Nuprl Lemma : wfd-tree-rec_wf
∀[X,T:Type]. ∀[b:X]. ∀[F:(T ⟶ X) ⟶ X]. ∀[t:wfd-tree(T)]. (wfd-tree-rec(b;r.F[r];t) ∈ X)
Proof
Definitions occuring in Statement :
wfd-tree-rec: wfd-tree-rec(b;r.F[r];t),
wfd-tree: wfd-tree(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
wfd-tree: wfd-tree(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
wfd-tree-rec: wfd-tree-rec(b;r.F[r];t),
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
prop: ℙ,
so_apply: x[s],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
so_apply: x[s1;s2;s3]
Lemmas referenced :
W-rec_wf,
bool_wf,
eqtt_to_assert,
equal_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
W_wf,
ifthenelse_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
applyEquality,
functionExtensionality,
functionEquality,
axiomEquality,
universeEquality,
isect_memberEquality
Latex:
\mforall{}[X,T:Type]. \mforall{}[b:X]. \mforall{}[F:(T {}\mrightarrow{} X) {}\mrightarrow{} X]. \mforall{}[t:wfd-tree(T)]. (wfd-tree-rec(b;r.F[r];t) \mmember{} X)
Date html generated:
2017_04_14-AM-07_45_11
Last ObjectModification:
2017_02_27-PM-03_16_11
Theory : co-recursion
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