### Nuprl Lemma : retraction-nat-nsub

`∀k:ℕ+. ∃r:ℕ ⟶ ℕk. ∀x:ℕk. ((r x) = x ∈ ℕk)`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat_plus: `ℕ+` nat: `ℕ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  so_apply: `x[s]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` less_than': `less_than'(a;b)` le: `A ≤ B` rev_implies: `P `` Q` not: `¬A` iff: `P `⇐⇒` Q` false: `False` assert: `↑b` bnot: `¬bb` guard: `{T}` sq_type: `SQType(T)` or: `P ∨ Q` bfalse: `ff` prop: `ℙ` lelt: `i ≤ j < k` int_seg: `{i..j-}` uimplies: `b supposing a` and: `P ∧ Q` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` implies: `P `` Q` nat_plus: `ℕ+` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` all: `∀x:A. B[x]`
Lemmas referenced :  nat_plus_wf int_seg_subtype_nat all_wf int_seg_wf nat_wf false_wf assert_of_bnot iff_weakening_uiff not_wf bnot_wf assert_wf iff_transitivity bool_subtype_base subtype_base_sq bool_cases_sqequal equal_wf eqff_to_assert less_than_wf le_wf and_wf assert_of_lt_int eqtt_to_assert bool_wf lt_int_wf
Rules used in proof :  functionExtensionality applyEquality impliesFunctionality voidElimination independent_functionElimination cumulativity instantiate dependent_functionElimination promote_hyp equalitySymmetry equalityTransitivity natural_numberEquality independent_pairFormation hypothesisEquality dependent_set_memberEquality independent_isectElimination productElimination sqequalRule equalityElimination unionElimination hypothesis because_Cache rename setElimination thin isectElimination sqequalHypSubstitution extract_by_obid introduction cut lambdaEquality dependent_pairFormation lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}k:\mBbbN{}\msupplus{}.  \mexists{}r:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}k.  \mforall{}x:\mBbbN{}k.  ((r  x)  =  x)

Date html generated: 2017_09_29-PM-05_47_03
Last ObjectModification: 2017_09_04-PM-00_14_44

Theory : arithmetic

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