Nuprl Lemma : fun-equiv-rel
∀[X,A:Type]. ∀[E:A ⟶ A ⟶ ℙ]. (EquivRel(A;a,b.E[a;b]) ⇒ EquivRel(X ⟶ A;f,g.fun-equiv(X;a,b.E[a;b];f;g)))
Proof
Definitions occuring in Statement :
fun-equiv: fun-equiv(X;a,b.E[a; b];f;g),
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 :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
fun-equiv: fun-equiv(X;a,b.E[a; b];f;g),
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
trans: Trans(T;x,y.E[x; y]),
so_lambda: λ2x y.t[x; y],
guard: {T}
Lemmas referenced :
all_wf,
equiv_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalRule,
hypothesisEquality,
functionEquality,
cut,
because_Cache,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
applyEquality,
hypothesis,
productElimination,
dependent_functionElimination,
independent_functionElimination,
cumulativity,
universeEquality
Latex:
\mforall{}[X,A:Type]. \mforall{}[E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
(EquivRel(A;a,b.E[a;b]) {}\mRightarrow{} EquivRel(X {}\mrightarrow{} A;f,g.fun-equiv(X;a,b.E[a;b];f;g)))
Date html generated:
2016_05_14-AM-06_09_09
Last ObjectModification:
2015_12_26-AM-11_48_14
Theory : quot_1
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