Nuprl Lemma : intlex-transitive
∀[l1,l2,l3:ℤ List]. (l1 ≤_lex l3 = tt) supposing (l2 ≤_lex l3 = tt and l1 ≤_lex l2 = tt)
Proof
Definitions occuring in Statement :
intlex: l1 ≤_lex l2,
list: T List,
btrue: tt,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
intlex: l1 ≤_lex l2,
has-value: (a)↓,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
top: Top,
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
false: False,
not: ¬A,
bor: p ∨bq,
band: p ∧b q,
le: A ≤ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
true: True,
rev_implies: P ⇐ Q,
squash: ↓T,
cand: A c∧ B,
less_than': less_than'(a;b),
subtract: n - m,
ge: i ≥ j
Lemmas referenced :
value-type-has-value,
nat_wf,
set-value-type,
le_wf,
int-value-type,
length_wf_nat,
equal-wf-base,
bool_wf,
list_subtype_base,
int_subtype_base,
list_wf,
lt_int_wf,
length_wf,
eqtt_to_assert,
assert_of_lt_int,
testxxx_lemma,
le_weakening2,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
eq_int_wf,
assert_of_eq_int,
le_weakening,
neg_assert_of_eq_int,
btrue_neq_bfalse,
decidable__lt,
iff_imp_equal_bool,
less_than_transitivity1,
true_wf,
assert_wf,
iff_wf,
squash_wf,
iff_weakening_equal,
decidable__equal_int,
false_wf,
not-equal-2,
not-lt-2,
add_functionality_wrt_le,
add-associates,
add-commutes,
le-add-cancel,
add-swap,
intlex-aux_wf,
and_wf,
eq_int_eq_true,
band_wf,
less_than_transitivity2,
set_subtype_base,
non_neg_length,
subtract_wf,
minus-one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
add-zero,
one-mul,
nat_properties,
intlex-aux-transitive,
bor_wf,
btrue_wf,
or_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_band
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
because_Cache,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_functionElimination,
voidElimination,
voidEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
independent_pairFormation,
addLevel,
impliesFunctionality,
imageElimination,
universeEquality,
imageMemberEquality,
addEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
equalityUniverse,
levelHypothesis,
sqequalIntensionalEquality,
multiplyEquality,
productEquality,
inrFormation,
orFunctionality
Latex:
\mforall{}[l1,l2,l3:\mBbbZ{} List]. (l1 \mleq{}\_lex l3 = tt) supposing (l2 \mleq{}\_lex l3 = tt and l1 \mleq{}\_lex l2 = tt)
Date html generated:
2017_09_29-PM-05_49_33
Last ObjectModification:
2017_07_26-PM-01_37_46
Theory : list_0
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