Nuprl Lemma : ratreduce_wf
∀[x:ℤ × ℕ+]. (ratreduce(x) ∈ {y:ℤ × ℕ+| ratreal(x) = ratreal(y)} )
Proof
Definitions occuring in Statement :
ratreduce: ratreduce(x),
ratreal: ratreal(r),
req: x = y,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ratreduce: ratreduce(x),
nat_plus: ℕ+,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
or: P ∨ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
divides: b | a,
has-value: (a)↓,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
sq_type: SQType(T),
guard: {T},
int_nzero: ℤ-o,
true: True,
rneq: x ≠ y
Lemmas referenced :
better-gcd-gcd,
gcd_is_divisor_1,
gcd_is_divisor_2,
absval-divides,
gcd_wf,
absval-positive,
gcd-non-zero,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
int_subtype_base,
nequal_wf,
value-type-has-value,
nat_wf,
set-value-type,
le_wf,
int-value-type,
absval_wf,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
istype-le,
subtype_base_sq,
decidable__lt,
itermMultiply_wf,
int_term_value_mul_lemma,
mul_positive_iff,
istype-less_than,
set_subtype_base,
less_than_wf,
nat_plus_wf,
decidable__equal_int,
divide-exact,
equal_wf,
squash_wf,
true_wf,
istype-universe,
divide_wfa,
int_nzero_wf,
subtype_rel_self,
iff_weakening_equal,
req_functionality,
ratreal_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
ratreal-req,
req_wf,
req-int-fractions,
nat_plus_inc_int_nzero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
productElimination,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
independent_pairFormation,
independent_functionElimination,
promote_hyp,
independent_isectElimination,
inrFormation_alt,
lambdaFormation_alt,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
equalityIstype,
inhabitedIsType,
applyEquality,
baseClosed,
sqequalBase,
equalitySymmetry,
intEquality,
callbyvalueReduce,
equalityTransitivity,
dependent_set_memberEquality_alt,
unionElimination,
imageElimination,
instantiate,
cumulativity,
multiplyEquality,
productIsType,
baseApply,
closedConclusion,
universeEquality,
imageMemberEquality,
independent_pairEquality,
applyLambdaEquality
Latex:
\mforall{}[x:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}]. (ratreduce(x) \mmember{} \{y:\mBbbZ{} \mtimes{} \mBbbN{}\msupplus{}| ratreal(x) = ratreal(y)\} )
Date html generated:
2019_10_30-AM-09_19_36
Last ObjectModification:
2019_10_10-AM-10_35_01
Theory : reals
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