Nuprl Lemma : unique-minimal-wellfounded-implies
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(decidable-non-minimal(T;x,y.R[x;y])
⇒ WellFnd{i}(T;x,y.R[x;y])
⇒ (∀m:T. (unique-minimal(T;x,y.R[x;y];m) ⇒ (∀y:T. (↓m ((λx,y. R[x;y])^*) y)))))
Proof
Definitions occuring in Statement :
decidable-non-minimal: decidable-non-minimal(T;x,y.R[x; y]),
unique-minimal: unique-minimal(T;x,y.R[x; y];m),
rel_star: R^*,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
squash: ↓T,
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
so_lambda: λ2x.t[x],
infix_ap: x f y,
so_apply: x[s1;s2],
so_apply: x[s],
prop: ℙ,
guard: {T},
squash: ↓T,
so_lambda: λ2x y.t[x; y],
decidable-non-minimal: decidable-non-minimal(T;x,y.R[x; y]),
decidable: Dec(P),
or: P ∨ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a,
unique-minimal: unique-minimal(T;x,y.R[x; y];m),
and: P ∧ Q,
not: ¬A,
false: False
Lemmas referenced :
rel_star_weakening,
rel_rel_star,
rel_star_transitivity,
decidable-non-minimal_wf,
wellfounded_wf,
unique-minimal_wf,
all_wf,
rel_star_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
lemma_by_obid,
applyEquality,
hypothesisEquality,
independent_functionElimination,
functionEquality,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
because_Cache,
cumulativity,
universeEquality,
isect_memberEquality,
unionElimination,
productElimination,
independent_isectElimination,
equalitySymmetry,
dependent_pairFormation,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(decidable-non-minimal(T;x,y.R[x;y])
{}\mRightarrow{} WellFnd\{i\}(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}m:T. (unique-minimal(T;x,y.R[x;y];m) {}\mRightarrow{} (\mforall{}y:T. (\mdownarrow{}m rel\_star(T; \mlambda{}x,y. R[x;y]) y)))))
Date html generated:
2016_05_15-PM-07_51_25
Last ObjectModification:
2016_01_16-AM-09_36_45
Theory : general
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