Nuprl Lemma : coW-trans_wf

∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w1,w2,w3:coW(A;a.B[a])]. ∀[n:ℕ]. ∀[X:win2strat(coW-game(a.B[a];w1;w2);n)].
∀[Y:win2strat(coW-game(a.B[a];w2;w3);n)].
  (coW-trans(X; Y) ∈ win2strat(coW-game(a.B[a];w1;w3);n))


Proof



Definitions occuring in Statement :  coW-trans: coW-trans(X; Y),  coW-game: coW-game(a.B[a];w;w'),  coW: coW(A;a.B[a]),  win2strat: win2strat(g;n),  nat: ℕ,  uall: ∀[x:A]. B[x],  so_apply: x[s],  member: t ∈ T,  function: x:A ⟶ B[x],  universe: Type
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  nat: ℕ,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  uimplies: b supposing a,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  all: ∀x:A. B[x],  top: Top,  and: P ∧ Q,  prop: ℙ,  win2strat: win2strat(g;n),  eq_int: (i =z j),  subtract: n - m,  ifthenelse: if b then t else f fi ,  btrue: tt,  so_lambda: λ2x.t[x],  so_apply: x[s],  decidable: Dec(P),  or: P ∨ Q,  subtype_rel: A ⊆r B,  guard: {T},  bool: 𝔹,  unit: Unit,  it: ⋅,  uiff: uiff(P;Q),  bfalse: ff,  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  sq_type: SQType(T),  transMoves: transMoves(X;Y;moves),  play-len: ||moves||,  seq-len: ||s||,  pi1: fst(t),  seq-truncate: seq-truncate(s;n),  strat2play: strat2play(g;n;s),  sequence: sequence(T),  play-item: moves[i],  int_seg: {i..j-},  lelt: i ≤ j < k,  coW-game: coW-game(a.B[a];w;w'),  sg-pos: Pos(g),  squash: ↓T,  true: True,  sg-init: InitialPos(g),  sg-legal1: Legal1(x;y),  pi2: snd(t),  lt_int: i <z j,  le: A ≤ B,  less_than': less_than'(a;b),  copath-length: copath-length(p),  copath-nil: (),  let: let,  seq-add: seq-add(s;x),  seq-nil: seq-nil(),  bnot: ¬bb,  assert: ↑b,  seq-item: s[i],  cand: A c∧ B,  sg-legal2: Legal2(x;y),  coWtransInvariant: coWtransInvariant(x.B[x];w1;w2;w3;k;X;Y;a;b;moves),  label: ...$L... t,  copath: copath(a.B[a];w),  copathAgree: copathAgree(a.B[a];w;x;y),  play-truncate: play-truncate(f;m),  coW-trans: coW-trans(X; Y),  rev_uimplies: rev_uimplies(P;Q),  nequal: a ≠ b ∈ T ,  nat_plus: ℕ+,  coW-pos-lens: coW-pos-lens(p;i;j)
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,  win2strat_wf,  coW-game_wf,  decidable__le,  intformnot_wf,  int_formula_prop_not_lemma,  istype-le,  subtract-1-ge-0,  istype-nat,  coW_wf,  istype-universe,  eq_int_wf,  equal-wf-base,  bool_wf,  int_subtype_base,  assert_wf,  bnot_wf,  not_wf,  istype-assert,  intformeq_wf,  int_formula_prop_eq_lemma,  win2strat_subtype,  strat2play_wf,  subtract_wf,  itermSubtract_wf,  int_term_value_subtract_lemma,  play-len_wf,  uiff_transitivity,  eqtt_to_assert,  assert_of_eq_int,  iff_transitivity,  iff_weakening_uiff,  eqff_to_assert,  assert_of_bnot,  bool_cases,  subtype_base_sq,  bool_subtype_base,  mul-distributes,  add-associates,  mul-commutes,  add-swap,  add-commutes,  zero-add,  decidable__equal_int,  itermAdd_wf,  itermMultiply_wf,  int_term_value_add_lemma,  int_term_value_mul_lemma,  copath_length_nil_lemma,  play-item_wf,  decidable__lt,  strat2play-invariant-1,  sg-legal1_wf,  squash_wf,  true_wf,  sg-pos_wf,  simple-game_wf,  subtype_rel_self,  iff_weakening_equal,  equal_wf,  nat_wf,  set_subtype_base,  le_wf,  istype-false,  copath-length_wf,  lt_int_wf,  assert_of_lt_int,  int_seg_properties,  bool_cases_sqequal,  assert-bnot,  less_than_wf,  copath-nil_wf,  int_seg_wf,  seq-len_wf,  seq-item_wf,  sg-init_wf,  copathAgree-nil,  sg-legal2_wf,  