Nuprl Lemma : raise-left-endpoint-rless
∀a,b:ℝ. ∀n:ℕ+. ((a < b) ⇒ ((a < raise-left-endpoint(a;b;n)) ∧ (raise-left-endpoint(a;b;n) < b)))
Proof
Definitions occuring in Statement :
raise-left-endpoint: raise-left-endpoint(a;b;n),
rless: x < y,
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
member: t ∈ T,
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
raise-left-endpoint: raise-left-endpoint(a;b;n),
int_nzero: ℤ-o,
nat_plus: ℕ+,
nequal: a ≠ b ∈ T ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Lemmas referenced :
real_wf,
req_weakening,
radd-int,
req_inversion,
rmul_functionality,
radd_functionality,
req_functionality,
equal_wf,
int-to-real_wf,
rmul_wf,
radd_wf,
rmul-distrib2,
rmul-identity1,
radd-assoc,
req_transitivity,
radd_comm,
rmul_comm,
rmul-one-both,
rmul-distrib,
uiff_transitivity,
req_wf,
rless_wf,
nat_plus_wf,
int-rdiv_wf,
nat_plus_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
nequal_wf,
int-rmul_wf,
rdiv_wf,
rless-int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
rmul_preserves_rless,
rinv_wf2,
rless_functionality,
int-rdiv-req,
rdiv_functionality,
int-rmul-req,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
real_term_value_add_lemma,
rmul-rinv3,
radd-preserves-rless,
rless-implies-rless,
rsub_wf
Rules used in proof :
cut,
hypothesis,
extract_by_obid,
introduction,
intEquality,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
because_Cache,
natural_numberEquality,
addEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
sqequalRule,
independent_pairFormation,
dependent_set_memberEquality,
setElimination,
rename,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
inrFormation,
unionElimination
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}n:\mBbbN{}\msupplus{}. ((a < b) {}\mRightarrow{} ((a < raise-left-endpoint(a;b;n)) \mwedge{} (raise-left-endpoint(a;b;n) < b)))
Date html generated:
2017_10_03-AM-09_32_25
Last ObjectModification:
2017_07_28-AM-07_50_46
Theory : reals
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