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15 include "ground_2/notation/functions/identity_0.ma".
16 include "ground_2/notation/relations/isidentity_1.ma".
17 include "ground_2/relocation/nstream_after.ma".
19 (* RELOCATION N-STREAM ******************************************************)
21 let corec id: rtmap ≝ ↑id.
23 interpretation "identity (nstream)"
26 definition isid: predicate rtmap ≝ λf. f ≐ 𝐈𝐝.
28 interpretation "test for identity (trace)"
29 'IsIdentity f = (isid f).
31 (* Basic properties on id ***************************************************)
33 lemma id_unfold: 𝐈𝐝 = ↑𝐈𝐝.
34 >(stream_expand … (𝐈𝐝)) in ⊢ (??%?); normalize //
37 (* Basic properties on isid *************************************************)
39 lemma isid_eq_repl_back: eq_stream_repl_back … isid.
40 /2 width=3 by eq_stream_canc_sn/ qed-.
42 lemma isid_eq_repl_fwd: eq_stream_repl_fwd … isid.
43 /3 width=3 by isid_eq_repl_back, eq_stream_repl_sym/ qed-.
48 lemma isid_push: ∀f. 𝐈⦃f⦄ → 𝐈⦃↑f⦄.
49 #f #H normalize >id_unfold /2 width=1 by eq_seq/
52 (* Basic inversion lemmas on isid *******************************************)
54 lemma isid_inv_seq: ∀f,n. 𝐈⦃n@f⦄ → 𝐈⦃f⦄ ∧ n = 0.
55 #f #n normalize >id_unfold in ⊢ (???%→?);
56 #H elim (eq_stream_inv_seq ????? H) -H /2 width=1 by conj/
59 lemma isid_inv_push: ∀f. 𝐈⦃↑f⦄ → 𝐈⦃f⦄.
60 * #n #f #H elim (isid_inv_seq … H) -H //
63 lemma isid_inv_next: ∀f. 𝐈⦃⫯f⦄ → ⊥.
64 * #n #f #H elim (isid_inv_seq … H) -H
68 lemma isid_inv_gen: ∀f. 𝐈⦃f⦄ → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
69 * #n #f #H elim (isid_inv_seq … H) -H
70 #Hf #H destruct /2 width=3 by ex2_intro/
73 lemma isid_inv_eq_repl: ∀f1,f2. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → f1 ≐ f2.
74 /2 width=3 by eq_stream_canc_dx/ qed-.
76 (* inversion lemmas on at ***************************************************)
78 let corec id_inv_at: ∀f. (∀i. @⦃i, f⦄ ≡ i) → f ≐ 𝐈𝐝 ≝ ?.
79 * #n #f #Ht lapply (Ht 0)
80 #H lapply (at_inv_O1 … H) -H
81 #H0 >id_unfold @eq_seq
83 | @id_inv_at -id_inv_at
84 #i lapply (Ht (⫯i)) -Ht cases H0 -n
85 #H elim (at_inv_SOx … H) -H //
89 lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i, f⦄ ≡ i.
91 [ * #n #f #H elim (isid_inv_seq … H) -H //
92 | #i #IH * #n #f #H elim (isid_inv_seq … H) -H
97 lemma isid_inv_at_mono: ∀f,i1,i2. 𝐈⦃f⦄ → @⦃i1, f⦄ ≡ i2 → i1 = i2.
98 /3 width=6 by isid_inv_at, at_mono/ qed-.
100 (* Properties on at *********************************************************)
102 lemma id_at: ∀i. @⦃i, 𝐈𝐝⦄ ≡ i.
103 /2 width=1 by isid_inv_at/ qed.
105 lemma isid_at: ∀f. (∀i. @⦃i, f⦄ ≡ i) → 𝐈⦃f⦄.
106 /2 width=1 by id_inv_at/ qed.
108 lemma isid_at_total: ∀f. (∀i1,i2. @⦃i1, f⦄ ≡ i2 → i1 = i2) → 𝐈⦃f⦄.
110 #i lapply (at_total i f)
111 #H >(Ht … H) in ⊢ (???%); -Ht //
114 (* Properties on after ******************************************************)
116 lemma after_isid_dx: ∀f2,f1,f. f2 ⊚ f1 ≡ f → f2 ≐ f → 𝐈⦃f1⦄.
117 #f2 #f1 #f #Ht #H2 @isid_at_total
118 #i1 #i2 #Hi12 elim (after_at1_fwd … Hi12 … Ht) -f1
119 /3 width=6 by at_inj, eq_stream_sym/
122 lemma after_isid_sn: ∀f2,f1,f. f2 ⊚ f1 ≡ f → f1 ≐ f → 𝐈⦃f2⦄.
123 #f2 #f1 #f #Ht #H1 @isid_at_total
124 #i2 #i #Hi2 lapply (at_total i2 f1)
125 #H0 lapply (at_increasing … H0)
126 #Ht1 lapply (after_fwd_at2 … (f1@❴i2❵) … H0 … Ht)
127 /3 width=7 by at_eq_repl_back, at_mono, at_id_le/
130 (* Inversion lemmas on after ************************************************)
132 let corec isid_after_sn: ∀f1,f2. 𝐈⦃f1⦄ → f1 ⊚ f2 ≡ f2 ≝ ?.
133 * #n1 #f1 * * [ | #n2 ] #f2 #H cases (isid_inv_seq … H) -H
134 /3 width=7 by after_push, after_refl/
137 let corec isid_after_dx: ∀f2,f1. 𝐈⦃f2⦄ → f1 ⊚ f2 ≡ f1 ≝ ?.
139 [ #f1 #H cases (isid_inv_seq … H) -H
140 /3 width=7 by after_refl/
141 | #n1 #f1 #H @after_next [4,5: // |1,2: skip ] /2 width=1 by/
145 lemma after_isid_inv_sn: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≐ f.
146 /3 width=8 by isid_after_sn, after_mono/
149 lemma after_isid_inv_dx: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f2⦄ → f1 ≐ f.
150 /3 width=8 by isid_after_dx, after_mono/
153 axiom after_inv_isid3: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ∧ 𝐈⦃f2⦄.