Nuprl Lemma : imin_lb
∀a,b,c:ℤ. (imin(a;b) ≤ c ⇐⇒ (a ≤ c) ∨ (b ≤ c))
Proof
Definitions occuring in Statement :
imin: imin(a;b),
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
int: ℤ
Definitions unfolded in proof :
imin: imin(a;b),
all: ∀x:A. B[x],
has-value: (a)↓,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
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 ,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
rev_implies: P ⇐ Q,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
value-type-has-value,
int-value-type,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
or_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
callbyvalueReduce,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_pairFormation,
inlFormation,
dependent_functionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
inrFormation
Latex:
\mforall{}a,b,c:\mBbbZ{}. (imin(a;b) \mleq{} c \mLeftarrow{}{}\mRightarrow{} (a \mleq{} c) \mvee{} (b \mleq{} c))
Date html generated:
2017_04_14-AM-09_14_45
Last ObjectModification:
2017_02_27-PM-03_52_51
Theory : int_2
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