Nuprl Lemma : l-ordered-append
∀[T:Type]
∀L1,L2:T List.
∀[R:T ⟶ T ⟶ ℙ]
(l-ordered(T;x,y.R[x;y];L1 @ L2)
⇐⇒ l-ordered(T;x,y.R[x;y];L1) ∧ l-ordered(T;x,y.R[x;y];L2) ∧ (∀x,y:T. ((x ∈ L1) ⇒ (y ∈ L2) ⇒ R[x;y])))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
l_member: (x ∈ l),
append: as @ bs,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
l-ordered: l-ordered(T;x,y.R[x; y];L),
or: P ∨ Q,
guard: {T}
Lemmas referenced :
l_member_wf,
l-ordered_wf,
append_wf,
and_wf,
all_wf,
list_wf,
l_before_append_iff,
or_wf,
l_before_wf,
l_before_append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
productElimination,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
inlFormation,
inrFormation,
unionElimination
Latex:
\mforall{}[T:Type]
\mforall{}L1,L2:T List.
\mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]
(l-ordered(T;x,y.R[x;y];L1 @ L2)
\mLeftarrow{}{}\mRightarrow{} l-ordered(T;x,y.R[x;y];L1)
\mwedge{} l-ordered(T;x,y.R[x;y];L2)
\mwedge{} (\mforall{}x,y:T. ((x \mmember{} L1) {}\mRightarrow{} (y \mmember{} L2) {}\mRightarrow{} R[x;y])))
Date html generated:
2016_05_15-PM-04_36_30
Last ObjectModification:
2015_12_27-PM-02_44_49
Theory : general
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