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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/rt_transition/cpr.ma".
16 include "basic_2/rt_computation/fpbg_fqup.ma".
17 include "basic_2/dynamic/cnv_fsb.ma".
19 (* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
21 (* Inversion lemmas with degree-based equivalence for terms *****************)
23 lemma cnv_cpr_tdeq_fwd_refl (a) (h) (o) (G) (L):
24 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → T1 ≛[h,o] T2 →
25 ⦃G, L⦄ ⊢ T1 ![a,h] → T1 = T2.
26 #a #h #o #G #L #T1 #T2 #H @(cpr_ind … H) -G -L -T1 -T2
28 | #G #K #V1 #V2 #X2 #_ #_ #_ #H1 #_ -a -G -K -V1 -V2
29 lapply (tdeq_inv_lref1 … H1) -H1 #H destruct //
30 | #I #G #K #T2 #X2 #i #_ #_ #_ #H1 #_ -a -I -G -K -T2
31 lapply (tdeq_inv_lref1 … H1) -H1 #H destruct //
32 | #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
33 elim (tdeq_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
34 elim (cnv_inv_bind … H2) -H2 #HV1 #HT1
36 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #H1 #H2
37 elim (tdeq_inv_pair1 … H1) -H1 #V0 #T0 #HV0 #HT0 #H destruct
38 elim (cnv_fwd_flat … H2) -H2 #HV1 #HT1
40 | #G #K #V #T1 #X1 #X2 #HXT1 #HX12 #_ #H1 #H2
41 elim (cnv_fpbg_refl_false … o … H2) -a
42 @(fpbg_tdeq_div … H1) -H1
43 /3 width=9 by cpm_tdneq_cpm_fpbg, cpm_zeta, tdeq_lifts_inv_pair_sn/
44 | #G #L #U #T1 #T2 #HT12 #_ #H1 #H2
45 elim (cnv_fpbg_refl_false … o … H2) -a
46 @(fpbg_tdeq_div … H1) -H1
47 /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_eps, tdeq_inv_pair_xy_y/
48 | #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H1 #_
49 elim (tdeq_inv_pair … H1) -H1 #H #_ #_ destruct
50 | #p #G #L #V1 #V2 #X2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #H1 #_
51 elim (tdeq_inv_pair … H1) -H1 #H #_ #_ destruct
55 lemma cpm_tdeq_inv_bind (a) (h) (o) (n) (p) (I) (G) (L):
56 ∀V,T1. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ![a,h] →
57 ∀X. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T1 ➡[n,h] X → ⓑ{p,I}V.T1 ≛[h,o] X →
58 ∃∃T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓑ{p,I}V.T2.
59 #a #h #o #n #p #I #G #L #V #T1 #H0 #X #H1 #H2
60 elim (cpm_inv_bind1 … H1) -H1 *
61 [ #XV #T2 #HXV #HT12 #H destruct
62 elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
63 elim (cnv_inv_bind … H0) -H0 #HV #_
64 lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV -HV
65 /2 width=4 by ex3_intro/
66 | #X1 #HXT1 #HX1 #H1 #H destruct
67 elim (cnv_fpbg_refl_false … o … H0) -a
68 @(fpbg_tdeq_div … H2) -H2
69 /3 width=9 by cpm_tdneq_cpm_fpbg, cpm_zeta, tdeq_lifts_inv_pair_sn/
73 lemma cpm_tdeq_inv_appl (a) (h) (o) (n) (G) (L):
74 ∀V,T1. ⦃G, L⦄ ⊢ ⓐV.T1 ![a,h] →
75 ∀X. ⦃G, L⦄ ⊢ ⓐV.T1 ➡[n,h] X → ⓐV.T1 ≛[h,o] X →
76 ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓐV.T2.
77 #a #h #o #n #G #L #V #T1 #H0 #X #H1 #H2
78 elim (cpm_inv_appl1 … H1) -H1 *
79 [ #XV #T2 #HXV #HT12 #H destruct
80 elim (tdeq_inv_pair … H2) -H2 #_ #H2XV #H2T12
81 elim (cnv_inv_appl … H0) -H0 #m #q #W #U #_ #HV #_ #_ #_ -m -q -W -U
82 lapply (cnv_cpr_tdeq_fwd_refl … HXV H2XV HV) #H destruct -HXV -H2XV -HV
83 /2 width=4 by ex3_intro/
84 | #q #V2 #W1 #W2 #XT #T2 #_ #_ #_ #H1 #H destruct -H0
85 elim (tdeq_inv_pair … H2) -H2 #H #_ #_ destruct
86 | #q #V2 #XV #W1 #W2 #XT #T2 #_ #_ #_ #_ #H1 #H destruct -H0
87 elim (tdeq_inv_pair … H2) -H2 #H #_ #_ destruct
91 lemma cpm_tdeq_inv_cast (a) (h) (o) (n) (G) (L):
92 ∀U1,T1. ⦃G, L⦄ ⊢ ⓝU1.T1 ![a,h] →
93 ∀X. ⦃G, L⦄ ⊢ ⓝU1.T1 ➡[n,h] X → ⓝU1.T1 ≛[h,o] X →
94 ∃∃U2,T2. ⦃G, L⦄ ⊢ U1 ➡[n,h] U2 & U1 ≛[h,o] U2 & ⦃G, L⦄ ⊢ T1 ➡[n,h] T2 & T1 ≛[h,o] T2 & X = ⓝU2.T2.
95 #a #h #o #n #G #L #U1 #T1 #H0 #X #H1 #H2
96 elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
97 [ #U2 #T2 #HU12 #HT12 #H destruct -H0
98 elim (tdeq_inv_pair … H2) -H2 #_ #H2U12 #H2T12
99 /2 width=7 by ex5_2_intro/
101 elim (cnv_fpbg_refl_false … o … H0) -a
102 @(fpbg_tdeq_div … H2) -H2
103 /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_eps, tdeq_inv_pair_xy_y/
104 | #m #HU1X #H destruct
105 elim (cnv_fpbg_refl_false … o … H0) -a
106 @(fpbg_tdeq_div … H2) -H2
107 /3 width=6 by cpm_tdneq_cpm_fpbg, cpm_ee, tdeq_inv_pair_xy_x/