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

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  14
  15 include "basic_2/syntax/lveq_lveq.ma".
  16 include "basic_2/static/fsle_fqup.ma".
  17
  18 (* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
  19
  20 (* Advanced inversion lemmas ************************************************)
  21
  22 lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
  23                         ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
  24                         ∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 &
  25                                     L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
  26 #L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
  27 lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
  28 lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
  29 lapply (sle_eq_repl_back2 … Hn … Hgf2) -g2
  30 /2 width=6 by ex3_3_intro/
  31 qed-.
  32
  33 lemma fsle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
  34                            ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
  35                            ∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & f1 ⊆ f2.
  36 #L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
  37 elim (fsle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
  38 elim (lveq_inj_length … H2L) // -L2 #H1 #H2 destruct
  39 /2 width=3 by ex2_intro/
  40 qed-.
  41
  42 lemma fsle_inv_frees_eq: ∀L1,L2. |L1| = |L2| →
  43                          ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
  44                          ∀f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
  45                          f1 ⊆ f2.
  46 #L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1 #f2 #Hf2
  47 elim (fsle_frees_trans_eq … H2L … Hf2) // -L2 -T2
  48 /3 width=6 by frees_mono, sle_eq_repl_back1/
  49 qed-.
  50
  51 (* Main properties **********************************************************)
  52
  53 theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
  54                        ∀T2. ⦃L2, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
  55 #L1 #L2 #T1 #T
  56 * #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
  57 #T2
  58 * #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
  59 lapply (frees_mono … Hf0 … Hg0) -Hf0 -Hg0 #Hfg0
  60 elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
  61 lapply (sle_eq_repl_back1 … Hf … Hfg0) -f0
  62 /4 width=10 by sle_tls, sle_trans, ex4_4_intro/
  63 qed-.
  64
  65 theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1, T1⦄ ⊆ ⦃L1, T⦄ →
  66                        ∀L2,T2. ⦃L1, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
  67 #L1 #T1 #T
  68 * #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
  69 #L2 #T2
  70 * #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
  71 lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
  72 elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
  73 lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
  74 /4 width=10 by sle_tls, sle_trans, ex4_4_intro/
  75 qed-.
  76
  77 theorem fsle_bind_sn_ge: ∀L1,L2. |L2| ≤ |L1| →
  78                          ∀V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
  79                          ∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
  80 #L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
  81 elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
  82 elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
  83 elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
  84 <tls_xn in H2n2; #H2n2
  85 /4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
  86 qed.
  87
  88 theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
  89                       ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⊆ ⦃L2, T⦄.
  90 #L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
  91 elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
  92 elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
  93 elim (sor_isfin_ex f1 f2) /2 width=3 by frees_fwd_isfin/ #f #Hf #_
  94 /4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
  95 qed.
  96
  97 theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
  98                       ∀I2,T1,T2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
  99                       ∀p,I1. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
 100 #L1 #L2 #HL #V1 #V2
 101 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I2 #T1 #T2
 102 * #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
 103 elim (lveq_inj_length … H1L) // #H1 #H2 destruct
 104 elim (lveq_inj_length … H2L) // -HL -H2L #H1 #H2 destruct
 105 elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
 106 elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
 107 /4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
 108 qed.
 109
 110 theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
 111                    ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
 112                    ∀p. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
 113 #L1 #L2 #V1 #V2
 114 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I1 #I2 #T1 #T2
 115 * #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
 116 elim (lveq_inv_pair_pair … H2L) -H2L #H2L #H1 #H2 destruct
 117 elim (lveq_inj … H2L … H1L) -H1L #H1 #H2 destruct
 118 elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
 119 elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
 120 /4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
 121 qed.
 122
 123 theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
 124                    ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
 125                    ∀I1,I2. ⦃L1, ⓕ{I1}V1.T1⦄ ⊆ ⦃L2, ⓕ{I2}V2.T2⦄.
 126 /3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn/ qed-.