matita/matita/contribs/lambdadelta/basic_2A/examples/ex_snv_eta.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2A/dynamic/snv.ma".
  16
  17 (* EXAMPLES *****************************************************************)
  18
  19 (* Extended validy (basic?_2) vs. restricted validity (basic_1) *************)
  20
  21 (* extended validity of a closure, last arg of snv_appl > 1 *)
  22 lemma snv_extended: ∀h,g,a,G,L,k. ⦃G, L.ⓛ⋆k.ⓛⓛ{a}⋆k.⋆k.ⓛ#0⦄ ⊢ ⓐ#2.#0 ¡[h, g].
  23 #h #g #a #G #L #k elim (deg_total h g k)
  24 #d #Hd @(snv_appl … a … (⋆k) … (⋆k) (0+1+1))
  25 [ /4 width=5 by snv_lref, drop_drop_lt/
  26 | /4 width=13 by snv_bind, snv_lref/
  27 | /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
  28 | @(lstas_scpds … (d+1+1))
  29   /5 width=11 by lstas_bind, lstas_succ, da_bind, da_ldec, da_sort, lift_bind/
  30 ]
  31 qed.
  32
  33 (* restricted validity of the η-expanded closure, last arg of snv_appl = 1 **)
  34 lemma snv_restricted: ∀h,g,a,G,L,k. ⦃G, L.ⓛ⋆k.ⓛⓛ{a}⋆k.⋆k.ⓛⓛ{a}⋆k.ⓐ#0.#1⦄ ⊢ ⓐ#2.#0 ¡[h, g].
  35 #h #g #a #G #L #k elim (deg_total h g k)
  36 #d #Hd @(snv_appl … a … (⋆k) … (ⓐ#0.#2) (0+1))
  37 [ /4 width=5 by snv_lref, drop_drop_lt/
  38 | @snv_lref [4: // |1,2,3: skip ]
  39   @snv_bind //
  40   @(snv_appl … a … (⋆k) … (⋆k) (0+1))
  41   [ @snv_lref [4: // |1,2,3: skip ] //
  42   | @snv_lref [4: /2 width=1 by drop_drop_lt/ |1,2,3: skip ] @snv_bind //
  43   | @(lstas_scpds … (d+1)) /3 width=6 by da_sort, da_ldec, lstas_succ/
  44   | @(lstas_scpds … (d+1)) /3 width=8 by lstas_succ, lstas_bind, drop_drop, lift_bind/
  45     @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ] /3 width=1 by da_sort, da_bind/
  46   ]
  47 | /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
  48 | @(lstas_scpds … (d+1+1)) //
  49   [ @da_ldec [3: // |1,2: skip ]
  50     @da_bind @da_flat @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ]
  51     /3 width=1 by da_sort, da_bind/
  52   | @lstas_succ [4: // |1,2: skip ]
  53     [2: @lstas_bind | skip ]
  54     [2: @lstas_appl | skip ]
  55     [2: @lstas_zero
  56         [4: /2 width=1 by drop_drop_lt/ |5: /2 width=2 by lstas_bind/ |*: skip ]
  57     |1: skip ]
  58     /4 width=2 by lift_flat, lift_bind, lift_lref_ge_minus, lift_lref_lt/
  59   ]
  60 ]
  61 qed.