Nuprl Lemma : gcd-positive
∀[y,x:ℕ]. (0 < gcd(x;y)) supposing ((0 ≤ y) and 0 < x)
Proof
Definitions occuring in Statement :
gcd: gcd(a;b),
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
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,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
member-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
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,
subtype_rel_self,
eq_int_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
le_wf,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
istype-assert,
remainder_wfa,
nequal_wf,
rem_bounds_1,
gcd_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
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,
productElimination,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
equalityElimination,
baseApply,
closedConclusion,
baseClosed,
intEquality,
Error :equalityIstype,
sqequalBase,
Error :functionIsType,
addEquality
Latex:
\mforall{}[y,x:\mBbbN{}]. (0 < gcd(x;y)) supposing ((0 \mleq{} y) and 0 < x)
Date html generated:
2019_06_20-PM-02_26_59
Last ObjectModification:
2019_03_06-AM-11_06_24
Theory : num_thy_1
Home
Index