copath_wf,  coWtransInvariant_wf,  pi1_wf,  pi2_wf,  subtype_rel_wf,  add_functionality_wrt_eq,  strat2play_subtype,  le-add-cancel,  le_antisymmetry_iff,  mul-associates,  equal-wf-T-base,  mul_preserves_le,  le-add-cancel2,  add-zero,  less-iff-le,  le-add-cancel-alt,  add_functionality_wrt_le,  minus-one-mul-top,  minus-minus,  minus-add,  minus-one-mul,  condition-implies-le,  not-le-2,  false_wf,  play-truncate_wf,  less_than_transitivity1,  le_weakening2,  strat2play_subtype_le,  not-lt-2,  add-mul-special,  zero-mul,  lelt_wf,  seq-len-truncate,  sequence_wf,  seq-truncate_wf,  add-is-int-iff,  subtract-add-cancel,  bfalse_wf,  not-equal-2,  eq_int_eq_false,  win2strat-properties,  seq-truncate-item,  le_reflexive,  coW-play-invariant,  mod2-2n,  mod2-2n-plus-1,  strat2play-invariant,  and_wf,  or_wf,  less_than_irreflexivity,  le_int_wf,  assert_functionality_wrt_uiff,  bnot_of_lt_int,  assert_of_le_int,  copathAgree_wf,  strat2play-add,  add-subtract-cancel,  mul-distributes-right,  copathAgree_refl,  istype_wf,  subtype_rel_set,  seq-add_wf,  seq-add-len,  strat2play-longer,  le_weakening,  coW-pos-lens_wf,  intformor_wf,  int_formula_prop_or_lemma
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :isect_memberFormation_alt,  introduction,  cut,  extract_by_obid,  sqequalHypSubstitution,  isectElimination,  thin,  hypothesisEquality,  hypothesis,  setElimination,  rename,  sqequalRule,  intWeakElimination,  Error :lambdaFormation_alt,  natural_numberEquality,  independent_isectElimination,  approximateComputation,  independent_functionElimination,  Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality,  dependent_functionElimination,  Error :isect_memberEquality_alt,  voidElimination,  independent_pairFormation,  Error :universeIsType,  axiomEquality,  equalityTransitivity,  equalitySymmetry,  Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  applyEquality,  Error :dependent_set_memberEquality_alt,  unionElimination,  because_Cache,  instantiate,  cumulativity,  Error :functionIsType,  universeEquality,  baseApply,  closedConclusion,  baseClosed,  intEquality,  Error :equalityIstype,  sqequalBase,  dependentIntersectionElimination,  dependentIntersection_memberEquality,  functionExtensionality,  setEquality,  equalityElimination,  productElimination,  multiplyEquality,  Error :setIsType,  Error :productIsType,  addEquality,  imageElimination,  imageMemberEquality,  hyp_replacement,  applyLambdaEquality,  independent_pairEquality,  Error :dependent_pairEquality_alt,  promote_hyp,  Error :inlFormation_alt,  dependentIntersectionEqElimination,  voidEquality,  Error :inrFormation_alt,  Error :equalityIsType4,  minusEquality,  Error :equalityIsType1,  dependent_set_memberEquality,  lambdaFormation,  lambdaEquality,  isect_memberEquality,  productEquality,  Error :unionIsType,  Error :equalityIsType2,  unionEquality,  Error :equalityIsType3,  pointwiseFunctionality
Latex:
\mforall{}[A:\mBbbU{}'].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[w1,w2,w3:coW(A;a.B[a])].  \mforall{}[n:\mBbbN{}].
\mforall{}[X:win2strat(coW-game(a.B[a];w1;w2);n)].  \mforall{}[Y:win2strat(coW-game(a.B[a];w2;w3);n)].
    (coW-trans(X;  Y)  \mmember{}  win2strat(coW-game(a.B[a];w1;w3);n))


Date html generated: 2019_06_20-PM-01_10_53
Last ObjectModification: 2019_01_02-PM-04_22_25

Theory : co-recursion-2


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