Nuprl Lemma : le-iff-imin
∀[a,b:ℤ]. uiff(a ≤ b;imin(a;b) = a ∈ ℤ)
Proof
Definitions occuring in Statement :
imin: imin(a;b),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
imin: imin(a;b),
uall: ∀[x:A]. B[x],
member: t ∈ T,
has-value: (a)↓,
uimplies: b supposing 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 ,
prop: ℙ,
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
value-type-has-value,
int-value-type,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
le_wf,
less_than'_wf,
equal-wf-base,
int_subtype_base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_pairFormation,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
applyEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
natural_numberEquality,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[a,b:\mBbbZ{}]. uiff(a \mleq{} b;imin(a;b) = a)
Date html generated:
2017_04_14-AM-09_13_57
Last ObjectModification:
2017_02_27-PM-03_51_10
Theory : int_2
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