matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/da_sta.etc

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  13 (**************************************************************************)
  14
  15 include "basic_2/static/sta.ma".
  16 include "basic_2/static/da_da.ma".
  17
  18 (* Properties on static type assignment for terms ***************************)
  19
  20 lemma da_sta_conf: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
  21                    ∀l. ⦃G, L⦄ ⊢ T ▪[h, g] l → ⦃G, L⦄ ⊢ U ▪[h, g] l-1.
  22 #h #g #G #L #T #U #H elim H -G -L -T -U
  23 [ #G #L #k #l #H
  24   lapply (da_inv_sort … H) -H /3 width=1 by da_sort, deg_next/
  25 | #G #L #K #V #U #W #i #HLK #_ #HWU #IHVW #l #H
  26   elim (da_inv_lref … H) -H * #K0 #V0 [| #l0] #HLK0 #HV0
  27   lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
  28   lapply (drop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
  29 | #G #L #K #W #V #U #i #HLK #_ #HWU #IHWV #l #H
  30   elim (da_inv_lref … H) -H * #K0 #V0 [| #l0] #HLK0 #HV0 [| #H0 ]
  31   lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
  32   lapply (drop_fwd_drop2 … HLK) -HLK /3 width=8 by da_lift/
  33 | #a #I #G #L #V #T #U #_ #IHTU #l #H
  34   lapply (da_inv_bind … H) -H /3 width=1 by da_bind/
  35 | #G #L #V #T #U #_ #IHTU #l #H
  36   lapply (da_inv_flat … H) -H /3 width=1 by da_flat/
  37 | #G #L #W #T #U #_ #IHTU #l #H
  38   lapply (da_inv_flat … H) -H /2 width=1 by/
  39 ]
  40 qed-.
  41
  42 lemma sta_da: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U →
  43               ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
  44 #h #g #G #L #T #U #H elim H -G -L -T -U
  45 [ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
  46 | #G #L #K #V #W #W0 #i #HLK #_ #_ * /3 width=5 by da_ldef, ex_intro/
  47 | #G #L #K #W #V #W0 #i #HLK #_ #_ * /3 width=5 by da_ldec, ex_intro/
  48 | #a #I #G #L #V #T #U #_ * /3 width=2 by da_bind, ex_intro/
  49 | #G #L #V #T #U #_ * /3 width=2 by da_flat, ex_intro/
  50 | #G #L #W #T #U #_ * /3 width=2 by da_flat, ex_intro/
  51 ]
  52 qed-.
  53
  54 lemma sta_da_ge: ∀h,G,L,T,U,l0. ⦃G, L⦄ ⊢ T •[h] U →
  55                  ∃∃g,l. ⦃G, L⦄ ⊢ T ▪[h, g] l & l0 ≤ l.
  56 #h #G #L #T #U #l0 #H elim H -G -L -T -U
  57 [ /3 width=4 by da_sort, ex2_2_intro/
  58 | #G #L #K #V #W #W0 #i #HLK #_ #_ * /3 width=5 by da_ldef, ex2_2_intro/
  59 | #G #L #K #W #V #W0 #i #HLK #_ #_ * /4 width=5 by da_ldec, lt_to_le, le_S_S, ex2_2_intro/
  60 | #a #I #G #L #V #T #U #_ * /3 width=4 by da_bind, ex2_2_intro/
  61 | #G #L #V #T #U #_ * /3 width=4 by da_flat, ex2_2_intro/
  62 | #G #L #W #T #U #_ * /3 width=4 by da_flat, ex2_2_intro/
  63 ]
  64 qed-.
  65
  66 (* Inversion lrmmas on static type assignment for terms *********************)
  67
  68 lemma da_inv_sta: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l →
  69                   ∃U. ⦃G, L⦄ ⊢ T •[h] U.
  70 #h #g #G #L #T #l #H elim H -G -L -T -l
  71 [ /2 width=2/
  72 | #G #L #K #V #i #l #HLK #_ * #W #HVW
  73   elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldef, ex_intro/
  74 | #G #L #K #W #i #l #HLK #_ * #V #HWV
  75   elim (lift_total W 0 (i+1)) /3 width=7 by sta_ldec, ex_intro/
  76 | #a #I #G #L #V #T #l #_ * /3 width=2 by sta_bind, ex_intro/
  77 | * #G #L #V #T #l #_ * /3 width=2 by sta_appl, sta_cast, ex_intro/
  78 ]
  79 qed-.
  80
  81 lemma sta_inv_refl_pos: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •[h] T → ⊥.
  82 #h #g #G #L #T #l #H1T #HTT
  83 lapply (da_sta_conf … HTT … H1T) -HTT <minus_plus_m_m #H2T
  84 lapply (da_mono … H2T … H1T) -h -G -L -T #H
  85 elim (plus_xySz_x_false 0 l 0) //
  86 qed-.