Nuprl Lemma : strong-continuity2-implies-weak-skolem-cantor-nat
∀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],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
pi1: fst(t),
isl: isl(x),
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
quotient: x,y:A//B[x; y],
squash: ↓T
Lemmas referenced :
nat_wf,
bool_wf,
strong-continuity2-no-inner-squash-cantor2,
exists_wf,
int_seg_wf,
unit_wf2,
all_wf,
equal_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
isect_wf,
assert_wf,
isl_wf,
and_wf,
btrue_wf,
subtype_base_sq,
bool_subtype_base,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
true_wf,
quotient_wf,
equiv_rel_true,
quotient-member-eq,
equal-wf-base,
member_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
unionEquality,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
independent_pairFormation,
inlEquality,
dependent_pairFormation,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate,
cumulativity,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
imageElimination,
universeEquality,
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:
2017_04_17-AM-09_57_00
Last ObjectModification:
2017_02_27-PM-05_51_30
Theory : continuity
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