Nuprl Lemma : cont-induction-lemma
∀[P:ℕ ⟶ ℙ]. (P[0] ⇒ (∀n:ℕ. (P[n] ⇒ P[n + 1])) ⇒ (∀n:ℕ. ∀[m:ℕ]. ((P[n] ⇒ P[n + m]) ⇒ P[n + m])))
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,
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
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),
guard: {T}
Lemmas referenced :
false_wf,
all_wf,
nat_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
uall_wf,
primrec-induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
because_Cache,
functionEquality,
applyEquality,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
universeEquality,
cumulativity
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]
(P[0] {}\mRightarrow{} (\mforall{}n:\mBbbN{}. (P[n] {}\mRightarrow{} P[n + 1])) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. \mforall{}[m:\mBbbN{}]. ((P[n] {}\mRightarrow{} P[n + m]) {}\mRightarrow{} P[n + m])))
Date html generated:
2016_05_15-PM-04_09_08
Last ObjectModification:
2016_01_16-AM-11_03_26
Theory : general
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