matita/matita/contribs/lambdadelta/basic_2/static/lsubf_frees.ma

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  14
  15 include "basic_2/static/lsubf.ma".
  16
  17 axiom lsubf_inv_sor_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
  18                         ∀x2,y2. x2⋓y2 ≡ f2 →
  19                         ∃∃x1,y1. ⦃L1, x1⦄ ⫃𝐅* ⦃L2, x2⦄ & ⦃L1, y1⦄ ⫃𝐅* ⦃L2, y2⦄ & x1⋓y1 ≡ f1.
  20
  21 (* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
  22
  23 (* Properties with context-sensitive free variables *************************)
  24
  25 lemma lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 →
  26                          ∀f1,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 ⊢ 𝐅*⦃T⦄ ≡ f1.
  27 #f2 #L2 #T #H elim H -f2 -L2 -T
  28 [ /3 width=5 by lsubf_fwd_isid_dx, frees_sort/
  29 | #f2 #i #Hf2 #g1 #Y1 #H
  30   elim (lsubf_inv_atom2 … H) -H #Hg1 #H destruct
  31   elim (eq_inv_pushs_dx … Hg1) -Hg1 #g #Hg #H destruct
  32   elim (eq_inv_xn … Hg) -Hg
  33   /3 width=3 by frees_atom, isid_eq_repl_fwd/
  34 | #f2 #I #K2 #W #_ #IH #g1 #Y1 #H elim (lsubf_inv_pair2 … H) -H *
  35   [ #f1 #K1 #H12 #H1 #H2 destruct /3 width=1 by frees_pair/
  36   | #f #f0 #f1 #K1 #V #H12 #Hf #Hf1 #H1 #H2 #H3 destruct
  37     /4 width=5 by frees_pair, frees_flat/
  38   ]
  39 | #f2 #I #L2 #Hf2 #g1 #Y1 #H elim (lsubf_inv_unit2 … H) -H *
  40   [ #f1 #L1 #H12 #H1 #H2 destruct
  41     /3 width=5 by lsubf_fwd_isid_dx, frees_unit/
  42   | #f #f0 #f1 #J #L1 #V #H12 #Hf #Hf1 #H1 #H2 destruct
  43     /5 width=9 by lsubf_fwd_isid_dx, frees_eq_repl_back, frees_pair, sor_isid_inv_sn/
  44   ]
  45 | #f2 #I #L2 #i #_ #IH #g1 #L1 #H elim (lsubf_inv_push2 … H) -H
  46   /3 width=1 by frees_lref/
  47 | /3 width=5 by lsubf_fwd_isid_dx, frees_gref/
  48 | #f2V #f2T #f2 #p #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f1 #L1 #H12
  49   elim (lsubf_inv_sor_dx … H12 … Hf2) -f2 #f1V #g1T #HV #HT #Hf1
  50   elim (lsubf_tl_dx … (BPair I V) … HT) -HT #f1T #HT #H destruct
  51   /3 width=5 by frees_bind/
  52 | #f2V #f2T #f2 #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f1 #L1 #H12
  53   elim (lsubf_inv_sor_dx … H12 … Hf2) -f2 /3 width=5 by frees_flat/
  54 ]
  55 qed-.