### Nuprl Lemma : cWO-rel-path-barred

`∀[T:Type]`
`  ∀[R:T ⟶ T ⟶ ℙ]`
`    (∀f:ℕ ⟶ T. (↓∃m:ℕ. ∃n:ℕm. (¬R[f n;f m])))`
`    `` (∀alpha:{f:ℕ ⟶ (T?)| ∀x:ℕ. (cWO-rel(R) x f (f x))} . (↓∃m:ℕ. (cWObar() m alpha))) `
`    supposing ∀a,b,c:T.  (R[a;b] `` R[b;c] `` R[a;c]) `
`  supposing T`

Proof

Definitions occuring in Statement :  cWObar: `cWObar()` cWO-rel: `cWO-rel(R)` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` squash: `↓T` implies: `P `` Q` unit: `Unit` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` union: `left + right` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` implies: `P `` Q` all: `∀x:A. B[x]` cWO-rel: `cWO-rel(R)` cWObar: `cWObar()` squash: `↓T` prop: `ℙ` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` so_apply: `x[s]` so_apply: `x[s1;s2]` int_seg: `{i..j-}` lelt: `i ≤ j < k` exists: `∃x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` top: `Top` true: `True` cand: `A c∧ B` isl: `isl(x)` isr: `isr(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff` guard: `{T}` sq_stable: `SqStable(P)` nat_plus: `ℕ+` less_than: `a < b` sq_type: `SQType(T)` outl: `outl(x)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation sqequalHypSubstitution sqequalRule hypothesis imageElimination imageMemberEquality hypothesisEquality thin baseClosed extract_by_obid isectElimination functionEquality unionEquality lambdaEquality applyEquality natural_numberEquality setElimination rename because_Cache independent_isectElimination independent_pairFormation dependent_set_memberEquality productElimination dependent_functionElimination setEquality isect_memberEquality equalityTransitivity equalitySymmetry cumulativity universeEquality unionElimination independent_functionElimination addEquality productEquality functionExtensionality voidElimination voidEquality minusEquality intEquality instantiate dependent_pairFormation multiplyEquality baseApply closedConclusion addLevel levelHypothesis hyp_replacement inlEquality applyLambdaEquality promote_hyp

Latex:
\mforall{}[T:Type]
\mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}]
(\mforall{}f:\mBbbN{}  {}\mrightarrow{}  T.  (\mdownarrow{}\mexists{}m:\mBbbN{}.  \mexists{}n:\mBbbN{}m.  (\mneg{}R[f  n;f  m])))
{}\mRightarrow{}  (\mforall{}alpha:\{f:\mBbbN{}  {}\mrightarrow{}  (T?)|  \mforall{}x:\mBbbN{}.  (cWO-rel(R)  x  f  (f  x))\}  .  (\mdownarrow{}\mexists{}m:\mBbbN{}.  (cWObar()  m  alpha)))
supposing  \mforall{}a,b,c:T.    (R[a;b]  {}\mRightarrow{}  R[b;c]  {}\mRightarrow{}  R[a;c])
supposing  T

Date html generated: 2019_06_20-AM-11_29_36
Last ObjectModification: 2018_08_21-PM-01_53_39

Theory : bar-induction

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