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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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11 (* v GNU General Public License Version 2 *)
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15 include "ground/lib/star.ma".
16 include "basic_2/notation/relations/predtystar_4.ma".
17 include "basic_2/rt_transition/cpx.ma".
19 (* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS *************)
21 definition cpxs (G): relation3 lenv term term ≝
25 "extended context-sensitive parallel rt-computation (term)"
26 'PRedTyStar G L T1 T2 = (cpxs G L T1 T2).
28 (* Basic eliminators ********************************************************)
30 lemma cpxs_ind (G) (L) (T1) (Q:predicate …):
32 (∀T,T2. ❪G,L❫ ⊢ T1 ⬈* T → ❪G,L❫ ⊢ T ⬈ T2 → Q T → Q T2) →
33 ∀T2. ❪G,L❫ ⊢ T1 ⬈* T2 → Q T2.
34 #L #G #T1 #Q #HT1 #IHT1 #T2 #HT12
35 @(TC_star_ind … HT1 IHT1 … HT12) //
38 lemma cpxs_ind_dx (G) (L) (T2) (Q:predicate …):
40 (∀T1,T. ❪G,L❫ ⊢ T1 ⬈ T → ❪G,L❫ ⊢ T ⬈* T2 → Q T → Q T1) →
41 ∀T1. ❪G,L❫ ⊢ T1 ⬈* T2 → Q T1.
42 #G #L #T2 #Q #HT2 #IHT2 #T1 #HT12
43 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
46 (* Basic properties *********************************************************)
48 lemma cpxs_refl (G) (L):
50 /2 width=1 by inj/ qed.
52 lemma cpx_cpxs (G) (L): ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈ T2 → ❪G,L❫ ⊢ T1 ⬈* T2.
53 /2 width=1 by inj/ qed.
55 lemma cpxs_strap1 (G) (L):
56 ∀T1,T. ❪G,L❫ ⊢ T1 ⬈* T →
57 ∀T2. ❪G,L❫ ⊢ T ⬈ T2 → ❪G,L❫ ⊢ T1 ⬈* T2.
58 normalize /2 width=3 by step/ qed-.
60 lemma cpxs_strap2 (G) (L):
61 ∀T1,T. ❪G,L❫ ⊢ T1 ⬈ T →
62 ∀T2. ❪G,L❫ ⊢ T ⬈* T2 → ❪G,L❫ ⊢ T1 ⬈* T2.
63 normalize /2 width=3 by TC_strap/ qed-.
65 (* Basic_2A1: was just: cpxs_sort *)
66 lemma cpxs_qu (G) (L):
67 ∀s1,s2. ❪G,L❫ ⊢ ⋆s1 ⬈* ⋆s2.
68 /2 width=1 by cpx_cpxs/ qed.
70 lemma cpxs_bind_dx (G) (L):
71 ∀V1,V2. ❪G,L❫ ⊢ V1 ⬈ V2 →
72 ∀I,T1,T2. ❪G,L. ⓑ[I]V1❫ ⊢ T1 ⬈* T2 →
73 ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈* ⓑ[p,I]V2.T2.
74 #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
75 /3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
78 lemma cpxs_flat_dx (G) (L):
79 ∀V1,V2. ❪G,L❫ ⊢ V1 ⬈ V2 →
80 ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈* T2 →
81 ∀I. ❪G,L❫ ⊢ ⓕ[I]V1.T1 ⬈* ⓕ[I]V2.T2.
82 #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
83 /3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
86 lemma cpxs_flat_sn (G) (L):
87 ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈ T2 →
88 ∀V1,V2. ❪G,L❫ ⊢ V1 ⬈* V2 →
89 ∀I. ❪G,L❫ ⊢ ⓕ[I]V1.T1 ⬈* ⓕ[I]V2.T2.
90 #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
91 /3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
94 lemma cpxs_pair_sn (G) (L):
95 ∀I,V1,V2. ❪G,L❫ ⊢ V1 ⬈* V2 →
96 ∀T. ❪G,L❫ ⊢ ②[I]V1.T ⬈* ②[I]V2.T.
97 #G #L #I #V1 #V2 #H @(cpxs_ind … H) -V2
98 /3 width=3 by cpxs_strap1, cpx_pair_sn/
101 lemma cpxs_zeta (G) (L) (V):
103 ∀T2. ❪G,L❫ ⊢ T ⬈* T2 → ❪G,L❫ ⊢ +ⓓV.T1 ⬈* T2.
104 #G #L #V #T1 #T #HT1 #T2 #H @(cpxs_ind … H) -T2
105 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_zeta/
108 (* Basic_2A1: was: cpxs_zeta *)
109 lemma cpxs_zeta_dx (G) (L) (V):
111 ∀T1. ❪G,L.ⓓV❫ ⊢ T1 ⬈* T → ❪G,L❫ ⊢ +ⓓV.T1 ⬈* T2.
112 #G #L #V #T2 #T #HT2 #T1 #H @(cpxs_ind_dx … H) -T1
113 /3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
116 lemma cpxs_eps (G) (L):
117 ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈* T2 →
118 ∀V. ❪G,L❫ ⊢ ⓝV.T1 ⬈* T2.
119 #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
120 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
123 (* Basic_2A1: was: cpxs_ct *)
124 lemma cpxs_ee (G) (L):
125 ∀V1,V2. ❪G,L❫ ⊢ V1 ⬈* V2 →
126 ∀T. ❪G,L❫ ⊢ ⓝV1.T ⬈* V2.
127 #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
128 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ee/
131 lemma cpxs_beta_dx (G) (L):
132 ∀p,V1,V2,W1,W2,T1,T2.
133 ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L.ⓛW1❫ ⊢ T1 ⬈* T2 → ❪G,L❫ ⊢ W1 ⬈ W2 →
134 ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ⬈* ⓓ[p]ⓝW2.V2.T2.
135 #G #L #p #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
136 /4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
139 lemma cpxs_theta_dx (G) (L):
140 ∀p,V1,V,V2,W1,W2,T1,T2.
141 ❪G,L❫ ⊢ V1 ⬈ V → ⇧[1] V ≘ V2 → ❪G,L.ⓓW1❫ ⊢ T1 ⬈* T2 →
142 ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ⬈* ⓓ[p]W2.ⓐV2.T2.
143 #G #L #p #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
144 /4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
147 (* Basic inversion lemmas ***************************************************)
149 (* Basic_2A1: wa just: cpxs_inv_sort1 *)
150 lemma cpxs_inv_sort1 (G) (L):
151 ∀X2,s1. ❪G,L❫ ⊢ ⋆s1 ⬈* X2 →
153 #G #L #X2 #s1 #H @(cpxs_ind … H) -X2 /2 width=2 by ex_intro/
154 #X #X2 #_ #HX2 * #s #H destruct
155 elim (cpx_inv_sort1 … HX2) -HX2 #s2 #H destruct /2 width=2 by ex_intro/
158 lemma cpxs_inv_cast1 (G) (L):
159 ∀W1,T1,U2. ❪G,L❫ ⊢ ⓝW1.T1 ⬈* U2 →
160 ∨∨ ∃∃W2,T2. ❪G,L❫ ⊢ W1 ⬈* W2 & ❪G,L❫ ⊢ T1 ⬈* T2 & U2 = ⓝW2.T2
163 #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
164 #U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
165 #W #T #HW1 #HT1 #H destruct
166 elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
167 #W2 #T2 #HW2 #HT2 #H destruct
168 lapply (cpxs_strap1 … HW1 … HW2) -W
169 lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5 by or3_intro0, ex3_2_intro/