Nuprl Lemma : poss-maj-unanimous
∀T:Type. ∀eq:EqDecider(T). ∀L:T List. ∀x,y:T. ((poss-maj(eq;L;x) = <||L||, y> ∈ (ℕ × T)) ⇒ (∀z∈L.z = y ∈ T))
Proof
Definitions occuring in Statement :
poss-maj: poss-maj(eq;L;x),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
deq: EqDecider(T),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
pi2: snd(t),
l_all: (∀x∈L.P[x]),
sq_type: SQType(T),
guard: {T},
deq: EqDecider(T),
int_seg: {i..j-},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
eqof: eqof(d),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
poss-maj-invariant,
poss-maj_wf,
nat_wf,
nat_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,
length_wf_nat,
and_wf,
equal_wf,
pi1_wf,
le_wf,
pi2_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
select_wf,
int_seg_properties,
length_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
assert_wf,
eqof_wf,
not_wf,
int_seg_wf,
subtract_wf,
count_wf,
bnot_wf,
all_wf,
list_wf,
deq_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
count-length-filter,
pos_length,
filter_wf5,
equal-wf-T-base,
assert_of_null,
filter_is_empty,
lelt_wf,
length-filter
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
cumulativity,
productEquality,
productElimination,
rename,
sqequalRule,
isectElimination,
setElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
applyEquality,
promote_hyp,
instantiate,
independent_functionElimination,
hyp_replacement,
independent_pairEquality,
equalityElimination,
addLevel,
impliesFunctionality,
levelHypothesis,
impliesLevelFunctionality,
functionEquality,
universeEquality,
baseClosed
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}L:T List. \mforall{}x,y:T. ((poss-maj(eq;L;x) = <||L||, y>) {}\mRightarrow{} (\mforall{}z\mmember{}L.z = y))
Date html generated:
2017_04_17-AM-09_08_39
Last ObjectModification:
2017_02_27-PM-05_18_06
Theory : decidable!equality
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