Nuprl Lemma : primrec-induction
∀[P:ℕ ⟶ ℙ]. (P[0] ⇒ (∀n:ℕ. (P[n] ⇒ P[n + 1])) ⇒ (∀n:ℕ. P[n]))
Proof
Definitions occuring in Statement :
nat: 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],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
guard: {T},
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_wf,
all_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
false_wf,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
primrec0_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
primrec-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
subtract-add-cancel,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
because_Cache,
universeEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
cumulativity,
intWeakElimination,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
productElimination,
promote_hyp,
instantiate
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. (P[0] {}\mRightarrow{} (\mforall{}n:\mBbbN{}. (P[n] {}\mRightarrow{} P[n + 1])) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. P[n]))
Date html generated:
2018_05_21-PM-06_59_19
Last ObjectModification:
2017_07_26-PM-05_02_03
Theory : general
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