### Nuprl Lemma : int-int-retraction-reals

`∃r:(ℤ ⟶ ℤ) ⟶ ℝ. ∀x:ℝ. (x = (r (λi.if i <z 1 then x 1 else x i fi )))`

Proof

Definitions occuring in Statement :  req: `x = y` real: `ℝ` ifthenelse: `if b then t else f fi ` lt_int: `i <z j` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` prop: `ℙ` int_upper: `{i...}` le: `A ≤ B` false: `False` not: `¬A` implies: `P `` Q` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` real: `ℝ` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` decidable: `Dec(P)` top: `Top`
Lemmas referenced :  real-regular less_than_wf real_wf int-int-retraction-reals-1 false_wf le_wf all_wf squash_wf true_wf req_wf iff_weakening_equal req-iff-bdd-diff accelerate_wf regular-int-seq_wf nat_plus_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot decidable__lt not-lt-2 add_functionality_wrt_le add-commutes zero-add le-add-cancel trivial-bdd-diff bdd-diff_functionality bdd-diff_weakening accelerate-bdd-diff
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation imageMemberEquality baseClosed hypothesis dependent_functionElimination productElimination dependent_pairFormation applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality because_Cache independent_isectElimination independent_functionElimination setElimination rename functionExtensionality intEquality unionElimination equalityElimination promote_hyp instantiate voidElimination isect_memberEquality voidEquality

Latex:
\mexists{}r:(\mBbbZ{}  {}\mrightarrow{}  \mBbbZ{})  {}\mrightarrow{}  \mBbbR{}.  \mforall{}x:\mBbbR{}.  (x  =  (r  (\mlambda{}i.if  i  <z  1  then  x  1  else  x  i  fi  )))

Date html generated: 2017_10_03-AM-10_07_18
Last ObjectModification: 2017_07_28-AM-08_53_12

Theory : reals

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