Nuprl Lemma : fset-closure-exists2
∀[T:Type]
∀eq:EqDecider(T). ∀r:T ⟶ ℕ. ∀fs:{f:T ⟶ T| ∀x:T. r (f x) < r x supposing ¬((f x) = x ∈ T)} List. ∀s:fset(T).
∃c:fset(T). (c = fs closure of s)
Proof
Definitions occuring in Statement :
fset-closure: (c = fs closure of s),
fset: fset(T),
list: T List,
deq: EqDecider(T),
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
prop: ℙ,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
fset-closure-exists,
subtype_rel_list,
not_wf,
equal_wf,
less_than_wf,
istype-void,
istype-less_than,
fset_wf,
list_wf,
istype-nat,
deq_wf,
istype-universe,
l_all_iff,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
setEquality,
functionEquality,
sqequalRule,
isectEquality,
hypothesis,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
setIsType,
functionIsType,
universeIsType,
isectIsType,
equalityIstype,
instantiate,
universeEquality,
inhabitedIsType,
productElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}r:T {}\mrightarrow{} \mBbbN{}. \mforall{}fs:\{f:T {}\mrightarrow{} T| \mforall{}x:T. r (f x) < r x supposing \mneg{}((f x) = x)\} List.
\mforall{}s:fset(T).
\mexists{}c:fset(T). (c = fs closure of s)
Date html generated:
2020_05_19-PM-09_52_43
Last ObjectModification:
2020_01_04-PM-08_00_05
Theory : finite!sets
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