Nuprl Lemma : alt-int-rdiv_wf
∀[x:ℝ]. ∀[k:ℕ+]. (alt-int-rdiv(x;k) ∈ {y:ℝ| k * y = x} )
Proof
Definitions occuring in Statement :
alt-int-rdiv: alt-int-rdiv(x;k),
int-rmul: k1 * a,
req: x = y,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
alt-int-rdiv: alt-int-rdiv(x;k),
nat_plus: ℕ+,
real: ℝ,
false: False,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
regular-int-seq: k-regular-seq(f),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
nat: ℕ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
sq_stable: SqStable(P),
le: A ≤ B,
bdd-diff: bdd-diff(f;g),
accelerate: accelerate(k;f),
int-rmul: k1 * a,
has-value: (a)↓,
less_than: a < b,
less_than': less_than'(a;b),
int_nzero: ℤ-o,
absval: |i|
Lemmas referenced :
rounding-div_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
real-regular,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
mul_cancel_in_le,
absval_wf,
subtract_wf,
absval_nat_plus,
intformand_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
set_subtype_base,
less_than_wf,
int_subtype_base,
le_wf,
squash_wf,
true_wf,
absval_mul,
subtype_rel_self,
iff_weakening_equal,
nat_wf,
absval-non-neg,
absval_pos,
nat_plus_subtype_nat,
istype-le,
regular-int-seq_wf,
nat_plus_wf,
real_wf,
rounding-div-property,
absval-diff-symmetry,
decidable__equal_int,
itermMultiply_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_functionality,
multiply_functionality_wrt_le,
le_weakening,
le_transitivity,
int-triangle-inequality,
add_functionality_wrt_le,
add_functionality_wrt_eq,
mul_preserves_le,
sq_stable__le,
multiply-is-int-iff,
false_wf,
req_wf,
int-rmul_wf,
int-rmul-one,
req-iff-bdd-diff,
accelerate-bdd-diff,
subtype_rel_sets_simple,
accelerate_wf,
bdd-diff_functionality,
bdd-diff_weakening,
bdd-diff_inversion,
nat_properties,
value-type-has-value,
int-value-type,
set-value-type,
istype-top,
mul_nat_plus,
divide_wfa,
nequal_wf,
div_rem_sum2,
remainder_wfa,
rem_bounds_absval,
itermMinus_wf,
int_term_value_minus_lemma,
absval_sym,
sq_stable__less_than,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
int_eqEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
lambdaEquality_alt,
extract_by_obid,
isectElimination,
applyEquality,
inhabitedIsType,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
because_Cache,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
int_eqReduceTrueSq,
dependent_functionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
universeIsType,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq,
multiplyEquality,
addEquality,
independent_pairFormation,
intEquality,
baseClosed,
sqequalBase,
imageElimination,
imageMemberEquality,
universeEquality,
axiomEquality,
isectIsTypeImplies,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionEquality,
functionIsType,
setEquality,
callbyvalueReduce,
sqleReflexivity,
lessCases,
axiomSqEquality,
applyLambdaEquality,
minusEquality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. (alt-int-rdiv(x;k) \mmember{} \{y:\mBbbR{}| k * y = x\} )
Date html generated:
2019_10_29-AM-09_33_25
Last ObjectModification:
2019_02_18-PM-03_34_55
Theory : reals
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