Nuprl Lemma : weak-antecedent-surjection-conditional2
∀es:EO
∀[P1,Q1,P2,Q2:E ⟶ ℙ].
∀dcd_P1:e:E ⟶ Dec(P1 e). ∀f:{e:E| P1 e} ⟶ {e:E| Q1 e} . ∀g:{e:E| P2 e} ⟶ {e:E| Q2 e} .
(∀e:E. Dec(Q1 e)) ⇒ (∀e:E. Dec(Q2 e)) ⇒ Q1 ←←= f== P1 ⇒ Q2 ←←= g== P2 ⇒ Q1 ∨ Q2 ←←= [P1? f : g]== P1 ∨ P2
supposing ∀e:E. ((P1 e) ⇒ (¬(P2 e)))
Proof
Definitions occuring in Statement :
weak-antecedent-surjection: Q ←←= f== P,
conditional: [P? f : g],
es-E: E,
event_ordering: EO,
predicate_or: P1 ∨ P2,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
predicate_or: P1 ∨ P2
Latex:
\mforall{}es:EO
\mforall{}[P1,Q1,P2,Q2:E {}\mrightarrow{} \mBbbP{}].
\mforall{}dcd$_{P1}$:e:E {}\mrightarrow{} Dec(P1 e). \mforall{}f:\{e:E| P1 e\} {}\mrightarrow{} \{e:E| Q1 e\} . \mforall{}g:\{e:E| P2 e\000C\} {}\mrightarrow{} \{e:E| Q2 e\} .
(\mforall{}e:E. Dec(Q1 e))
{}\mRightarrow{} (\mforall{}e:E. Dec(Q2 e))
{}\mRightarrow{} Q1 \mleftarrow{}\mleftarrow{}= f== P1
{}\mRightarrow{} Q2 \mleftarrow{}\mleftarrow{}= g== P2
{}\mRightarrow{} Q1 \mvee{} Q2 \mleftarrow{}\mleftarrow{}= [P1? f : g]== P1 \mvee{} P2
supposing \mforall{}e:E. ((P1 e) {}\mRightarrow{} (\mneg{}(P2 e)))
Date html generated:
2016_05_16-AM-10_18_17
Last ObjectModification:
2015_12_28-PM-09_22_55
Theory : new!event-ordering
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