Nuprl Lemma : l-ordered-reorder
∀[A:Type]
∀R:A ⟶ A ⟶ 𝔹. ∀L:A List.
(Trans(A;x,y.↑R[x;y])
⇒ (∀x∈L.(∀y∈L.(¬↑R[x;y]) ⇒ (↑R[y;x])))
⇒ (∃L':A List. (l-ordered(A;x,y.↑R[x;y];L') ∧ permutation(A;L;L'))))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
permutation: permutation(T;L1;L2),
l_all: (∀x∈L.P[x]),
list: T List,
trans: Trans(T;x,y.E[x; y]),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
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],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
and: P ∧ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
sq_type: SQType(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
trans: Trans(T;x,y.E[x; y]),
squash: ↓T,
top: Top
Lemmas referenced :
list_induction,
trans_wf,
assert_wf,
l_all_wf2,
l_member_wf,
not_wf,
exists_wf,
list_wf,
l-ordered_wf,
permutation_wf,
nil_wf,
l-ordered-nil-true,
permutation-nil-iff,
true_wf,
equal-wf-base,
l_all_iff,
cons_wf,
cons_member,
equal_wf,
assert_witness,
all_wf,
l-ordered-decomp,
append_wf,
filter_wf5,
bnot_wf,
bool_wf,
l-ordered-append,
l-ordered-filter,
l-ordered-cons,
member_filter_2,
assert_of_bnot,
member-permutation,
and_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base,
permutation-cons,
squash_wf,
iff_weakening_equal,
length_wf,
length-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
applyEquality,
functionExtensionality,
because_Cache,
hypothesis,
setElimination,
rename,
setEquality,
dependent_functionElimination,
productEquality,
independent_functionElimination,
dependent_pairFormation,
natural_numberEquality,
independent_pairFormation,
addLevel,
productElimination,
baseClosed,
voidEquality,
voidElimination,
allFunctionality,
promote_hyp,
inrFormation,
impliesFunctionality,
independent_isectElimination,
universeEquality,
inlFormation,
unionElimination,
dependent_set_memberEquality,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
imageElimination,
imageMemberEquality,
isect_memberEquality
Latex:
\mforall{}[A:Type]
\mforall{}R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List.
(Trans(A;x,y.\muparrow{}R[x;y])
{}\mRightarrow{} (\mforall{}x\mmember{}L.(\mforall{}y\mmember{}L.(\mneg{}\muparrow{}R[x;y]) {}\mRightarrow{} (\muparrow{}R[y;x])))
{}\mRightarrow{} (\mexists{}L':A List. (l-ordered(A;x,y.\muparrow{}R[x;y];L') \mwedge{} permutation(A;L;L'))))
Date html generated:
2018_05_21-PM-07_39_58
Last ObjectModification:
2017_07_26-PM-05_14_08
Theory : general
Home
Index