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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