Nuprl Lemma : alist-domain-first
∀[A:Type]
∀d:A List. ∀f1:a:{a:A| (a ∈ d)} ⟶ Top. ∀x:A. ∀eq:EqDecider(A).
((x ∈ d)
⇒ (∃i:ℕ||d||. ((∀j:ℕi. (¬((fst(map(λx.<x, f1 x>;d)[j])) = x ∈ A))) ∧ ((fst(map(λx.<x, f1 x>;d)[i])) = x ∈ A))))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
select: L[n],
length: ||as||,
map: map(f;as),
list: T List,
deq: EqDecider(T),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
top: Top,
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
and: P ∧ Q,
int_seg: {i..j-},
top: Top,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
lelt: i ≤ j < k,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
pi1: fst(t)
Lemmas referenced :
l_member-first,
l_member_wf,
deq_wf,
istype-top,
list_wf,
istype-universe,
int_seg_wf,
select-map,
istype-void,
subtype_rel_list,
top_wf,
int_seg_properties,
length_wf,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
pi1_wf_top,
select_wf,
map_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
map-length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
hypothesis,
productElimination,
dependent_pairFormation_alt,
universeIsType,
functionIsType,
setIsType,
instantiate,
universeEquality,
independent_pairFormation,
promote_hyp,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
unionElimination,
imageElimination,
approximateComputation,
int_eqEquality,
productIsType,
because_Cache,
equalityIstype,
productEquality,
independent_pairEquality
Latex:
\mforall{}[A:Type]
\mforall{}d:A List. \mforall{}f1:a:\{a:A| (a \mmember{} d)\} {}\mrightarrow{} Top. \mforall{}x:A. \mforall{}eq:EqDecider(A).
((x \mmember{} d)
{}\mRightarrow{} (\mexists{}i:\mBbbN{}||d||
((\mforall{}j:\mBbbN{}i. (\mneg{}((fst(map(\mlambda{}x.<x, f1 x>d)[j])) = x))) \mwedge{} ((fst(map(\mlambda{}x.<x, f1 x>d)[i])) = x))))
Date html generated:
2019_10_16-AM-11_25_15
Last ObjectModification:
2018_11_30-AM-10_16_56
Theory : finite!partial!functions
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