Nuprl Lemma : cWObar_wf

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  (cWObar() ∈ n:ℕ ⟶ cWO-rel(R)-consistent-seq(n) ⟶ ℙ)`

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` unit: `Unit` member: `t ∈ T` function: `x:A ⟶ B[x]` union: `left + right` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` cWObar: `cWObar()` prop: `ℙ` and: `P ∧ Q` nat: `ℕ` consistent-seq: `R-consistent-seq(n)` int_seg: `{i..j-}` lelt: `i ≤ j < k` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` 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)` true: `True`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule lambdaEquality productEquality lemma_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis cumulativity applyEquality dependent_set_memberEquality independent_pairFormation dependent_functionElimination unionElimination lambdaFormation voidElimination productElimination independent_functionElimination independent_isectElimination addEquality isect_memberEquality voidEquality minusEquality intEquality because_Cache unionEquality axiomEquality equalityTransitivity equalitySymmetry functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    (cWObar()  \mmember{}  n:\mBbbN{}  {}\mrightarrow{}  cWO-rel(R)-consistent-seq(n)  {}\mrightarrow{}  \mBbbP{})

Date html generated: 2016_05_13-PM-03_52_13
Last ObjectModification: 2015_12_26-AM-10_16_57

Theory : bar-induction

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