Nuprl Lemma : param-W-induction
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type].
∀C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P
∀[Q:par:P ⟶ (pW par) ⟶ ℙ]
((∀par:P. ∀a:A[par]. ∀f:b:B[par;a] ⟶ (pW C[par;a;b]). ((∀b:B[par;a]. Q[C[par;a;b];f b]) ⇒ Q[par;pW-sup(a;f)]))
⇒ (∀par:P. ∀w:pW par. Q[par;w]))
Proof
Definitions occuring in Statement :
pW-sup: pW-sup(a;f),
param-W: pW,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_apply: x[s1;s2],
so_apply: x[s1;s2;s3],
so_apply: x[s],
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
prop: ℙ,
subtype_rel: A ⊆r B
Lemmas referenced :
pW-rec_wf,
param-W_wf,
all_wf,
pW-sup_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
sqequalRule,
hypothesis,
applyEquality,
lambdaEquality,
cumulativity,
functionEquality,
universeEquality
Latex:
\mforall{}[P:Type]. \mforall{}[A:P {}\mrightarrow{} Type]. \mforall{}[B:p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type].
\mforall{}C:p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P
\mforall{}[Q:par:P {}\mrightarrow{} (pW par) {}\mrightarrow{} \mBbbP{}]
((\mforall{}par:P. \mforall{}a:A[par]. \mforall{}f:b:B[par;a] {}\mrightarrow{} (pW C[par;a;b]).
((\mforall{}b:B[par;a]. Q[C[par;a;b];f b]) {}\mRightarrow{} Q[par;pW-sup(a;f)]))
{}\mRightarrow{} (\mforall{}par:P. \mforall{}w:pW par. Q[par;w]))
Date html generated:
2016_05_14-AM-06_14_33
Last ObjectModification:
2015_12_26-PM-00_05_35
Theory : co-recursion
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