Nuprl Lemma : intlex-total
∀as,bs:ℤ List. ((↑as ≤_lex bs) ∨ (↑bs ≤_lex as))
Proof
Definitions occuring in Statement :
intlex: l1 ≤_lex l2,
list: T List,
assert: ↑b,
all: ∀x:A. B[x],
or: P ∨ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
intlex: l1 ≤_lex l2,
has-value: (a)↓,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
true: True,
prop: ℙ,
band: p ∧b q,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
not: ¬A,
bor: p ∨bq,
nequal: a ≠ b ∈ T ,
le: A ≤ B,
less_than': less_than'(a;b),
intlex-aux: intlex-aux(l1;l2),
nil: [],
cons: [a / b],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
exposed-it: exposed-it,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
value-type-has-value,
nat_wf,
set-value-type,
le_wf,
int-value-type,
length_wf_nat,
lt_int_wf,
length_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
testxxx_lemma,
assert_wf,
bor_wf,
eq_int_wf,
assert_of_eq_int,
intlex-aux_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
equal_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
list_wf,
neg_assert_of_eq_int,
not-equal-2,
not-lt-2,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
list_induction,
all_wf,
or_wf,
nil_wf,
list-cases,
length_of_nil_lemma,
length_of_cons_lemma,
cons_wf,
product_subtype_list,
spread_cons_lemma,
top_wf,
less_than_transitivity2,
le_weakening2,
less_than_irreflexivity,
less_than_transitivity1,
le_weakening,
unit_wf2,
decidable__equal_int,
false_wf,
le_antisymmetry_iff,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
minus-one-mul-top,
zero-add,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
callbyvalueReduce,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
inlFormation,
dependent_set_memberEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
inrFormation,
addEquality,
functionEquality,
rename,
hypothesis_subsumption,
lessCases,
isect_memberFormation,
sqequalAxiom,
independent_pairFormation,
imageMemberEquality,
imageElimination,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
inrEquality,
axiomEquality,
minusEquality
Latex:
\mforall{}as,bs:\mBbbZ{} List. ((\muparrow{}as \mleq{}\_lex bs) \mvee{} (\muparrow{}bs \mleq{}\_lex as))
Date html generated:
2017_09_29-PM-05_49_38
Last ObjectModification:
2017_07_26-PM-01_37_50
Theory : list_0
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