Nuprl Lemma : equal-Sierpinski-bottom
∀[x:ℕ ⟶ 𝔹]. uiff(x = ⊥ ∈ (ℕ ⟶ 𝔹);∀n:ℕ. (¬↑(x n)))
Proof
Definitions occuring in Statement :
Sierpinski-bottom: ⊥,
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
Sierpinski-bottom: ⊥,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
assert_wf,
bfalse_wf,
top_wf,
nat_wf,
assert_functionality_wrt_uiff,
not_wf,
equal_wf,
bool_wf,
iff_imp_equal_bool,
false_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
isectElimination,
applyEquality,
lambdaEquality,
hypothesisEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
addLevel,
impliesFunctionality,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
functionEquality,
functionExtensionality,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. uiff(x = \mbot{};\mforall{}n:\mBbbN{}. (\mneg{}\muparrow{}(x n)))
Date html generated:
2019_10_31-AM-06_35_16
Last ObjectModification:
2015_12_28-AM-11_21_54
Theory : synthetic!topology
Home
Index