matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma

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  14
  15 include "basic_2/notation/relations/suptermstar_6.ma".
  16 include "basic_2/s_transition/fquq.ma".
  17
  18 (* STAR-ITERATED SUPCLOSURE *************************************************)
  19
  20 definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq.
  21
  22 interpretation "star-iterated structural successor (closure)"
  23    'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2).
  24
  25 (* Basic eliminators ********************************************************)
  26
  27 lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
  28                 (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
  29                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2.
  30 #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
  31 @(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
  32 qed-.
  33
  34 lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
  35                    (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
  36                    ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1.
  37 #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
  38 @(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
  39 qed-.
  40
  41 (* Basic properties *********************************************************)
  42
  43 lemma fqus_refl: tri_reflexive … fqus.
  44 /2 width=1 by tri_inj/ qed.
  45
  46 lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
  47 /2 width=1 by tri_inj/ qed.
  48
  49 lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
  50                    ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
  51 /2 width=5 by tri_step/ qed-.
  52
  53 lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
  54                    ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
  55 /2 width=5 by tri_TC_strap/ qed-.
  56
  57 (* Basic inversion lemmas ***************************************************)
  58
  59 lemma fqus_inv_fqu_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
  60                        (∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨
  61                        ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
  62 #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/
  63 #G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/
  64 * #HG #HL #HT #_ destruct //
  65 qed-.
  66
  67 lemma fqus_inv_atom1: ∀I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐* ⦃G2, L2, T2⦄ →
  68                       ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2.
  69 #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/
  70 #G #L #T #H elim (fqu_inv_atom1 … H)
  71 qed-.
  72
  73 lemma fqus_inv_sort1: ∀I,G1,G2,L1,L2,V1,T2,s. ⦃G1, L1.ⓑ{I}V1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄ →
  74                       (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄.
  75 #I #G1 #G2 #L1 #L2 #V #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
  76 #G #L #T #H elim (fqu_inv_sort1 … H) -H
  77 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
  78 qed-.
  79
  80 lemma fqus_inv_zero1: ∀I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐* ⦃G2, L2, T2⦄ →
  81                       (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄.
  82 #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
  83 #G #L #T #H elim (fqu_inv_zero1 … H) -H
  84 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
  85 qed-.
  86
  87 lemma fqus_inv_lref1: ∀I,G1,G2,L1,L2,V1,T2,i. ⦃G1, L1.ⓑ{I}V1, #⫯i⦄ ⊐* ⦃G2, L2, T2⦄ →
  88                       (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #(⫯i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐* ⦃G2, L2, T2⦄.
  89 #I #G1 #G2 #L1 #L2 #V #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
  90 #G #L #T #H elim (fqu_inv_lref1 … H) -H
  91 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
  92 qed-.
  93
  94 lemma fqus_inv_gref1: ∀I,G1,G2,L1,L2,V1,T2,l. ⦃G1, L1.ⓑ{I}V1, §l⦄ ⊐* ⦃G2, L2, T2⦄ →
  95                       (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐* ⦃G2, L2, T2⦄.
  96 #I #G1 #G2 #L1 #L2 #V #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
  97 #G #L #T #H elim (fqu_inv_gref1 … H) -H
  98 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
  99 qed-.
 100
 101 lemma fqus_inv_bind1: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
 102                       ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
 103                        | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
 104                        | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
 105 #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
 106 #G #L #T #H elim (fqu_inv_bind1 … H) -H *
 107 #H1 #H2 #H3 #H destruct /2 width=1 by or3_intro1, or3_intro2/
 108 qed-.
 109
 110 lemma fqus_inv_flat1: ∀I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
 111                       ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2
 112                        | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
 113                        | ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
 114 #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
 115 #G #L #T #H elim (fqu_inv_flat1 … H) -H *
 116 #H1 #H2 #H3 #H destruct /2 width=1 by or3_intro1, or3_intro2/
 117 qed-.
 118
 119 (* Basic_2A1: removed theorems 1: fqus_drop *)