Nuprl Lemma : red-black-example-ext
∀[A,D:Type]. ∀[Red,Black:D ⟶ ℙ]. ∀[R:D ⟶ D ⟶ ℙ].
((∀x:D. (Red[x] ∨ Black[x]))
⇒ (∀x,y,z:D. (R[x;y] ⇒ R[y;z] ⇒ R[x;z]))
⇒ (∀x:D. (R[x;x] ⇒ A))
⇒ (∀x:D. (Red[x] ⇒ (∃y:D. (Black[y] ∧ R[x;y]))))
⇒ (∀x:D. (Black[x] ⇒ (∃y:D. (Red[y] ∧ R[x;y]))))
⇒ (∃m:D. ((∀x:D. (Red[x] ⇒ R[x;m])) ∨ (∀x:D. (Black[x] ⇒ R[x;m]))))
⇒ A)
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
spreadn: spread3,
red-black-example
Lemmas referenced :
red-black-example
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A,D:Type]. \mforall{}[Red,Black:D {}\mrightarrow{} \mBbbP{}]. \mforall{}[R:D {}\mrightarrow{} D {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:D. (Red[x] \mvee{} Black[x]))
{}\mRightarrow{} (\mforall{}x,y,z:D. (R[x;y] {}\mRightarrow{} R[y;z] {}\mRightarrow{} R[x;z]))
{}\mRightarrow{} (\mforall{}x:D. (R[x;x] {}\mRightarrow{} A))
{}\mRightarrow{} (\mforall{}x:D. (Red[x] {}\mRightarrow{} (\mexists{}y:D. (Black[y] \mwedge{} R[x;y]))))
{}\mRightarrow{} (\mforall{}x:D. (Black[x] {}\mRightarrow{} (\mexists{}y:D. (Red[y] \mwedge{} R[x;y]))))
{}\mRightarrow{} (\mexists{}m:D. ((\mforall{}x:D. (Red[x] {}\mRightarrow{} R[x;m])) \mvee{} (\mforall{}x:D. (Black[x] {}\mRightarrow{} R[x;m]))))
{}\mRightarrow{} A)
Date html generated:
2018_05_21-PM-08_55_37
Last ObjectModification:
2018_05_19-PM-05_07_58
Theory : general
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