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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/psubststar_6.ma".
16 include "basic_2/relocation/cpy.ma".
18 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
20 definition cpys: ynat → ynat → relation4 genv lenv term term ≝
21 λd,e,G. LTC … (cpy d e G).
23 interpretation "context-sensitive extended multiple substritution (term)"
24 'PSubstStar G L T1 d e T2 = (cpys d e G L T1 T2).
26 (* Basic eliminators ********************************************************)
28 lemma cpys_ind: ∀G,L,T1,d,e. ∀R:predicate term. R T1 →
29 (∀T,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶[d, e] T2 → R T → R T2) →
30 ∀T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R T2.
31 #G #L #T1 #d #e #R #HT1 #IHT1 #T2 #HT12
32 @(TC_star_ind … HT1 IHT1 … HT12) //
35 lemma cpys_ind_dx: ∀G,L,T2,d,e. ∀R:predicate term. R T2 →
36 (∀T1,T. ⦃G, L⦄ ⊢ T1 ▶[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → R T → R T1) →
37 ∀T1. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R T1.
38 #G #L #T2 #d #e #R #HT2 #IHT2 #T1 #HT12
39 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
42 (* Basic properties *********************************************************)
44 lemma cpy_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
45 /2 width=1 by inj/ qed.
47 lemma cpys_strap1: ∀G,L,T1,T,T2,d,e.
48 ⦃G, L⦄ ⊢ T1 ▶*[d, e] T → ⦃G, L⦄ ⊢ T ▶[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
49 normalize /2 width=3 by step/ qed-.
51 lemma cpys_strap2: ∀G,L,T1,T,T2,d,e.
52 ⦃G, L⦄ ⊢ T1 ▶[d, e] T → ⦃G, L⦄ ⊢ T ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
53 normalize /2 width=3 by TC_strap/ qed-.
55 lemma lsuby_cpys_trans: ∀G,d,e. lsub_trans … (cpys d e G) (lsuby d e).
56 /3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
59 lemma cpys_refl: ∀G,L,d,e. reflexive … (cpys d e G L).
60 /2 width=1 by cpy_cpys/ qed.
62 lemma cpys_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
63 ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 →
64 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] ⓑ{a,I}V2.T2.
65 #G #L #V1 #V2 #d #e #HV12 @(cpys_ind … HV12) -V2
66 [ #I #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
67 | /3 width=5 by cpys_strap1, cpy_bind/
71 lemma cpys_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 →
72 ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 →
73 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] ⓕ{I}V2.T2.
74 #G #L #V1 #V2 #d #e #HV12 @(cpys_ind … HV12) -V2
75 [ #T1 #T2 #HT12 @(cpys_ind … HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
76 | /3 width=5 by cpys_strap1, cpy_flat/
79 lemma cpys_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶*[d1, e1] T2 →
80 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
81 ⦃G, L⦄ ⊢ T1 ▶*[d2, e2] T2.
82 #G #L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(cpys_ind … H) -T2
83 /3 width=7 by cpys_strap1, cpy_weak/
86 lemma cpys_weak_top: ∀G,L,T1,T2,d,e.
87 ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, |L| - d] T2.
88 #G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2
89 /3 width=4 by cpys_strap1, cpy_weak_top/
92 lemma cpys_weak_full: ∀G,L,T1,T2,d,e.
93 ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[0, |L|] T2.
94 #G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2
95 /3 width=5 by cpys_strap1, cpy_weak_full/
98 lemma cpys_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
99 ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
100 d ≤ dt → d + e ≤ dt + et →
101 ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶*[d+e, dt+et-(d+e)] U2 & ⇧[d, e] T2 ≡ U2.
102 #G #L #U1 #U2 #dt #et #H #T1 #d #e #HTU1 #Hddt #Hdedet @(cpys_ind … H) -U2
103 [ /2 width=3 by ex2_intro/
104 | -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
105 elim (cpy_up … HU2 … HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
109 lemma cpys_append: ∀G,d,e. l_appendable_sn … (cpys d e G).
110 #G #d #e #K #T1 #T2 #H @(cpys_ind … H) -T2
111 /3 width=3 by cpys_strap1, cpy_append/
114 (* Basic inversion lemmas ***************************************************)
116 (* Note: this can be derived from cpys_inv_atom1 *)
117 lemma cpys_inv_sort1: ∀G,L,T2,k,d,e. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] T2 → T2 = ⋆k.
118 #G #L #T2 #k #d #e #H @(cpys_ind … H) -T2 //
119 #T #T2 #_ #HT2 #IHT1 destruct
120 >(cpy_inv_sort1 … HT2) -HT2 //
123 (* Note: this can be derived from cpys_inv_atom1 *)
124 lemma cpys_inv_gref1: ∀G,L,T2,p,d,e. ⦃G, L⦄ ⊢ §p ▶*[d, e] T2 → T2 = §p.
125 #G #L #T2 #p #d #e #H @(cpys_ind … H) -T2 //
126 #T #T2 #_ #HT2 #IHT1 destruct
127 >(cpy_inv_gref1 … HT2) -HT2 //
130 lemma cpys_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] U2 →
131 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 &
132 ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] T2 &
134 #a #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind … H) -U2
135 [ /2 width=5 by ex3_2_intro/
136 | #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
137 elim (cpy_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
138 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V1) ?) -HT2
139 /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/
143 lemma cpys_inv_flat1: ∀I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] U2 →
144 ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] V2 & ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 &
146 #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind … H) -U2
147 [ /2 width=5 by ex3_2_intro/
148 | #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
149 elim (cpy_inv_flat1 … HU2) -HU2
150 /3 width=5 by cpys_strap1, ex3_2_intro/
154 lemma cpys_inv_refl_O2: ∀G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▶*[d, 0] T2 → T1 = T2.
155 #G #L #T1 #T2 #d #H @(cpys_ind … H) -T2 //
156 #T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 … HT2) -HT2 //
159 lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀d,e:nat.
160 ⦃G, L⦄ ⊢ U1 ▶*[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
161 #G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2
162 /2 width=7 by cpy_inv_lift1_eq/
165 (* Basic forward lemmas *****************************************************)
167 lemma cpys_fwd_tw: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → ♯{T1} ≤ ♯{T2}.
168 #G #L #T1 #T2 #d #e #H @(cpys_ind … H) -T2 //
169 #T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw … HT2) -HT2
170 /2 width=3 by transitive_le/
173 lemma cpys_fwd_shift1: ∀G,L,L1,T1,T,d,e. ⦃G, L⦄ ⊢ L1 @@ T1 ▶*[d, e] T →
174 ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
175 #G #L #L1 #T1 #T #d #e #H @(cpys_ind … H) -T
176 [ /2 width=4 by ex2_2_intro/
177 | #T #X #_ #HX * #L0 #T0 #HL10 #H destruct
178 elim (cpy_fwd_shift1 … HX) -HX #L2 #T2 #HL02 #H destruct
179 /2 width=4 by ex2_2_intro/