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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/notation/relations/pred_6.ma".
16 include "basic_2/notation/relations/pred_5.ma".
17 include "basic_2/rt_transition/cpg.ma".
19 (* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
21 (* Basic_2A1: includes: cpr *)
22 definition cpm (h) (G) (L) (n): relation2 term term ≝
23 λT1,T2. ∃∃c. 𝐑𝐓⦃n,c⦄ & ⦃G,L⦄ ⊢ T1 ⬈[eq_t,c,h] T2.
26 "t-bound context-sensitive parallel rt-transition (term)"
27 'PRed n h G L T1 T2 = (cpm h G L n T1 T2).
30 "context-sensitive parallel r-transition (term)"
31 'PRed h G L T1 T2 = (cpm h G L O T1 T2).
33 (* Basic properties *********************************************************)
35 lemma cpm_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ➡[1,h] ⋆(⫯[h]s).
36 /2 width=3 by cpg_ess, ex2_intro/ qed.
38 lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
39 ⇧*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ➡[n,h] W2.
40 #n #h #G #K #V1 #V2 #W2 *
41 /3 width=5 by cpg_delta, ex2_intro/
44 lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
45 ⇧*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ➡[↑n,h] W2.
46 #n #h #G #K #V1 #V2 #W2 *
47 /3 width=5 by cpg_ell, ex2_intro, isrt_succ/
50 lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T →
51 ⇧*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ➡[n,h] U.
52 #n #h #I #G #K #T #U #i *
53 /3 width=5 by cpg_lref, ex2_intro/
56 (* Basic_2A1: includes: cpr_bind *)
57 lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2.
58 ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 →
59 ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2.
60 #n #h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
61 /5 width=5 by cpg_bind, isrt_max_O1, isr_shift, ex2_intro/
64 lemma cpm_appl: ∀n,h,G,L,V1,V2,T1,T2.
65 ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
66 ⦃G,L⦄ ⊢ ⓐV1.T1 ➡[n,h] ⓐV2.T2.
67 #n #h #G #L #V1 #V2 #T1 #T2 * #cV #HcV #HV12 *
68 /5 width=5 by isrt_max_O1, isr_shift, cpg_appl, ex2_intro/
71 lemma cpm_cast: ∀n,h,G,L,U1,U2,T1,T2.
72 ⦃G,L⦄ ⊢ U1 ➡[n,h] U2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
73 ⦃G,L⦄ ⊢ ⓝU1.T1 ➡[n,h] ⓝU2.T2.
74 #n #h #G #L #U1 #U2 #T1 #T2 * #cU #HcU #HU12 *
75 /4 width=6 by cpg_cast, isrt_max_idem1, isrt_mono, ex2_intro/
78 (* Basic_2A1: includes: cpr_zeta *)
79 lemma cpm_zeta (n) (h) (G) (L):
80 ∀T1,T. ⇧*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
81 ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ➡[n,h] T2.
82 #n #h #G #L #T1 #T #HT1 #T2 *
83 /3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
86 (* Basic_2A1: includes: cpr_eps *)
87 lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ➡[n,h] T2.
88 #n #h #G #L #V #T1 #T2 *
89 /3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/
92 lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ➡[↑n,h] V2.
93 #n #h #G #L #V1 #V2 #T *
94 /3 width=3 by cpg_ee, isrt_succ, ex2_intro/
97 (* Basic_2A1: includes: cpr_beta *)
98 lemma cpm_beta: ∀n,h,p,G,L,V1,V2,W1,W2,T1,T2.
99 ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 →
100 ⦃G,L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n,h] ⓓ{p}ⓝW2.V2.T2.
101 #n #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 *
102 /6 width=7 by cpg_beta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
105 (* Basic_2A1: includes: cpr_theta *)
106 lemma cpm_theta: ∀n,h,p,G,L,V1,V,V2,W1,W2,T1,T2.
107 ⦃G,L⦄ ⊢ V1 ➡[h] V → ⇧*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
108 ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
109 ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n,h] ⓓ{p}W2.ⓐV2.T2.
