Nuprl Lemma : es-bound-list2
∀es:EO
∀[T:Type]
∀i:Id
∀[P:T ⟶ ℙ]
∀L:T List
∀[Q:E ⟶ {x:T| (x ∈ L)} ⟶ ℙ]
((∀x:T. Dec(P[x]))
⇒ (∀x∈L.P[x] ⇒ (∃e:E. Q[e;x]))
⇒ ∃e'@i.True supposing ¬(∃x∈L. P[x])
⇒ ∃e'@i.(∀x∈L.P[x] ⇒ (∃e:E. (e ≤loc e' ∧ Q[e;x])))
supposing (∀x∈L.∀e:E. (Q[e;x] ⇒ (loc(e) = i ∈ Id))))
Proof
Definitions occuring in Statement :
existse-at: ∃e@i.P[e],
es-le: e ≤loc e' ,
es-loc: loc(e),
es-E: E,
event_ordering: EO,
Id: Id,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
l_all: (∀x∈L.P[x]),
so_apply: x[s1;s2],
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Latex:
\mforall{}es:EO
\mforall{}[T:Type]
\mforall{}i:Id
\mforall{}[P:T {}\mrightarrow{} \mBbbP{}]
\mforall{}L:T List
\mforall{}[Q:E {}\mrightarrow{} \{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]
((\mforall{}x:T. Dec(P[x]))
{}\mRightarrow{} (\mforall{}x\mmember{}L.P[x] {}\mRightarrow{} (\mexists{}e:E. Q[e;x]))
{}\mRightarrow{} \mexists{}e'@i.True supposing \mneg{}(\mexists{}x\mmember{}L. P[x])
{}\mRightarrow{} \mexists{}e'@i.(\mforall{}x\mmember{}L.P[x] {}\mRightarrow{} (\mexists{}e:E. (e \mleq{}loc e' \mwedge{} Q[e;x])))
supposing (\mforall{}x\mmember{}L.\mforall{}e:E. (Q[e;x] {}\mRightarrow{} (loc(e) = i))))
Date html generated:
2016_05_16-AM-09_50_36
Last ObjectModification:
2016_01_17-PM-01_23_33
Theory : new!event-ordering
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