### Nuprl Lemma : at_AFbar

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  ∀n:ℕ. ∀s:AF-spread-law(x,y.R[x;y])-consistent-seq(n).`
`    ((AFbar() n s) `` (¬{a:T| AF-spread-law(x,y.R[x;y]) n s (inl a)} ))`

Proof

Definitions occuring in Statement :  AFbar: `AFbar()` AF-spread-law: `AF-spread-law(x,y.R[x; y])` consistent-seq: `R-consistent-seq(n)` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` 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` AFbar: `AFbar()` AF-spread-law: `AF-spread-law(x,y.R[x; y])` 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` le: `A ≤ B` less_than': `less_than'(a;b)` isr: `isr(x)` bfalse: `ff` so_lambda: `λ2x.t[x]` so_apply: `x[s]` so_apply: `x[s1;s2]` guard: `{T}` top: `Top`
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaFormation sqequalRule setElimination rename productElimination independent_pairFormation independent_functionElimination natural_numberEquality applyEquality because_Cache dependent_set_memberEquality dependent_functionElimination unionElimination voidElimination independent_isectElimination addEquality minusEquality unionEquality cumulativity productEquality equalityTransitivity equalitySymmetry lambdaEquality inlEquality universeEquality functionEquality isect_memberEquality voidEquality intEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].