Nuprl Lemma : coW-game-reachable

∀[A:𝕌']
  ∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]). ∀p,q:Pos(coW-game(a.B[a];w;w')).
    (sg-reachable(coW-game(a.B[a];w;w');p;q) ⇒ coW-pos-agree(a.B[a];w;w';p;q))


Proof



Definitions occuring in Statement :  coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),  coW-game: coW-game(a.B[a];w;w'),  coW: coW(A;a.B[a]),  sg-reachable: sg-reachable(g;x;y),  sg-pos: Pos(g),  uall: ∀[x:A]. B[x],  so_apply: x[s],  all: ∀x:A. B[x],  implies: P ⇒ Q,  function: x:A ⟶ B[x],  universe: Type
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  implies: P ⇒ Q,  sg-reachable: sg-reachable(g;x;y),  exists: ∃x:A. B[x],  and: P ∧ Q,  member: t ∈ T,  squash: ↓T,  prop: ℙ,  so_lambda: λ2x.t[x],  so_apply: x[s],  true: True,  subtype_rel: A ⊆r B,  uimplies: b supposing a,  guard: {T},  iff: P ⇐⇒ Q,  rev_implies: P ⇐ Q,  less_than: a < b,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  false: False,  top: Top,  nat_plus: ℕ+,  int_seg: {i..j-},  lelt: i ≤ j < k,  decidable: Dec(P),  or: P ∨ Q,  nat: ℕ,  ge: i ≥ j ,  sq_type: SQType(T),  subtract: n - m,  int_nzero: ℤ-o,  nequal: a ≠ b ∈ T ,  sq_stable: SqStable(P),  coW-game: coW-game(a.B[a];w;w'),  sg-pos: Pos(g),  pi1: fst(t),  coW-pos-agree: coW-pos-agree(a.B[a];w;w';p;q),  sg-legal1: Legal1(x;y),  pi2: snd(t),  cand: A c∧ B,  uiff: uiff(P;Q),  sg-legal2: Legal2(x;y)
Lemmas referenced :  coW-pos-agree_wf,  squash_wf,  true_wf,  subtype_rel_self,  iff_weakening_equal,  sg-reachable_wf,  coW-game_wf,  sg-pos_wf,  coW_wf,  istype-universe,  full-omega-unsat,  intformand_wf,  intformless_wf,  itermConstant_wf,  itermVar_wf,  intformle_wf,  istype-int,  int_formula_prop_and_lemma,  istype-void,  int_formula_prop_less_lemma,  int_term_value_constant_lemma,  int_term_value_var_lemma,  int_formula_prop_le_lemma,  int_formula_prop_wf,  istype-less_than,  seq-len_wf,  istype-le,  sequence_wf,  nat_plus_wf,  sg-legal2_wf,  seq-item_wf,  subtract_wf,  nat_plus_properties,  decidable__le,  intformnot_wf,  itermSubtract_wf,  itermMultiply_wf,  int_formula_prop_not_lemma,  int_term_value_subtract_lemma,  int_term_value_mul_lemma,  decidable__lt,  sg-legal1_wf,  nat_properties,  itermAdd_wf,  int_term_value_add_lemma,  primrec-wf2,  all_wf,  le_wf,  less_than_wf,  nat_wf,  istype-nat,  decidable__equal_int,  intformeq_wf,  int_formula_prop_eq_lemma,  seq-truncate_wf,  seq-len-truncate,  seq-truncate-item,  subtype_base_sq,  int_subtype_base,  coW-pos-agree_refl,  coW-pos-agree_transitivity,  div_rem_sum,  nequal_wf,  rem_bounds_1,  div_bounds_1,  set_subtype_base,  decidable__or,  equal-wf-base,  decidable__and2,  intformor_wf,  int_formula_prop_or_lemma,  sq_stable__coW-pos-agree,  copath-length_wf,  add-is-int-iff,  false_wf,  copath_wf,  copathAgree_wf,  copathAgree_refl
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  sqequalHypSubstitution,  productElimination,  thin,  cut,  applyEquality,  Error :lambdaEquality_alt,  imageElimination,  introduction,  extract_by_obid,  isectElimination,  hypothesisEquality,  equalityTransitivity,  hypothesis,  equalitySymmetry,  Error :universeIsType,  because_Cache,  sqequalRule,  natural_numberEquality,  imageMemberEquality,  baseClosed,  instantiate,  universeEquality,  independent_isectElimination,  independent_functionElimination,  Error :inhabitedIsType,  cumulativity,  Error :functionIsType,  approximateComputation,  Error :dependent_pairFormation_alt,  int_eqEquality,  dependent_functionElimination,  Error :isect_memberEquality_alt,  voidElimination,  independent_pairFormation,  rename,  setElimination,  multiplyEquality,  Error :dependent_set_memberEquality_alt,  unionElimination,  Error :productIsType,  addEquality,  Error :setIsType,  functionEquality,  closedConclusion,  intEquality,  promote_hyp,  Error :equalityIstype,  sqequalBase,  divideEquality,  Error :unionIsType,  baseApply,  remainderEquality,  productEquality,  Error :inlFormation_alt,  Error :inrFormation_alt,  pointwiseFunctionality,  hyp_replacement,  applyLambdaEquality
Latex:
\mforall{}[A:\mBbbU{}']
    \mforall{}B:A  {}\mrightarrow{}  Type.  \mforall{}w,w':coW(A;a.B[a]).  \mforall{}p,q:Pos(coW-game(a.B[a];w;w')).
        (sg-reachable(coW-game(a.B[a];w;w');p;q)  {}\mRightarrow{}  coW-pos-agree(a.B[a];w;w';p;q))


Date html generated: 2019_06_20-PM-01_11_04
Last ObjectModification: 2019_01_02-PM-03_59_18

Theory : co-recursion-2


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