### Nuprl Lemma : strong-continuity2-implies-weak-skolem2

`∀F:(ℕ ⟶ 𝔹) ⟶ ℕ. ⇃(∃M:(ℕ ⟶ 𝔹) ⟶ ℕ. ∀f,g:ℕ ⟶ 𝔹.  ((f = g ∈ (ℕM f ⟶ 𝔹)) `` ((F f) = (F g) ∈ ℕ)))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` bool: `𝔹` 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 :  all: `∀x:A. B[x]` weak-continuity-skolem: `weak-continuity-skolem(T;F)` uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` equipollent: `A ~ B` exists: `∃x:A. B[x]` pi1: `fst(t)` and: `P ∧ Q` cand: `A c∧ B` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` prop: `ℙ` squash: `↓T` subtype_rel: `A ⊆r B` guard: `{T}`
Lemmas referenced :  equipollent_inversion equipollent-two istype-nat bool_wf compose_wf nat_wf int_seg_wf Kleene-M_wf int_seg_subtype_nat istype-false decidable__le full-omega-unsat intformnot_wf intformle_wf itermConstant_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_le_lemma int_term_value_constant_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma istype-le istype-less_than subtype_rel_wf squash_wf implies-quotient-true basic-strong-continuity_wf weak-continuity-skolem_wf basic-implies-strong-continuity2 strong-continuity2-weak-skolem weak-continuity-skolem_functionality
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache independent_functionElimination hypothesis rename Error :functionIsType,  Error :universeIsType,  Error :lambdaEquality_alt,  applyEquality hypothesisEquality closedConclusion natural_numberEquality productElimination sqequalRule Error :inhabitedIsType,  Error :equalityIstype,  equalityTransitivity equalitySymmetry dependent_functionElimination Error :dependent_set_memberEquality_alt,  independent_isectElimination independent_pairFormation unionElimination approximateComputation Error :dependent_pairFormation_alt,  Error :isect_memberEquality_alt,  voidElimination Error :productIsType,  imageMemberEquality baseClosed

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}M:(\mBbbN{}  {}\mrightarrow{}  \mBbbB{})  {}\mrightarrow{}  \mBbbN{}.  \mforall{}f,g:\mBbbN{}  {}\mrightarrow{}  \mBbbB{}.    ((f  =  g)  {}\mRightarrow{}  ((F  f)  =  (F  g))))

Date html generated: 2019_06_20-PM-02_51_42
Last ObjectModification: 2019_02_09-PM-11_57_48

Theory : continuity

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