### Nuprl Lemma : simple_fan_theorem-ext

`∀[X:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ ℙ]`
`  (∀n:ℕ. ∀s:ℕn ⟶ 𝔹.  Dec(X[n;s])) `` (∃k:ℕ [(∀f:ℕ ⟶ 𝔹. ∃n:ℕk. X[n;f])]) supposing ∀f:ℕ ⟶ 𝔹. (↓∃n:ℕ. X[n;f])`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` exists: `∃x:A. B[x]` squash: `↓T` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  member: `t ∈ T` bottom: `⊥` seq-normalize: `seq-normalize(n;s)` uall: `∀[x:A]. B[x]` top: `Top` has-value: `(a)↓` not: `¬A` implies: `P `` Q` false: `False` and: `P ∧ Q` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` less_than: `a < b` less_than': `less_than'(a;b)` true: `True` squash: `↓T` prop: `ℙ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` simple_fan_theorem basic_bar_induction so_lambda: `so_lambda(x,y,z,w.t[x; y; z; w])` so_apply: `x[s1;s2;s3;s4]` strict4: `strict4(F)`
Lemmas referenced :  simple_fan_theorem strictness-apply bottom_diverge exception-not-bottom has-value_wf_base is-exception_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf bottom-sqle eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot exception-not-value value-type-has-value int-value-type lifting-strict-less base_wf basic_bar_induction
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution equalityTransitivity equalitySymmetry isectElimination isect_memberEquality voidElimination voidEquality sqequalSqle sqleRule sqleReflexivity divergentSqle callbyvalueCallbyvalue callbyvalueReduce independent_functionElimination callbyvalueExceptionCases axiomSqleEquality exceptionSqequal baseApply closedConclusion baseClosed hypothesisEquality callbyvalueLess productElimination lambdaFormation unionElimination equalityElimination because_Cache independent_isectElimination lessCases isect_memberFormation sqequalAxiom independent_pairFormation natural_numberEquality imageMemberEquality imageElimination dependent_pairFormation promote_hyp dependent_functionElimination cumulativity lessExceptionCases intEquality callbyvalueAdd addExceptionCases inlFormation

Latex:
\mforall{}[X:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbP{}]
(\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbB{}.    Dec(X[n;s]))  {}\mRightarrow{}  (\mexists{}k:\mBbbN{}  [(\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  \mexists{}n:\mBbbN{}k.  X[n;f])])
supposing  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.  (\mdownarrow{}\mexists{}n:\mBbbN{}.  X[n;f])

Date html generated: 2018_05_21-PM-00_03_36
Last ObjectModification: 2018_05_19-AM-07_10_58

Theory : bool_1

Home Index