Nuprl Lemma : extended-fan-theorem

C:ℕ ⟶ (ℕ ⟶ 𝔹) ⟶ ℙ
  ((∀a:ℕ ⟶ 𝔹. ∃n:ℕ(C a))  ⇃(∃m:ℕ. ∀a:ℕ ⟶ 𝔹. ∃n:ℕ. ∀b:ℕ ⟶ 𝔹((a b ∈ (ℕm ⟶ 𝔹))  (C b))))


Proof




Definitions occuring in Statement :  quotient: x,y:A//B[x; y] int_seg: {i..j-} nat: bool: 𝔹 prop: all: x:A. B[x] exists: x:A. B[x] implies:  Q true: True apply: a function: x:A ⟶ B[x] natural_number: $n equal: t ∈ T
Definitions unfolded in proof :  all: x:A. B[x] implies:  Q member: t ∈ T prop: uall: [x:A]. B[x] so_lambda: λ2x.t[x] so_apply: x[s] subtype_rel: A ⊆B exists: x:A. B[x] nat: and: P ∧ Q uimplies: supposing a le: A ≤ B less_than': less_than'(a;b) false: False not: ¬A true: True so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] cand: c∧ B quotient: x,y:A//B[x; y] squash: T isl: isl(x) sq_type: SQType(T) guard: {T} assert: b ifthenelse: if then else fi  btrue: tt pi1: fst(t) int_seg: {i..j-} ge: i ≥  lelt: i ≤ j < k decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) top: Top uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :  all_wf nat_wf bool_wf exists_wf strong-continuity2-no-inner-squash-unique-bool pi1_wf equal_wf int_seg_wf unit_wf2 subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self assert_wf isl_wf true_wf quotient_wf equiv_rel_true quotient-member-eq equal-wf-base member_wf squash_wf fan_theorem decidable__assert and_wf btrue_wf subtype_base_sq bool_subtype_base set_subtype_base le_wf int_subtype_base int_seg_subtype int_seg_properties nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf assert_functionality_wrt_uiff itermConstant_wf int_term_value_constant_lemma decidable__equal_int intformeq_wf int_formula_prop_eq_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation rename cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin functionEquality hypothesis sqequalRule lambdaEquality applyEquality functionExtensionality hypothesisEquality cumulativity universeEquality dependent_functionElimination because_Cache productElimination dependent_pairEquality equalityTransitivity equalitySymmetry independent_functionElimination natural_numberEquality setElimination unionEquality productEquality independent_isectElimination independent_pairFormation inlEquality promote_hyp pointwiseFunctionality pertypeElimination imageElimination imageMemberEquality baseClosed dependent_pairFormation dependent_set_memberEquality applyLambdaEquality instantiate intEquality unionElimination int_eqEquality isect_memberEquality voidElimination voidEquality computeAll

Latex:
\mforall{}C:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbP{}
    ((\mforall{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  \mexists{}n:\mBbbN{}.  (C  n  a))  {}\mRightarrow{}  \00D9(\mexists{}m:\mBbbN{}.  \mforall{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  \mexists{}n:\mBbbN{}.  \mforall{}b:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  ((a  =  b)  {}\mRightarrow{}  (C  n  b))))



Date html generated: 2017_04_20-AM-07_22_19
Last ObjectModification: 2017_02_27-PM-05_57_48

Theory : continuity


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