Nuprl Lemma : locally-ranked-induction
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Trans(T;x,y.R[y;x])
⇒ (∀k:ℕ. ∀rank:T ⟶ ℕ. ∀l:T ⟶ ℕk.
((∀x,y:T. (((l x) = (l y) ∈ ℤ) ⇒ R[x;y] ⇒ rank x < rank y)) ⇒ (∀[Q:T ⟶ ℙ]. TI(T;x,y.R[y;x];x.Q[x])))))
Proof
Definitions occuring in Statement :
trans: Trans(T;x,y.E[x; y]),
TI: TI(T;x,y.R[x; y];t.Q[t]),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
uimplies: b supposing a
Lemmas referenced :
locally-ranked-is-well-founded,
all_wf,
equal_wf,
less_than_wf,
nat_wf,
int_seg_wf,
trans_wf,
tcWO-induction
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
cumulativity,
sqequalRule,
lambdaEquality,
because_Cache,
functionEquality,
intEquality,
applyEquality,
functionExtensionality,
universeEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T;x,y.R[y;x])
{}\mRightarrow{} (\mforall{}k:\mBbbN{}. \mforall{}rank:T {}\mrightarrow{} \mBbbN{}. \mforall{}l:T {}\mrightarrow{} \mBbbN{}k.
((\mforall{}x,y:T. (((l x) = (l y)) {}\mRightarrow{} R[x;y] {}\mRightarrow{} rank x < rank y))
{}\mRightarrow{} (\mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. TI(T;x,y.R[y;x];x.Q[x])))))
Date html generated:
2017_04_14-AM-07_38_02
Last ObjectModification:
2017_02_27-PM-03_09_50
Theory : rel_1
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