Nuprl Lemma : radd*_functionality_wrt_infinitesmal
∀x,y:ℝ*. (is-infinitesmal(x) ⇒ is-infinitesmal(y) ⇒ is-infinitesmal(x + y))
Proof
Definitions occuring in Statement :
is-infinitesmal: is-infinitesmal(x),
radd*: x + y,
real*: ℝ*,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
rleq*: x ≤ y,
rrel*: R*(x,y),
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
rabs*: |x|,
radd*: x + y,
rfun*: f*(x),
rfun*2: f*(x;y),
real*: ℝ*,
subtype_rel: A ⊆r B,
int_upper: {i...},
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
sq_type: SQType(T),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rge*: x ≥ y
Lemmas referenced :
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rmul_preserves_rless,
set_wf,
real_wf,
all_wf,
rless*_wf,
rabs*_wf,
rstar_wf,
real*_wf,
infinitesmal-iff,
radd*_wf,
is-infinitesmal_wf,
rmul_wf,
rmul-zero-both,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rless_functionality,
req_transitivity,
rmul-rinv3,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
false_wf,
le_wf,
r-triangle-inequality,
subtype_rel_self,
nat_wf,
int_upper_wf,
rleq_wf,
int_upper_subtype_nat,
radd_wf,
rstar_functionality,
rmul_preserves_req,
itermAdd_wf,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
equal-wf-base,
true_wf,
nequal_wf,
rmul_comm,
req_weakening,
req*_functionality,
rstar-radd,
req*_weakening,
req_functionality,
int-rinv-cancel,
real_term_value_add_lemma,
rless*_functionality,
req*_inversion,
rless*_functionality_wrt_implies,
rleq*_weakening_rless,
rless*_transitivity2,
radd*_functionality_wrt_rless*_2,
rleq*_weakening_equal,
radd*_functionality_wrt_rless*_1,
rleq_weakening_equal,
rstar-rleq
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
setElimination,
thin,
rename,
dependent_set_memberEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
natural_numberEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
setEquality,
addLevel,
allFunctionality,
impliesFunctionality,
functionEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairFormation,
applyEquality,
instantiate,
cumulativity,
unionElimination
Latex:
\mforall{}x,y:\mBbbR{}*. (is-infinitesmal(x) {}\mRightarrow{} is-infinitesmal(y) {}\mRightarrow{} is-infinitesmal(x + y))
Date html generated:
2018_05_22-PM-09_29_31
Last ObjectModification:
2017_10_10-PM-01_59_11
Theory : reals_2
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