Nuprl Lemma : fpf-compatible-join-cap
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f,g:a:A fp-> B[a]]. ∀[x:A]. ∀[z:B[x]].
f ⊕ g(x)?z = g(x)?f(x)?z ∈ B[x] supposing f || g
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-compatible: f || g,
fpf-cap: f(x)?z,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uimplies: b supposing a,
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,
uimplies: b supposing a,
fpf-compatible: f || g,
fpf-cap: f(x)?z,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
cand: A c∧ B,
top: Top
Lemmas referenced :
fpf-compatible_wf,
fpf_wf,
deq_wf,
fpf-dom_wf,
fpf-join_wf,
top_wf,
subtype-fpf2,
fpf-join-dom,
equal-wf-T-base,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
equal_wf,
squash_wf,
true_wf,
fpf-join-ap-left,
fpf-ap_wf,
subtype_rel_self,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
istype-assert,
uiff_transitivity,
assert_of_bnot,
fpf-join-ap-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
Error :memTop,
dependent_functionElimination,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
equalityElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
instantiate,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
cumulativity,
voidElimination,
unionIsType,
independent_pairFormation,
universeEquality,
voidEquality,
isect_memberEquality,
lambdaFormation,
functionExtensionality,
lambdaEquality,
inrFormation_alt,
inlFormation_alt
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f,g:a:A fp-> B[a]]. \mforall{}[x:A]. \mforall{}[z:B[x]].
f \moplus{} g(x)?z = g(x)?f(x)?z supposing f || g
Date html generated:
2020_05_20-AM-09_03_18
Last ObjectModification:
2019_12_26-PM-04_07_17
Theory : finite!partial!functions
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