Nuprl Lemma : fpf-compatible-join-iff
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f,g,h:a:A fp-> B[a]].
uiff(h || f ⊕ g;h || f ∧ h || g) supposing f || g
Proof
Definitions occuring in Statement :
fpf-join: f ⊕ g,
fpf-compatible: f || g,
fpf: a:A fp-> B[a],
deq: EqDecider(T),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
fpf-compatible: f || g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
prop: ℙ,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
squash: ↓T,
true: True,
guard: {T},
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
fpf-compatible_wf,
fpf-join_wf,
fpf_wf,
deq_wf,
fpf-join-dom,
equal_wf,
squash_wf,
true_wf,
fpf-join-ap-left,
subtype_rel_self,
iff_weakening_equal,
fpf-join-ap,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
fpf-ap_wf,
bool_cases_sqequal,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
axiomEquality,
hypothesis,
productEquality,
extract_by_obid,
isectElimination,
applyEquality,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
inlFormation,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
inrFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f,g,h:a:A fp-> B[a]].
uiff(h || f \moplus{} g;h || f \mwedge{} h || g) supposing f || g
Date html generated:
2018_05_21-PM-09_28_29
Last ObjectModification:
2018_05_19-PM-04_38_27
Theory : finite!partial!functions
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