Nuprl Lemma : square_non_neg
∀x:ℤ. (0 ≤ (x * x))
Proof
Definitions occuring in Statement :
le: A ≤ B,
all: ∀x:A. B[x],
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
and: P ∧ Q,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
top: Top,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
less_than': less_than'(a;b),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
mul_preserves_le,
absval_wf,
nat_wf,
absval-non-neg,
le_wf,
squash_wf,
true_wf,
decidable__equal_int,
multiply-is-int-iff,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
false_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma,
absval_unfold
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
independent_isectElimination,
because_Cache,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
intEquality,
dependent_functionElimination,
unionElimination,
pointwiseFunctionality,
promote_hyp,
productElimination,
baseApply,
closedConclusion,
baseClosed,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageMemberEquality,
minusEquality,
equalityElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
independent_pairFormation,
independent_functionElimination,
multiplyEquality,
instantiate,
cumulativity
Latex:
\mforall{}x:\mBbbZ{}. (0 \mleq{} (x * x))
Date html generated:
2017_04_14-AM-09_15_33
Last ObjectModification:
2017_02_27-PM-03_53_21
Theory : int_2
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