Nuprl Lemma : l-ordered-eq-rels
∀T:Type. ∀R1,R2:T ⟶ T ⟶ ℙ. ∀L:T List.
((∀x∈L.(∀y∈L.R1[x;y] ⇐⇒ R2[x;y])) ⇒ l-ordered(T;x,y.R1[x;y];L) ⇒ l-ordered(T;x,y.R2[x;y];L))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];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,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
l-ordered: l-ordered(T;x,y.R[x; y];L),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
l_all_iff,
l_all_wf2,
iff_wf,
l_member_wf,
l_before_member2,
l_before_member,
l_before_wf,
l-ordered_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
lemma_by_obid,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
productElimination,
allFunctionality,
promote_hyp,
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{} l-ordered(T;x,y.R1[x;y];L) {}\mRightarrow{} l-ordered(T;x,y.R2[x;y];L))
Date html generated:
2016_05_15-PM-04_38_09
Last ObjectModification:
2015_12_27-PM-02_43_32
Theory : general
Home
Index