Nuprl Lemma : rat-rleq-cases
∀x,y:ℤ × ℕ+. ((↓ratreal(x) ≤ ratreal(y)) ∨ (↓ratreal(y) ≤ ratreal(x)))
Proof
Definitions occuring in Statement :
ratreal: ratreal(r),
rleq: x ≤ y,
nat_plus: ℕ+,
all: ∀x:A. B[x],
squash: ↓T,
or: P ∨ Q,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
or: P ∨ Q,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
rleq_functionality,
ratreal_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
ratreal-req,
rleq-int-fractions,
squash_wf,
rleq_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
le_wf,
istype-le,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
productElimination,
thin,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
multiplyEquality,
hypothesisEquality,
setElimination,
rename,
hypothesis,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
inlEquality_alt,
independent_pairEquality,
because_Cache,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
imageMemberEquality,
baseClosed,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
inrEquality_alt,
productIsType
Latex:
\mforall{}x,y:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}. ((\mdownarrow{}ratreal(x) \mleq{} ratreal(y)) \mvee{} (\mdownarrow{}ratreal(y) \mleq{} ratreal(x)))
Date html generated:
2019_10_30-AM-09_28_00
Last ObjectModification:
2019_01_11-AM-11_39_39
Theory : reals
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