### Nuprl Lemma : monotone-bar-induction4

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

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` seq-add: `s.x@n` int_upper: `{i...}` 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` and: `P ∧ Q` true: `True` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` and: `P ∧ Q` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` so_lambda: `λ2x.t[x]` int_upper: `{i...}` so_apply: `x[s1;s2]` sq_stable: `SqStable(P)` squash: `↓T` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` so_lambda: `λ2x y.t[x; y]` guard: `{T}` isl: `isl(x)` 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` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` true: `True` outl: `outl(x)` less_than: `a < b` subtract: `n - m` seq-add: `s.x@n` nequal: `a ≠ b ∈ T ` seq-adjoin: `s++t` seq-append: `seq-append(n;m;s1;s2)` cand: `A c∧ B` quotient: `x,y:A//B[x; y]`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  rename cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin Error :lambdaEquality_alt,  productEquality applyEquality hypothesisEquality isectElimination hypothesis because_Cache natural_numberEquality setElimination independent_isectElimination sqequalRule independent_pairFormation functionEquality closedConclusion independent_functionElimination imageMemberEquality baseClosed imageElimination Error :dependent_set_memberEquality_alt,  addEquality unionElimination approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination Error :universeIsType,  Error :functionIsType,  productElimination Error :inhabitedIsType,  Error :productIsType,  Error :equalityIstype,  universeEquality Error :unionIsType,  instantiate equalityTransitivity equalitySymmetry intEquality baseApply sqequalBase hyp_replacement Error :functionExtensionality_alt,  equalityElimination promote_hyp cumulativity applyLambdaEquality Error :setIsType,  minusEquality functionExtensionality int_eqReduceTrueSq int_eqReduceFalseSq lessCases Error :isect_memberFormation_alt,  axiomSqEquality Error :isectIsTypeImplies,  pointwiseFunctionality pertypeElimination

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

Date html generated: 2019_06_20-PM-02_55_53
Last ObjectModification: 2018_12_06-PM-11_35_00

Theory : continuity

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