Nuprl Lemma : l_all_assert_iff_reduce
∀[A:Type]. ∀[P:A ⟶ 𝔹]. ∀[L:A List].  uiff((∀x∈L.↑P[x]);↑reduce(λx,b. (P[x] ∧b b);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-}
, 
guard: {T}
, 
lelt: i ≤ j < k
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
not: ¬A
, 
less_than: a < b
, 
squash: ↓T
, 
true: True
, 
subtype_rel: A ⊆r B
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
assert_of_band, 
iff_weakening_uiff, 
false_wf, 
int_term_value_add_lemma, 
itermAdd_wf, 
add-is-int-iff, 
length_of_cons_lemma, 
cons_wf, 
l_all_cons, 
and_wf, 
true_wf, 
length_of_nil_lemma, 
nil_wf, 
l_all_nil, 
l_all_wf_nil, 
int_seg_wf, 
int_formula_prop_less_lemma, 
intformless_wf, 
decidable__lt, 
int_formula_prop_wf, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
intformnot_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
decidable__le, 
length_wf, 
int_seg_properties, 
select_wf, 
assert_witness, 
reduce_cons_lemma, 
reduce_nil_lemma, 
list_wf, 
btrue_wf, 
band_wf, 
bool_wf, 
reduce_wf, 
l_member_wf, 
assert_wf, 
l_all_wf, 
uiff_wf, 
list_induction
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
applyEquality, 
setElimination, 
rename, 
hypothesis, 
setEquality, 
independent_functionElimination, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
lambdaFormation, 
because_Cache, 
productElimination, 
independent_pairEquality, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
cumulativity, 
independent_isectElimination, 
natural_numberEquality, 
unionElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
independent_pairFormation, 
computeAll, 
imageElimination, 
universeEquality, 
axiomEquality, 
addLevel, 
addEquality, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
closedConclusion, 
baseClosed
Latex:
\mforall{}[A:Type].  \mforall{}[P:A  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[L:A  List].    uiff((\mforall{}x\mmember{}L.\muparrow{}P[x]);\muparrow{}reduce(\mlambda{}x,b.  (P[x]  \mwedge{}\msubb{}  b);tt;L))
Date html generated:
2016_05_14-PM-02_45_25
Last ObjectModification:
2016_01_15-AM-07_36_50
Theory : list_1
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