### Nuprl Lemma : at_cWObar

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  ∀n:ℕ. ∀s:cWO-rel(R)-consistent-seq(n).  ((cWObar() n s) `` (¬{a:T| cWO-rel(R) n s (inl a)} ))`

Proof

Definitions occuring in Statement :  cWObar: `cWObar()` cWO-rel: `cWO-rel(R)` consistent-seq: `R-consistent-seq(n)` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` unit: `Unit` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q` not: `¬A` false: `False` cWObar: `cWObar()` cWO-rel: `cWO-rel(R)` isl: `isl(x)` outl: `outl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` and: `P ∧ Q` cand: `A c∧ B` true: `True` consistent-seq: `R-consistent-seq(n)` int_seg: `{i..j-}` nat: `ℕ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` prop: `ℙ` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` isr: `isr(x)` bfalse: `ff` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaFormation setElimination rename sqequalRule independent_functionElimination productElimination independent_pairFormation natural_numberEquality applyEquality because_Cache dependent_set_memberEquality dependent_functionElimination unionElimination voidElimination independent_isectElimination addEquality lambdaEquality isect_memberEquality voidEquality minusEquality intEquality unionEquality cumulativity productEquality equalityTransitivity equalitySymmetry functionExtensionality inlEquality universeEquality functionEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}n:\mBbbN{}.  \mforall{}s:cWO-rel(R)-consistent-seq(n).    ((cWObar()  n  s)  {}\mRightarrow{}  (\mneg{}\{a:T|  cWO-rel(R)  n  s  (inl  a)\}  ))

Date html generated: 2017_04_14-AM-07_28_20
Last ObjectModification: 2017_02_27-PM-02_56_44

Theory : bar-induction

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