Nuprl Lemma : fpf-union-contains2
∀[A,V:Type]. ∀[B:A ⟶ Type].
∀eq:EqDecider(A). ∀f,g:x:A fp-> B[x] List.
∀x:A. ∀R:(V List) ⟶ V ⟶ 𝔹. (fpf-union-compatible(A;V;x.B[x];eq;R;f;g) ⇒ g(x)?[] ⊆ fpf-union(f;g;eq;R;x))
supposing fpf-single-valued(A;eq;x.B[x];V;g)
supposing ∀a:A. (B[a] ⊆r V)
Proof
Definitions occuring in Statement :
fpf-single-valued: fpf-single-valued(A;eq;x.B[x];V;g),
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g),
fpf-union: fpf-union(f;g;eq;R;x),
fpf-cap: f(x)?z,
fpf: a:A fp-> B[a],
l_contains: A ⊆ B,
nil: [],
list: T List,
deq: EqDecider(T),
bool: 𝔹,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
fpf-single-valued: fpf-single-valued(A;eq;x.B[x];V;g),
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
fpf-union: fpf-union(f;g;eq;R;x),
fpf-cap: f(x)?z,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
l_contains: A ⊆ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
cand: A c∧ B,
fpf-union-compatible: fpf-union-compatible(A;C;x.B[x];eq;R;f;g),
label: ...$L... t
Lemmas referenced :
equal_wf,
l_member_wf,
fpf-ap_wf,
list_wf,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_contains_weakening,
l_contains_nil,
nil_wf,
fpf-union-compatible_wf,
fpf-single-valued_wf,
fpf_wf,
deq_wf,
all_wf,
subtype_rel_wf,
l_all_iff,
append_wf,
filter_wf5,
subtype_rel_list,
subtype_rel_dep_function,
subtype_rel_transitivity,
subtype_rel_self,
set_wf,
member_append,
decidable__assert,
member_filter,
or_wf,
l_member_subtype,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
rename,
lambdaFormation,
extract_by_obid,
isectElimination,
cumulativity,
applyEquality,
functionExtensionality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
functionEquality,
universeEquality,
setElimination,
setEquality,
inrFormation,
independent_pairFormation,
inlFormation,
productEquality,
addLevel,
orFunctionality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A,V:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
\mforall{}eq:EqDecider(A). \mforall{}f,g:x:A fp-> B[x] List.
\mforall{}x:A. \mforall{}R:(V List) {}\mrightarrow{} V {}\mrightarrow{} \mBbbB{}.
(fpf-union-compatible(A;V;x.B[x];eq;R;f;g) {}\mRightarrow{} g(x)?[] \msubseteq{} fpf-union(f;g;eq;R;x))
supposing fpf-single-valued(A;eq;x.B[x];V;g)
supposing \mforall{}a:A. (B[a] \msubseteq{}r V)
Date html generated:
2018_05_21-PM-09_18_31
Last ObjectModification:
2018_02_09-AM-10_17_10
Theory : finite!partial!functions
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