### Nuprl Lemma : increasing_is_id

`∀[k:ℕ]. ∀[f:ℕk ⟶ ℕk].  ∀[i:ℕk]. ((f i) = i ∈ ℤ) supposing increasing(f;k)`

Proof

Definitions occuring in Statement :  increasing: `increasing(f;k)` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` top: `Top` true: `True` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` increasing: `increasing(f;k)` sq_type: `SQType(T)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination Error :lambdaFormation_alt,  natural_numberEquality independent_isectElimination independent_functionElimination voidElimination Error :universeIsType,  sqequalRule Error :lambdaEquality_alt,  dependent_functionElimination Error :isect_memberEquality_alt,  axiomEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  productElimination Error :dependent_set_memberEquality_alt,  independent_pairFormation because_Cache functionExtensionality applyEquality closedConclusion Error :functionIsType,  equalityTransitivity equalitySymmetry unionElimination addEquality minusEquality Error :productIsType,  Error :dependent_pairFormation_alt,  baseApply baseClosed intEquality Error :equalityIsType1,  promote_hyp voidEquality isect_memberEquality lambdaEquality dependent_set_memberEquality lambdaFormation cumulativity instantiate sqequalIntensionalEquality dependent_pairFormation multiplyEquality

Latex:
\mforall{}[k:\mBbbN{}].  \mforall{}[f:\mBbbN{}k  {}\mrightarrow{}  \mBbbN{}k].    \mforall{}[i:\mBbbN{}k].  ((f  i)  =  i)  supposing  increasing(f;k)

Date html generated: 2019_06_20-AM-11_33_32
Last ObjectModification: 2018_10_18-PM-04_02_07

Theory : int_1

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