### Nuprl Lemma : bar_induction

`∀[T:Type]. ∀[R,A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ].`
`  ((∀n:ℕ. ∀s:ℕn ⟶ T.  Dec(R[n;s]))`
`  `` (∀n:ℕ. ∀s:ℕn ⟶ T.  (R[n;s] `` A[n;s]))`
`  `` (∀n:ℕ. ∀s:ℕn ⟶ T.  ((∀t:T. A[n + 1;s++t]) `` A[n;s]))`
`  `` (∀n:ℕ. ∀s:ℕn ⟶ T.  ((∀alpha:ℕ ⟶ T. (↓∃m:ℕ. R[n + m;seq-append(n;m;s;alpha)])) `` A[n;s])))`

Proof

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

Latex:
\mforall{}[T:Type].  \mforall{}[R,A:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    Dec(R[n;s]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    (R[n;s]  {}\mRightarrow{}  A[n;s]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    ((\mforall{}t:T.  A[n  +  1;s++t])  {}\mRightarrow{}  A[n;s]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    ((\mforall{}alpha:\mBbbN{}  {}\mrightarrow{}  T.  (\mdownarrow{}\mexists{}m:\mBbbN{}.  R[n  +  m;seq-append(n;m;s;alpha)]))  {}\mRightarrow{}  A[n;s])))

Date html generated: 2017_04_14-AM-07_27_16
Last ObjectModification: 2017_02_27-PM-02_56_29

Theory : bar-induction

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