### Nuprl Lemma : monotone-bar-induction3-2

`∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.`
`  ((∀n:ℕ. ∀s:ℕn ⟶ ℕ.  (B[n;s] `` (∀m:ℕ. B[n + 1;s.m@n])))`
`  `` (∀n:ℕ. ∀s:ℕn ⟶ ℕ.  (B[n;s] `` Q[n;s]))`
`  `` (∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕ. Q[n + 1;s.m@n]) `` Q[n;s]))`
`  `` (∀alpha:ℕ ⟶ ℕ. ⇃(∃m:ℕ. B[m;alpha]))`
`  `` ⇃(Q[0;λx.⊥]))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` seq-add: `s.x@n` int_seg: `{i..j-}` nat: `ℕ` bottom: `⊥` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` so_apply: `x[s1;s2]` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_lambda: `λ2x y.t[x; y]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` squash: `↓T` true: `True` ext2Baire: `ext2Baire(n;f;d)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` cand: `A c∧ B` outl: `outl(x)` isl: `isl(x)` less_than: `a < b` seq-add: `s.x@n` nequal: `a ≠ b ∈ T ` subtract: `n - m` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation rename cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin lambdaEquality applyEquality functionExtensionality hypothesisEquality because_Cache isectElimination hypothesis sqequalRule natural_numberEquality setElimination independent_isectElimination independent_pairFormation functionEquality unionEquality productEquality inlEquality dependent_set_memberEquality productElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination addEquality unionElimination universeEquality cumulativity imageElimination imageMemberEquality baseClosed addLevel hyp_replacement equalitySymmetry equalityTransitivity levelHypothesis equalityElimination promote_hyp instantiate applyLambdaEquality baseApply closedConclusion inrEquality int_eqReduceTrueSq int_eqReduceFalseSq

Latex:
\mforall{}B,Q:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    (B[n;s]  {}\mRightarrow{}  (\mforall{}m:\mBbbN{}.  B[n  +  1;s.m@n])))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    (B[n;s]  {}\mRightarrow{}  Q[n;s]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    ((\mforall{}m:\mBbbN{}.  Q[n  +  1;s.m@n])  {}\mRightarrow{}  Q[n;s]))
{}\mRightarrow{}  (\mforall{}alpha:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}m:\mBbbN{}.  B[m;alpha]))
{}\mRightarrow{}  \00D9(Q[0;\mlambda{}x.\mbot{}]))

Date html generated: 2017_04_17-AM-10_03_45
Last ObjectModification: 2017_02_27-PM-05_55_58

Theory : continuity

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