Nuprl Lemma : l_all_reduce
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ 𝔹]. uiff((∀x∈L.↑P[x]);↑reduce(λx,y. (P[x] ∧b y);tt;L))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
reduce: reduce(f;k;as),
list: T List,
band: p ∧b q,
assert: ↑b,
btrue: tt,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
guard: {T},
false: False,
true: True,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
or: P ∨ Q,
band: p ∧b q,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
uall_wf,
bool_wf,
uiff_wf,
l_all_wf,
assert_wf,
l_member_wf,
reduce_wf,
band_wf,
btrue_wf,
list_wf,
reduce_nil_lemma,
reduce_cons_lemma,
assert_witness,
select_wf,
sq_stable__le,
int_seg_wf,
length_wf,
true_wf,
less_than_irreflexivity,
less_than_transitivity1,
all_wf,
base_wf,
stuck-spread,
length_of_nil_lemma,
assert_of_band,
bool_cases_sqequal,
l_all_cons,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
applyEquality,
setElimination,
rename,
setEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
because_Cache,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
cumulativity,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
universeEquality,
axiomEquality,
independent_pairFormation,
productEquality,
unionElimination,
addLevel
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. uiff((\mforall{}x\mmember{}L.\muparrow{}P[x]);\muparrow{}reduce(\mlambda{}x,y. (P[x] \mwedge{}\msubb{} y);tt;L))
Date html generated:
2019_06_20-PM-00_41_28
Last ObjectModification:
2018_08_24-PM-11_01_12
Theory : list_0
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