### Nuprl Lemma : AF-path-barred

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  (AFx,y:T.R[x;y] `` (∀alpha:{f:ℕ ⟶ (T?)| ∀x:ℕ. (AF-spread-law(x,y.R[x;y]) x f (f x))} . (↓∃m:ℕ. (AFbar() m alpha))))`

Proof

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

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(AFx,y:T.R[x;y]
{}\mRightarrow{}  (\mforall{}alpha:\{f:\mBbbN{}  {}\mrightarrow{}  (T?)|  \mforall{}x:\mBbbN{}.  (AF-spread-law(x,y.R[x;y])  x  f  (f  x))\}
(\mdownarrow{}\mexists{}m:\mBbbN{}.  (AFbar()  m  alpha))))

Date html generated: 2019_06_20-AM-11_29_23
Last ObjectModification: 2018_08_21-PM-01_52_50

Theory : bar-induction

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