Nuprl Lemma : ndiff_zero
∀[a,b:ℤ]. uiff((a -- b) = 0 ∈ ℤ;a ≤ b)
Proof
Definitions occuring in Statement :
ndiff: a -- b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
ndiff: a -- b,
imax: imax(a;b),
has-value: (a)↓,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
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),
guard: {T},
bnot: ¬bb,
assert: ↑b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
value-type-has-value,
int-value-type,
subtract_wf,
less_than'_wf,
equal-wf-base,
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,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
int_subtype_base,
decidable__le,
assert_wf,
bnot_wf,
not_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
callbyvalueReduce,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
independent_pairFormation,
isect_memberFormation,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
because_Cache,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
impliesFunctionality
Latex:
\mforall{}[a,b:\mBbbZ{}]. uiff((a -- b) = 0;a \mleq{} b)
Date html generated:
2017_04_14-AM-09_15_06
Last ObjectModification:
2017_02_27-PM-03_52_53
Theory : int_2
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