Nuprl Lemma : weak-antecedent-surjection-conditional
∀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
⇒ λe.((Q1 e) ∨ (Q2 e)) ←←= [P1? f : g]== λe.((P1 e) ∨ (P2 e))
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,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x]
Lemmas :
es-E_wf,
weak-antecedent-surjection_wf,
subtype_rel_dep_function,
set_wf,
all_wf,
decidable_wf,
not_wf,
event_ordering_wf,
weak-antecedent-conditional,
or_wf,
decidable__or,
equal_wf,
subtype_rel_sets
\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{} \mlambda{}e.((Q1 e) \mvee{} (Q2 e)) \mleftarrow{}\mleftarrow{}= [P1? f : g]== \mlambda{}e.((P1 e) \mvee{} (P2 e))
supposing \mforall{}e:E. ((P1 e) {}\mRightarrow{} (\mneg{}(P2 e)))
Date html generated:
2015_07_17-AM-09_02_45
Last ObjectModification:
2015_01_27-PM-00_56_27
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