Nuprl Lemma : wellfounded-irreflexive
∀[A:Type]. ∀[r:A ⟶ A ⟶ ℙ]. ∀a:A. (¬r[a;a]) supposing WellFnd{i}(A;x,y.r[x;y])
Proof
Definitions occuring in Statement :
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
wellfounded_wf,
equal_wf,
false_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
universeEquality,
extract_by_obid,
isectElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
hyp_replacement
Latex:
\mforall{}[A:Type]. \mforall{}[r:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}a:A. (\mneg{}r[a;a]) supposing WellFnd\{i\}(A;x,y.r[x;y])
Date html generated:
2016_10_21-AM-09_35_41
Last ObjectModification:
2016_07_12-AM-04_59_42
Theory : well_fnd
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