Nuprl Lemma : fpf-join-idempotent
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[eq:EqDecider(A)]. (f ⊕ f = f ∈ a:A fp-> B[a])
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
fpf-ap: f(x),
fpf-cap: f(x)?z,
fpf-dom: x ∈ dom(f),
pi1: fst(t),
pi2: snd(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
not: ¬A,
false: False,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
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
Lemmas referenced :
deq_wf,
fpf_wf,
filter_is_nil,
bnot_wf,
deq-member_wf,
l_all_iff,
not_wf,
assert_wf,
l_member_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
list-subtype,
false_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert-deq-member,
append_nil_sq,
subtype_rel_list,
top_wf,
bool_wf,
eqtt_to_assert,
strong-subtype-set3,
strong-subtype-self,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
independent_isectElimination,
dependent_functionElimination,
setEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
dependent_set_memberEquality,
independent_functionElimination,
lambdaFormation,
voidElimination,
addLevel,
impliesFunctionality,
independent_pairFormation,
voidEquality,
dependent_pairEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[eq:EqDecider(A)]. (f \moplus{} f = f)
Date html generated:
2018_05_21-PM-09_21_12
Last ObjectModification:
2018_02_09-AM-10_18_12
Theory : finite!partial!functions
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