Nuprl Lemma : nat-plus-ind-primes
∀[P:ℕ+ ⟶ ℙ]. (P[1] ⇒ (∀p:Prime. P[p]) ⇒ (∀n,m:ℕ+. (P[n] ⇒ P[m] ⇒ P[n * m])) ⇒ (∀n:ℕ+. P[n]))
Proof
Definitions occuring in Statement :
Prime: Prime,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n
Definitions unfolded in proof :
Prime: Prime,
subtype_rel: A ⊆r B,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
prop: ℙ,
and: P ∧ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
int_upper: {i...},
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
le: A ≤ B,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
mul-list: Π(ns) ,
reduce: reduce(f;k;as),
list_ind: list_ind,
cons: [a / b]
Lemmas referenced :
istype-less_than,
decidable__lt,
int_upper_properties,
Prime_wf,
mul_nat_plus,
subtype_rel_self,
nat_plus_wf,
sq_stable__equal,
istype-le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_plus_properties,
prime-factors,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
list_induction,
int_upper_wf,
prime_wf,
mul-list_wf,
subtype_rel_list,
istype-int_upper,
mul-list-positive,
subtype_rel_set,
subtype_rel_sets_simple,
le_wf,
less_than_wf,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
list_wf,
mul_list_nil_lemma
Rules used in proof :
universeEquality,
applyEquality,
functionIsType,
equalitySymmetry,
equalityTransitivity,
voidElimination,
universeIsType,
independent_pairFormation,
sqequalRule,
Error :memTop,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
dependent_set_memberEquality_alt,
independent_functionElimination,
because_Cache,
independent_isectElimination,
intEquality,
cumulativity,
isectElimination,
instantiate,
unionElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
setEquality,
setIsType,
productElimination,
multiplyEquality
Latex:
\mforall{}[P:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbP{}]. (P[1] {}\mRightarrow{} (\mforall{}p:Prime. P[p]) {}\mRightarrow{} (\mforall{}n,m:\mBbbN{}\msupplus{}. (P[n] {}\mRightarrow{} P[m] {}\mRightarrow{} P[n * m])) {}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. P[n]))
Date html generated:
2020_05_20-AM-08_08_13
Last ObjectModification:
2019_12_13-AM-10_08_47
Theory : general
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