Nuprl Lemma : intlex-aux-transitive
∀[l1:ℤ List]. ∀[l2,l3:{as:ℤ List| ||as|| = ||l1|| ∈ ℤ} ].
(intlex-aux(l1;l3) = tt) supposing (intlex-aux(l2;l3) = tt and intlex-aux(l1;l2) = tt)
Proof
Definitions occuring in Statement :
intlex-aux: intlex-aux(l1;l2),
length: ||as||,
list: T List,
btrue: tt,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
intlex-aux: intlex-aux(l1;l2),
nil: [],
it: ⋅,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
cons: [a / b],
top: Top,
less_than': less_than'(a;b),
not: ¬A,
colength: colength(L),
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
subtract: n - m,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
decidable: Dec(P),
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
bfalse: ff,
bnot: ¬bb,
isl: isl(x),
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
list-cases,
iff_imp_equal_bool,
intlex-aux_wf,
nil_wf,
istype-assert,
true_wf,
set_subtype_base,
list_wf,
equal_wf,
length_wf,
list_subtype_base,
int_subtype_base,
bool_wf,
istype-int,
product_subtype_list,
colength-cons-not-zero,
istype-void,
istype-nat,
colength_wf_list,
istype-false,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
le_wf,
spread_cons_lemma,
sq_stable__le,
add-associates,
add-commutes,
add-swap,
zero-add,
length_of_cons_lemma,
length_of_nil_lemma,
le_weakening2,
non_neg_length,
length_wf_nat,
istype-sqequal,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
cons_wf,
subtract_wf,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
one-mul,
decidable__lt,
istype-top,
less_than_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
not-lt-2,
less-iff-le,
decidable__equal_int,
bfalse_wf,
btrue_wf,
btrue_neq_bfalse,
eq_int_wf,
assert_of_eq_int,
not-equal-2,
neg_assert_of_eq_int,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :universeIsType,
sqequalRule,
Error :lambdaEquality_alt,
dependent_functionElimination,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
intEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
voidEquality,
sqequalBase,
Error :setIsType,
promote_hyp,
hypothesis_subsumption,
Error :dependent_set_memberEquality_alt,
instantiate,
cumulativity,
imageMemberEquality,
applyLambdaEquality,
Error :dependent_pairFormation_alt,
addEquality,
minusEquality,
multiplyEquality,
lessCases,
axiomSqEquality,
equalityElimination,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
Error :productIsType,
hyp_replacement,
int_eqEquality
Latex:
\mforall{}[l1:\mBbbZ{} List]. \mforall{}[l2,l3:\{as:\mBbbZ{} List| ||as|| = ||l1||\} ].
(intlex-aux(l1;l3) = tt) supposing (intlex-aux(l2;l3) = tt and intlex-aux(l1;l2) = tt)
Date html generated:
2019_06_20-PM-00_42_33
Last ObjectModification:
2019_01_17-PM-04_26_10
Theory : list_0
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