Nuprl Lemma : all-but-one
∀[T:Type]. ∀[P:T ⟶ ℙ].
∀L:T List
(∀x,y:T. Dec(x = y ∈ T))
⇒ ((∀x∈L.(∀y∈L.P[x] ∨ P[y] supposing ¬(x = y ∈ T))) ⇐⇒ (∃x∈L. (∀y∈L.P[y] supposing ¬(x = y ∈ T))))
supposing 0 < ||L||
Proof
Definitions occuring in Statement :
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
top: Top,
decidable: Dec(P),
l_all: (∀x∈L.P[x]),
not: ¬A,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
subtract: n - m,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
true: True,
l_exists: (∃x∈L. P[x]),
select: L[n],
cons: [a / b],
subtype_rel: A ⊆r B
Lemmas referenced :
member-less_than,
length_wf,
l_all_wf,
isect_wf,
not_wf,
equal_wf,
or_wf,
l_member_wf,
l_exists_wf,
all_wf,
decidable_wf,
less_than_wf,
list_wf,
list_induction,
length_of_nil_lemma,
length_of_cons_lemma,
cons_wf,
l_all_cons,
decidable__lt,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
int_formula_prop_less_lemma,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
int_seg_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
add-subtract-cancel,
select-cons-tl,
squash_wf,
le_wf,
true_wf,
and_wf,
false_wf,
iff_weakening_equal,
list-cases,
product_subtype_list,
nil_wf,
l_all_nil,
l_exists_iff,
l_all_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
independent_isectElimination,
rename,
independent_pairFormation,
sqequalRule,
lambdaEquality,
setElimination,
applyEquality,
functionExtensionality,
because_Cache,
setEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
productElimination,
voidElimination,
isect_memberEquality,
voidEquality,
addEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
productEquality,
imageMemberEquality,
baseClosed,
instantiate,
applyLambdaEquality,
promote_hyp,
hypothesis_subsumption,
inrFormation,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
\mforall{}L:T List
(\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} ((\mforall{}x\mmember{}L.(\mforall{}y\mmember{}L.P[x] \mvee{} P[y] supposing \mneg{}(x = y))) \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L. (\mforall{}y\mmember{}L.P[y] supposing \mneg{}(x = y))))
supposing 0 < ||L||
Date html generated:
2017_04_17-AM-07_50_17
Last ObjectModification:
2017_02_27-PM-04_24_43
Theory : list_1
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