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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