Nuprl Lemma : red-black-example
∀[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 :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s1;s2],
and: P ∧ Q,
guard: {T}
Lemmas referenced :
subtype_rel_self,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
productIsType,
universeIsType,
unionIsType,
functionIsType,
because_Cache,
applyEquality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
universeEquality,
inhabitedIsType,
independent_functionElimination
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:
2020_05_20-AM-08_19_34
Last ObjectModification:
2020_02_01-AM-11_42_37
Theory : general
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