Nuprl Lemma : fpf-join-assoc
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f,g,h:a:A fp-> B[a]]. (f ⊕ g ⊕ h = f ⊕ g ⊕ h ∈ 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 :
fpf-join: f ⊕ g,
fpf: a:A fp-> B[a],
fpf-ap: f(x),
fpf-cap: f(x)?z,
fpf-dom: x ∈ dom(f),
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
squash: ↓T,
all: ∀x:A. B[x],
true: True,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bor: p ∨bq,
bfalse: ff,
band: p ∧b q,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
l_member_wf,
list_wf,
deq_wf,
istype-universe,
append_assoc,
append_wf,
squash_wf,
true_wf,
filter_append,
bnot_wf,
deq-member_wf,
filter_wf5,
filter_filter,
bool_wf,
deq-member-append,
eqtt_to_assert,
assert-deq-member,
bfalse_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_imp_equal_bool,
istype-assert,
iff_transitivity,
assert_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-void,
member_filter,
member_append
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
productElimination,
thin,
dependent_pairEquality_alt,
functionIsType,
setIsType,
because_Cache,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
setElimination,
rename,
productIsType,
inhabitedIsType,
instantiate,
universeEquality,
Error :memTop,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
cumulativity,
voidElimination,
independent_pairFormation,
functionExtensionality_alt,
inlFormation_alt,
unionIsType,
inrFormation_alt,
dependent_set_memberEquality_alt
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f,g,h:a:A fp-> B[a]]. (f \moplus{} g \moplus{} h = f \moplus{} g \moplus{} h)
Date html generated:
2020_05_20-AM-09_02_33
Last ObjectModification:
2020_01_09-AM-00_05_17
Theory : finite!partial!functions
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