Nuprl Lemma : strong-subtype-b-union
∀[A,B:Type]. strong-subtype(A;A ⋃ B) ∧ strong-subtype(B;A ⋃ B) supposing ¬A ⋂ B
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
isect2: T1 ⋂ T2,
b-union: A ⋃ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
strong-subtype: strong-subtype(A;B),
subtype_rel: A ⊆r B,
not: ¬A,
implies: P ⇒ Q,
false: False,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
isect2: T1 ⋂ T2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q)
Lemmas referenced :
isect2_wf,
subtype_rel_set,
exists_wf,
equal_wf,
strong-subtype_witness,
b-union_wf,
not_wf,
strong-subtype-b-union-better,
strong-subtype_transitivity,
strong-subtype-void,
strong-subtype-ext-equal,
isect2_subtype_rel2,
isect2_subtype_rel,
bool_wf,
subtype_rel_b-union_iff,
subtype_rel_b-union-right,
subtype_rel_b-union-left
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
independent_functionElimination,
thin,
hypothesis,
voidElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_pairFormation,
voidEquality,
sqequalRule,
applyEquality,
independent_isectElimination,
productElimination,
independent_pairEquality,
because_Cache,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
unionElimination,
equalityElimination
Latex:
\mforall{}[A,B:Type]. strong-subtype(A;A \mcup{} B) \mwedge{} strong-subtype(B;A \mcup{} B) supposing \mneg{}A \mcap{} B
Date html generated:
2016_05_13-PM-04_12_00
Last ObjectModification:
2015_12_26-AM-11_21_32
Theory : subtype_1
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