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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/unfold/tpss_lift.ma".
16 include "basic_2/reducibility/tpr_lift.ma".
17 include "basic_2/reducibility/cpr.ma".
19 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
21 (* Advanced properties ******************************************************)
23 lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
24 ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 ▶* [0, |L| - i - 1] W1 →
25 ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
26 #L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
27 lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
28 @ex2_1_intro [2: // | skip | @tpss_subst /width=6/ ] (**) (* /3 width=6/ is too slow *)
31 lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2.
32 L.ⓛV ⊢ T1 ➡ T2 → L ⊢ ⓛV1. T1 ➡ ⓛV2. T2.
33 #L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02
34 lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
35 @(ex2_1_intro … (ⓛV0.T0)) /2 width=1/ -V1 -T1 (**) (* explicit constructors *)
37 @(tpss_lsubs_trans (L.ⓛV)) // -T0 -T2 /2 width=1/
40 (* Advanced inversion lemmas ************************************************)
42 (* Basic_1: was: pr2_gen_lref *)
43 lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
45 ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
46 K ⊢ V1 ▶* [0, |L| - i - 1] T1 &
50 >(tpr_inv_atom1 … H) -H #H
51 elim (tpss_inv_lref1 … H) -H /2 width=1/
55 (* Basic_1: was: pr2_gen_abst *)
56 lemma cpr_inv_abst1: ∀L,V1,T1,U2. L ⊢ ⓛV1. T1 ➡ U2 → ∀I,W.
57 ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛV2. T2.
58 #L #V1 #T1 #Y * #X #H1 #H2 #I #W
59 elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
60 elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
61 lapply (tpss_lsubs_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
64 (* Basic_1: was pr2_gen_appl *)
65 lemma cpr_inv_appl1: ∀L,V1,U0,U2. L ⊢ ⓐV1. U0 ➡ U2 →
66 ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U0 ➡ T2 &
68 | ∃∃V2,W,T1,T2. L ⊢ V1 ➡ V2 & L. ⓓV2 ⊢ T1 ➡ T2 &
71 | ∃∃V2,V,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 & L. ⓓW2 ⊢ T1 ➡ T2 &
75 #L #V1 #U0 #Y * #X #H1 #H2
76 elim (tpr_inv_appl1 … H1) -H1 *
77 [ #V #U #HV1 #HU0 #H destruct
78 elim (tpss_inv_flat1 … H2) -H2 #V2 #U2 #HV2 #HU2 #H destruct /4 width=5/
79 | #V #W #T0 #T #HV1 #HT0 #H #H1 destruct
80 elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
81 lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=8/
82 | #V0 #V #W #W0 #T #T0 #HV10 #HW0 #HT0 #HV0 #H #H1 destruct
83 elim (tpss_inv_bind1 … H2) -H2 #W2 #X #HW02 #HX #HY destruct
84 elim (tpss_inv_flat1 … HX) -HX #V2 #T2 #HV2 #HT2 #H destruct
85 elim (tpss_inv_lift1_ge … HV2 … HV0 ?) -V // [3: /2 width=1/ |2: skip ] #V <minus_plus_m_m
86 lapply (tpss_weak … HT2 0 (|L|+1) ? ?) -HT2 // /4 width=12/
90 (* Note: the main property of simple terms *)
91 lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
92 ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
95 elim (cpr_inv_appl1 … H) -H *
97 | #V2 #W #W1 #W2 #_ #_ #H #_ destruct
98 elim (simple_inv_bind … HT1)
99 | #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
100 elim (simple_inv_bind … HT1)
104 (* Relocation properties ****************************************************)
106 (* Basic_1: was: pr2_lift *)
107 lemma cpr_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
108 ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
109 K ⊢ T1 ➡ T2 → L ⊢ U1 ➡ U2.
110 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 * #T #HT1 #HT2
111 elim (lift_total T d e) #U #HTU
112 lapply (tpr_lift … HT1 … HTU1 … HTU) -T1 #HU1
113 elim (lt_or_ge (|K|) d) #HKd
114 [ lapply (tpss_lift_le … HT2 … HLK HTU … HTU2) -T2 -T -HLK [ /2 width=2/ | /3 width=4/ ]
115 | lapply (tpss_lift_be … HT2 … HLK HTU … HTU2) -T2 -T -HLK // /3 width=4/
119 (* Basic_1: was: pr2_gen_lift *)
120 lemma cpr_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
121 ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
122 ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
123 #L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
124 elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
125 elim (lt_or_ge (|L|) d) #HLd
126 [ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK [ /5 width=4/ | /2 width=2/ ]
127 | elim (lt_or_ge (|L|) (d + e)) #HLde
128 [ elim (tpss_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -U -HLK // [ /5 width=4/ | /2 width=2/ ]
129 | elim (tpss_inv_lift1_be … HU2 … HLK … HTU ? ?) -U -HLK // /5 width=4/