110 #n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 *
111 /6 width=9 by cpg_theta, isrt_plus_O2, isrt_max, isr_shift, ex2_intro/
114 (* Basic properties with r-transition ***************************************)
116 (* Note: this is needed by cpms_ind_sn and cpms_ind_dx *)
117 (* Basic_1: includes by definition: pr0_refl *)
118 (* Basic_2A1: includes: cpr_atom *)
119 lemma cpr_refl: ∀h,G,L. reflexive … (cpm h G L 0).
120 /3 width=3 by cpg_refl, ex2_intro/ qed.
122 (* Advanced properties ******************************************************)
124 lemma cpm_sort (h) (G) (L):
125 ∀n. n ≤ 1 → ∀s. ⦃G,L⦄ ⊢ ⋆s ➡[n,h] ⋆((next h)^n s).
127 #n #H #s <(le_n_O_to_eq n) /2 width=1 by le_S_S_to_le/
130 (* Basic inversion lemmas ***************************************************)
132 lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[n,h] T2 →
134 | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s & n = 1
135 | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⇧*[1] V2 ≘ T2 &
136 L = K.ⓓV1 & J = LRef 0
137 | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⇧*[1] V2 ≘ T2 &
138 L = K.ⓛV1 & J = LRef 0 & n = ↑m
139 | ∃∃I,K,T,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⇧*[1] T ≘ T2 &
140 L = K.ⓘ{I} & J = LRef (↑i).
141 #n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
142 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/
143 | #s #H1 #H2 #H3 destruct /4 width=3 by isrt_inv_01, or5_intro1, ex3_intro/
144 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
145 /4 width=6 by or5_intro2, ex4_3_intro, ex2_intro/
146 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
147 elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
148 /4 width=9 by or5_intro3, ex5_4_intro, ex2_intro/
149 | #I #K #V2 #i #HV2 #HVT2 #H1 #H2 destruct
150 /4 width=8 by or5_intro4, ex4_4_intro, ex2_intro/
154 lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ➡[n,h] T2 →
155 ∧∧ T2 = ⋆(((next h)^n) s) & n ≤ 1.
156 #n #h #G #L #T2 #s * #c #Hc #H
157 elim (cpg_inv_sort1 … H) -H * #H1 #H2 destruct
158 [ lapply (isrt_inv_00 … Hc) | lapply (isrt_inv_01 … Hc) ] -Hc
159 #H destruct /2 width=1 by conj/
162 lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[n,h] T2 →
164 | ∃∃K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 & ⇧*[1] V2 ≘ T2 &
166 | ∃∃m,K,V1,V2. ⦃G,K⦄ ⊢ V1 ➡[m,h] V2 & ⇧*[1] V2 ≘ T2 &
168 #n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
169 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
170 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
171 /4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
172 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
173 elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
174 /4 width=8 by or3_intro2, ex4_4_intro, ex2_intro/
178 lemma cpm_inv_zero1_unit (n) (h) (I) (K) (G):
179 ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #0 ➡[n,h] X2 → ∧∧ X2 = #0 & n = 0.
180 #n #h #I #G #K #X2 #H
181 elim (cpm_inv_zero1 … H) -H *
182 [ #H1 #H2 destruct /2 width=1 by conj/
183 | #Y #X1 #X2 #_ #_ #H destruct
184 | #m #Y #X1 #X2 #_ #_ #H destruct
188 lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[n,h] T2 →
189 ∨∨ T2 = #(↑i) ∧ n = 0
190 | ∃∃I,K,T. ⦃G,K⦄ ⊢ #i ➡[n,h] T & ⇧*[1] T ≘ T2 & L = K.ⓘ{I}.
191 #n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
192 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
193 | #I #K #V2 #HV2 #HVT2 #H destruct
194 /4 width=6 by ex3_3_intro, ex2_intro, or_intror/
198 lemma cpm_inv_lref1_ctop (n) (h) (G):
199 ∀X2,i. ⦃G,⋆⦄ ⊢ #i ➡[n,h] X2 → ∧∧ X2 = #i & n = 0.
200 #n #h #G #X2 * [| #i ] #H
201 [ elim (cpm_inv_zero1 … H) -H *
202 [ #H1 #H2 destruct /2 width=1 by conj/
203 | #Y #X1 #X2 #_ #_ #H destruct
204 | #m #Y #X1 #X2 #_ #_ #H destruct
206 | elim (cpm_inv_lref1 … H) -H *
207 [ #H1 #H2 destruct /2 width=1 by conj/
208 | #Z #Y #X0 #_ #_ #H destruct
213 lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ➡[n,h] T2 → T2 = §l ∧ n = 0.
