Nuprl Lemma : test-model
∀[Dom:Type]. ∀[A,B:Dom ⟶ ℙ]. ∀[R:Dom ⟶ Dom ⟶ ℙ].
((∀x:Dom. ∃y:Dom. (R[x;y] ∧ (A[y] ∨ B[y])))
⇒ (∀x,y,z:Dom. ((R[x;y] ∧ R[x;z] ∧ A[y] ∧ A[z]) ⇒ A[x]))
⇒ (∀x,y,z:Dom. ((R[x;y] ∧ R[x;z] ∧ B[y] ∧ B[z]) ⇒ A[x]))
⇒ (∀x:Dom. A[x]))
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,
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s]
Lemmas referenced :
all_wf,
and_wf,
exists_wf,
or_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
unionElimination,
because_Cache,
independent_functionElimination,
independent_pairFormation,
lemma_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[Dom:Type]. \mforall{}[A,B:Dom {}\mrightarrow{} \mBbbP{}]. \mforall{}[R:Dom {}\mrightarrow{} Dom {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:Dom. \mexists{}y:Dom. (R[x;y] \mwedge{} (A[y] \mvee{} B[y])))
{}\mRightarrow{} (\mforall{}x,y,z:Dom. ((R[x;y] \mwedge{} R[x;z] \mwedge{} A[y] \mwedge{} A[z]) {}\mRightarrow{} A[x]))
{}\mRightarrow{} (\mforall{}x,y,z:Dom. ((R[x;y] \mwedge{} R[x;z] \mwedge{} B[y] \mwedge{} B[z]) {}\mRightarrow{} A[x]))
{}\mRightarrow{} (\mforall{}x:Dom. A[x]))
Date html generated:
2016_05_15-PM-03_19_25
Last ObjectModification:
2015_12_27-PM-01_03_52
Theory : general
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