Nuprl Lemma : subtype_rel_Formco
∀[A,B:Type]. Formco(A) ⊆r Formco(B) supposing A ⊆r B
Proof
Definitions occuring in Statement :
Formco: Formco(C),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
Formco: Formco(C),
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_apply: x[s],
subtype_rel: A ⊆r B,
type-monotone: Monotone(T.F[T])
Lemmas referenced :
corec-subtype-corec,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
subtype_rel_product,
subtype_rel_ifthenelse,
ifthenelse_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
productEquality,
atomEquality,
hypothesisEquality,
tokenEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
voidEquality,
universeEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[A,B:Type]. Formco(A) \msubseteq{}r Formco(B) supposing A \msubseteq{}r B
Date html generated:
2018_05_21-PM-11_26_31
Last ObjectModification:
2017_10_11-AM-11_25_32
Theory : PZF
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