214 #n #h #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
215 #H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/
218 (* Basic_2A1: includes: cpr_inv_bind1 *)
219 lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] U2 →
220 ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 &
222 | ∃∃T. ⇧*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 &
224 #n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
225 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
226 elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
227 elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
228 /4 width=5 by ex3_2_intro, ex2_intro, or_introl/
229 | #cT #T2 #HT21 #HTU2 #H1 #H2 #H3 destruct
230 /5 width=5 by isrt_inv_plus_O_dx, ex4_intro, ex2_intro, or_intror/
234 (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
235 (* Basic_2A1: includes: cpr_inv_abbr1 *)
236 lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡[n,h] U2 →
237 ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡[n,h] T2 &
239 | ∃∃T. ⇧*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 & p = true.
240 #n #h #p #G #L #V1 #T1 #U2 #H
241 elim (cpm_inv_bind1 … H) -H
242 [ /3 width=1 by or_introl/
243 | * /3 width=3 by ex3_intro, or_intror/
247 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
248 (* Basic_2A1: includes: cpr_inv_abst1 *)
249 lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡[n,h] U2 →
250 ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡[n,h] T2 &
252 #n #h #p #G #L #V1 #T1 #U2 #H
253 elim (cpm_inv_bind1 … H) -H
254 [ /3 width=1 by or_introl/
255 | * #T #_ #_ #_ #H destruct
259 lemma cpm_inv_abst_bi: ∀n,h,p1,p2,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}V1.T1 ➡[n,h] ⓛ{p2}V2.T2 →
260 ∧∧ ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡[n,h] T2 & p1 = p2.
261 #n #h #p1 #p2 #G #L #V1 #V2 #T1 #T2 #H
262 elim (cpm_inv_abst1 … H) -H #XV #XT #HV #HT #H destruct
263 /2 width=1 by and3_intro/
266 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
267 (* Basic_2A1: includes: cpr_inv_appl1 *)
268 lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ➡[n,h] U2 →
269 ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 &
271 | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 &
272 ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 &
273 U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
274 | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V & ⇧*[1] V ≘ V2 &
275 ⦃G,L⦄ ⊢ W1 ➡[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 &
276 U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
277 #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
278 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
279 elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
280 elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
281 /4 width=5 by or3_intro0, ex3_2_intro, ex2_intro/
282 | #cV #cW #cT #p #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #H1 #H2 #H3 destruct
283 lapply (isrt_inv_plus_O_dx … Hc ?) -Hc // #Hc
284 elim (isrt_inv_max … Hc) -Hc #n0 #nT #Hc #HcT #H destruct
285 elim (isrt_inv_max … Hc) -Hc #nV #nW #HcV #HcW #H destruct
286 elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
287 elim (isrt_inv_shift … HcW) -HcW #HcW #H destruct
288 /4 width=11 by or3_intro1, ex5_6_intro, ex2_intro/
289 | #cV #cW #cT #p #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #H1 #H2 #H3 destruct
290 lapply (isrt_inv_plus_O_dx … Hc ?) -Hc // #Hc
291 elim (isrt_inv_max … Hc) -Hc #n0 #nT #Hc #HcT #H destruct
292 elim (isrt_inv_max … Hc) -Hc #nV #nW #HcV #HcW #H destruct
293 elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
294 elim (isrt_inv_shift … HcW) -HcW #HcW #H destruct
295 /4 width=13 by or3_intro2, ex6_7_intro, ex2_intro/
299 lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓝV1.U1 ➡[n,h] U2 →
300 ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 & ⦃G,L⦄ ⊢ U1 ➡[n,h] T2 &
302 | ⦃G,L⦄ ⊢ U1 ➡[n,h] U2
303 | ∃∃m. ⦃G,L⦄ ⊢ V1 ➡[m,h] U2 & n = ↑m.
