Nuprl Lemma : implies-qle2
∀[a,b:ℚ]. a ≤ b supposing ∀e:ℚ. (0 < e ⇒ (a ≤ (b + e)))
Proof
Definitions occuring in Statement :
qle: r ≤ s,
qless: r < s,
qadd: r + s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
uiff: uiff(P;Q),
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
qadd_wf,
qle_wf,
qless_wf,
all_wf,
qle_witness,
nat_plus_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
qless-int,
qinv-positive,
not_wf,
equal-wf-T-base,
int-equal-in-rationals,
equal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_plus_properties,
int-subtype-rationals,
less_than_wf,
rationals_wf,
subtype_rel_set,
qdiv_wf,
implies-qle
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation,
hypothesis,
dependent_functionElimination,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
intEquality,
lambdaEquality,
setElimination,
rename,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addLevel,
impliesFunctionality,
productElimination,
baseClosed,
independent_functionElimination,
unionElimination,
functionEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a,b:\mBbbQ{}]. a \mleq{} b supposing \mforall{}e:\mBbbQ{}. (0 < e {}\mRightarrow{} (a \mleq{} (b + e)))
Date html generated:
2016_05_15-PM-11_32_46
Last ObjectModification:
2016_01_16-PM-09_13_55
Theory : rationals
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