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
Lemmas :
es-E_wf,
antecedent-surjection_wf,
all_wf,
decidable_wf,
not_wf,
event_ordering_wf,
btrue_wf,
bfalse_wf,
equal_wf,
iff_wf,
assert_wf,
true_wf,
false_wf,
or_wf,
bool_wf,
subtype_rel_sets,
equal-wf-T-base,
bnot_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
set_wf,
isect_wf,
es-causl_wf,
sq_stable_from_decidable,
iff_weakening_equal,
squash_wf
\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:
2015_07_17-AM-09_10_10
Last ObjectModification:
2015_02_04-PM-06_30_36
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