Nuprl Lemma : rless-witness_wf
∀[x,y:ℝ]. ∀[p:x < y]. (rless-witness(x;y;p) ∈ ℕ+)
Proof
Definitions occuring in Statement :
rless-witness: rless-witness(x;y;p),
rless: x < y,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
sq_exists: ∃x:A [B[x]],
rless: x < y,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat_plus: ℕ+,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rless-witness: rless-witness(x;y;p)
Lemmas referenced :
equal_wf,
pi1_wf_top,
subtype_rel_function,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
rsub_wf,
rleq_wf,
nat_plus_wf,
exists_wf,
rless_wf,
all_wf,
real_wf,
subtype_rel_self,
rless-implies-rleq
Rules used in proof :
axiomEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairEquality,
lambdaFormation,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
unionElimination,
independent_functionElimination,
productElimination,
dependent_functionElimination,
inrFormation,
independent_isectElimination,
rename,
setElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
lambdaEquality,
functionEquality,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
instantiate,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[p:x < y]. (rless-witness(x;y;p) \mmember{} \mBbbN{}\msupplus{})
Date html generated:
2018_05_22-PM-02_01_58
Last ObjectModification:
2018_05_21-AM-00_14_53
Theory : reals
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