Nuprl Lemma : combine-antecedent-surjections
∀es:EO
∀[A,B,P,Q:E ⟶ ℙ].
((∀e:E. Dec(P e))
⇒ (∀e:E. Dec(A e))
⇒ (∀e:E. Dec(B e))
⇒ (∀f:{e:E| A e} ⟶ {e:E| P e} . ∀g:{e:E| B e} ⟶ {e:E| Q e} .
(P ←⟵ f── A
⇒ Q ←⟵ g── B
⇒ (∃h:{e:E| (A e) ∨ (B e)} ⟶ {e:E| (P e) ∨ (Q e)}
(λe.((P e) ∨ (Q e)) ←⟵ h── λe.((A e) ∨ (B e))
∧ (∀e:{e:E| (A e) ∨ (B e)} . ((A e) ⇒ ((h e) = (f e) ∈ E)))
∧ (∀e:{e:E| (A e) ∨ (B e)} . (h e) = (g e) ∈ E supposing ¬(A e))))))) supposing
((∀e:E. (¬((P e) ∧ (Q e)))) and
(∀e:E. (¬((A e) ∧ (B e)))))
Proof
Definitions occuring in Statement :
antecedent-surjection: Q ←⟵ f── P,
es-E: E,
event_ordering: EO,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
bfalse: ff,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
cand: A c∧ B,
sq_type: SQType(T),
bnot: ¬bb,
antecedent-surjection: Q ←⟵ f── P,
antecedent-function: Q ⟵─f── P,
sq_stable: SqStable(P),
squash: ↓T
Latex:
\mforall{}es:EO
\mforall{}[A,B,P,Q:E {}\mrightarrow{} \mBbbP{}].
((\mforall{}e:E. Dec(P e))
{}\mRightarrow{} (\mforall{}e:E. Dec(A e))
{}\mRightarrow{} (\mforall{}e:E. Dec(B e))
{}\mRightarrow{} (\mforall{}f:\{e:E| A e\} {}\mrightarrow{} \{e:E| P e\} . \mforall{}g:\{e:E| B e\} {}\mrightarrow{} \{e:E| Q e\} .
(P \mleftarrow{}\mleftarrow{}{} f{}{} A
{}\mRightarrow{} Q \mleftarrow{}\mleftarrow{}{} g{}{} B
{}\mRightarrow{} (\mexists{}h:\{e:E| (A e) \mvee{} (B e)\} {}\mrightarrow{} \{e:E| (P e) \mvee{} (Q e)\}
(\mlambda{}e.((P e) \mvee{} (Q e)) \mleftarrow{}\mleftarrow{}{} h{}{} \mlambda{}e.((A e) \mvee{} (B e))
\mwedge{} (\mforall{}e:\{e:E| (A e) \mvee{} (B e)\} . ((A e) {}\mRightarrow{} ((h e) = (f e))))
\mwedge{} (\mforall{}e:\{e:E| (A e) \mvee{} (B e)\} . (h e) = (g e) supposing \mneg{}(A e))))))) supposing
((\mforall{}e:E. (\mneg{}((P e) \mwedge{} (Q e)))) and
(\mforall{}e:E. (\mneg{}((A e) \mwedge{} (B e)))))
Date html generated:
2016_05_16-AM-10_35_59
Last ObjectModification:
2016_01_17-PM-01_33_13
Theory : new!event-ordering
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