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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/substitution/ltps.ma".
16 include "basic_2/unfold/tpss.ma".
18 (* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
20 definition ltpss: nat → nat → relation lenv ≝
21 λd,e. TC … (ltps d e).
23 interpretation "partial unfold (local environment)"
24 'PSubstStar L1 d e L2 = (ltpss d e L1 L2).
26 (* Basic eliminators ********************************************************)
28 lemma ltpss_ind: ∀d,e,L1. ∀R:predicate lenv. R L1 →
29 (∀L,L2. L1 [d, e] ▶* L → L [d, e] ▶ L2 → R L → R L2) →
30 ∀L2. L1 [d, e] ▶* L2 → R L2.
31 #d #e #L1 #R #HL1 #IHL1 #L2 #HL12 @(TC_star_ind … HL1 IHL1 … HL12) //
34 lemma ltpss_ind_dx: ∀d,e,L2. ∀R:predicate lenv. R L2 →
35 (∀L1,L. L1 [d, e] ▶ L → L [d, e] ▶* L2 → R L → R L1) →
36 ∀L1. L1 [d, e] ▶* L2 → R L1.
37 #d #e #L2 #R #HL2 #IHL2 #L1 #HL12 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
40 (* Basic properties *********************************************************)
42 lemma ltpss_strap: ∀L1,L,L2,d,e.
43 L1 [d, e] ▶ L → L [d, e] ▶* L2 → L1 [d, e] ▶* L2.
46 lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L.
49 lemma ltpss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → L1 [0, |L2|] ▶* L2.
50 #L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
52 >(ltps_fwd_length … HL2) /3 width=5/
55 (* Basic forward lemmas *****************************************************)
57 lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → |L1| = |L2|.
58 #L1 #L2 #d #e #H @(ltpss_ind … H) -L2 //
59 #L #L2 #_ #HL2 #IHL12 >IHL12 -IHL12
60 /2 width=3 by ltps_fwd_length/
63 (* Basic inversion lemmas ***************************************************)
65 lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶* L2 → L1 = L2.
66 #d #L1 #L2 #H @(ltpss_ind … H) -L2 //
67 #L #L2 #_ #HL2 #IHL <(ltps_inv_refl_O2 … HL2) -HL2 //
70 lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶* L2 → L2 = ⋆.
71 #d #e #L2 #H @(ltpss_ind … H) -L2 //
72 #L #L2 #_ #HL2 #IHL destruct
73 >(ltps_inv_atom1 … HL2) -HL2 //
76 fact ltpss_inv_atom2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆.
77 #d #e #L1 #L2 #H @(ltpss_ind … H) -L2 //
78 #L2 #L #_ #HL2 #IHL2 #H destruct
79 lapply (ltps_inv_atom2 … HL2) -HL2 /2 width=1/
82 lemma ltpss_inv_atom2: ∀d,e,L1. L1 [d, e] ▶* ⋆ → L1 = ⋆.
85 fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → d = 0 → 0 < e →
86 ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
87 ∃∃K1,V1. K1 [0, e - 1] ▶ K2 &
88 K2 ⊢ V1 [0, e - 1] ▶ V2 &
90 #d #e #L1 #L2 * -d e L1 L2
91 [ #d #e #_ #_ #K1 #I #V1 #H destruct
92 | #L1 #I #V #_ #H elim (lt_refl_false … H)
93 | #L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #_ #_ #K2 #J #W2 #H destruct /2 width=5/
94 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
98 lemma ltps_inv_tps22: ∀e,L1,K2,I,V2. L1 [0, e] ▶ K2. ⓑ{I} V2 → 0 < e →
99 ∃∃K1,V1. K1 [0, e - 1] ▶ K2 & K2 ⊢ V1 [0, e - 1] ▶ V2 &
103 fact ltps_inv_tps12_aux: ∀d,e,L1,L2. L1 [d, e] ▶ L2 → 0 < d →
104 ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
105 ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
106 K2 ⊢ V1 [d - 1, e] ▶ V2 &
108 #d #e #L1 #L2 * -d e L1 L2
109 [ #d #e #_ #I #K2 #V2 #H destruct
110 | #L #I #V #H elim (lt_refl_false … H)
111 | #L1 #L2 #I #V1 #V2 #e #_ #_ #H elim (lt_refl_false … H)
112 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #_ #J #K2 #W2 #H destruct /2 width=5/
116 lemma ltps_inv_tps12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶ K2. ⓑ{I} V2 → 0 < d →
117 ∃∃K1,V1. K1 [d - 1, e] ▶ K2 &
118 K2 ⊢ V1 [d - 1, e] ▶ V2 &