Nuprl Lemma : list-index-cmp-zero
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:T List]. ∀[A:Type]. ∀[f:A ⟶ T]. ∀[x,y:{x:A| (f x ∈ L)} ].
uiff((list-index-cmp(eq;L;f) x y) = 0 ∈ ℤ;(f x) = (f y) ∈ T)
Proof
Definitions occuring in Statement :
list-index-cmp: list-index-cmp(eq;L;f),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
comparison: comparison(T),
prop: ℙ,
int-minus-comparison: int-minus-comparison(f),
list-index-cmp: list-index-cmp(eq;L;f),
guard: {T},
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_seg: {i..j-},
squash: ↓T,
less_than: a < b,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True
Lemmas referenced :
list-index-cmp_wf,
int_subtype_base,
l_member_wf,
list_wf,
deq_wf,
list-index-property,
lelt_wf,
set_subtype_base,
length_wf,
int_seg_wf,
subtype_base_sq,
subtract_wf,
equal-wf-T-base,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
int_seg_properties,
equal_wf,
isl-list-index,
list-index_wf,
top_wf,
outl_wf,
full-omega-unsat,
istype-int,
istype-void,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
hypothesis,
Error :equalityIsType4,
Error :universeIsType,
intEquality,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
baseClosed,
Error :equalityIsType1,
productElimination,
independent_pairEquality,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :setIsType,
Error :functionIsType,
universeEquality,
because_Cache,
independent_isectElimination,
cumulativity,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
lambdaEquality,
natural_numberEquality,
instantiate,
imageElimination,
dependent_set_memberEquality,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
lambdaFormation,
Error :dependent_set_memberEquality_alt,
closedConclusion,
Error :lambdaFormation_alt,
applyLambdaEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
hyp_replacement,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:T List]. \mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} T]. \mforall{}[x,y:\{x:A| (f x \mmember{} L)\} ].
uiff((list-index-cmp(eq;L;f) x y) = 0;(f x) = (f y))
Date html generated:
2019_06_20-PM-01_56_41
Last ObjectModification:
2018_10_15-PM-02_32_50
Theory : decidable!equality
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