Nuprl Lemma : Cauchy-equation-1-iff
∀[f:ℝ ⟶ ℝ]
uiff(∀x,y:ℝ. (f(x + y) = (f(x) + f(y)));∀x:ℝ. (f(x) = (x * f(r1)))) supposing ∀x,y:ℝ. ((x = y) ⇒ ((f x) = (f y)))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
nequal: a ≠ b ∈ T ,
less_than': less_than'(a;b),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
unit: Unit,
bool: 𝔹,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
int-to-real: r(n),
btrue: tt,
eq_int: (i =z j),
length: ||as||,
radd-list: radd-list(L),
it: ⋅,
nil: [],
bfalse: ff,
lt_int: i <z j,
ifthenelse: if b then t else f fi ,
from-upto: [n, m),
list_ind: list_ind,
map: map(f;as),
evalall: evalall(t),
callbyvalueall: callbyvalueall,
subtract: n - m,
rsum: Σ{x[k] | n≤k≤m},
lelt: i ≤ j < k,
int_seg: {i..j-},
le: A ≤ B,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
false: False,
nat: ℕ,
rdiv: (x/y),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
squash: ↓T,
true: True,
rfun-ap: f(x),
r-ap: f(x),
rfun: I ⟶ℝ,
less_than: a < b
Lemmas referenced :
radd_wf,
req_wf,
all_wf,
int-to-real_wf,
rmul_wf,
rfun-ap_wf,
req_witness,
real_wf,
radd_functionality,
req_transitivity,
rsum_unroll,
real_term_value_minus_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req_weakening,
int_term_value_add_lemma,
neg_assert_of_eq_int,
false_wf,
assert_of_eq_int,
eq_int_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
itermMinus_wf,
req-iff-rsub-is-0,
itermMultiply_wf,
itermAdd_wf,
req_inversion,
radd-rminus-both,
rminus_wf,
radd-preserves-req,
radd-zero,
rfun-ap_functionality,
req_functionality,
nat_wf,
decidable__lt,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
decidable__le,
lelt_wf,
subtract-add-cancel,
int_seg_wf,
subtract_wf,
rsum_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
rinv-mul-as-rdiv,
rmul_functionality,
rmul-rinv3,
uiff_transitivity,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
bool_cases,
rsum-constant2,
rmul_preserves_req,
rinv_wf2,
not_wf,
bnot_wf,
assert_wf,
rsub_wf,
req-implies-req,
le_wf,
set_subtype_base,
rmul_comm,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
ifthenelse_wf,
nat_plus_wf,
rless_wf,
nat_plus_properties,
rless-int,
rdiv_wf,
nat_plus_subtype_nat,
rdiv_functionality,
rmul-one,
radd_comm,
radd-rminus,
rminus-int,
true_wf,
squash_wf,
rminus_functionality,
rmul-rinv,
int_term_value_minus_lemma,
rminus-rminus,
set_wf,
subtype_rel_self,
subtype_rel_dep_function,
i-member_wf,
exists_wf,
rneq_wf,
member_riiint_lemma,
riiint_wf,
functions-equal-on-rationals,
rmul-distrib2
Rules used in proof :
functionEquality,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
independent_pairEquality,
productElimination,
because_Cache,
independent_functionElimination,
natural_numberEquality,
applyEquality,
functionExtensionality,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
extract_by_obid,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cumulativity,
instantiate,
promote_hyp,
equalityElimination,
unionElimination,
addEquality,
dependent_set_memberEquality,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
intWeakElimination,
rename,
setElimination,
impliesFunctionality,
inrFormation,
baseClosed,
imageMemberEquality,
imageElimination,
minusEquality,
setEquality,
inlFormation
Latex:
\mforall{}[f:\mBbbR{} {}\mrightarrow{} \mBbbR{}]
uiff(\mforall{}x,y:\mBbbR{}. (f(x + y) = (f(x) + f(y)));\mforall{}x:\mBbbR{}. (f(x) = (x * f(r1))))
supposing \mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} ((f x) = (f y)))
Date html generated:
2017_10_04-PM-11_02_35
Last ObjectModification:
2017_07_31-PM-00_33_16
Theory : reals_2
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