### Nuprl Lemma : fan-realizer_wf

`fan-realizer ∈ ∀[X:(𝔹 List) ⟶ ℙ]. (tbar(𝔹;X) `` Decidable(X) `` (∃k:ℕ. ∀f:ℕ ⟶ 𝔹. ∃n:ℕk. (X map(f;upto(n)))))`

Proof

Definitions occuring in Statement :  fan-realizer: `fan-realizer` tbar: `tbar(T;X)` dec-predicate: `Decidable(X)` upto: `upto(n)` map: `map(f;as)` list: `T List` int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` member: `t ∈ T` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  member: `t ∈ T` fan-theorem simple-fan-theorem simple_fan_theorem basic_bar_induction uall: `∀[x:A]. B[x]` so_lambda: so_lambda4 so_apply: `x[s1;s2;s3;s4]` uimplies: `b supposing a` strict4: `strict4(F)` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` has-value: `(a)↓` prop: `ℙ` or: `P ∨ Q` squash: `↓T` false: `False` seq-normalize: `seq-normalize(n;s)` fan-realizer: `fan-realizer` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` not: `¬A` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` pi1: `fst(t)` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B`
Lemmas referenced :  fan-theorem lifting-strict-less value-type-has-value int-value-type has-value_wf_base istype-base istype-universe exception-not-value is-exception_wf strictness-apply lt_int_wf eqtt_to_assert assert_of_lt_int istype-top bottom-sqle eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf less_than_wf istype-less_than bottom_diverge exception-not-bottom dec-predicate_wf list_wf tbar_wf nat_wf set-value-type le_wf istype-int istype-nat int_seg_wf map_wf upto_wf subtype_rel_function int_seg_subtype_nat istype-false subtype_rel_self simple-fan-theorem simple_fan_theorem basic_bar_induction
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity instantiate extract_by_obid hypothesis sqequalHypSubstitution sqequalRule introduction isectElimination thin baseClosed Error :memTop,  independent_isectElimination independent_pairFormation lambdaFormation_alt callbyvalueAdd baseApply closedConclusion hypothesisEquality productElimination intEquality because_Cache universeIsType addExceptionCases exceptionSqequal inlFormation_alt imageMemberEquality imageElimination sqleReflexivity independent_functionElimination voidElimination isect_memberEquality_alt lambdaEquality_alt sqequalSqle divergentSqle callbyvalueLess inhabitedIsType unionElimination equalityElimination equalityTransitivity equalitySymmetry lessCases isect_memberFormation_alt axiomSqEquality isectIsTypeImplies natural_numberEquality dependent_pairFormation_alt equalityIstype promote_hyp dependent_functionElimination cumulativity lessExceptionCases axiomSqleEquality callbyvalueCallbyvalue callbyvalueReduce callbyvalueExceptionCases functionIsType universeEquality applyEquality isectIsType productIsType setElimination rename productEquality functionEquality

Latex:
fan-realizer  \mmember{}  \mforall{}[X:(\mBbbB{}  List)  {}\mrightarrow{}  \mBbbP{}]
(tbar(\mBbbB{};X)  {}\mRightarrow{}  Decidable(X)  {}\mRightarrow{}  (\mexists{}k:\mBbbN{}.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  \mexists{}n:\mBbbN{}k.  (X  map(f;upto(n)))))

Date html generated: 2020_05_20-AM-09_07_38
Last ObjectModification: 2020_01_10-PM-03_32_46

Theory : bar!induction

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