Nuprl Lemma : radd-positive-implies
∀x,y:ℝ. ((r0 < (x + y)) ⇒ ((r0 < x) ∨ (r0 < y)))
Proof
Definitions occuring in Statement :
rless: x < y,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
member: t ∈ T,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
int-to-real: r(n),
sq_type: SQType(T),
guard: {T},
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
uiff: uiff(P;Q),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
absval: |i|,
less_than: a < b,
squash: ↓T
Lemmas referenced :
decidable__lt,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int-to-real_wf,
rless_wf,
radd_wf,
real_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
radd-approx,
divide_wfa,
nequal_wf,
istype-le,
div_rem_sum,
rem_bounds_absval,
absval_strict_ubound,
remainder_wfa,
absval_wf,
nat_wf,
set_subtype_base,
le_wf,
istype-false,
absval_pos
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
natural_numberEquality,
applyEquality,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality_alt,
multiplyEquality,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
inlFormation_alt,
dependent_set_memberFormation_alt,
because_Cache,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
inhabitedIsType,
inrFormation_alt,
equalityIstype,
baseClosed,
sqequalBase,
productElimination,
imageElimination,
minusEquality
Latex:
\mforall{}x,y:\mBbbR{}. ((r0 < (x + y)) {}\mRightarrow{} ((r0 < x) \mvee{} (r0 < y)))
Date html generated:
2019_10_29-AM-10_00_07
Last ObjectModification:
2019_05_24-AM-11_13_35
Theory : reals
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