Nuprl Lemma : radd-rmin
∀[x,y,z:ℝ]. ((x + rmin(y;z)) = rmin(x + y;x + z))
Proof
Definitions occuring in Statement :
rmin: rmin(x;y),
req: x = y,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
real: ℝ,
rmin: rmin(x;y),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
le: A ≤ B,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
nat_plus: ℕ+,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T
Lemmas referenced :
req-iff-bdd-diff,
radd_wf,
rmin_wf,
req_witness,
real_wf,
nat_plus_wf,
imin_wf,
trivial-bdd-diff,
bdd-diff_functionality,
radd-bdd-diff,
rmin_functionality_wrt_bdd-diff,
ifthenelse_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
le_wf,
nat_plus_properties,
less_than_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermVar_wf,
intformnot_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
add-is-int-iff,
false_wf,
squash_wf,
true_wf,
add_functionality_wrt_eq,
imin_unfold,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
isect_memberEquality,
because_Cache,
applyEquality,
lambdaEquality,
setElimination,
rename,
addEquality,
lambdaFormation,
dependent_functionElimination,
intEquality,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
dependent_set_memberEquality,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[x,y,z:\mBbbR{}]. ((x + rmin(y;z)) = rmin(x + y;x + z))
Date html generated:
2017_10_03-AM-08_29_12
Last ObjectModification:
2017_07_28-AM-07_25_45
Theory : reals
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