Nuprl Lemma : p-conditional-domain
∀[A,B:Type]. ∀f,g:A ⟶ (B + Top). ∀x:A. (↑can-apply([f?g];x) ⇐⇒ (↑can-apply(f;x)) ∨ (↑can-apply(g;x)))
Proof
Definitions occuring in Statement :
p-conditional: [f?g],
can-apply: can-apply(f;x),
assert: ↑b,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
union: left + right,
universe: Type
Definitions unfolded in proof :
can-apply: can-apply(f;x),
p-conditional: [f?g],
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
isl: isl(x),
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
istype-assert,
ifthenelse_wf,
btrue_wf,
bfalse_wf,
top_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
assert_witness,
istype-top,
bool_cases,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
inhabitedIsType,
hypothesis,
unionElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
unionEquality,
because_Cache,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
unionIsType,
functionIsType,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[A,B:Type].
\mforall{}f,g:A {}\mrightarrow{} (B + Top). \mforall{}x:A. (\muparrow{}can-apply([f?g];x) \mLeftarrow{}{}\mRightarrow{} (\muparrow{}can-apply(f;x)) \mvee{} (\muparrow{}can-apply(g;x)))
Date html generated:
2020_05_20-AM-08_06_11
Last ObjectModification:
2019_12_26-PM-04_07_22
Theory : general
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