Nuprl Lemma : strong-continuity2-implies-uniform-continuity2-nat
∀F:(ℕ ⟶ 𝔹) ⟶ ℕ. ∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕn ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ ℕ))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
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,
subtype_rel: A ⊆r B,
nat: ℕ,
implies: P ⇒ Q,
uniform-continuity-pi: ucA(T;F;n),
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
exists: ∃x:A. B[x],
true: True,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
guard: {T},
quotient: x,y:A//B[x; y],
squash: ↓T,
sq_type: SQType(T),
uniform-continuity-pi-pi: ucpB(T;F;n)
Lemmas referenced :
nat_wf,
bool_wf,
strong-continuity2-implies-uniform-continuity-int,
uniform-continuity-pi-pi-prop2,
decidable__equal_nat,
exists_wf,
all_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
true_wf,
quotient_wf,
uniform-continuity-pi-pi_wf,
equiv_rel_true,
quotient-member-eq,
equal_subtype,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
le_wf,
equal-wf-base,
member_wf,
squash_wf,
prop-truncation-implies,
uniform-continuity-pi-pi-prop,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
uniform-continuity-pi_wf,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
functionExtensionality,
applyEquality,
hypothesisEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
independent_functionElimination,
because_Cache,
productElimination,
independent_pairFormation,
isectElimination,
natural_numberEquality,
independent_isectElimination,
intEquality,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
dependent_pairEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
applyLambdaEquality,
pointwiseFunctionality,
pertypeElimination,
productEquality,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
independent_pairEquality,
equalityElimination,
axiomEquality
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((f = g) {}\mRightarrow{} ((F f) = (F g)))
Date html generated:
2017_04_17-AM-09_59_33
Last ObjectModification:
2017_02_27-PM-05_52_48
Theory : continuity
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