Nuprl Lemma : imin-imax-distributive
∀[x,y,z:ℤ]. (imin(imax(x;y);imax(x;z)) = imax(x;imin(y;z)) ∈ ℤ)
Proof
Definitions occuring in Statement :
imin: imin(a;b),
imax: imax(a;b),
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
imin: imin(a;b),
imax: imax(a;b),
uimplies: b supposing a,
has-value: (a)↓,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P)
Lemmas referenced :
value-type-has-value,
int-value-type,
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,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
intEquality,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
extract_by_obid,
independent_isectElimination,
callbyvalueReduce,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
natural_numberEquality,
lambdaEquality,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[x,y,z:\mBbbZ{}]. (imin(imax(x;y);imax(x;z)) = imax(x;imin(y;z)))
Date html generated:
2017_04_14-AM-09_14_36
Last ObjectModification:
2017_02_27-PM-03_52_17
Theory : int_2
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