### Nuprl Lemma : weak-continuity-rel-fun

`∀P:(ℕ ⟶ ℕ) ⟶ ℙ. ((∀f:ℕ ⟶ ℕ. ⇃(P f)) `` (∀f:ℕ ⟶ ℕ. ⇃(∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) `` (P g)))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  guard: `{T}` not: `¬A` false: `False` less_than': `less_than'(a;b)` and: `P ∧ Q` le: `A ≤ B` nat: `ℕ` exists: `∃x:A. B[x]` so_apply: `x[s]` uimplies: `b supposing a` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` uall: `∀[x:A]. B[x]` prop: `ℙ` member: `t ∈ T` implies: `P `` Q` all: `∀x:A. B[x]`
Lemmas referenced :  subtype_rel_self int_seg_subtype_nat subtype_rel_dep_function int_seg_wf equal_wf le_wf false_wf exists_wf implies-quotient-true weak-continuity-rel equiv_rel_true true_wf quotient_wf all_wf nat_wf
Rules used in proof :  productElimination rename setElimination independent_pairFormation natural_numberEquality dependent_set_memberEquality dependent_pairFormation independent_functionElimination dependent_functionElimination universeEquality cumulativity independent_isectElimination hypothesisEquality functionExtensionality applyEquality lambdaEquality sqequalRule because_Cache thin isectElimination sqequalHypSubstitution hypothesis extract_by_obid introduction cut functionEquality lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}P:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.  ((\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(P  f))  {}\mRightarrow{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}k:\mBbbN{}.  \mforall{}g:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  ((f  =  g)  {}\mRightarrow{}  (P  g)))))

Date html generated: 2017_04_17-AM-10_02_29
Last ObjectModification: 2017_04_15-PM-05_16_11

Theory : continuity

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