Nuprl Lemma : gcd_eq_args
∀a:ℤ. (gcd(a;a) ~ a)
Proof
Definitions occuring in Statement :
gcd: gcd(a;b),
all: ∀x:A. B[x],
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
gcd: gcd(a;b),
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],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
eq_int: (i =z j),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
prop: ℙ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
istype-int,
div_rem_sum,
nequal_wf,
div-self,
decidable__equal_int,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
hypothesisEquality,
natural_numberEquality,
Error :inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
voidElimination,
Error :dependent_set_memberEquality_alt,
Error :universeIsType,
pointwiseFunctionality,
rename,
baseApply,
closedConclusion,
baseClosed,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
independent_pairFormation
Latex:
\mforall{}a:\mBbbZ{}. (gcd(a;a) \msim{} a)
Date html generated:
2019_06_20-PM-02_22_02
Last ObjectModification:
2019_03_06-AM-11_06_19
Theory : num_thy_1
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