Nuprl Lemma : rec-nat-induction
∀[P:ℕ ⟶ ℙ]. (∀[n:ℕ]. (P[n] ⇒ P[n + 1])) ⇒ (∀[n:ℕ]. P[n]) supposing Top ⊆r P[0]
Proof
Definitions occuring in Statement :
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
ge: i ≥ j ,
guard: {T}
Lemmas referenced :
equal_wf,
subtract-add-cancel,
minus-minus,
less-iff-le,
not-ge-2,
subtract_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
top_wf,
subtype_rel_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le,
nat_wf,
uall_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
axiomEquality,
hypothesis,
thin,
rename,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
lambdaEquality,
functionEquality,
applyEquality,
because_Cache,
hypothesisEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
cumulativity,
universeEquality,
intWeakElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[P:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. (\mforall{}[n:\mBbbN{}]. (P[n] {}\mRightarrow{} P[n + 1])) {}\mRightarrow{} (\mforall{}[n:\mBbbN{}]. P[n]) supposing Top \msubseteq{}r P[0]
Date html generated:
2016_05_13-PM-04_02_53
Last ObjectModification:
2016_01_14-PM-07_24_38
Theory : int_1
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