Nuprl Lemma : accessible-induction
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ].
((∀t:T. ((∀x:T. (R[x;t] ⇒ P[x])) ⇒ P[t])) ⇒ (∀t:T. (accessible(T;x,y.R[x;y];t) ⇒ P[t])))
Proof
Definitions occuring in Statement :
accessible: accessible(T;x,y.R[x; y];t),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
accessible: accessible(T;x,y.R[x; y];t),
pi1: fst(t)
Lemmas referenced :
param-W-induction,
unit_wf2,
pi1_wf,
param-W_wf,
all_wf,
accessible_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality,
hypothesis,
hypothesisEquality,
productEquality,
applyEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
cumulativity,
productElimination,
rename,
dependent_pairEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}t:T. ((\mforall{}x:T. (R[x;t] {}\mRightarrow{} P[x])) {}\mRightarrow{} P[t])) {}\mRightarrow{} (\mforall{}t:T. (accessible(T;x,y.R[x;y];t) {}\mRightarrow{} P[t])))
Date html generated:
2016_05_14-AM-06_18_47
Last ObjectModification:
2015_12_26-PM-00_02_57
Theory : co-recursion
Home
Index