Nuprl Lemma : alist-domain-first
∀[A:Type]
∀d:A List. ∀f1:a:{a:A| (a ∈ d)} ⟶ Top. ∀x:A. ∀eq:EqDecider(A).
((x ∈ d)
⇒ (∃i:ℕ||d||. ((∀j:ℕi. (¬((fst(map(λx.<x, f1 x>;d)[j])) = x ∈ A))) ∧ ((fst(map(λx.<x, f1 x>;d)[i])) = x ∈ A))))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
deq: EqDecider(T),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
top: Top,
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
and: P ∧ Q,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
subtype_rel: A ⊆r B,
le: A ≤ B,
pi1: fst(t)
Latex:
\mforall{}[A:Type]
\mforall{}d:A List. \mforall{}f1:a:\{a:A| (a \mmember{} d)\} {}\mrightarrow{} Top. \mforall{}x:A. \mforall{}eq:EqDecider(A).
((x \mmember{} d)
{}\mRightarrow{} (\mexists{}i:\mBbbN{}||d||
((\mforall{}j:\mBbbN{}i. (\mneg{}((fst(map(\mlambda{}x.<x, f1 x>d)[j])) = x))) \mwedge{} ((fst(map(\mlambda{}x.<x, f1 x>d)[i])) = x))))
Date html generated:
2016_05_16-AM-11_05_02
Last ObjectModification:
2016_01_17-PM-03_48_19
Theory : event-ordering
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