Nuprl Lemma : gcd_subtract
∀a,b:ℕ. gcd(a - b;b) ~ gcd(a;b) supposing b ≤ a
Proof
Definitions occuring in Statement :
gcd: gcd(a;b),
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
subtract: n - m,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sq_type: SQType(T),
guard: {T},
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
le: A ≤ B,
subtract: n - m,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
assoced_nelim,
gcd_wf,
subtract_wf,
gcd-non-neg,
subtract_nat_wf,
le_wf,
nat_wf,
gcd_elim,
assoced_wf,
squash_wf,
true_wf,
istype-int,
subtype_rel_self,
iff_weakening_equal,
gcd_unique,
gcd_sat_pred,
gcd_p_sym,
gcd_p_shift,
add-associates,
istype-void,
minus-one-mul,
one-mul,
add-commutes,
add-mul-special,
zero-mul,
add-zero,
set_subtype_base,
gcd_p_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
setElimination,
rename,
because_Cache,
hypothesisEquality,
Error :universeIsType,
natural_numberEquality,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
axiomSqEquality,
Error :inhabitedIsType,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
addEquality,
multiplyEquality,
hyp_replacement,
Error :isect_memberEquality_alt,
voidElimination,
minusEquality,
independent_pairFormation,
Error :productIsType,
Error :equalityIsType4,
baseApply,
closedConclusion,
applyLambdaEquality
Latex:
\mforall{}a,b:\mBbbN{}. gcd(a - b;b) \msim{} gcd(a;b) supposing b \mleq{} a
Date html generated:
2019_06_20-PM-02_27_04
Last ObjectModification:
2018_10_03-AM-10_23_49
Theory : num_thy_1
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