Nuprl Lemma : strong-subtype-implies
∀[A,B:Type]. (strong-subtype(A;B) ⇒ {∀b:B. ∀a:A. ((b = a ∈ B) ⇒ (b = a ∈ A))})
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
exists_wf,
strong-subtype_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
dependent_pairFormation,
because_Cache,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
productElimination,
sqequalRule,
dependent_set_memberEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
universeEquality,
isect_memberEquality
Latex:
\mforall{}[A,B:Type]. (strong-subtype(A;B) {}\mRightarrow{} \{\mforall{}b:B. \mforall{}a:A. ((b = a) {}\mRightarrow{} (b = a))\})
Date html generated:
2017_04_14-AM-07_36_46
Last ObjectModification:
2017_02_27-PM-03_09_05
Theory : subtype_1
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