Nuprl Lemma : accessible-iff
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[t:T]. (accessible(T;x,y.R[x;y];t) ⇐⇒ ∀u:T. (R[u;t] ⇒ accessible(T;x,y.R[x;y];u)))
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],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
accessible: accessible(T;x,y.R[x; y];t),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ext-family: F ≡ G,
all: ∀x:A. B[x],
ext-eq: A ≡ B,
pi1: fst(t),
pi2: snd(t),
guard: {T},
uimplies: b supposing a
Lemmas referenced :
accessible_wf,
all_wf,
param-W-ext,
unit_wf2,
pi1_wf,
it_wf,
param-W_wf,
ext-eq_inversion,
subtype_rel_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
lemma_by_obid,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
hypothesis_subsumption,
productEquality,
because_Cache,
dependent_functionElimination,
productElimination,
rename,
introduction,
dependent_pairEquality,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[t:T].
(accessible(T;x,y.R[x;y];t) \mLeftarrow{}{}\mRightarrow{} \mforall{}u:T. (R[u;t] {}\mRightarrow{} accessible(T;x,y.R[x;y];u)))
Date html generated:
2016_05_14-AM-06_18_41
Last ObjectModification:
2015_12_26-PM-00_03_03
Theory : co-recursion
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