Nuprl Lemma : complete_nat_ind
∀[P:ℕ ⟶ ℙ]. ((∀i:ℕ. ((∀j:ℕi. P[j]) ⇒ P[i])) ⇒ (∀i:ℕ. P[i]))
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,
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
true: True,
less_than: a < b,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
zero-mul,
add-mul-special,
le_wf,
add-zero,
zero-add,
sq_stable__le,
not-le-2,
decidable__le,
lelt_wf,
le-add-cancel,
add-associates,
add_functionality_wrt_le,
less-iff-le,
add-commutes,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-minus,
minus-add,
condition-implies-le,
not-lt-2,
decidable__lt,
primrec-wf2,
less_than_wf,
set_wf,
subtract_wf,
less_than_irreflexivity,
less_than_transitivity1,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
all_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
independent_isectElimination,
independent_pairFormation,
because_Cache,
universeEquality,
cumulativity,
productElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
intEquality,
introduction,
dependent_set_memberEquality,
unionElimination,
addEquality,
minusEquality,
isect_memberEquality,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
multiplyEquality
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}i:\mBbbN{}. ((\mforall{}j:\mBbbN{}i. P[j]) {}\mRightarrow{} P[i])) {}\mRightarrow{} (\mforall{}i:\mBbbN{}. P[i]))
Date html generated:
2016_05_13-PM-04_02_56
Last ObjectModification:
2016_01_14-PM-07_24_48
Theory : int_1
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