Nuprl Lemma : permr_upto_functionality_wrt_permr_upto
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
(EquivRel(T;x,y.R[x;y])
⇒ (∀as,as',bs,bs':T List.
(as ≡ bs upto x,y.R[x;y]
⇒ as' ≡ bs' upto x,y.R[x;y]
⇒ (as ≡ as' upto x,y.R[x;y] ⇐⇒ bs ≡ bs' 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],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
guard: {T}
Lemmas referenced :
equiv_rel_wf,
istype-universe,
equiv_rel_self_functionality,
list_wf,
permr_upto_wf,
permr_upto_equiv_rel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
hypothesis,
functionIsType,
universeEquality,
dependent_functionElimination,
because_Cache,
independent_functionElimination
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(EquivRel(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}as,as',bs,bs':T List.
(as \mequiv{} bs upto x,y.R[x;y]
{}\mRightarrow{} as' \mequiv{} bs' upto x,y.R[x;y]
{}\mRightarrow{} (as \mequiv{} as' upto x,y.R[x;y] \mLeftarrow{}{}\mRightarrow{} bs \mequiv{} bs' upto x,y.R[x;y] ))))
Date html generated:
2019_10_16-PM-01_01_23
Last ObjectModification:
2018_10_08-PM-00_48_41
Theory : perms_2
Home
Index