Nuprl Lemma : free-from-atom-bool-subtype
∀[a:Atom1]. ∀[T:Type]. ∀[n:T]. a#n:T supposing T ⊆r 𝔹
Proof
Definitions occuring in Statement :
free-from-atom: a#x:T,
atom: Atom$n,
bool: 𝔹,
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,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
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,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False,
not: ¬A,
true: True
Lemmas referenced :
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
subtype_rel_wf,
true_wf,
equal-wf-T-base,
false_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
thin,
extract_by_obid,
lambdaFormation,
because_Cache,
unionElimination,
equalityElimination,
isectElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
freeFromAtomAxiom,
isect_memberEquality,
universeEquality,
atomnEquality,
freeFromAtomTriviality,
baseClosed,
natural_numberEquality
Latex:
\mforall{}[a:Atom1]. \mforall{}[T:Type]. \mforall{}[n:T]. a\#n:T supposing T \msubseteq{}r \mBbbB{}
Date html generated:
2017_04_14-AM-07_31_00
Last ObjectModification:
2017_02_27-PM-02_59_39
Theory : bool_1
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