Nuprl Lemma : complete_nat_measure_ind
∀[T:Type]. ∀[measure:T ⟶ ℕ]. ∀[P:T ⟶ ℙ].
((∀i:T. ((∀j:{j:T| measure[j] < measure[i]} . P[j]) ⇒ P[i])) ⇒ (∀i:T. P[i]))
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
genrec: genrec,
nat: ℕ,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
sq_type: SQType(T)
Lemmas referenced :
all_wf,
less_than_wf,
nat_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
le_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-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,
subtype_rel-equal,
base_wf,
equal_wf,
subtype_base_sq,
int_subtype_base,
not-le-2,
le_reflexive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
lambdaEquality,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
functionEquality,
setEquality,
because_Cache,
applyEquality,
functionExtensionality,
hypothesis,
lambdaFormation,
setElimination,
rename,
universeEquality,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
axiomEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
sqequalIntensionalEquality,
promote_hyp,
applyLambdaEquality,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[measure:T {}\mrightarrow{} \mBbbN{}]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}i:T. ((\mforall{}j:\{j:T| measure[j] < measure[i]\} . P[j]) {}\mRightarrow{} P[i])) {}\mRightarrow{} (\mforall{}i:T. P[i]))
Date html generated:
2017_04_14-AM-07_32_42
Last ObjectModification:
2017_02_27-PM-03_07_25
Theory : int_1
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