Nuprl Lemma : strat2play-longer

∀[g:SimpleGame]. ∀[n:ℕ]. ∀[s:win2strat(g;n)]. ∀[moves:strat2play(g;n;s)]. ∀[x:sequence(Pos(g))].
  ((x ∈ strat2play(g;n;s)) ∧ (seq-truncate(x;||moves||) = moves ∈ strat2play(g;n;s))) supposing 
     ((seq-truncate(x;||moves||) = moves ∈ sequence(Pos(g))) and 
     (||moves|| ≤ ||x||))


Proof



Definitions occuring in Statement :  strat2play: strat2play(g;n;s),  win2strat: win2strat(g;n),  sg-pos: Pos(g),  simple-game: SimpleGame,  seq-truncate: seq-truncate(s;n),  seq-len: ||s||,  sequence: sequence(T),  nat: ℕ,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  le: A ≤ B,  and: P ∧ Q,  member: t ∈ T,  equal: s = t ∈ T
Definitions unfolded in proof :  label: ...$L... t,  less_than: a < b,  nat_plus: ℕ+,  so_apply: x[s],  so_lambda: λ2x.t[x],  exists: ∃x:A. B[x],  win2strat: win2strat(g;n),  play-truncate: play-truncate(f;m),  play-len: ||moves||,  it: ⋅,  unit: Unit,  bool: 𝔹,  bfalse: ff,  sq_type: SQType(T),  sq_stable: SqStable(P),  squash: ↓T,  play-item: moves[i],  btrue: tt,  ifthenelse: if b then t else f fi ,  eq_int: (i =z j),  strat2play: strat2play(g;n;s),  true: True,  subtract: n - m,  uiff: uiff(P;Q),  rev_implies: P ⇐ Q,  iff: P ⇐⇒ Q,  or: P ∨ Q,  decidable: Dec(P),  not: ¬A,  less_than': less_than'(a;b),  le: A ≤ B,  top: Top,  lelt: i ≤ j < k,  int_seg: {i..j-},  cand: A c∧ B,  all: ∀x:A. B[x],  subtype_rel: A ⊆r B,  and: P ∧ Q,  prop: ℙ,  uimplies: b supposing a,  guard: {T},  ge: i ≥ j ,  false: False,  implies: P ⇒ Q,  nat: ℕ,  member: t ∈ T,  uall: ∀[x:A]. B[x]
Lemmas referenced :  omega-shadow,  minus-zero,  two-mul,  one-mul,  not-equal-implies-less,  le-add-cancel-alt,  mul-commutes,  mul-distributes,  mul-distributes-right,  set_subtype_base,  play-item_wf,  sg-legal2_wf,  seq-truncate-truncate,  le_reflexive,  seq-len-truncate,  play-len_wf,  equal-wf-T-base,  play-truncate_wf,  uiff_transitivity,  assert_of_bnot,  iff_weakening_uiff,  iff_transitivity,  eqff_to_assert,  assert_of_eq_int,  eqtt_to_assert,  bool_subtype_base,  subtype_base_sq,  bool_cases,  sq_stable__le,  multiply_nat_wf,  add_nat_wf,  mul-associates,  mul_bounds_1a,  add-is-int-iff,  zero-mul,  add-mul-special,  not-le-2,  win2strat_subtype,  bool_wf,  int_subtype_base,  equal-wf-base,  not_wf,  bnot_wf,  assert_wf,  le_weakening,  eq_int_wf,  le-add-cancel2,  sg-legal1_wf,  iff_weakening_equal,  subtype_rel_self,  not-lt-2,  decidable__lt,  true_wf,  squash_wf,  le_transitivity,  simple-game_wf,  le_weakening2,  le-add-cancel,  add-zero,  add_functionality_wrt_le,  add-commutes,  add-swap,  add-associates,  minus-minus,  minus-add,  minus-one-mul-top,  zero-add,  minus-one-mul,  condition-implies-le,  less-iff-le,  not-ge-2,  subtract_wf,  decidable__le,  false_wf,  win2strat_wf,  strat2play_wf,  le_wf,  seq-truncate_wf,  int_seg_wf,  seq-truncate-item,  lelt_wf,  nat_wf,  seq-len_wf,  sg-pos_wf,  sequence_wf,  and_wf,  seq-item_wf,  equal_wf,  strat2play_subtype,  less_than_wf,  ge_wf,  less_than_irreflexivity,  less_than_transitivity1,  nat_properties
Rules used in proof :  sqequalIntensionalEquality,  dependent_pairFormation,  equalityElimination,  impliesFunctionality,  cumulativity,  dependentIntersection_memberEquality,  closedConclusion,  baseApply,  dependentIntersectionElimination,  promote_hyp,  instantiate,  baseClosed,  imageMemberEquality,  universeEquality,  imageElimination,  minusEquality,  intEquality,  unionElimination,  multiplyEquality,  addEquality,  setEquality,  independent_pairEquality,  voidEquality,  levelHypothesis,  hyp_replacement,  addLevel,  productElimination,  applyLambdaEquality,  productEquality,  independent_pairFormation,  equalitySymmetry,  equalityTransitivity,  dependent_set_memberEquality,  applyEquality,  because_Cache,  isect_memberEquality,  axiomEquality,  dependent_functionElimination,  lambdaEquality,  voidElimination,  independent_functionElimination,  independent_isectElimination,  natural_numberEquality,  lambdaFormation,  intWeakElimination,  sqequalRule,  rename,  setElimination,  hypothesis,  hypothesisEquality,  thin,  isectElimination,  sqequalHypSubstitution,  extract_by_obid,  cut,  introduction,  isect_memberFormation,  sqequalReflexivity,  computationStep,  sqequalTransitivity,  sqequalSubstitution
Latex:
\mforall{}[g:SimpleGame].  \mforall{}[n:\mBbbN{}].  \mforall{}[s:win2strat(g;n)].  \mforall{}[moves:strat2play(g;n;s)].  \mforall{}[x:sequence(Pos(g))].
    ((x  \mmember{}  strat2play(g;n;s))  \mwedge{}  (seq-truncate(x;||moves||)  =  moves))  supposing 
          ((seq-truncate(x;||moves||)  =  moves)  and 
          (||moves||  \mleq{}  ||x||))


Date html generated: 2018_07_25-PM-01_33_23
Last ObjectModification: 2018_06_25-AM-10_43_52

Theory : co-recursion


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