Nuprl Lemma : good-sg-win2

∀g:SimpleGame
  ((∃Good:Pos(g) ⟶ ℙ'
     (Good[InitialPos(g)] ∧ (∀p:Pos(g). ∀q:{q:Pos(g)| Legal1(p;q)} . ∀gd:Good[p].  ∃r:{r:Pos(g)| Legal2(q;r)} . Good[r])\000C))
  ⇒ win2(g))


Proof



Definitions occuring in Statement :  win2: win2(g),  sg-legal2: Legal2(x;y),  sg-legal1: Legal1(x;y),  sg-init: InitialPos(g),  sg-pos: Pos(g),  simple-game: SimpleGame,  prop: ℙ,  so_apply: x[s],  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  implies: P ⇒ Q,  and: P ∧ Q,  set: {x:A| B[x]} ,  function: x:A ⟶ B[x]
Definitions unfolded in proof :  all: ∀x:A. B[x],  implies: P ⇒ Q,  exists: ∃x:A. B[x],  and: P ∧ Q,  win2: win2(g),  uall: ∀[x:A]. B[x],  member: t ∈ T,  nat: ℕ,  false: False,  ge: i ≥ j ,  uimplies: b supposing a,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  top: Top,  prop: ℙ,  win2strat: win2strat(g;n),  so_apply: x[s],  subtype_rel: A ⊆r B,  sq_type: SQType(T),  guard: {T},  decidable: Dec(P),  or: P ∨ Q,  pi1: fst(t),  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  uiff: uiff(P;Q),  ifthenelse: if b then t else f fi ,  bfalse: ff,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  squash: ↓T,  int_seg: {i..j-},  lelt: i ≤ j < k,  subtract: n - m,  goodAux: goodAux(g0;G;moves),  eq_int: (i =z j),  play-len: ||moves||,  play-truncate: play-truncate(f;m),  true: True,  less_than': less_than'(a;b),  le: A ≤ B,  less_than: a < b,  nat_plus: ℕ+,  play-item: moves[i],  pi2: snd(t),  seq-item: s[i],  nequal: a ≠ b ∈ T ,  let: let,  sq_stable: SqStable(P),  seq-len: ||s||,  seq-truncate: seq-truncate(s;n),  sequence: sequence(T)
Lemmas referenced :  nat_properties,  full-omega-unsat,  intformand_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  istype-int,  int_formula_prop_and_lemma,  istype-void,  int_formula_prop_le_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_less_lemma,  int_formula_prop_wf,  ge_wf,  istype-less_than,  subtract-1-ge-0,  istype-nat,  sg-pos_wf,  sg-init_wf,  subtype_rel_self,  sg-legal1_wf,  sg-legal2_wf,  simple-game_wf,  eq_int_wf,  equal-wf-base,  bool_wf,  assert_wf,  bnot_wf,  not_wf,  istype-assert,  intformnot_wf,  intformeq_wf,  int_formula_prop_not_lemma,  int_formula_prop_eq_lemma,  int_subtype_base,  subtype_base_sq,  strat2play_wf,  subtract_wf,  decidable__le,  itermSubtract_wf,  int_term_value_subtract_lemma,  istype-le,  play-len_wf,  subtype_rel_function,  uiff_transitivity,  eqtt_to_assert,  assert_of_eq_int,  iff_transitivity,  iff_weakening_uiff,  eqff_to_assert,  assert_of_bnot,  play-item_wf,  itermMultiply_wf,  int_term_value_mul_lemma,  decidable__lt,  itermAdd_wf,  int_term_value_add_lemma,  strat2play-invariant-1,  seq-len-truncate,  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-le-2,  istype-false,  strat2play_subtype,  mul_bounds_1b,  mul-commutes,  mul-distributes,  subtract-add-cancel,  bool_subtype_base,  equal_wf,  squash_wf,  true_wf,  istype-universe,  eq_int_eq_false,  bfalse_wf,  iff_weakening_equal,  seq-truncate-truncate,  seq-truncate-item,  mul-swap,  mul-associates,  not-lt-2,  omega-shadow,  minus-zero,  zero-mul,  mul-distributes-right,  two-mul,  add-mul-special,  one-mul,  le_reflexive,  not-equal-implies-less,  istype-sqequal,  sq_stable__le,  subtract_nat_wf,  subtype_rel-equal,  le-add-cancel-alt
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :lambdaFormation_alt,  sqequalHypSubstitution,  productElimination,  thin,  rename,  cut,  hypothesis,  promote_hyp,  Error :isect_memberFormation_alt,  introduction,  extract_by_obid,  isectElimination,  hypothesisEquality,  setElimination,  intWeakElimination,  natural_numberEquality,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality,  dependent_functionElimination,  Error :isect_memberEquality_alt,  voidElimination,  sqequalRule,  independent_pairFormation,  Error :universeIsType,  axiomEquality,  equalityTransitivity,  equalitySymmetry,  Error :functionIsTypeImplies,  Error :inhabitedIsType,  Error :productIsType,  Error :functionIsType,  universeEquality,  applyEquality,  instantiate,  because_Cache,  Error :setIsType,  baseClosed,  intEquality,  Error :equalityIstype,  sqequalBase,  baseApply,  closedConclusion,  dependentIntersection_memberEquality,  cumulativity,  Error :dependent_set_memberEquality_alt,  unionElimination,  functionExtensionality,  functionEquality,  setEquality,  productEquality,  equalityElimination,  imageElimination,  multiplyEquality,  addEquality,  applyLambdaEquality,  minusEquality,  hyp_replacement,  imageMemberEquality
Latex:
\mforall{}g:SimpleGame
    ((\mexists{}Good:Pos(g)  {}\mrightarrow{}  \mBbbP{}'
          (Good[InitialPos(g)]
          \mwedge{}  (\mforall{}p:Pos(g).  \mforall{}q:\{q:Pos(g)|  Legal1(p;q)\}  .  \mforall{}gd:Good[p].    \mexists{}r:\{r:Pos(g)|  Legal2(q;r)\}  .  Good[r]))\000C)
    {}\mRightarrow{}  win2(g))


Date html generated: 2019_06_20-PM-00_53_46
Last ObjectModification: 2019_01_02-PM-03_35_18

Theory : co-recursion-2


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