### Nuprl Lemma : bar_recursion_wf0

`∀[T:Type]. ∀[R,A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ]. ∀[d:∀n:ℕ. ∀s:ℕn ⟶ T.  Dec(R[n;s])].`
`∀[b:∀n:ℕ. ∀s:ℕn ⟶ T.  (R[n;s] `` A[n;s])]. ∀[i:∀n:ℕ. ∀s:ℕn ⟶ T.  ((∀t:T. A[n + 1;seq-append(n;1;s;λi.t)]) `` A[n;s])].`
`  ((∀alpha:ℕ ⟶ T. (↓∃m:ℕ. R[m;alpha])) `` (bar_recursion(d;b;i;0;λm.eval x = m in ⊥) ∈ A[0;λm.⊥]))`

Proof

Definitions occuring in Statement :  bar_recursion: bar_recursion seq-append: `seq-append(n;m;s1;s2)` int_seg: `{i..j-}` nat: `ℕ` bottom: `⊥` callbyvalue: callbyvalue decidable: `Dec(P)` 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` member: `t ∈ T` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  label: `...\$L... t` guard: `{T}` lelt: `i ≤ j < k` int_seg: `{i..j-}` true: `True` top: `Top` subtract: `n - m` squash: `↓T` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` or: `P ∨ Q` decidable: `Dec(P)` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` prop: `ℙ` not: `¬A` false: `False` less_than': `less_than'(a;b)` and: `P ∧ Q` le: `A ≤ B` uimplies: `b supposing a` nat: `ℕ` so_apply: `x[s]` subtype_rel: `A ⊆r B` so_apply: `x[s1;s2]` so_lambda: `λ2x.t[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]`
Rules used in proof :  equalitySymmetry equalityTransitivity instantiate universeEquality minusEquality intEquality voidEquality isect_memberEquality imageElimination baseClosed imageMemberEquality productElimination voidElimination unionElimination dependent_functionElimination addEquality dependent_set_memberEquality independent_pairFormation independent_isectElimination rename setElimination natural_numberEquality functionExtensionality applyEquality because_Cache lambdaEquality sqequalRule cumulativity functionEquality independent_functionElimination lambdaFormation hypothesisEquality thin isectElimination sqequalHypSubstitution hypothesis isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution extract_by_obid introduction cut

Latex:
\mforall{}[T:Type].  \mforall{}[R,A:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[d:\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    Dec(R[n;s])].  \mforall{}[b:\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.
(R[n;s]
{}\mRightarrow{}  A[n;s])].
\mforall{}[i:\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    ((\mforall{}t:T.  A[n  +  1;seq-append(n;1;s;\mlambda{}i.t)])  {}\mRightarrow{}  A[n;s])].
((\mforall{}alpha:\mBbbN{}  {}\mrightarrow{}  T.  (\mdownarrow{}\mexists{}m:\mBbbN{}.  R[m;alpha]))  {}\mRightarrow{}  (bar\_recursion(d;b;i;0;\mlambda{}m.eval  x  =  m  in  \mbot{})  \mmember{}  A[0;\mlambda{}m.\mbot{}]))

Date html generated: 2017_09_29-PM-05_47_35
Last ObjectModification: 2017_09_01-PM-11_45_59

Theory : bar-induction

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