### Nuprl Lemma : increasing_implies

`∀[k:ℕ]. ∀[f:ℕk ⟶ ℤ].  {∀[x,y:ℕk].  f x < f y supposing x < y} supposing increasing(f;k)`

Proof

Definitions occuring in Statement :  increasing: `increasing(f;k)` int_seg: `{i..j-}` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` guard: `{T}` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  guard: `{T}` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` increasing: `increasing(f;k)` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` false: `False` and: `P ∧ Q` ge: `i ≥ j ` le: `A ≤ B` cand: `A c∧ B` less_than: `a < b` squash: `↓T` lelt: `i ≤ j < k` true: `True` less_than': `less_than'(a;b)` subtype_rel: `A ⊆r B` uiff: `uiff(P;Q)` rev_implies: `P `` Q` not: `¬A` iff: `P `⇐⇒` Q` or: `P ∨ Q` decidable: `Dec(P)` sq_type: `SQType(T)` top: `Top` subtract: `n - m` sq_stable: `SqStable(P)` rev_uimplies: `rev_uimplies(P;Q)` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat_plus: `ℕ+`
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation_alt introduction cut sqequalHypSubstitution hypothesis extract_by_obid isectElimination thin setElimination rename hypothesisEquality isect_memberEquality_alt applyEquality independent_isectElimination isectIsTypeImplies inhabitedIsType because_Cache universeIsType natural_numberEquality functionIsType lambdaFormation_alt intWeakElimination independent_pairFormation productElimination imageElimination independent_functionElimination voidElimination lambdaEquality_alt dependent_functionElimination functionIsTypeImplies lambdaEquality unionElimination cumulativity instantiate promote_hyp equalityTransitivity minusEquality multiplyEquality addEquality voidEquality isect_memberEquality dependent_set_memberEquality baseClosed intEquality functionExtensionality applyLambdaEquality equalitySymmetry hyp_replacement lambdaFormation imageMemberEquality closedConclusion dependent_set_memberEquality_alt Error :memTop,  productIsType dependent_pairFormation_alt equalityIstype baseApply sqequalBase

Latex:
\mforall{}[k:\mBbbN{}].  \mforall{}[f:\mBbbN{}k  {}\mrightarrow{}  \mBbbZ{}].    \{\mforall{}[x,y:\mBbbN{}k].    f  x  <  f  y  supposing  x  <  y\}  supposing  increasing(f;k)

Date html generated: 2020_05_19-PM-09_36_05
Last ObjectModification: 2020_01_04-PM-07_56_48

Theory : int_1

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