Nuprl Lemma : AF-induction2
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(∀[Q:T ⟶ ℙ]. TI(T;x,y.R[x;y];t.Q[t])) supposing (AFx,y:T.¬R[x;y] and (∀x,y,z:T. (R[x;y] ⇒ R[y;z] ⇒ R[x;z])))
Proof
Definitions occuring in Statement :
almost-full: AFx,y:T.R[x; y],
TI: TI(T;x,y.R[x; y];t.Q[t]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
almost-full: AFx,y:T.R[x; y],
all: ∀x:A. B[x],
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
TI: TI(T;x,y.R[x; y];t.Q[t]),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
almost-full_wf,
all_wf,
false_wf,
subtype_rel_sets,
not_wf,
AF-induction,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
imageElimination,
hypothesis,
imageMemberEquality,
baseClosed,
functionEquality,
lemma_by_obid,
rename,
isectElimination,
applyEquality,
independent_isectElimination,
lambdaFormation,
independent_functionElimination,
voidElimination,
because_Cache,
setElimination,
setEquality,
universeEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(\mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. TI(T;x,y.R[x;y];t.Q[t])) supposing
(AFx,y:T.\mneg{}R[x;y] and
(\mforall{}x,y,z:T. (R[x;y] {}\mRightarrow{} R[y;z] {}\mRightarrow{} R[x;z])))
Date html generated:
2016_05_13-PM-03_51_25
Last ObjectModification:
2016_01_14-PM-06_59_49
Theory : bar-induction
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