Nuprl Lemma : in-bar-equal
∀[T:Type]. ∀x:bar-base(T). ∀b:T. (bar-equal(T;x;in-bar(b)) ⇐⇒ x↓b)
Proof
Definitions occuring in Statement :
bar-equal: bar-equal(T;x;y),
bar-converges: x↓a,
in-bar: in-bar(b),
bar-base: bar-base(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
bar-equal: bar-equal(T;x;y),
bar-converges: x↓a,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
bar-val: bar-val(n;x),
in-bar: in-bar(b)
Lemmas referenced :
bar-equal_wf,
in-bar_wf,
bar-converges_wf,
bar-base_wf,
false_wf,
le_wf,
unit_wf2,
equal_wf,
bar-val_wf,
in-bar-converges,
bar-converges-unique,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
universeEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
inlEquality,
unionEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}x:bar-base(T). \mforall{}b:T. (bar-equal(T;x;in-bar(b)) \mLeftarrow{}{}\mRightarrow{} x\mdownarrow{}b)
Date html generated:
2016_10_21-AM-09_47_40
Last ObjectModification:
2016_07_12-AM-05_07_45
Theory : co-recursion
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