### Nuprl Lemma : extended-fan-theorem

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

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` exists: `∃x:A. B[x]` nat: `ℕ` and: `P ∧ Q` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` true: `True` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cand: `A c∧ B` quotient: `x,y:A//B[x; y]` squash: `↓T` isl: `isl(x)` sq_type: `SQType(T)` guard: `{T}` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` pi1: `fst(t)` int_seg: `{i..j-}` ge: `i ≥ j ` 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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