Nuprl Lemma : respond-implies-win2

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


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],  implies: P ⇒ Q,  set: {x:A| B[x]} 
Definitions unfolded in proof :  all: ∀x:A. B[x],  implies: P ⇒ Q,  exists: ∃x:A. B[x],  win2: win2(g),  uall: ∀[x:A]. B[x],  member: t ∈ T,  nat: ℕ,  false: False,  ge: i ≥ j ,  guard: {T},  uimplies: b supposing a,  prop: ℙ,  win2strat: win2strat(g;n),  subtype_rel: A ⊆r B,  top: Top,  not: ¬A,  decidable: Dec(P),  or: P ∨ Q,  iff: P ⇐⇒ Q,  and: P ∧ Q,  rev_implies: P ⇐ Q,  uiff: uiff(P;Q),  subtract: n - m,  le: A ≤ B,  less_than': less_than'(a;b),  true: True,  pi1: fst(t),  bool: 𝔹,  unit: Unit,  it: ⋅,  btrue: tt,  ifthenelse: if b then t else f fi ,  bfalse: ff,  int_seg: {i..j-},  lelt: i ≤ j < k,  squash: ↓T,  sq_type: SQType(T),  play-item: moves[i],  nat_plus: ℕ+,  less_than: a < b,  spreadn: spread4,  sg-change-init: g@j,  sg-pos: Pos(g),  simple-game: SimpleGame,  sequence: sequence(T),  play-truncate: play-truncate(f;m),  play-len: ||moves||,  seq-item: s[i],  seq-truncate: seq-truncate(s;n),  seq-len: ||s||,  pi2: snd(t),  eq_int: (i =z j),  sq_stable: SqStable(P),  sg-reachable: sg-reachable(g;x;y),  so_lambda: λ2x.t[x],  so_apply: x[s],  istype: istype(T),  cand: A c∧ B,  seq-tl: seq-tl(s),  strat2play: strat2play(g;n;s),  sg-init: InitialPos(g),  nequal: a ≠ b ∈ T ,  int_nzero: ℤ-o,  sg-legal2: Legal2(x;y)
Lemmas referenced :  nat_properties,  less_than_transitivity1,  less_than_irreflexivity,  ge_wf,  istype-less_than,  subtract-1-ge-0,  istype-nat,  sg-pos_wf,  sg-legal1_wf,  sg-init_wf,  sg-legal2_wf,  win2_wf,  sg-change-init_wf,  simple-game_wf,  eq_int_wf,  equal-wf-base,  bool_wf,  assert_wf,  istype-void,  bnot_wf,  not_wf,  istype-assert,  int_subtype_base,  strat2play_wf,  subtract_wf,  decidable__le,  istype-false,  not-le-2,  less-iff-le,  condition-implies-le,  minus-one-mul,  zero-add,  minus-one-mul-top,  minus-add,  minus-minus,  add-associates,  add-swap,  add-commutes,  add_functionality_wrt_le,  add-zero,  le-add-cancel,  istype-le,  play-len_wf,  uiff_transitivity,  eqtt_to_assert,  assert_of_eq_int,  iff_transitivity,  iff_weakening_uiff,  eqff_to_assert,  assert_of_bnot,  strat2play-invariant-1,  play-item_wf,  istype-sqequal,  mul-distributes,  mul-commutes,  mul-associates,  mul-distributes-right,  zero-mul,  subtype_base_sq,  not-equal-implies-less,  squash_wf,  true_wf,  subtype_rel_self,  iff_weakening_equal,  le_reflexive,  one-mul,  add-mul-special,  two-mul,  not-lt-2,  omega-shadow,  decidable__lt,  equal-wf-T-base,  decidable__int_equal,  not-equal-2,  minus-zero,  le-add-cancel2,  mul_preserves_le,  le_wf,  le_antisymmetry_iff,  istype-int,  win2strat-properties,  win2strat_wf,  less_than_wf,  strat2play_subtype,  sequence_wf,  seq-len_wf,  sg-reachable_wf,  seq-item_wf,  le_weakening,  seq-tl-len,  seq-tl_wf,  sq_stable__le,  le-add-cancel-alt,  int_seg_wf,  strat2play-invariant,  seq-truncate_wf,  nat_wf,  set_subtype_base,  nat_plus_wf,  int_seg_properties,  nat_plus_properties,  seq-len-truncate,  seq-truncate-item,  seq-tl-item,  mul_bounds_1a,  mod2-2n,  add_nat_wf,  multiply_nat_wf,  nat_plus_subtype_nat,  mod2-2n-plus-1,  seq-settype,  add-member-int_seg2,  add-is-int-iff,  sg-legal1-change-init,  sg-reachable_self,  le_weakening2,  mul-swap,  bool_subtype_base,  equal_wf,  istype-universe,  eq_int_eq_false,  multiply-is-int-iff,  bfalse_wf,  trivial-cancel,  nequal_wf,  le_transitivity,  add_nat_plus,  trivial-equal,  play-truncate_wf
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :lambdaFormation_alt,  cut,  hypothesis,  promote_hyp,  thin,  sqequalHypSubstitution,  productElimination,  sqequalRule,  Error :isect_memberFormation_alt,  rename,  introduction,  extract_by_obid,  isectElimination,  hypothesisEquality,  setElimination,  intWeakElimination,  natural_numberEquality,  independent_isectElimination,  independent_functionElimination,  voidElimination,  Error :universeIsType,  Error :lambdaEquality_alt,  dependent_functionElimination,  axiomEquality,  equalityTransitivity,  equalitySymmetry,  Error :functionIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsType,  Error :setIsType,  Error :productIsType,  because_Cache,  applyEquality,  baseClosed,  intEquality,  Error :isect_memberEquality_alt,  Error :equalityIstype,  baseApply,  closedConclusion,  dependentIntersection_memberEquality,  Error :dependent_set_memberEquality_alt,  unionElimination,  independent_pairFormation,  addEquality,  minusEquality,  sqequalBase,  Error :dependent_pairFormation_alt,  functionExtensionality,  setEquality,  equalityElimination,  imageElimination,  multiplyEquality,  instantiate,  cumulativity,  imageMemberEquality,  universeEquality,  Error :isectIsType,  Error :equalityIsType1,  sqequalIntensionalEquality,  applyLambdaEquality,  Error :equalityIsType4
Latex:
\mforall{}g:SimpleGame
    ((\mforall{}p:\{p:Pos(g)|  Legal1(InitialPos(g);p)\}  .  \mexists{}q:\{q:Pos(g)|  Legal2(p;q)\}  .  win2(g@q))  {}\mRightarrow{}  win2(g))


Date html generated: 2019_06_20-PM-00_55_22
Last ObjectModification: 2019_01_02-PM-01_32_32

Theory : co-recursion-2


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