Nuprl Lemma : permr_upto_equiv_rel
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. (EquivRel(T;x,y.R[x;y]) ⇒ EquivRel(T List;xs,ys.xs ≡ ys upto x,y.R[x;y] ))
Proof
Definitions occuring in Statement :
permr_upto: as ≡ bs upto x,y.R[x; y] ,
list: T List,
equiv_rel: EquivRel(T;x,y.E[x; y]),
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
implies: P ⇒ Q,
and: P ∧ Q
Lemmas referenced :
list_wf,
permr_upto_wf,
istype-universe,
equiv_rel_wf,
permr_upto_weakening,
permr_weakening,
permr_upto_inversion,
permr_upto_transitivity
Rules used in proof :
universeIsType,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
because_Cache,
functionIsType,
universeEquality,
lambdaFormation_alt,
independent_pairFormation,
independent_functionElimination
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(EquivRel(T;x,y.R[x;y]) {}\mRightarrow{} EquivRel(T List;xs,ys.xs \mequiv{} ys upto x,y.R[x;y] ))
Date html generated:
2019_10_16-PM-01_01_22
Last ObjectModification:
2018_10_08-PM-05_34_45
Theory : perms_2
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