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14
15 include "basic_2/notation/relations/predty_5.ma".
16 include "basic_2/rt_transition/cpg.ma".
17
18 (* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
19
20 definition cpx (h): relation4 genv lenv term term ≝
21                     λG,L,T1,T2. ∃c. ⦃G,L⦄ ⊢ T1 ⬈[eq_f,c,h] T2.
22
23 interpretation
24    "unbound context-sensitive parallel rt-transition (term)"
25    'PRedTy h G L T1 T2 = (cpx h G L T1 T2).
26
27 (* Basic properties *********************************************************)
28
29 (* Basic_2A1: was: cpx_st *)
30 lemma cpx_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] ⋆(⫯[h]s).
31 /2 width=2 by cpg_ess, ex_intro/ qed.
32
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/
37 qed.
38
39 lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T →
40                 ⇧*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U.
41 #h #I #G #K #T #U #i *
42 /3 width=4 by cpg_lref, ex_intro/
43 qed.
44
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/
50 qed.
51
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/
57 qed.
58
59 lemma cpx_zeta (h) (G) (L):
60                ∀T1,T. ⇧*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ⬈[h] T2 →
61                ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
62 #h #G #L #T1 #T #HT1 #T2 *
63 /3 width=4 by cpg_zeta, ex_intro/
64 qed.
65
66 lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
67 #h #G #L #V #T1 #T2 *
68 /3 width=2 by cpg_eps, ex_intro/
69 qed.
70
71 (* Basic_2A1: was: cpx_ct *)
72 lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ⬈[h] V2.
73 #h #G #L #V1 #V2 #T *
74 /3 width=2 by cpg_ee, ex_intro/
75 qed.
76
77 lemma cpx_beta: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
78                 ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
79                 ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2.
80 #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
81 /3 width=2 by cpg_beta, ex_intro/
82 qed.
83
84 lemma cpx_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
85                  ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⇧*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ⬈[h] W2 →
86                  ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
87                  ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
88 #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
89 /3 width=4 by cpg_theta, ex_intro/
90 qed.
91
92 (* Basic_2A1: includes: cpx_atom *)
93 lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L).
94 /3 width=2 by cpg_refl, ex_intro/ qed.
95
96 (* Advanced properties ******************************************************)
97
98 lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 →
99                    ∀T. ⦃G,L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
100 #h * /2 width=2 by cpx_flat, cpx_bind/
101 qed.
102
103 lemma cpg_cpx (h) (Rt) (c) (G) (L):
104               ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2.
105 #h #Rt #c #G #L #T1 #T2 #H elim H -c -G -L -T1 -T2
106 /2 width=3 by cpx_theta, cpx_beta, cpx_ee, cpx_eps, cpx_zeta, cpx_flat, cpx_bind, cpx_lref, cpx_delta/
107 qed.
108
109 (* Basic inversion lemmas ***************************************************)
110
111 lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ⬈[h] T2 →
112                      ∨∨ T2 = ⓪{J}
113                       | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s
114                       | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2 &
115                                      L = K.ⓑ{I}V1 & J = LRef 0
116                       | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⇧*[1] T ≘ T2 &
117                                    L = K.ⓘ{I} & J = LRef (↑i).
118 #h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
119 /4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_intro, ex_intro/
120 qed-.
121
122 lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] T2 →
123                      ∨∨ T2 = ⋆s | T2 = ⋆(⫯[h]s).
124 #h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
125 /2 width=1 by or_introl, or_intror/
126 qed-.
127
128 lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G,L⦄ ⊢ #0 ⬈[h] T2 →
129                      ∨∨ T2 = #0
130                       | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2 &
131                                      L = K.ⓑ{I}V1.
132 #h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
133 /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
134 qed-.
135
136 lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[h] T2 →
137                      ∨∨ T2 = #(↑i)
138                       | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⇧*[1] T ≘ T2 & L = K.ⓘ{I}.
139 #h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
140 /4 width=6 by ex3_3_intro, ex_intro, or_introl, or_intror/
141 qed-.
142
143 lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
144 #h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
145 qed-.
