Nuprl Lemma : weak-antecedent-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} .
((Q1 ←==f== P1 ∧ Q2 ←==g== P2) ⇒ λe.((Q1 e) ∨ (Q2 e)) ←==[P1? f : g]== λe.((P1 e) ∨ (P2 e)))
Proof
Definitions occuring in Statement :
weak-antecedent-function: Q ←==f== P,
conditional: [P? f : g],
es-E: E,
event_ordering: EO,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
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]
Lemmas :
weak-antecedent-function_wf,
subtype_rel_dep_function,
es-E_wf,
set_wf,
decidable_wf,
event_ordering_wf,
or_wf,
equal_wf
\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\} .
((Q1 \mleftarrow{}==f== P1 \mwedge{} Q2 \mleftarrow{}==g== P2) {}\mRightarrow{} \mlambda{}e.((Q1 e) \mvee{} (Q2 e)) \mleftarrow{}==[P1? f : g]== \mlambda{}e.((P1 e) \mvee{} (P2 e)))
Date html generated:
2015_07_17-AM-09_02_09
Last ObjectModification:
2015_01_27-PM-00_56_34
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