91 /5 width=3 by at_increasing, plt_succ_dx, ex2_intro/
95 (* Advanced properties on isid **********************************************)
97 lemma isid_at_total: ∀f. 𝐓❪f❫ → (∀i1,i2. @❪i1,f❫ ≘ i2 → i1 = i2) → 𝐈❪f❫.
99 #i lapply (H1f i) -H1f *
100 #j #Hf >(H2f … Hf) in ⊢ (???%); -H2f //