Nuprl Lemma : rmin-int
∀[a,b:ℤ]. (rmin(r(a);r(b)) = r(imin(a;b)))
Proof
Definitions occuring in Statement :
rmin: rmin(x;y),
req: x = y,
int-to-real: r(n),
imin: imin(a;b),
uall: ∀[x:A]. B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
real: ℝ,
all: ∀x:A. B[x],
int-to-real: r(n),
rmin: rmin(x;y),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
nat_plus: ℕ+,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
less_than': less_than'(a;b),
nat: ℕ
Lemmas referenced :
req-iff-bdd-diff,
rmin_wf,
int-to-real_wf,
imin_wf,
trivial-bdd-diff,
real_wf,
nat_plus_wf,
req_witness,
equal_wf,
squash_wf,
true_wf,
imin_unfold,
iff_weakening_equal,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
mul_preserves_lt,
mul_nat_plus,
less_than_wf,
itermMultiply_wf,
itermConstant_wf,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
mul_preserves_le,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
lambdaFormation,
independent_functionElimination,
intEquality,
isect_memberEquality,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
multiplyEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
voidElimination,
cumulativity,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. (rmin(r(a);r(b)) = r(imin(a;b)))
Date html generated:
2017_10_03-AM-08_28_51
Last ObjectModification:
2017_07_28-AM-07_25_32
Theory : reals
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