### Nuprl Lemma : weak-continuity-truncated

`∀[T:{T:Type| (T ⊆r ℕ) ∧ (↓T)} ]`
`  ∀F:(ℕ ⟶ T) ⟶ ℕ. ⇃(∀f:ℕ ⟶ T. ∃n:ℕ. ∀g:ℕ ⟶ T. ((f = g ∈ (ℕn ⟶ T)) `` ((F f) = (F g) ∈ ℕ)))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` squash: `↓T` implies: `P `` Q` and: `P ∧ Q` true: `True` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` and: `P ∧ Q` so_apply: `x[s]` implies: `P `` Q` nat: `ℕ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` weak-continuity-skolem: `weak-continuity-skolem(T;F)` exists: `∃x:A. B[x]` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top`
Lemmas referenced :  weak-continuity-skolem-truncated nat_wf set_wf subtype_rel_wf squash_wf weak-continuity-skolem_wf all_wf exists_wf equal_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self implies-quotient-true decidable__le nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf le_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_functionElimination functionEquality hypothesis setElimination rename instantiate universeEquality sqequalRule lambdaEquality productEquality cumulativity functionExtensionality applyEquality because_Cache natural_numberEquality independent_isectElimination independent_pairFormation productElimination independent_functionElimination dependent_pairFormation unionElimination equalityTransitivity equalitySymmetry applyLambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll dependent_set_memberEquality

Latex:
\mforall{}[T:\{T:Type|  (T  \msubseteq{}r  \mBbbN{})  \mwedge{}  (\mdownarrow{}T)\}  ]
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  T)  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mforall{}f:\mBbbN{}  {}\mrightarrow{}  T.  \mexists{}n:\mBbbN{}.  \mforall{}g:\mBbbN{}  {}\mrightarrow{}  T.  ((f  =  g)  {}\mRightarrow{}  ((F  f)  =  (F  g))))

Date html generated: 2017_04_17-AM-09_54_06
Last ObjectModification: 2017_02_27-PM-05_48_50

Theory : continuity

Home Index