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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/grammar/tstc_tstc.ma".
16 include "basic_2/computation/cpxs_cpxs.ma".
17 include "basic_2/computation/csx_alt.ma".
18 include "basic_2/computation/csx_lift.ma".
20 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
22 (* Advanced properties ******************************************************)
24 lemma csx_lpx_conf: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
25 ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
26 #h #g #G #L1 #L2 #HL12 #T #H @(csx_ind_alt … H) -T #T #_ #IHT
27 @csx_intro #T0 #HLT0 #HT0
28 @IHT /2 width=2/ -IHT -HT0 /2 width=3 by lpx_cpx_trans/
31 lemma csx_abst: ∀h,g,a,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
32 ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓛ{a}W.T.
33 #h #g #a #G #L #W #HW @(csx_ind … HW) -W #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
35 elim (cpx_inv_abst1 … H1) -H1
36 #W0 #T0 #HLW0 #HLT0 #H destruct
37 elim (eq_false_inv_tpair_sn … H2) -H2
38 [ -IHT #H lapply (csx_cpx_trans … HLT0) // -HT
39 #HT0 lapply (csx_lpx_conf … (L.ⓛW0) … HT0) -HT0 /2 width=1/ /3 width=1/
40 | -IHW -HLW0 -HT * #H destruct /3 width=1/
44 lemma csx_abbr: ∀h,g,a,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V →
45 ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V. T.
46 #h #g #a #G #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csx_ind_alt … HT) -T #T #HT #IHT
48 elim (cpx_inv_abbr1 … H1) -H1 *
49 [ #V1 #T1 #HLV1 #HLT1 #H destruct
50 elim (eq_false_inv_tpair_sn … H2) -H2
51 [ #HV1 @IHV // /2 width=1/ -HV1
52 @(csx_lpx_conf … (L.ⓓV)) /2 width=1/ -HLV1 /2 width=3 by csx_cpx_trans/
53 | -IHV -HLV1 * #H destruct /3 width=1/
55 | -IHV -IHT -H2 #T0 #HLT0 #HT0
56 lapply (csx_cpx_trans … HT … HLT0) -T #HLT0
57 lapply (csx_inv_lift … HLT0 … HT0) -T0 /2 width=3/
61 fact csx_appl_beta_aux: ∀h,g,a,G,L,U1. ⦃G, L⦄ ⊢ ⬊*[h, g] U1 →
62 ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T1.
63 #h #g #a #G #L #X #H @(csx_ind … H) -X
64 #X #HT1 #IHT1 #V #W #T1 #H1 destruct
66 elim (cpx_inv_appl1 … H1) -H1 *
67 [ -HT1 #V0 #Y #HLV0 #H #H0 destruct
68 elim (cpx_inv_abst1 … H) -H #W0 #T0 #HLW0 #HLT0 #H destruct
69 @IHT1 -IHT1 [4: // | skip |3: #H destruct /2 width=1/ ] -H2
70 lapply (lsubr_cpx_trans … HLT0 (L.ⓓⓝW.V) ?) -HLT0 [ /2 width=1/ ] /3 width=1/
71 | -IHT1 -H2 #b #V0 #W0 #W2 #T0 #T2 #HLV0 #HLW02 #HLT02 #H1 #H3 destruct
72 lapply (lsubr_cpx_trans … HLT02 (L.ⓓⓝW0.V) ?) -HLT02 [ /2 width=1/ ] #HT02
73 @(csx_cpx_trans … HT1) -HT1 /3 width=1/
74 | -HT1 -IHT1 -H2 #b #V0 #V1 #W0 #W1 #T0 #T3 #_ #_ #_ #_ #H destruct
78 (* Basic_1: was just: sn3_beta *)
79 lemma csx_appl_beta: ∀h,g,a,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T.
80 /2 width=3 by csx_appl_beta_aux/ qed.
82 fact csx_appl_theta_aux: ∀h,g,a,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
83 ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
84 #h #g #a #G #L #X #H @(csx_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
85 lapply (csx_fwd_pair_sn … HVT) #HV
86 lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
88 elim (cpx_inv_appl1 … HL) -HL *
89 [ -HV #V0 #Y #HLV10 #HL #H0 destruct
90 elim (cpx_inv_abbr1 … HL) -HL *
91 [ #V3 #T3 #HV3 #HLT3 #H0 destruct
92 elim (lift_total V0 0 1) #V4 #HV04
93 elim (eq_term_dec (ⓓ{a}V.ⓐV2.T) (ⓓ{a}V3.ⓐV4.T3))
95 elim (eq_false_inv_tpair_sn … H) -H
96 [ -HLV10 -HV3 -HLT3 -HVT
97 >(lift_inj … HV12 … HV04) -V4
102 lapply (cpx_lift … HLV10 (L. ⓓV) … HV12 … HV04) -HLV10 -HV12 /2 width=1/ #HV24
103 @(IHVT … H … HV04) -IHVT // -H -HV04 /4 width=1/
105 | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct
106 lapply (csx_cpx_trans … HVT (ⓐV2.T0) ?) /2 width=1/ -T #HVT0
107 lapply (csx_inv_lift … L … 1 HVT0 ? ? ?) -HVT0 [ /2 width=4/ |2,3: skip | /2 width=1/ ] -V2 -T0 #HVY
108 @(csx_cpx_trans … HVY) /2 width=1/
110 | -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #W1 #T0 #T1 #_ #_ #_ #H destruct
111 | -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
112 lapply (cpx_lift … HLV10 (L. ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /2 width=1/ #HLV23
113 @csx_abbr /2 width=3 by csx_cpx_trans/ -HV
114 @(csx_lpx_conf … (L. ⓓW0)) /2 width=1/ -W1
115 @(csx_cpxs_trans … HVT) -HVT /3 width=1/
119 lemma csx_appl_theta: ∀h,g,a,V1,V2. ⇧[0, 1] V1 ≡ V2 →
120 ∀G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
121 /2 width=5 by csx_appl_theta_aux/ qed.
123 (* Basic_1: was just: sn3_appl_appl *)
124 lemma csx_appl_simple_tstc: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
125 (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 ≃ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
126 𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
127 #h #g #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #H @(csx_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
129 elim (cpx_inv_appl1_simple … HL) -HL //
130 #V0 #T0 #HLV0 #HLT10 #H0 destruct
131 elim (eq_false_inv_tpair_sn … H) -H
133 @(csx_cpx_trans … (ⓐV0.T1)) /2 width=1/ -HLT10
134 @IHV -IHV // -H1T1 -H3T1 /2 width=1/ -HV0
136 @(csx_cpx_trans … (ⓐV.T2)) /2 width=1/ -HLV0
137 @H2T1 -H2T1 // -HLT12 /2 width=1/
138 | -IHV -H1T1 -HLV0 * #H #H1T10 destruct
139 elim (tstc_dec T1 T0) #H2T10
140 [ @IHT1 -IHT1 // /2 width=1/ -H1T10 /2 width=3/ -H3T1
142 @H2T1 -H2T1 /2 width=3/ -HLT10 -HLT02 /3 width=3/
144 @H2T1 -H2T1 /2 width=1/