Nuprl Lemma : symmetrized_preorder
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (Preorder(T;x,y.R[x;y]) ⇒ EquivRel(T;a,b.Symmetrize(x,y.R[x;y];a;b)))
Proof
Definitions occuring in Statement :
symmetrize: Symmetrize(x,y.R[x; y];a;b),
preorder: Preorder(T;x,y.R[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
symmetrize: Symmetrize(x,y.R[x; y];a;b),
equiv_rel: EquivRel(T;x,y.E[x; y]),
preorder: Preorder(T;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
sym: Sym(T;x,y.E[x; y]),
subtype_rel: A ⊆r B,
trans: Trans(T;x,y.E[x; y]),
guard: {T}
Lemmas referenced :
refl_wf,
trans_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
independent_pairFormation,
hypothesis,
productEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
instantiate,
because_Cache,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (Preorder(T;x,y.R[x;y]) {}\mRightarrow{} EquivRel(T;a,b.Symmetrize(x,y.R[x;y];a;b)))
Date html generated:
2019_06_20-PM-00_28_58
Last ObjectModification:
2018_08_25-PM-10_14_33
Theory : rel_1
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