Nuprl Lemma : sorted-by-eq-rels
∀T:Type. ∀R1,R2:T ⟶ T ⟶ ℙ. ∀L:T List.
((∀x∈L.(∀y∈L.R1[x;y] ⇐⇒ R2[x;y])) ⇒ sorted-by(λx,y. R1[x;y];L) ⇒ sorted-by(λx,y. R2[x;y];L))
Proof
Definitions occuring in Statement :
sorted-by: sorted-by(R;L),
l_all: (∀x∈L.P[x]),
list: T List,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
l-ordered-is-sorted-by,
l-ordered-eq-rels,
sorted-by_wf,
l_member_wf,
l_all_wf2,
iff_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
productElimination,
independent_pairFormation,
independent_functionElimination,
because_Cache,
setElimination,
rename,
setEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}T:Type. \mforall{}R1,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}L:T List.
((\mforall{}x\mmember{}L.(\mforall{}y\mmember{}L.R1[x;y] \mLeftarrow{}{}\mRightarrow{} R2[x;y])) {}\mRightarrow{} sorted-by(\mlambda{}x,y. R1[x;y];L) {}\mRightarrow{} sorted-by(\mlambda{}x,y. R2[x;y];L))
Date html generated:
2016_05_15-PM-04_38_16
Last ObjectModification:
2015_12_27-PM-02_43_06
Theory : general
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