### Nuprl Lemma : decidable__AFbar

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀n:ℕ. ∀s:AF-spread-law(x,y.R[x;y])-consistent-seq(n).  Dec(AFbar() n s)`

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` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` union: `left + right` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` AFbar: `AFbar()` member: `t ∈ T` nat: `ℕ` consistent-seq: `R-consistent-seq(n)` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` 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)` true: `True` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]`
Lemmas referenced :  decidable__and2 less_than_wf assert_wf isr_wf unit_wf2 subtract_wf decidable__le false_wf not-le-2 less-iff-le condition-implies-le minus-one-mul zero-add minus-one-mul-top nat_wf minus-add minus-minus add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel decidable__lt not-lt-2 add-mul-special zero-mul le-add-cancel-alt and_wf le_wf decidable__assert consistent-seq_wf AF-spread-law_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalRule cut lemma_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis isect_memberEquality cumulativity applyEquality dependent_set_memberEquality independent_pairFormation dependent_functionElimination unionElimination voidElimination productElimination independent_functionElimination independent_isectElimination addEquality lambdaEquality voidEquality minusEquality intEquality because_Cache unionEquality functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
\mforall{}n:\mBbbN{}.  \mforall{}s:AF-spread-law(x,y.R[x;y])-consistent-seq(n).    Dec(AFbar()  n  s)

Date html generated: 2016_05_13-PM-03_51_01
Last ObjectModification: 2015_12_26-AM-10_17_26

Theory : bar-induction

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