Nuprl Lemma : imax-list-as-reduce
∀[L:ℤ List]
imax-list(L) = outl(reduce(λx,y. case y of inl(z) => inl imax(x;z) | inr(z) => inl x;inr ⋅ ;L)) ∈ ℤ supposing 0 < ||L|\000C|
Proof
Definitions occuring in Statement :
imax-list: imax-list(L),
length: ||as||,
reduce: reduce(f;k;as),
list: T List,
imax: imax(a;b),
outl: outl(x),
less_than: a < b,
it: ⋅,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inr: inr x ,
inl: inl x,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_apply: x[s1;s2],
assoc: Assoc(T;op),
infix_ap: x f y,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
comm: Comm(T;op),
sq_type: SQType(T),
all: ∀x:A. B[x],
imax-list: imax-list(L)
Lemmas referenced :
combine-list-as-reduce,
imax_wf,
equal_wf,
squash_wf,
true_wf,
imax_assoc,
iff_weakening_equal,
imax_com,
subtype_base_sq,
int_subtype_base,
imax-list_wf,
less_than_wf,
length_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
lambdaEquality,
hypothesisEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
because_Cache,
isect_memberEquality,
axiomEquality,
instantiate,
cumulativity,
dependent_functionElimination
Latex:
\mforall{}[L:\mBbbZ{} List]
imax-list(L) = outl(reduce(\mlambda{}x,y. case y of inl(z) => inl imax(x;z) | inr(z) => inl x;inr \mcdot{} ;L))
supposing 0 < ||L||
Date html generated:
2017_04_17-AM-07_39_59
Last ObjectModification:
2017_02_27-PM-04_12_41
Theory : list_1
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