Nuprl Lemma : wellfounded-minimal-field
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(WellFnd{i}(T;x,y.R x y)
⇒ (∀x,y:T. Dec(R x y))
⇒ (∀y:T. ∃L:T List. ∀x:T. ((R x y) ⇒ (x ∈ L)))
⇒ (∀y:T. ∃x:T. (((R^*) x y) ∧ (∀z:T. (¬(R z x))))))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
rel_star: R^*,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
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,
true: True,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
infix_ap: x f y
Lemmas referenced :
wellfounded-minimal,
true_wf,
decidable__true,
rel_star_wf,
all_wf,
not_wf,
exists_wf,
list_wf,
l_member_wf,
decidable_wf,
wellfounded_wf,
rel-rel-plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
lambdaEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
productElimination,
dependent_pairFormation,
independent_pairFormation,
voidElimination,
applyEquality,
productEquality,
universeEquality,
functionEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(WellFnd\{i\}(T;x,y.R x y)
{}\mRightarrow{} (\mforall{}x,y:T. Dec(R x y))
{}\mRightarrow{} (\mforall{}y:T. \mexists{}L:T List. \mforall{}x:T. ((R x y) {}\mRightarrow{} (x \mmember{} L)))
{}\mRightarrow{} (\mforall{}y:T. \mexists{}x:T. ((rel\_star(T; R) x y) \mwedge{} (\mforall{}z:T. (\mneg{}(R z x))))))
Date html generated:
2016_05_15-PM-04_52_29
Last ObjectModification:
2015_12_27-PM-02_32_42
Theory : general
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