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,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
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
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:
2016_05_16-AM-11_05_59
Last ObjectModification:
2015_12_29-AM-09_18_30
Theory : event-ordering
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