Nuprl Lemma : strong-continuity2-implies-weak
∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕn ⟶ ℕ)) ⇒ ((F f) = (F g) ∈ ℕ)))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
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],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
istype-nat,
strong-continuity2-implies-weak-skolem,
implies-quotient-true,
nat_wf,
equal_wf,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
equal-wf-base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
decidable__le,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
istype-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :functionIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
because_Cache,
Error :inhabitedIsType,
hypothesisEquality,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
sqequalRule,
productEquality,
functionEquality,
natural_numberEquality,
applyEquality,
Error :lambdaEquality_alt,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
intEquality,
closedConclusion,
independent_functionElimination,
productElimination,
Error :productIsType,
Error :equalityIsType1,
Error :universeIsType,
Error :equalityIsType4,
Error :dependent_pairFormation_alt,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
approximateComputation,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :dependent_set_memberEquality_alt
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((f = g) {}\mRightarrow{} ((F f) = (F g))))
Date html generated:
2019_06_20-PM-02_51_45
Last ObjectModification:
2018_11_21-AM-09_56_24
Theory : continuity
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