Nuprl Lemma : AF-induction3
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(∀[Q:T ⟶ ℙ]. TI(T;x,y.R[x;y];t.Q[t])) supposing
((∃R':T ⟶ T ⟶ ℙ. (AFx,y:T.R'[x;y] ∧ (∀x,y:T. (R[x;y] ⇒ (¬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],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: 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,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
almost-full: AFx,y:T.R[x; y],
all: ∀x:A. B[x],
squash: ↓T,
cand: A c∧ B,
not: ¬A,
nat: ℕ,
guard: {T}
Lemmas referenced :
nat_wf,
less_than_wf,
not_wf,
all_wf,
almost-full_wf,
exists_wf,
AF-induction2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
instantiate,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
productElimination,
lambdaFormation,
dependent_functionElimination,
imageElimination,
introduction,
dependent_pairFormation,
independent_pairFormation,
setElimination,
rename,
imageMemberEquality,
baseClosed,
independent_functionElimination
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
((\mexists{}R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. (AFx,y:T.R'[x;y] \mwedge{} (\mforall{}x,y:T. (R[x;y] {}\mRightarrow{} (\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_29
Last ObjectModification:
2016_01_14-PM-06_59_46
Theory : bar-induction
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