Nuprl Lemma : bar-converges-unique
∀[T:Type]. ∀[x:bar-base(T)]. ∀[a,b:T]. (x↓a ⇒ x↓b ⇒ (a = b ∈ T))
Proof
Definitions occuring in Statement :
bar-converges: x↓a,
bar-base: bar-base(T),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
bar-converges: x↓a,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
all: ∀x:A. B[x],
uimplies: b supposing a,
true: True,
btrue: tt,
isl: isl(x),
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
bar-base_wf,
bar-val_wf,
unit_wf2,
equal_wf,
nat_wf,
exists_wf,
decidable__le,
bar-val-stable,
isl_wf,
assert_wf,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel
Rules used in proof :
universeEquality,
because_Cache,
isect_memberEquality,
axiomEquality,
dependent_functionElimination,
inlEquality,
hypothesisEquality,
cumulativity,
unionEquality,
lambdaEquality,
isectElimination,
extract_by_obid,
hypothesis,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
unionElimination,
rename,
setElimination,
independent_isectElimination,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
natural_numberEquality,
applyEquality,
independent_pairFormation,
voidElimination,
independent_functionElimination,
addEquality,
voidEquality,
intEquality,
minusEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:bar-base(T)]. \mforall{}[a,b:T]. (x\mdownarrow{}a {}\mRightarrow{} x\mdownarrow{}b {}\mRightarrow{} (a = b))
Date html generated:
2019_06_20-PM-00_37_02
Last ObjectModification:
2018_09_16-PM-01_32_22
Theory : co-recursion
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