304 #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
305 [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
306 elim (isrt_inv_max … Hc) -Hc #nV #nT #HcV #HcT #H destruct
307 lapply (isrt_eq_t_trans … HcV HcVT) -HcVT #H
308 lapply (isrt_inj … H HcT) -H #H destruct <idempotent_max
309 /4 width=5 by or3_intro0, ex3_2_intro, ex2_intro/
310 | #cU #U12 #H destruct
311 /4 width=3 by isrt_inv_plus_O_dx, or3_intro1, ex2_intro/
312 | #cU #H12 #H destruct
313 elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
314 /4 width=3 by or3_intro2, ex2_intro/
318 (* Basic forward lemmas *****************************************************)
320 (* Basic_2A1: includes: cpr_fwd_bind1_minus *)
321 lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n,h] T → ∀p.
322 ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] ⓑ{p,I}V2.T2 &
324 #n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
325 /3 width=4 by ex2_2_intro, ex2_intro/
328 (* Basic eliminators ********************************************************)
330 lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
331 (∀I,G,L. Q 0 G L (⓪{I}) (⓪{I})) →
332 (∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
333 (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
334 ⇧*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
335 ) → (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
336 ⇧*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
337 ) → (∀n,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T → Q n G K (#i) T →
338 ⇧*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U)
339 ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 →
340 Q 0 G L V1 V2 → Q n G (L.ⓑ{I}V1) T1 T2 → Q n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
341 ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
342 Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
343 ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
344 Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
345 ) → (∀n,G,L,V,T1,T,T2. ⇧*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
346 Q n G L T T2 → Q n G L (+ⓓV.T1) T2
347 ) → (∀n,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
348 Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
349 ) → (∀n,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 →
350 Q n G L V1 V2 → Q (↑n) G L (ⓝV1.T) V2
351 ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 →
352 Q 0 G L V1 V2 → Q 0 G L W1 W2 → Q n G (L.ⓛW1) T1 T2 →
353 Q n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
354 ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
355 Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 →
356 ⇧*[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
358 ∀n,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → Q n G L T1 T2.
359 #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2
360 * #c #HC #H generalize in match HC; -HC generalize in match n; -n
361 elim H -c -G -L -T1 -T2
362 [ #I #G #L #n #H <(isrt_inv_00 … H) -H //
363 | #G #L #s #n #H <(isrt_inv_01 … H) -H //
364 | /3 width=4 by ex2_intro/
365 | #c #G #L #V1 #V2 #W2 #HV12 #HVW2 #IH #x #H
366 elim (isrt_inv_plus_SO_dx … H) -H // #n #Hc #H destruct
367 /3 width=4 by ex2_intro/
368 | /3 width=4 by ex2_intro/
369 | #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #n #H
370 elim (isrt_inv_max_shift_sn … H) -H #HcV #HcT
371 /3 width=3 by ex2_intro/
372 | #cV #cT #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #n #H
373 elim (isrt_inv_max_shift_sn … H) -H #HcV #HcT
374 /3 width=3 by ex2_intro/
375 | #cU #cT #G #L #U1 #U2 #T1 #T2 #HUT #HU12 #HT12 #IHU #IHT #n #H
376 elim (isrt_inv_max_eq_t … H) -H // #HcV #HcT
377 /3 width=3 by ex2_intro/
378 | #c #G #L #V #T1 #T #T2 #HT1 #HT2 #IH #n #H
379 lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
380 /3 width=4 by ex2_intro/
381 | #c #G #L #U #T1 #T2 #HT12 #IH #n #H
382 lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
383 /3 width=3 by ex2_intro/
384 | #c #G #L #U1 #U2 #T #HU12 #IH #x #H
385 elim (isrt_inv_plus_SO_dx … H) -H // #n #Hc #H destruct
386 /3 width=3 by ex2_intro/
387 | #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #IHV #IHW #IHT #n #H
388 lapply (isrt_inv_plus_O_dx … H ?) -H // >max_shift #H
389 elim (isrt_inv_max_shift_sn … H) -H #H #HcT
390 elim (isrt_O_inv_max … H) -H #HcV #HcW
391 /3 width=3 by ex2_intro/
392 | #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #IHV #IHW #IHT #n #H
393 lapply (isrt_inv_plus_O_dx … H ?) -H // >max_shift #H
394 elim (isrt_inv_max_shift_sn … H) -H #H #HcT
395 elim (isrt_O_inv_max … H) -H #HcV #HcW
396 /3 width=4 by ex2_intro/