Nuprl Lemma : bottom-pair-member-approx-type
∀[A,B:Type]. (A ⇒ B ⇒ (<⊥, ⊥> ∈ approx-type(A × B)))
Proof
Definitions occuring in Statement :
approx-type: approx-type(T),
bottom: ⊥,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
false: False,
not: ¬A,
cand: A c∧ B,
exists: ∃x:A. B[x]
Lemmas referenced :
approx-type_wf,
member-approx-type,
equal-wf-base,
sqle_wf_base,
is-exception_wf,
has-value_wf_base,
exception-not-bottom,
bottom_diverge
Rules used in proof :
isect_memberEquality,
universeEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
dependent_functionElimination,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
cumulativity,
productEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
pointwiseFunctionalityForEquality,
rename,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination,
productElimination,
baseClosed,
imageMemberEquality,
independent_pairEquality,
independent_pairFormation,
voidElimination,
independent_functionElimination,
sqleRule,
divergentSqle,
closedConclusion,
baseApply,
dependent_pairFormation
Latex:
\mforall{}[A,B:Type]. (A {}\mRightarrow{} B {}\mRightarrow{} (<\mbot{}, \mbot{}> \mmember{} approx-type(A \mtimes{} B)))
Date html generated:
2018_05_21-PM-00_05_27
Last ObjectModification:
2017_12_30-PM-02_28_40
Theory : partial_1
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