### Nuprl Lemma : WCP_wf

`∀F:(ℕ+ ⟶ ℤ) ⟶ 𝔹. ∀f:ℕ+ ⟶ ℤ. ∀G:n:ℕ+ ⟶ {g:ℕ+ ⟶ ℤ| f = g ∈ (ℕ+n ⟶ ℤ)} .  (WCP(F;f;G) ∈ {n:ℕ+| F f = F (G n)} )`

Proof

Definitions occuring in Statement :  WCP: `WCP(F;f;G)` int_seg: `{i..j-}` nat_plus: `ℕ+` bool: `𝔹` all: `∀x:A. B[x]` member: `t ∈ T` set: `{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 :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` so_apply: `x[s]` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` exists: `∃x:A. B[x]` pi1: `fst(t)` weak-continuity-principle-nat+-int-bool-ext WCP: `WCP(F;f;G)`
Lemmas referenced :  weak-continuity-principle-nat+-int-bool-ext set_wf nat_plus_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat_plus false_wf subtype_rel_self bool_wf all_wf exists_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin instantiate extract_by_obid hypothesis functionEquality because_Cache introduction sqequalHypSubstitution isectElimination intEquality sqequalRule lambdaEquality natural_numberEquality setElimination rename hypothesisEquality applyEquality independent_isectElimination independent_pairFormation functionExtensionality setEquality productElimination dependent_set_memberEquality equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination

Latex:
\mforall{}F:(\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{})  {}\mrightarrow{}  \mBbbB{}.  \mforall{}f:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}.  \mforall{}G:n:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \{g:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}|  f  =  g\}  .    (WCP(F;f;G)  \mmember{}  \{n:\mBbbN{}\msupplus{}|  F  f  =  F  (G  n)\}  \000C)

Date html generated: 2017_09_29-PM-06_06_30
Last ObjectModification: 2017_07_08-PM-01_12_36

Theory : continuity

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