Nuprl Lemma : strong-subtype-b-union-better
∀[A,B:Type]. strong-subtype(A;A ⋃ B) supposing strong-subtype(A ⋂ B;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],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
subtype_rel: A ⊆r B,
b-union: A ⋃ B,
tunion: ⋃x:A.B[x],
bool: 𝔹,
unit: Unit,
ifthenelse: if b then t else f fi ,
pi2: snd(t),
exists: ∃x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
btrue: tt,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
guard: {T},
all: ∀x:A. B[x],
pi1: fst(t),
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
isect2: T1 ⋂ T2,
it: ⋅,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
subtype_rel_b-union-left,
b-union_wf,
strong-subtype_witness,
strong-subtype_wf,
isect2_wf,
istype-universe,
btrue_wf,
ifthenelse_wf,
pi2_wf,
top_wf,
istype-top,
istype-void,
pair-eta,
subtype_rel_product,
bool_wf,
pi1_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
strong-subtype-implies,
equal_wf,
squash_wf,
true_wf,
isect2_subtype_rel,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_pairFormation,
Error :lambdaEquality_alt,
setElimination,
rename,
imageElimination,
productElimination,
unionElimination,
equalityElimination,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
Error :setIsType,
Error :universeIsType,
Error :productIsType,
Error :equalityIstype,
because_Cache,
applyEquality,
independent_functionElimination,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
instantiate,
universeEquality,
imageEqInduction,
baseClosed,
Error :dependent_pairEquality_alt,
independent_pairEquality,
voidElimination,
independent_isectElimination,
Error :lambdaFormation_alt,
applyLambdaEquality,
dependent_functionElimination,
cumulativity,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}[A,B:Type]. strong-subtype(A;A \mcup{} B) supposing strong-subtype(A \mcap{} B;B)
Date html generated:
2019_06_20-PM-00_28_04
Last ObjectModification:
2019_01_02-PM-03_17_43
Theory : subtype_1
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