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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/notation/relations/supterm_6.ma".
16 include "basic_2/grammar/cl_weight.ma".
17 include "basic_2/relocation/ldrop.ma".
19 (* SUPCLOSURE ***************************************************************)
22 inductive fsup: tri_relation genv lenv term ≝
23 | fsup_lref_O : ∀I,G,L,V. fsup G (L.ⓑ{I}V) (#0) G L V
24 | fsup_pair_sn : ∀I,G,L,V,T. fsup G L (②{I}V.T) G L V
25 | fsup_bind_dx : ∀a,I,G,L,V,T. fsup G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
26 | fsup_flat_dx : ∀I,G,L,V,T. fsup G L (ⓕ{I}V.T) G L T
27 | fsup_ldrop_lt: ∀G,L,K,T,U,d,e.
28 ⇩[d, e] L ≡ K → ⇧[d, e] T ≡ U → 0 < e → fsup G L U G K T
29 | fsup_ldrop : ∀G1,G2,L1,K1,K2,T1,T2,U1,d,e.
30 ⇩[d, e] L1 ≡ K1 → ⇧[d, e] T1 ≡ U1 →
31 fsup G1 K1 T1 G2 K2 T2 → fsup G1 L1 U1 G2 K2 T2
35 "structural successor (closure)"
36 'SupTerm G1 L1 T1 G2 L2 T2 = (fsup G1 L1 T1 G2 L2 T2).
38 (* Basic properties *********************************************************)
40 lemma fsup_lref_S_lt: ∀I,G1,G2,L,K,V,T,i. 0 < i → ⦃G1, L, #(i-1)⦄ ⊃ ⦃G2, K, T⦄ → ⦃G1, L.ⓑ{I}V, #i⦄ ⊃ ⦃G2, K, T⦄.
41 #I #G1 #G2 #L #K #V #T #i #Hi #H /3 width=7 by fsup_ldrop, ldrop_ldrop, lift_lref_ge_minus/ (**) (* auto too slow without trace *)
44 lemma fsup_lref: ∀I,G,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊃ ⦃G, K, V⦄.
45 #I #G #K #V #i @(nat_elim1 i) -i #i #IH #L #H
46 elim (ldrop_inv_O1_pair2 … H) -H *
48 | #I1 #K1 #V1 #HK1 #H #Hi destruct
49 lapply (IH … HK1) /2 width=1/
53 (* Basic forward lemmas *****************************************************)
55 lemma fsup_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
56 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
57 [ #G #L #K #T #U #d #e #HLK #HTU #HKL
58 lapply (ldrop_fwd_lw_lt … HLK HKL) -HKL -HLK #HKL
59 lapply (lift_fwd_tw … HTU) -d -e #H
60 normalize in ⊢ (?%%); /2 width=1/
61 | #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12
62 lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1
63 lapply (lift_fwd_tw … HTU1) -HTU1 #HTU1
64 @(lt_to_le_to_lt … IHT12) -IHT12 /2 width=1/
68 fact fsup_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
69 ∀i. T1 = #i → |L2| < |L1|.
70 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
73 |5: /2 width=4 by ldrop_fwd_length_lt4/
74 |6: #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #i #H destruct
75 lapply (ldrop_fwd_length_le4 … HLK1) -HLK1 #HLK1
76 elim (lift_inv_lref2 … HTU1) -HTU1 * #Hdei #H destruct
77 @(lt_to_le_to_lt … HLK1) /2 width=2/
78 ] #I #G #L #V #T #j #H destruct
81 lemma fsup_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊃ ⦃G2, L2, T2⦄ → |L2| < |L1|.
82 /2 width=7 by fsup_fwd_length_lref1_aux/