Nuprl Lemma : fpf-decompose
∀[A:Type]
∀eq:EqDecider(A)
∀[B:A ⟶ Type]
∀f:a:A fp-> B[a]
∃g:a:A fp-> B[a]
∃a:A
∃b:B[a]
((f ⊆ g ⊕ a : b ∧ g ⊕ a : b ⊆ f)
∧ (∀a':A. ¬(a' = a ∈ A) supposing ↑a' ∈ dom(g))
∧ ||fpf-domain(g)|| < ||fpf-domain(f)||)
supposing 0 < ||fpf-domain(f)||
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-join: f ⊕ g,
fpf-sub: f ⊆ g,
fpf-domain: fpf-domain(f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
length: ||as||,
deq: EqDecider(T),
assert: ↑b,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
le: A ≤ B,
fpf: a:A fp-> B[a],
fpf-domain: fpf-domain(f),
pi1: fst(t),
eqof: eqof(d),
fpf-dom: x ∈ dom(f),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
less_than': less_than'(a;b),
cons: [a / b],
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
cand: A c∧ B,
fpf-compatible: f || g,
fpf-sub: f ⊆ g,
rev_uimplies: rev_uimplies(P;Q),
true: True,
guard: {T},
sq_type: SQType(T),
btrue: tt
Latex:
\mforall{}[A:Type]
\mforall{}eq:EqDecider(A)
\mforall{}[B:A {}\mrightarrow{} Type]
\mforall{}f:a:A fp-> B[a]
\mexists{}g:a:A fp-> B[a]
\mexists{}a:A
\mexists{}b:B[a]
((f \msubseteq{} g \moplus{} a : b \mwedge{} g \moplus{} a : b \msubseteq{} f)
\mwedge{} (\mforall{}a':A. \mneg{}(a' = a) supposing \muparrow{}a' \mmember{} dom(g))
\mwedge{} ||fpf-domain(g)|| < ||fpf-domain(f)||)
supposing 0 < ||fpf-domain(f)||
Date html generated:
2016_05_16-AM-11_30_37
Last ObjectModification:
2016_01_17-PM-03_58_51
Theory : event-ordering
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