1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "Basic_2/substitution/tps.ma".
17 (* PARTIAL UNFOLD ON TERMS **************************************************)
19 definition tpss: nat → nat → lenv → relation term ≝
20 λd,e,L. TC … (tps d e L).
22 interpretation "partial unfold (term)"
23 'PSubstStar L T1 d e T2 = (tpss d e L T1 T2).
25 (* Basic eliminators ********************************************************)
27 lemma tpss_ind: ∀d,e,L,T1. ∀R: term → Prop. R T1 →
28 (∀T,T2. L ⊢ T1 [d, e] ≫* T → L ⊢ T [d, e] ≫ T2 → R T → R T2) →
29 ∀T2. L ⊢ T1 [d, e] ≫* T2 → R T2.
30 #d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
33 (* Basic properties *********************************************************)
35 lemma tpss_strap: ∀L,T1,T,T2,d,e.
36 L ⊢ T1 [d, e] ≫ T → L ⊢ T [d, e] ≫* T2 → L ⊢ T1 [d, e] ≫* T2.
39 lemma tpss_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫* T2 →
40 ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ≫* T2.
43 lemma tpss_refl: ∀d,e,L,T. L ⊢ T [d, e] ≫* T.
46 lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 [d, e] ≫* V2 →
47 ∀I,T1,T2. L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫* T2 →
48 L ⊢ 𝕓{I} V1. T1 [d, e] ≫* 𝕓{I} V2. T2.
49 #L #V1 #V2 #d #e #HV12 elim HV12 -HV12 V2
50 [ #V2 #HV12 #I #T1 #T2 #HT12 elim HT12 -HT12 T2
52 | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
54 | #V #V2 #_ #HV12 #IHV #I #T1 #T2 #HT12
55 lapply (tpss_lsubs_conf … HT12 (L. 𝕓{I} V) ?) -HT12 /2/ #HT12
56 lapply (IHV … HT12) -IHV HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
60 lemma tpss_flat: ∀L,I,V1,V2,T1,T2,d,e.
61 L ⊢ V1 [d, e] ≫ * V2 → L ⊢ T1 [d, e] ≫* T2 →
62 L ⊢ 𝕗{I} V1. T1 [d, e] ≫* 𝕗{I} V2. T2.
63 #L #I #V1 #V2 #T1 #T2 #d #e #HV12 elim HV12 -HV12 V2
64 [ #V2 #HV12 #HT12 elim HT12 -HT12 T2
66 | #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
68 | #V #V2 #_ #HV12 #IHV #HT12
69 lapply (IHV … HT12) -IHV HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
73 lemma tpss_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫* T2 →
74 ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
75 L ⊢ T1 [d2, e2] ≫* T2.
76 #L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(tpss_ind … H) -H T2
78 | #T #T2 #_ #HT12 #IHT
79 lapply (tps_weak … HT12 … Hd21 Hde12) -HT12 Hd21 Hde12 /2/
83 lemma tpss_weak_top: ∀L,T1,T2,d,e.
84 L ⊢ T1 [d, e] ≫* T2 → L ⊢ T1 [d, |L| - d] ≫* T2.
85 #L #T1 #T2 #d #e #H @(tpss_ind … H) -H T2
87 | #T #T2 #_ #HT12 #IHT
88 lapply (tps_weak_top … HT12) -HT12 /2/
92 lemma tpss_weak_all: ∀L,T1,T2,d,e.
93 L ⊢ T1 [d, e] ≫* T2 → L ⊢ T1 [0, |L|] ≫* T2.
94 #L #T1 #T2 #d #e #HT12
95 lapply (tpss_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
96 lapply (tpss_weak_top … HT12) //
99 (* Basic inversion lemmas ***************************************************)
101 (* Note: this can be derived from tpss_inv_atom1 *)
102 lemma tpss_inv_sort1: ∀L,T2,k,d,e. L ⊢ ⋆k [d, e] ≫* T2 → T2 = ⋆k.
103 #L #T2 #k #d #e #H @(tpss_ind … H) -H T2
105 | #T #T2 #_ #HT2 #IHT destruct -T
106 >(tps_inv_sort1 … HT2) -HT2 //
110 lemma tpss_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫* U2 →
111 ∃∃V2,T2. L ⊢ V1 [d, e] ≫* V2 &
112 L. 𝕓{I} V2 ⊢ T1 [d + 1, e] ≫* T2 &
114 #d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -H U2
116 | #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct -U;
117 elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
118 lapply (tpss_lsubs_conf … HT1 (L. 𝕓{I} V2) ?) -HT1 /3 width=5/
122 lemma tpss_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫* U2 →
123 ∃∃V2,T2. L ⊢ V1 [d, e] ≫* V2 & L ⊢ T1 [d, e] ≫* T2 &
125 #d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -H U2
127 | #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct -U;
128 elim (tps_inv_flat1 … HU2) -HU2 /3 width=5/
132 lemma tpss_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≫* T2 → T1 = T2.
133 #L #T1 #T2 #d #H @(tpss_ind … H) -H T2
135 | #T #T2 #_ #HT2 #IHT <(tps_inv_refl_O2 … HT2) -HT2 //