Nuprl Lemma : uniform-comp-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],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
all: ∀x:A. B[x],
ge: i ≥ j ,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
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,
sq_stable: SqStable(P)
Lemmas referenced :
istype-nat,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
subtract-1-ge-0,
decidable__lt,
subtract_wf,
not-lt-2,
condition-implies-le,
minus-add,
istype-void,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
less-iff-le,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
istype-le,
decidable__le,
not-le-2,
sq_stable__le,
zero-add,
add-zero,
add-mul-special,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalRule,
Error :isectIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
Error :functionIsType,
Error :universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
productElimination,
independent_isectElimination,
independent_pairFormation,
instantiate,
universeEquality,
because_Cache,
intWeakElimination,
imageElimination,
independent_functionElimination,
voidElimination,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :dependent_set_memberEquality_alt,
unionElimination,
addEquality,
minusEquality,
Error :productIsType,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
imageMemberEquality,
baseClosed,
multiplyEquality
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:
2019_06_20-AM-11_33_40
Last ObjectModification:
2019_03_12-PM-05_31_41
Theory : int_1
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