Nuprl Lemma : first_index_equal
∀[T:Type]. ∀[L1,L2:T List]. ∀[P,Q:T ⟶ 𝔹].
index-of-first a in L1.P[a] ∧b Q[a] = index-of-first a in L2.P[a] ∧b Q[a] ∈ ℤ supposing L1 agree_on(T;a.↑P[a]) L2
Proof
Definitions occuring in Statement :
agree_on: agree_on(T;x.P[x]),
first_index: index-of-first x in L.P[x],
list: T List,
band: p ∧b q,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
or: P ∨ Q,
agree_on: agree_on(T;x.P[x]),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
infix_ap: x f y,
select: L[n],
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
first_index_wf,
nil_wf,
void_wf,
infix_ap_wf,
list_wf,
equal_wf,
length_wf,
all_wf,
int_seg_wf,
or_wf,
assert_wf,
select_wf,
subtype_rel_list,
int_seg_properties,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
istype-universe,
bool_wf,
product_subtype_list,
colength-cons-not-zero,
agree_on_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
length_of_nil_lemma,
length_of_cons_lemma,
stuck-spread,
istype-base,
non_neg_length,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
cons_wf,
colength_wf_list,
le_wf,
istype-false,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
first_index_cons,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
bfalse_wf,
nat_wf,
add-is-int-iff,
false_wf,
add-member-int_seg2,
add-associates,
add-swap,
add-commutes,
zero-add,
add-subtract-cancel,
iff_weakening_uiff,
assert_functionality_wrt_uiff,
select_cons_tl,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
decidable__or,
decidable__assert,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
ifthenelse_wf,
lt_int_wf,
equal-wf-base,
ite_rw_false,
not_wf,
assert_of_lt_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
because_Cache,
voidEquality,
functionExtensionality,
instantiate,
applyEquality,
cumulativity,
universeEquality,
productEquality,
intEquality,
functionEquality,
productElimination,
functionIsType,
promote_hyp,
hypothesis_subsumption,
equalityIsType1,
equalityIsType4,
baseClosed,
dependent_set_memberEquality_alt,
imageElimination,
baseApply,
closedConclusion,
equalityElimination,
pointwiseFunctionality,
unionIsType,
addEquality,
productIsType,
inlFormation_alt,
inrFormation_alt,
imageMemberEquality,
hyp_replacement,
equalityIsType2
Latex:
\mforall{}[T:Type]. \mforall{}[L1,L2:T List]. \mforall{}[P,Q:T {}\mrightarrow{} \mBbbB{}].
index-of-first a in L1.P[a] \mwedge{}\msubb{} Q[a] = index-of-first a in L2.P[a] \mwedge{}\msubb{} Q[a]
supposing L1 agree\_on(T;a.\muparrow{}P[a]) L2
Date html generated:
2019_10_15-AM-10_58_20
Last ObjectModification:
2018_10_09-AM-09_57_25
Theory : list!
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