Nuprl Lemma : win2-iff

∀g:SimpleGame. (win2(g) ⇐⇒ ∀p:{p:Pos(g)| Legal1(InitialPos(g);p)} . ∃q:{q:Pos(g)| Legal2(p;q)} . win2(g@q))


Proof



Definitions occuring in Statement :  sg-change-init: g@j,  win2: win2(g),  sg-legal2: Legal2(x;y),  sg-legal1: Legal1(x;y),  sg-init: InitialPos(g),  sg-pos: Pos(g),  simple-game: SimpleGame,  all: ∀x:A. B[x],  exists: ∃x:A. B[x],  iff: P ⇐⇒ Q,  set: {x:A| B[x]} 
Definitions unfolded in proof :  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  and: P ∧ Q,  implies: P ⇒ Q,  win2: win2(g),  member: t ∈ T,  nat_plus: ℕ+,  decidable: Dec(P),  or: P ∨ Q,  uall: ∀[x:A]. B[x],  uimplies: b supposing a,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  top: Top,  prop: ℙ,  false: False,  subtype_rel: A ⊆r B,  nat: ℕ,  subtract: n - m,  rev_implies: P ⇐ Q,  strat2play: strat2play(g;n;s),  eq_int: (i =z j),  ifthenelse: if b then t else f fi ,  btrue: tt,  cand: A c∧ B,  le: A ≤ B,  less_than': less_than'(a;b),  seq-len: ||s||,  pi1: fst(t),  seq-cons: seq-cons(a;s),  seq-nil: seq-nil(),  seq-item: s[i],  pi2: snd(t),  sg-init: InitialPos(g),  play-item: moves[i],  bfalse: ff,  int_seg: {i..j-},  lelt: i ≤ j < k,  play-len: ||moves||,  so_lambda: λ2x.t[x],  so_apply: x[s],  ge: i ≥ j ,  win2strat: win2strat(g;n),  bool: 𝔹,  unit: Unit,  it: ⋅,  uiff: uiff(P;Q),  simple-game: SimpleGame,  sg-pos: Pos(g),  sg-change-init: g@j,  spreadn: spread4,  true: True,  guard: {T},  squash: ↓T,  less_than: a < b,  sq_type: SQType(T),  sequence: sequence(T),  seq-truncate: seq-truncate(s;n),  play-truncate: play-truncate(f;m),  bnot: ¬bb,  assert: ↑b,  sg-legal1: Legal1(x;y),  nequal: a ≠ b ∈ T ,  sq_stable: SqStable(P),  sg-reachable: sg-reachable(g;x;y),  sg-legal2: Legal2(x;y)
Lemmas referenced :  win2strat-properties,  decidable__lt,  full-omega-unsat,  intformnot_wf,  intformless_wf,  itermConstant_wf,  istype-int,  int_formula_prop_not_lemma,  istype-void,  int_formula_prop_less_lemma,  int_term_value_constant_lemma,  int_formula_prop_wf,  istype-less_than,  decidable__le,  intformle_wf,  int_formula_prop_le_lemma,  istype-le,  istype-nat,  win2strat_wf,  sg-pos_wf,  sg-legal1_wf,  sg-init_wf,  win2_wf,  respond-implies-win2,  sg-legal2_wf,  sg-change-init_wf,  simple-game_wf,  seq-cons_wf,  seq-nil_wf,  istype-false,  seq-len_wf,  seq-item_wf,  intformand_wf,  itermVar_wf,  int_formula_prop_and_lemma,  int_term_value_var_lemma,  play-len_wf,  int_subtype_base,  subtype_rel_sets_simple,  play-item_wf,  nat_properties,  ge_wf,  subtract-1-ge-0,  eq_int_wf,  equal-wf-base,  bool_wf,  assert_wf,  bnot_wf,  not_wf,  istype-assert,  intformeq_wf,  int_formula_prop_eq_lemma,  strat2play_wf,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  uiff_transitivity,  eqtt_to_assert,  assert_of_eq_int,  iff_transitivity,  iff_weakening_uiff,  eqff_to_assert,  assert_of_bnot,  itermAdd_wf,  int_term_value_add_lemma,  add-subtract-cancel,  sg-reachable_wf,  le_wf,  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,  strat2play_subtype,  sequence_subtype,  omega-shadow,  not-lt-2,  two-mul,  add-mul-special,  one-mul,  le_reflexive,  not-equal-implies-less,  zero-mul,  mul-distributes-right,  mul-associates,  set_subtype_base,  le_weakening2,  seq-cons-item,  nat_wf,  subtype_base_sq,  strat2play-invariant-1,  mul-commutes,  mul-distributes,  le-add-cancel-alt,  mul-swap,  mul_preserves_le,  itermMultiply_wf,  int_term_value_mul_lemma,  bool_cases_sqequal,  bool_subtype_base,  assert-bnot,  neg_assert_of_eq_int,  decidable__equal_int,  le-add-cancel2,  add-is-int-iff,  not-equal-2,  le_antisymmetry_iff,  decidable__int_equal,  istype-universe,  true_wf,  squash_wf,  equal_wf,  int_seg_wf,  minus-zero,  sq_stable__le,  lelt_wf,  subtype_rel_self,  iff_weakening_equal,  subtract-add-cancel,  strat2play-invariant,  seq-tl-len,  seq-add_wf,  seq-tl_wf,  nat_plus_wf,  nat_plus_properties,  lt_int_wf,  assert_of_lt_int,  less_than_wf,  seq-add-item,  seq-add-len,  seq-tl-item,  mod2-2n,  mod2-2n-plus-1,  nat_plus_subtype_nat,  multiply_nat_wf,  add_nat_wf,  multiply-is-int-iff
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :lambdaFormation_alt,  independent_pairFormation,  rename,  sqequalHypSubstitution,  cut,  introduction,  extract_by_obid,  dependent_functionElimination,  thin,  hypothesisEquality,  Error :dependent_set_memberEquality_alt,  natural_numberEquality,  hypothesis,  unionElimination,  isectElimination,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  Error :isect_memberEquality_alt,  voidElimination,  sqequalRule,  Error :universeIsType,  applyEquality,  equalityTransitivity,  equalitySymmetry,  Error :isectIsType,  productElimination,  Error :setIsType,  Error :functionIsType,  Error :productIsType,  because_Cache,  setElimination,  Error :inhabitedIsType,  Error :equalityIstype,  int_eqEquality,  baseClosed,  sqequalBase,  applyLambdaEquality,  Error :isect_memberFormation_alt,  intWeakElimination,  axiomEquality,  Error :functionIsTypeImplies,  intEquality,  baseApply,  closedConclusion,  dependentIntersection_memberEquality,  equalityElimination,  addEquality,  minusEquality,  imageElimination,  imageMemberEquality,  multiplyEquality,  Error :equalityIsType4,  promote_hyp,  sqequalIntensionalEquality,  Error :equalityIsType1,  cumulativity,  instantiate,  Error :equalityIsType2,  universeEquality,  hyp_replacement,  Error :functionExtensionality_alt,  Error :dependent_pairEquality_alt
Latex:
\mforall{}g:SimpleGame
    (win2(g)  \mLeftarrow{}{}\mRightarrow{}  \mforall{}p:\{p:Pos(g)|  Legal1(InitialPos(g);p)\}  .  \mexists{}q:\{q:Pos(g)|  Legal2(p;q)\}  .  win2(g@q))


Date html generated: 2019_06_20-PM-00_55_57
Last ObjectModification: 2019_01_02-PM-03_42_52

Theory : co-recursion-2


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