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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 "basic_2/notation/relations/predty_5.ma".
16 include "basic_2/rt_transition/cpg.ma".
18 (* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
20 definition cpx (h): relation4 genv lenv term term ≝
21 λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[eq_f, c, h] T2.
24 "uncounted context-sensitive parallel rt-transition (term)"
25 'PRedTy h G L T1 T2 = (cpx h G L T1 T2).
27 (* Basic properties *********************************************************)
29 (* Basic_2A1: was: cpx_st *)
30 lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s).
31 /2 width=2 by cpg_ess, ex_intro/ qed.
33 lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 →
34 ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
35 #h * #G #K #V1 #V2 #W2 *
36 /3 width=4 by cpg_delta, cpg_ell, ex_intro/
39 lemma cpx_lref: ∀h,I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
40 ⬆*[1] T ≡ U → ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ⬈[h] U.
41 #h #I #G #K #V #T #U #i *
42 /3 width=4 by cpg_lref, ex_intro/
45 lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
46 ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
47 ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
48 #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
49 /3 width=2 by cpg_bind, ex_intro/
52 lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
53 ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
54 ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
55 #h * #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
56 /3 width=5 by cpg_appl, cpg_cast, ex_intro/
59 lemma cpx_zeta: ∀h,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈[h] T →
60 ⬆*[1] T2 ≡ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
61 #h #G #L #V #T1 #T #T2 *
62 /3 width=4 by cpg_zeta, ex_intro/
65 lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
67 /3 width=2 by cpg_eps, ex_intro/
70 (* Basic_2A1: was: cpx_ct *)
71 lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ⬈[h] V2.
73 /3 width=2 by cpg_ee, ex_intro/
76 lemma cpx_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
77 ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
78 ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2.
79 #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
80 /3 width=2 by cpg_beta, ex_intro/
83 lemma cpx_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
84 ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
85 ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
86 ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
87 #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
88 /3 width=4 by cpg_theta, ex_intro/
91 (* Basic_2A1: includes: cpx_atom *)
92 lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L).
93 /3 width=2 by cpg_refl, ex_intro/ qed.
95 (* Advanced properties ******************************************************)
97 lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
98 ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
99 #h * /2 width=2 by cpx_flat, cpx_bind/
102 (* Basic inversion lemmas ***************************************************)
104 lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[h] T2 →
106 | ∃∃s. T2 = ⋆(next h s) & J = Sort s
107 | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2 &
108 L = K.ⓑ{I}V1 & J = LRef 0
109 | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2 &
110 L = K.ⓑ{I}V & J = LRef (⫯i).
111 #h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
112 /4 width=9 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_5_intro, ex4_4_intro, ex2_intro, ex_intro/
115 lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] T2 →
116 T2 = ⋆s ∨ T2 = ⋆(next h s).
117 #h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
118 /2 width=1 by or_introl, or_intror/
121 lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[h] T2 →
123 ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2 &
125 #h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
126 /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
129 lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ⬈[h] T2 →
131 ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
132 #h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
133 /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
136 lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
137 #h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
140 lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 → (
141 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
144 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T & ⬆*[1] U2 ≡ T &
146 #h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
147 /4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
150 lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 → (
151 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
154 ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T & ⬆*[1] U2 ≡ T & p = true.
155 #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
156 /4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
159 lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 →
160 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 &
162 #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
163 /3 width=5 by ex3_2_intro, ex_intro/
166 lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
167 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
169 | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
170 ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
171 U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
172 | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≡ V2 &
173 ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
174 U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
175 #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
176 /4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/
179 lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
180 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
182 | ⦃G, L⦄ ⊢ U1 ⬈[h] U2
183 | ⦃G, L⦄ ⊢ V1 ⬈[h] U2.
184 #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
185 /4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
188 (* Advanced inversion lemmas ************************************************)
190 lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
192 ∃∃V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2.
193 #h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
194 /4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
197 lemma cpx_inv_lref1_pair: ∀h,I,G,K,V,T2,i. ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ⬈[h] T2 →
199 ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2.
200 #h #I #G #L #V #T2 #i * #c #H elim (cpg_inv_lref1_pair … H) -H *
201 /4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
204 lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
205 ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
207 | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
208 | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
209 | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
210 ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
212 U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
213 | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≡ V2 &
214 ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
216 U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
217 #h * #G #L #V1 #U1 #U2 #H
218 [ elim (cpx_inv_appl1 … H) -H *
219 /3 width=14 by or5_intro0, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro/
220 | elim (cpx_inv_cast1 … H) -H [ * ]
221 /3 width=14 by or5_intro0, or5_intro1, or5_intro2, ex3_2_intro, conj/
225 (* Basic forward lemmas *****************************************************)
227 lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
228 ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
230 #h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
231 /3 width=4 by ex2_2_intro, ex_intro/
234 (* Basic eliminators ********************************************************)
236 lemma cpx_ind: ∀h. ∀R:relation4 genv lenv term term.
237 (∀I,G,L. R G L (⓪{I}) (⓪{I})) →
238 (∀G,L,s. R G L (⋆s) (⋆(next h s))) →
239 (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → R G K V1 V2 →
240 ⬆*[1] V2 ≡ W2 → R G (K.ⓑ{I}V1) (#0) W2
241 ) → (∀I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → R G K (#i) T →
242 ⬆*[1] T ≡ U → R G (K.ⓑ{I}V) (#⫯i) (U)
243 ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
244 R G L V1 V2 → R G (L.ⓑ{I}V1) T1 T2 → R G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
245 ) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
246 R G L V1 V2 → R G L T1 T2 → R G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
247 ) → (∀G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈[h] T → R G (L.ⓓV) T1 T →
248 ⬆*[1] T2 ≡ T → R G L (+ⓓV.T1) T2
249 ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → R G L T1 T2 →
251 ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → R G L V1 V2 →
253 ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
254 R G L V1 V2 → R G L W1 W2 → R G (L.ⓛW1) T1 T2 →
255 R G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
256 ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
257 R G L V1 V → R G L W1 W2 → R G (L.ⓓW1) T1 T2 →
258 ⬆*[1] V ≡ V2 → R G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
260 ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → R G L T1 T2.
261 #h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
262 * #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/