Nuprl Lemma : complete-nat-induction
∀[P:ℕ ⟶ ℙ]. ((∀n:ℕ. ((∀m:ℕn. P[m]) ⇒ P[n])) ⇒ (∀n:ℕ. P[n]))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
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],
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],
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
all_wf,
nat_wf,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
natrec_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
rename,
cut,
lemma_by_obid,
isectElimination,
thin,
hypothesis,
lambdaEquality,
functionEquality,
natural_numberEquality,
setElimination,
hypothesisEquality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
because_Cache,
universeEquality,
cumulativity,
introduction
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}n:\mBbbN{}. ((\mforall{}m:\mBbbN{}n. P[m]) {}\mRightarrow{} P[n])) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. P[n]))
Date html generated:
2016_05_13-PM-04_03_17
Last ObjectModification:
2015_12_26-AM-10_56_08
Theory : int_1
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