Nuprl Lemma : nat_ind
∀[P:ℕ ⟶ ℙ{k}]. (P[0] ⇒ (∀i:ℕ+. (P[i - 1] ⇒ P[i])) ⇒ {∀j:ℕ. P[j]})
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
nat_wf,
all_wf,
nat_plus_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
nat_plus_subtype_nat,
set_wf,
less_than_wf,
primrec-wf2
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
hypothesis,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
applyEquality,
lambdaEquality,
cumulativity,
hypothesisEquality,
universeEquality,
functionEquality,
dependent_set_memberEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
because_Cache,
introduction
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}\{k\}]. (P[0] {}\mRightarrow{} (\mforall{}i:\mBbbN{}\msupplus{}. (P[i - 1] {}\mRightarrow{} P[i])) {}\mRightarrow{} \{\mforall{}j:\mBbbN{}. P[j]\})
Date html generated:
2016_05_13-PM-04_02_50
Last ObjectModification:
2015_12_26-AM-10_56_26
Theory : int_1
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