Nuprl Lemma : intlex-aux_wf
∀[l1:ℤ List]. ∀[l2:{as:ℤ List| ||as|| = ||l1|| ∈ ℤ} ]. (intlex-aux(l1;l2) ∈ 𝔹)
Proof
Definitions occuring in Statement :
intlex-aux: intlex-aux(l1;l2),
length: ||as||,
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
intlex-aux: intlex-aux(l1;l2),
all: ∀x:A. B[x],
or: P ∨ Q,
nil: [],
it: ⋅,
bool: 𝔹,
unit: Unit,
cons: [a / b],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
subtract: n - m,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
exposed-it: exposed-it,
btrue: tt,
less_than: a < b,
squash: ↓T,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
decidable: Dec(P)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
list_wf,
list_subtype_base,
int_subtype_base,
list-cases,
length_of_nil_lemma,
unit_wf2,
product_subtype_list,
length_of_cons_lemma,
istype-void,
spread_cons_lemma,
le_weakening2,
length_wf,
non_neg_length,
length_wf_nat,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
istype-int,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
eq_int_wf,
assert_of_eq_int,
set_subtype_base,
le_wf,
neg_assert_of_eq_int,
subtract-1-ge-0,
le_weakening,
decidable__le,
istype-false,
not-le-2,
less-iff-le,
add-associates,
add-swap,
le-add-cancel2,
decidable__equal_int,
subtract_wf,
not-equal-2,
minus-minus,
nat_wf,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
one-mul
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
Error :lambdaEquality_alt,
dependent_functionElimination,
Error :isect_memberEquality_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :inhabitedIsType,
Error :setIsType,
intEquality,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
Error :functionIsTypeImplies,
unionElimination,
Error :inlEquality_alt,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :dependent_pairFormation_alt,
sqequalIntensionalEquality,
addEquality,
because_Cache,
minusEquality,
Error :equalityIsType1,
equalityElimination,
lessCases,
axiomSqEquality,
independent_pairFormation,
imageMemberEquality,
imageElimination,
Error :equalityIsType2,
instantiate,
cumulativity,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
Error :inrEquality_alt,
Error :dependent_set_memberEquality_alt,
multiplyEquality
Latex:
\mforall{}[l1:\mBbbZ{} List]. \mforall{}[l2:\{as:\mBbbZ{} List| ||as|| = ||l1||\} ]. (intlex-aux(l1;l2) \mmember{} \mBbbB{})
Date html generated:
2019_06_20-PM-00_42_09
Last ObjectModification:
2018_10_07-PM-08_51_40
Theory : list_0
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