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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/relocation/cpy_lift.ma".
16 include "basic_2/substitution/cpys.ma".
18 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
20 (* Advanced properties ******************************************************)
22 lemma cpys_subst: ∀I,G,L,K,V,U1,i,d,e.
24 ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, d+e-i-1] U1 →
25 ∀U2. ⇧[0, i + 1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2.
26 #I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind … H) -U1
27 [ /3 width=5 by cpy_cpys, cpy_subst/
28 | #U #U1 #_ #HU1 #IHU #U2 #HU12
29 elim (lift_total U 0 (i+1)) #U0 #HU0
30 lapply (IHU … HU0) -IHU #H
31 lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
32 lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
33 lapply (cpy_weak … HU02 d e ? ?) -HU02 [2,3: /2 width=3 by cpys_strap1, le_S/ ]
34 >minus_plus >commutative_plus /2 width=1 by le_minus_to_plus_r/
38 (* Advanced inverion lemmas *************************************************)
40 lemma cpys_inv_atom1: ∀G,L,T2,I,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
42 ∃∃J,K,V1,V2,i. d ≤ i & i < d + e &
43 ⇩[O, i] L ≡ K.ⓑ{J}V1 &
44 ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
47 #G #L #T2 #I #d #e #H @(cpys_ind … H) -T2
48 [ /2 width=1 by or_introl/
51 elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
52 | * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
53 lapply (ldrop_fwd_ldrop2 … HLK) #H
54 elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) normalize -HT2 -H -HVT [2,3,4: /2 width=1 by le_S/ ]
55 <minus_plus /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
60 lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*×[d, e] T2 →
62 ∃∃I,K,V1,V2. d ≤ i & i < d + e &
63 ⇩[O, i] L ≡ K.ⓑ{I}V1 &
64 ⦃G, K⦄ ⊢ V1 ▶*×[0, d + e - i - 1] V2 &
66 #G #L #T2 #i #d #e #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
67 * #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
70 (* Relocation properties ****************************************************)
72 lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
73 ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
74 ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
75 ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2.
76 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2
77 [ #U2 #H >(lift_mono … HTU1 … H) -H //
78 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
79 elim (lift_total T d e) #U #HTU
80 lapply (IHT … HTU) -IHT #HU1
81 lapply (cpy_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
85 lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
86 ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
87 ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
88 ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*×[dt, et + e] U2.
89 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2
90 [ #U2 #H >(lift_mono … HTU1 … H) -H //
91 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
92 elim (lift_total T d e) #U #HTU
93 lapply (IHT … HTU) -IHT #HU1
94 lapply (cpy_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
98 lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
99 ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
100 ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
101 ⦃G, L⦄ ⊢ U1 ▶*×[dt + e, et] U2.
102 #G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2
103 [ #U2 #H >(lift_mono … HTU1 … H) -H //
104 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
105 elim (lift_total T d e) #U #HTU
106 lapply (IHT … HTU) -IHT #HU1
107 lapply (cpy_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
111 lemma cpys_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
112 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
114 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
115 #G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(cpys_ind … H) -U2
116 [ /2 width=3 by ex2_intro/
117 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
118 elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
122 lemma cpys_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
123 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
124 dt ≤ d → d + e ≤ dt + et →
125 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
126 #G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(cpys_ind … H) -U2
127 [ /2 width=3 by ex2_intro/
128 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
129 elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
133 lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
134 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
136 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
137 #G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2
138 [ /2 width=3 by ex2_intro/
139 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
140 elim (cpy_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
144 lemma cpys_inv_lift1_eq: ∀G,L,U1,U2,d,e.
145 ⦃G, L⦄ ⊢ U1 ▶*×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
146 #G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2 //
147 #U #U2 #_ #HU2 #IHU destruct
148 <(cpy_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
151 lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
152 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
153 d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
154 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[d, dt + et - (d + e)] T2 &
156 #G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(cpys_ind … H) -U2
157 [ /2 width=3 by ex2_intro/
158 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
159 elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
163 lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
164 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
165 dt ≤ d → dt + et ≤ d + e →
166 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
167 #G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(cpys_ind … H) -U2
168 [ /2 width=3 by ex2_intro/
169 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
170 elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
174 lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
175 ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
176 dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
177 ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
178 #G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(cpys_ind … H) -U2
179 [ /2 width=3 by ex2_intro/
180 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
181 elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/