Nuprl Lemma : prec-induction
∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[Q:i:P ⟶ prec(lbl,p.a[lbl;p];i) ⟶ TYPE].
((∀i:P. ∀x:prec(lbl,p.a[lbl;p];i). ((∀j:P. ∀z:{z:prec(lbl,p.a[lbl;p];j)| prec_sub+(P;lbl,p.a[lbl;p]) <j, z> <i, x>} .\000C Q[j;z]) ⇒ Q[i;x]))
⇒ (∀i:P. ∀x:prec(lbl,p.a[lbl;p];i). Q[i;x]))
Proof
Definitions occuring in Statement :
prec_sub+: prec_sub+(P;lbl,p.a[lbl; p]),
prec: prec(lbl,p.a[lbl; p];i),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
union: left + right,
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
uimplies: b supposing a
Lemmas referenced :
prec-size-induction-ext,
prec_wf,
istype-atom,
prec_sub+_wf,
istype-less_than,
prec-size_wf,
subtype_rel_self,
list_wf,
istype-universe,
prec-sub+-size
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaFormation_alt,
independent_functionElimination,
dependent_functionElimination,
Error :setIsType,
Error :universeIsType,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
Error :dependent_pairEquality_alt,
because_Cache,
Error :functionIsType,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
Error :TYPEMemberIsType,
instantiate,
universeEquality,
Error :TYPEIsType,
unionEquality,
cumulativity,
Error :dependent_set_memberEquality_alt,
independent_isectElimination
Latex:
\mforall{}[P:Type]. \mforall{}[a:Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)]. \mforall{}[Q:i:P {}\mrightarrow{} prec(lbl,p.a[lbl;p];i) {}\mrightarrow{} TYPE].
((\mforall{}i:P. \mforall{}x:prec(lbl,p.a[lbl;p];i).
((\mforall{}j:P. \mforall{}z:\{z:prec(lbl,p.a[lbl;p];j)| prec\_sub+(P;lbl,p.a[lbl;p]) <j, z> <i, x>\} . Q[j;z]) {}\mRightarrow{}\000C Q[i;x]))
{}\mRightarrow{} (\mforall{}i:P. \mforall{}x:prec(lbl,p.a[lbl;p];i). Q[i;x]))
Date html generated:
2019_06_20-PM-02_14_28
Last ObjectModification:
2019_03_12-PM-06_04_12
Theory : tuples
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