Nuprl Lemma : insert-int-lex
∀as,bs:ℤ List. ∀x:ℤ. ((↑as ≤_lex bs) ⇒ (↑insert-int(x;as) ≤_lex insert-int(x;bs)))
Proof
Definitions occuring in Statement :
intlex: l1 ≤_lex l2,
insert-int: insert-int(x;l),
list: T List,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cons: [a / b],
subtype_rel: A ⊆r B,
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
insert-int: insert-int(x;l),
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
btrue: tt,
less_than': less_than'(a;b),
true: True,
not: ¬A,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
decidable: Dec(P),
subtract: n - m,
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
assert_witness,
intlex_wf,
insert-int_wf,
list-cases,
nil_wf,
insert_int_nil_lemma,
intlex-cons-same,
istype-assert,
istype-int,
product_subtype_list,
colength-cons-not-zero,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
spread_cons_lemma,
list_ind_cons_lemma,
value-type-has-value,
list_wf,
list-value-type,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
cons_wf,
istype-nat,
colength_wf_list,
le_weakening2,
istype-le,
istype-void,
sq_stable__le,
decidable__equal_int,
subtract_wf,
istype-false,
not-equal-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
minus-minus,
intlex-by-length,
length_of_cons_lemma,
length_of_nil_lemma,
length_wf,
non_neg_length,
length_wf_nat,
istype-sqequal,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
not-lt-2,
omega-shadow,
squash_wf,
true_wf,
add_functionality_wrt_eq,
length-insert-int,
subtype_rel_self,
iff_weakening_equal,
decidable__lt,
intlex-length,
add-zero,
minus-zero,
intlex-cons,
less-iff-le,
le-add-cancel2,
list_subtype_base,
le_weakening,
equal_wf,
istype-universe,
length_cons,
mul-distributes,
mul-associates,
less_than_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
universeIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
closedConclusion,
intEquality,
because_Cache,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
Error :memTop,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
equalityElimination,
lessCases,
isect_memberFormation_alt,
axiomSqEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
applyLambdaEquality,
addEquality,
minusEquality,
baseApply,
applyEquality,
sqequalBase,
multiplyEquality,
universeEquality,
inlFormation_alt,
unionIsType,
productIsType,
inrFormation_alt
Latex:
\mforall{}as,bs:\mBbbZ{} List. \mforall{}x:\mBbbZ{}. ((\muparrow{}as \mleq{}\_lex bs) {}\mRightarrow{} (\muparrow{}insert-int(x;as) \mleq{}\_lex insert-int(x;bs)))
Date html generated:
2020_05_19-PM-09_37_41
Last ObjectModification:
2019_12_26-PM-00_32_18
Theory : list_0
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