146
147 lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 →
148                      ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
149                                  U2 = ⓑ{p,I}V2.T2
150                       | ∃∃T. ⇧*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[h] U2 &
151                              p = true & I = Abbr.
152 #h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
153 /4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
154 qed-.
155
156 lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 →
157                      ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
158                                  U2 = ⓓ{p}V2.T2
159                       | ∃∃T. ⇧*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ⬈[h] U2 & p = true.
160 #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
161 /4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
162 qed-.
163
164 lemma cpx_inv_abst1: ∀h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈[h] U2 →
165                      ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈[h] T2 &
166                               U2 = ⓛ{p}V2.T2.
167 #h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
168 /3 width=5 by ex3_2_intro, ex_intro/
169 qed-.
170
171 lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
172                      ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
173                                  U2 = ⓐV2.T2
174                       | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[h] W2 &
175                                             ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
176                                             U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
177                       | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V & ⇧*[1] V ≘ V2 &
178                                               ⦃G,L⦄ ⊢ W1 ⬈[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
179                                               U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
180 #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
181 /4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/
182 qed-.
183
184 lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
185                      ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
186                                  U2 = ⓝV2.T2
187                       | ⦃G,L⦄ ⊢ U1 ⬈[h] U2
188                       | ⦃G,L⦄ ⊢ V1 ⬈[h] U2.
189 #h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
190 /4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
191 qed-.
192
193 (* Advanced inversion lemmas ************************************************)
194
195 lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G,K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
196                           ∨∨ T2 = #0
197                            | ∃∃V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⇧*[1] V2 ≘ T2.
198 #h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
199 /4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
200 qed-.
201
202 lemma cpx_inv_lref1_bind: ∀h,I,G,K,T2,i. ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 →
203                           ∨∨ T2 = #(↑i)
204                            | ∃∃T. ⦃G,K⦄ ⊢ #i ⬈[h] T & ⇧*[1] T ≘ T2.
205 #h #I #G #L #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H *
206 /4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
207 qed-.
208
209 lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
210                      ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ U1 ⬈[h] T2 &
211                                  U2 = ⓕ{I}V2.T2
212                       | (⦃G,L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
213                       | (⦃G,L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
214                       | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 & ⦃G,L⦄ ⊢ W1 ⬈[h] W2 &
215                                             ⦃G,L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
216                                             U1 = ⓛ{p}W1.T1 &
217                                             U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
218                       | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V & ⇧*[1] V ≘ V2 &
219                                               ⦃G,L⦄ ⊢ W1 ⬈[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
220                                               U1 = ⓓ{p}W1.T1 &
221                                               U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
222 #h * #G #L #V1 #U1 #U2 #H
223 [ elim (cpx_inv_appl1 … H) -H *
224   /3 width=14 by or5_intro0, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro/
225 | elim (cpx_inv_cast1 … H) -H [ * ]
226   /3 width=14 by or5_intro0, or5_intro1, or5_intro2, ex3_2_intro, conj/
227 ]
228 qed-.
229
230 (* Basic forward lemmas *****************************************************)
231
232 lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
233                            ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
234                                     T = -ⓑ{I}V2.T2.
235 #h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
236 /3 width=4 by ex2_2_intro, ex_intro/
237 qed-.
238
239 (* Basic eliminators ********************************************************)
240
241 lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term.
242                (∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
243                (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
244                (∀I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
245                  ⇧*[1] V2 ≘ W2 → Q G (K.ⓑ{I}V1) (#0) W2
246                ) → (∀I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T →
247                  ⇧*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
248                ) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
249                   Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
250                ) → (∀I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
251                   Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
252                ) → (∀G,L,V,T1,T,T2. ⇧*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ⬈[h] T2 → Q G L T T2 →
253                   Q G L (+ⓓV.T1) T2
254                ) → (∀G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
255                   Q G L (ⓝV.T1) T2
256                ) → (∀G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
257                   Q G L (ⓝV1.T) V2
258                ) → (∀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 →
259                   Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
260                   Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
261                ) → (∀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 →
262                   Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
263                   ⇧*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
264                ) →
265                ∀G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2.
266 #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
267 * #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
268 qed-.