1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "ground/xoa/ex_5_3.ma".
16 include "ground/xoa/ex_6_4.ma".
17 include "basic_2A/notation/relations/lrsubeqa_3.ma".
18 include "basic_2A/static/lsubr.ma".
19 include "basic_2A/static/aaa.ma".
21 (* LOCAL ENVIRONMENT REFINEMENT FOR ATOMIC ARITY ASSIGNMENT *****************)
23 inductive lsuba (G:genv): relation lenv ≝
24 | lsuba_atom: lsuba G (⋆) (⋆)
25 | lsuba_pair: ∀I,L1,L2,V. lsuba G L1 L2 → lsuba G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
26 | lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
27 lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
31 "local environment refinement (atomic arity assignment)"
32 'LRSubEqA G L1 L2 = (lsuba G L1 L2).
34 (* Basic inversion lemmas ***************************************************)
36 fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 = ⋆ → L2 = ⋆.
39 | #I #L1 #L2 #V #_ #H destruct
40 | #L1 #L2 #W #V #A #_ #_ #_ #H destruct
44 lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⫃⁝ L2 → L2 = ⋆.
45 /2 width=4 by lsuba_inv_atom1_aux/ qed-.
47 fact lsuba_inv_pair1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
48 (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓑ{I}X) ∨
49 ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
50 G ⊢ K1 ⫃⁝ K2 & I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
52 [ #J #K1 #X #H destruct
53 | #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
54 | #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by or_intror, ex6_4_intro/
58 lemma lsuba_inv_pair1: ∀I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃⁝ L2 →
59 (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓑ{I}X) ∨
60 ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
61 I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
62 /2 width=3 by lsuba_inv_pair1_aux/ qed-.
64 fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L2 = ⋆ → L1 = ⋆.
67 | #I #L1 #L2 #V #_ #H destruct
68 | #L1 #L2 #W #V #A #_ #_ #_ #H destruct
72 lemma lsuba_inv_atom2: ∀G,L1. G ⊢ L1 ⫃⁝ ⋆ → L1 = ⋆.
73 /2 width=4 by lsuba_inv_atom2_aux/ qed-.
75 fact lsuba_inv_pair2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2,W. L2 = K2.ⓑ{I}W →
76 (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓑ{I}W) ∨
77 ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
78 G ⊢ K1 ⫃⁝ K2 & I = Abst & L1 = K1.ⓓⓝW.V.
80 [ #J #K2 #U #H destruct
81 | #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
82 | #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by or_intror, ex5_3_intro/
86 lemma lsuba_inv_pair2: ∀I,G,L1,K2,W. G ⊢ L1 ⫃⁝ K2.ⓑ{I}W →
87 (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓑ{I}W) ∨
88 ∃∃K1,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
89 I = Abst & L1 = K1.ⓓⓝW.V.
90 /2 width=3 by lsuba_inv_pair2_aux/ qed-.
92 (* Basic forward lemmas *****************************************************)
94 lemma lsuba_fwd_lsubr: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 ⫃ L2.
95 #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
98 (* Basic properties *********************************************************)
100 lemma lsuba_refl: ∀G,L. G ⊢ L ⫃⁝ L.
101 #G #L elim L -L /2 width=1 by lsuba_atom, lsuba_pair/
104 (* Note: the constant 0 cannot be generalized *)
105 lemma lsuba_drop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
106 ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, m] L2 ≡ K2.
107 #G #L1 #L2 #H elim H -L1 -L2
108 [ /2 width=3 by ex2_intro/
109 | #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
110 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
112 elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
113 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
114 | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
116 | #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #m #H
117 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
119 elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
120 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
121 | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
126 (* Note: the constant 0 cannot be generalized *)
127 lemma lsuba_drop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
128 ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, m] L1 ≡ K1.
129 #G #L1 #L2 #H elim H -L1 -L2
130 [ /2 width=3 by ex2_intro/
131 | #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
132 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
134 elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
135 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
136 | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
138 | #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #m #H
139 elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
141 elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
142 <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
143 | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/