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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 *)
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15 include "ground_2/xoa/ex_3_2.ma".
16 include "ground_2/notation/relations/ideq_2.ma".
17 include "ground_2/relocation/rtmap.ma".
19 (* RELOCATION MAP ***********************************************************)
21 coinductive eq: relation rtmap ≝
22 | eq_push: ∀f1,f2,g1,g2. eq f1 f2 → ⫯f1 = g1 → ⫯f2 = g2 → eq g1 g2
23 | eq_next: ∀f1,f2,g1,g2. eq f1 f2 → ↑f1 = g1 → ↑f2 = g2 → eq g1 g2
26 interpretation "extensional equivalence (rtmap)"
27 'IdEq f1 f2 = (eq f1 f2).
29 definition eq_repl (R:relation …) ≝
30 ∀f1,f2. f1 ≡ f2 → R f1 f2.
32 definition eq_repl_back (R:predicate …) ≝
33 ∀f1. R f1 → ∀f2. f1 ≡ f2 → R f2.
35 definition eq_repl_fwd (R:predicate …) ≝
36 ∀f1. R f1 → ∀f2. f2 ≡ f1 → R f2.
38 (* Basic properties *********************************************************)
40 corec lemma eq_refl: reflexive … eq.
41 #f cases (pn_split f) *
42 #g #Hg [ @(eq_push … Hg Hg) | @(eq_next … Hg Hg) ] -Hg //
45 corec lemma eq_sym: symmetric … eq.
47 #f1 #f2 #g1 #g2 #Hf #H1 #H2
48 [ @(eq_push … H2 H1) | @(eq_next … H2 H1) ] -g2 -g1 /2 width=1 by/
51 lemma eq_repl_sym: ∀R. eq_repl_back R → eq_repl_fwd R.
52 /3 width=3 by eq_sym/ qed-.
54 (* Basic inversion lemmas ***************************************************)
56 lemma eq_inv_px: ∀g1,g2. g1 ≡ g2 → ∀f1. ⫯f1 = g1 →
57 ∃∃f2. f1 ≡ f2 & ⫯f2 = g2.
59 #f1 #f2 #g1 #g2 #Hf * * -g1 -g2
61 [ lapply (injective_push … H) -H /2 width=3 by ex2_intro/
62 | elim (discr_push_next … H)
66 lemma eq_inv_nx: ∀g1,g2. g1 ≡ g2 → ∀f1. ↑f1 = g1 →
67 ∃∃f2. f1 ≡ f2 & ↑f2 = g2.
69 #f1 #f2 #g1 #g2 #Hf * * -g1 -g2
71 [ elim (discr_next_push … H)
72 | lapply (injective_next … H) -H /2 width=3 by ex2_intro/
76 lemma eq_inv_xp: ∀g1,g2. g1 ≡ g2 → ∀f2. ⫯f2 = g2 →
77 ∃∃f1. f1 ≡ f2 & ⫯f1 = g1.
79 #f1 #f2 #g1 #g2 #Hf * * -g1 -g2
81 [ lapply (injective_push … H) -H /2 width=3 by ex2_intro/
82 | elim (discr_push_next … H)
86 lemma eq_inv_xn: ∀g1,g2. g1 ≡ g2 → ∀f2. ↑f2 = g2 →
87 ∃∃f1. f1 ≡ f2 & ↑f1 = g1.
89 #f1 #f2 #g1 #g2 #Hf * * -g1 -g2
91 [ elim (discr_next_push … H)
92 | lapply (injective_next … H) -H /2 width=3 by ex2_intro/
96 (* Advanced inversion lemmas ************************************************)
98 lemma eq_inv_pp: ∀g1,g2. g1 ≡ g2 → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 → f1 ≡ f2.
99 #g1 #g2 #H #f1 #f2 #H1 elim (eq_inv_px … H … H1) -g1
101 #H lapply (injective_push … H) -H //
104 lemma eq_inv_nn: ∀g1,g2. g1 ≡ g2 → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 → f1 ≡ f2.
105 #g1 #g2 #H #f1 #f2 #H1 elim (eq_inv_nx … H … H1) -g1
107 #H lapply (injective_next … H) -H //
110 lemma eq_inv_pn: ∀g1,g2. g1 ≡ g2 → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 → ⊥.
111 #g1 #g2 #H #f1 #f2 #H1 elim (eq_inv_px … H … H1) -g1
113 #H elim (discr_next_push … H)
116 lemma eq_inv_np: ∀g1,g2. g1 ≡ g2 → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 → ⊥.
117 #g1 #g2 #H #f1 #f2 #H1 elim (eq_inv_nx … H … H1) -g1
119 #H elim (discr_push_next … H)
122 lemma eq_inv_gen: ∀f1,f2. f1 ≡ f2 →
123 (∃∃g1,g2. g1 ≡ g2 & ⫯g1 = f1 & ⫯g2 = f2) ∨
124 ∃∃g1,g2. g1 ≡ g2 & ↑g1 = f1 & ↑g2 = f2.
125 #f1 elim (pn_split f1) * #g1 #H1 #f2 #Hf
126 [ elim (eq_inv_px … Hf … H1) -Hf /3 width=5 by or_introl, ex3_2_intro/
127 | elim (eq_inv_nx … Hf … H1) -Hf /3 width=5 by or_intror, ex3_2_intro/
131 (* Main properties **********************************************************)
133 corec theorem eq_trans: Transitive … eq.
135 #f1 #f #g1 #g #Hf1 #H1 #H #f2 #Hf2
136 [ cases (eq_inv_px … Hf2 … H) | cases (eq_inv_nx … Hf2 … H) ] -g
137 /3 width=5 by eq_push, eq_next/
140 theorem eq_canc_sn: ∀f2. eq_repl_back (λf. f ≡ f2).
141 /3 width=3 by eq_trans, eq_sym/ qed-.
143 theorem eq_canc_dx: ∀f1. eq_repl_fwd (λf. f1 ≡ f).
144 /3 width=3 by eq_trans, eq_sym/ qed-.