Nuprl Lemma : intlex-cons
∀l1,l2:ℤ List. ∀x,y:ℤ.
uiff(↑[x / l1] ≤_lex [y / l2];||l1|| < ||l2|| ∨ (x < y ∧ (||l1|| = ||l2|| ∈ ℤ)) ∨ ((x = y ∈ ℤ) ∧ (↑l1 ≤_lex l2)))
Proof
Definitions occuring in Statement :
intlex: l1 ≤_lex l2,
length: ||as||,
cons: [a / b],
list: T List,
assert: ↑b,
less_than: a < b,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
intlex: l1 ≤_lex l2,
member: t ∈ T,
top: Top,
has-value: (a)↓,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
true: True,
prop: ℙ,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
sq_type: SQType(T),
guard: {T},
not: ¬A,
false: False,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
bfalse: ff,
bor: p ∨bq,
squash: ↓T,
band: p ∧b q,
intlex-aux: intlex-aux(l1;l2),
cons: [a / b],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
less_than: a < b,
cand: A c∧ B,
exists: ∃x:A. B[x],
bnot: ¬bb,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
length_of_cons_lemma,
value-type-has-value,
int-value-type,
length_wf,
nat_wf,
set-value-type,
le_wf,
length_wf_nat,
decidable__lt,
list_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
lt_int_wf,
btrue_wf,
less_than_wf,
true_wf,
assert_of_lt_int,
assert_wf,
iff_wf,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
zero-add,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
testxxx_lemma,
or_wf,
and_wf,
equal_wf,
le-add-cancel,
equal-wf-base,
list_subtype_base,
int_subtype_base,
bfalse_wf,
decidable__equal_int,
squash_wf,
eq_int_eq_true,
iff_weakening_equal,
spread_cons_lemma,
eqtt_to_assert,
top_wf,
assert_witness,
unit_wf2,
intlex-aux_wf,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
eq_int_eq_false,
not-equal-2,
decidable__le,
not-le-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
callbyvalueReduce,
isectElimination,
intEquality,
independent_isectElimination,
addEquality,
hypothesisEquality,
natural_numberEquality,
because_Cache,
lambdaEquality,
unionElimination,
instantiate,
cumulativity,
independent_pairFormation,
addLevel,
productElimination,
impliesFunctionality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
minusEquality,
isect_memberFormation,
axiomEquality,
rename,
inlFormation,
orFunctionality,
productEquality,
baseApply,
closedConclusion,
baseClosed,
imageElimination,
universeEquality,
imageMemberEquality,
equalityElimination,
lessCases,
sqequalAxiom,
inlEquality,
inrFormation,
dependent_set_memberEquality,
dependent_pairFormation,
promote_hyp,
int_eqReduceTrueSq,
int_eqReduceFalseSq
Latex:
\mforall{}l1,l2:\mBbbZ{} List. \mforall{}x,y:\mBbbZ{}.
uiff(\muparrow{}[x / l1] \mleq{}\_lex [y / l2];||l1|| < ||l2||
\mvee{} (x < y \mwedge{} (||l1|| = ||l2||))
\mvee{} ((x = y) \mwedge{} (\muparrow{}l1 \mleq{}\_lex l2)))
Date html generated:
2017_09_29-PM-05_49_43
Last ObjectModification:
2017_07_26-PM-01_37_54
Theory : list_0
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