Nuprl Lemma : int-rmul-req
∀[k:ℤ]. ∀[a:ℝ]. (k * a = (r(k) * a))
Proof
Definitions occuring in Statement :
int-rmul: k1 * a,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
real: ℝ,
int-to-real: r(n),
reg-seq-mul: reg-seq-mul(x;y),
int-rmul: k1 * a,
bdd-diff: bdd-diff(f;g),
has-value: (a)↓,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x.t[x],
less_than: a < b,
true: True,
squash: ↓T,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_nzero: ℤ-o,
regular-int-seq: k-regular-seq(f),
sq_stable: SqStable(P),
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
absval: |i|
Lemmas referenced :
req-iff-bdd-diff,
int-rmul_wf,
rmul_wf,
int-to-real_wf,
req_witness,
real_wf,
reg-seq-mul_wf,
value-type-has-value,
int-value-type,
absval_wf,
mul-non-neg1,
false_wf,
decidable__le,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
le_wf,
nat_plus_wf,
all_wf,
subtract_wf,
less_than_wf,
mul_nat_plus,
nat_plus_properties,
intformeq_wf,
itermMultiply_wf,
intformless_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_formula_prop_less_lemma,
equal-wf-base,
bdd-diff_functionality,
bdd-diff_weakening,
rmul-bdd-diff-reg-seq-mul,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
top_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
iff_weakening_equal,
subtype_rel_self,
absval_mul,
true_wf,
squash_wf,
equal-wf-T-base,
absval_nat_plus,
nat_wf,
int_term_value_minus_lemma,
itermMinus_wf,
decidable__lt,
mul_cancel_in_le,
nequal_wf,
div-cancel2,
decidable__equal_int,
int_subtype_base,
sq_stable__le,
mul-associates,
mul-distributes,
minus-one-mul,
mul-swap,
one-mul,
add-commutes,
absval_sym,
add_functionality_wrt_eq,
minus-add,
mul-commutes,
le_functionality,
le_weakening,
absval_unfold,
add-is-int-iff,
set_subtype_base,
multiply-is-int-iff,
nat_plus_subtype_nat,
absval_pos,
zero-add,
zero-mul,
zero-div-rem
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
because_Cache,
intEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
callbyvalueReduce,
dependent_pairFormation,
dependent_set_memberEquality,
multiplyEquality,
natural_numberEquality,
addEquality,
independent_pairFormation,
lambdaFormation,
dependent_functionElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
approximateComputation,
int_eqEquality,
voidElimination,
voidEquality,
lessCases,
baseClosed,
imageMemberEquality,
axiomSqEquality,
imageElimination,
minusEquality,
divideEquality,
baseApply,
closedConclusion,
cumulativity,
instantiate,
promote_hyp,
equalityElimination,
universeEquality
Latex:
\mforall{}[k:\mBbbZ{}]. \mforall{}[a:\mBbbR{}]. (k * a = (r(k) * a))
Date html generated:
2019_10_29-AM-09_32_26
Last ObjectModification:
2018_08_23-PM-01_45_09
Theory : reals
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