Nuprl Lemma : strong-subtype-union
∀[A,B,C,D:Type]. (strong-subtype(A + B;C + D)) supposing (strong-subtype(B;D) and strong-subtype(A;C))
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
guard: {T},
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
sq_type: SQType(T),
all: ∀x:A. B[x],
true: True,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ
Lemmas referenced :
strong-subtype-implies,
subtype_rel_union,
subtype_base_sq,
int_subtype_base,
subtype_rel_transitivity,
exists_wf,
equal_wf,
istype-universe,
strong-subtype_witness,
strong-subtype_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
independent_isectElimination,
independent_pairFormation,
Error :lambdaEquality_alt,
setElimination,
rename,
unionElimination,
Error :inlEquality_alt,
sqequalRule,
applyLambdaEquality,
applyEquality,
promote_hyp,
natural_numberEquality,
instantiate,
cumulativity,
intEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
setEquality,
because_Cache,
Error :inrEquality_alt,
Error :setIsType,
Error :unionIsType,
Error :productIsType,
Error :equalityIsType1,
Error :inhabitedIsType,
unionEquality,
Error :universeIsType,
Error :isect_memberEquality_alt,
universeEquality,
lambdaEquality,
dependent_set_memberEquality,
dependent_pairFormation
Latex:
\mforall{}[A,B,C,D:Type].
(strong-subtype(A + B;C + D)) supposing (strong-subtype(B;D) and strong-subtype(A;C))
Date html generated:
2019_06_20-PM-00_27_59
Last ObjectModification:
2018_10_11-PM-04_13_33
Theory : subtype_1
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