Nuprl Lemma : no-descending-chain-implies-wellfounded
∀[T:Type]. ∀[<:T ⟶ T ⟶ ℙ].
((∀x,y,z:T. ((x < y) ⇒ (y < z) ⇒ (x < z)))
⇒ (∀x,y:T. Dec(x < y))
⇒ no-descending-chain(T;<)
⇒ WellFnd{i}(T;x,y.x < y))
Proof
Definitions occuring in Statement :
no-descending-chain: no-descending-chain(T;<),
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
infix_ap: x f y,
all: ∀x:A. B[x],
prop: ℙ,
no-descending-chain: no-descending-chain(T;<),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
descending-chain-barred-implies-wellfounded,
not_wf,
decidable__not,
nat_wf,
no-descending-chain_wf,
all_wf,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
dependent_functionElimination,
because_Cache,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[<:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y,z:T. ((x < y) {}\mRightarrow{} (y < z) {}\mRightarrow{} (x < z)))
{}\mRightarrow{} (\mforall{}x,y:T. Dec(x < y))
{}\mRightarrow{} no-descending-chain(T;<)
{}\mRightarrow{} WellFnd\{i\}(T;x,y.x < y))
Date html generated:
2016_05_13-PM-03_52_06
Last ObjectModification:
2015_12_26-AM-10_17_15
Theory : bar-induction
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