Nuprl Lemma : gcd_sat_gcd_p
∀a,b:ℤ. GCD(a;b;gcd(a;b))
Proof
Definitions occuring in Statement :
gcd_p: GCD(a;b;y),
gcd: gcd(a;b),
all: ∀x:A. B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
gcd: gcd(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
bnot: ¬bb,
assert: ↑b,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtract: n - m
Lemmas referenced :
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
istype-less_than,
subtype_rel_self,
absval_wf,
le_wf,
gcd_p_wf,
gcd_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
eq_int_wf,
equal-wf-base,
bool_wf,
assert_wf,
gcd_p_zero,
bnot_wf,
not_wf,
istype-assert,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
absval_unfold,
lt_int_wf,
assert_of_lt_int,
istype-top,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
less_than_wf,
itermMinus_wf,
int_term_value_minus_lemma,
rem_bounds_absval,
nequal_wf,
remainder_wfa,
divide_wfa,
squash_wf,
true_wf,
rem_to_div,
iff_weakening_equal,
gcd_p_shift,
add-associates,
minus-one-mul,
mul-commutes,
add-commutes,
add-mul-special,
zero-mul,
add-zero,
gcd_p_sym
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
Error :functionIsType,
functionEquality,
intEquality,
Error :inhabitedIsType,
Error :setIsType,
addEquality,
baseApply,
closedConclusion,
baseClosed,
cumulativity,
Error :equalityIstype,
sqequalBase,
equalityElimination,
minusEquality,
lessCases,
Error :isect_memberFormation_alt,
axiomSqEquality,
Error :isectIsTypeImplies,
imageMemberEquality,
imageElimination,
promote_hyp,
multiplyEquality,
universeEquality
Latex:
\mforall{}a,b:\mBbbZ{}. GCD(a;b;gcd(a;b))
Date html generated:
2019_06_20-PM-02_21_54
Last ObjectModification:
2019_03_06-AM-11_06_20
Theory : num_thy_1
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