Nuprl Lemma : prime-factors-unique
∀ps:{m:ℕ| prime(m)} List. ∀qs:{qs:{m:ℕ| prime(m)} List| Π(ps) = Π(qs) ∈ ℤ} . permutation(ℤ;ps;qs)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
mul-list: Π(ns) ,
prime: prime(a),
permutation: permutation(T;L1;L2),
list: T List,
nat: ℕ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
mul-list: Π(ns) ,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
top: Top,
exists: ∃x:A. B[x],
divides: b | a,
false: False,
and: P ∧ Q,
prime: prime(a),
assoced: a ~ b,
not: ¬A,
label: ...$L... t,
guard: {T},
permutation: permutation(T;L1;L2),
cand: A c∧ B,
sq_type: SQType(T),
so_apply: x[s1;s2],
assoc: Assoc(T;op),
infix_ap: x f y,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
comm: Comm(T;op),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
list_induction,
nat_wf,
prime_wf,
list_wf,
equal-wf-base,
list_subtype_base,
set_subtype_base,
istype-nat,
le_wf,
istype-int,
int_subtype_base,
permutation_wf,
subtype_rel_list,
reduce_nil_lemma,
reduce_cons_lemma,
set_wf,
nil_wf,
reduce_wf,
equal-wf-base-T,
permutation-nil,
one_divs_any,
positive-prime-divides-product,
l_member-permutation,
cons_wf,
permutation_inversion,
inject_wf,
int_seg_wf,
length_wf,
permute_list_wf,
subtype_base_sq,
reduce-permutation,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
permutation_functionality_wrt_permutation,
cons_functionality_wrt_permutation,
permutation_weakening,
mul_cancel_in_eq,
nequal_wf,
mul-list_wf,
equal_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
setEquality,
hypothesis,
setElimination,
rename,
because_Cache,
lambdaEquality_alt,
functionEquality,
hypothesisEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_isectElimination,
natural_numberEquality,
setIsType,
universeIsType,
independent_functionElimination,
dependent_functionElimination,
Error :memTop,
functionIsType,
equalityIstype,
sqequalBase,
equalitySymmetry,
voidEquality,
voidElimination,
isect_memberEquality,
multiplyEquality,
lambdaEquality,
lambdaFormation,
dependent_pairFormation,
productElimination,
independent_pairFormation,
inhabitedIsType,
dependent_pairFormation_alt,
equalityTransitivity,
promote_hyp,
productIsType,
instantiate,
cumulativity,
isect_memberFormation_alt,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
dependent_set_memberEquality_alt,
levelHypothesis,
addLevel
Latex:
\mforall{}ps:\{m:\mBbbN{}| prime(m)\} List. \mforall{}qs:\{qs:\{m:\mBbbN{}| prime(m)\} List| \mPi{}(ps) = \mPi{}(qs) \} . permutation(\mBbbZ{};ps;qs)
Date html generated:
2020_05_20-AM-08_08_03
Last ObjectModification:
2020_01_04-PM-11_11_31
Theory : general
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