Nuprl Lemma : infinite-domain-example
∀[A,D:Type]. ∀[R,Eq:D ⟶ D ⟶ ℙ].
((∀x,y,z:D. (R[x;y] ⇒ (R[y;z] ∨ Eq[y;z]) ⇒ R[x;z]))
⇒ (∀x:D. (R[x;x] ⇒ A))
⇒ (∀x:D. ∃y:D. R[x;y])
⇒ (∃m:D. ∀x:D. ((Eq[x;m] ⇒ A) ⇒ R[x;m]))
⇒ A)
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: 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],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
or: P ∨ Q
Lemmas referenced :
exists_wf,
all_wf,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
applyEquality,
lemma_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
inrFormation,
inlFormation
Latex:
\mforall{}[A,D:Type]. \mforall{}[R,Eq:D {}\mrightarrow{} D {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y,z:D. (R[x;y] {}\mRightarrow{} (R[y;z] \mvee{} Eq[y;z]) {}\mRightarrow{} R[x;z]))
{}\mRightarrow{} (\mforall{}x:D. (R[x;x] {}\mRightarrow{} A))
{}\mRightarrow{} (\mforall{}x:D. \mexists{}y:D. R[x;y])
{}\mRightarrow{} (\mexists{}m:D. \mforall{}x:D. ((Eq[x;m] {}\mRightarrow{} A) {}\mRightarrow{} R[x;m]))
{}\mRightarrow{} A)
Date html generated:
2016_05_15-PM-07_21_57
Last ObjectModification:
2015_12_27-AM-11_25_53
Theory : general
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