Nuprl Lemma : list-eq-set-type
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[A,B:T List].
(A = B ∈ ({x:T| P[x]} List)) supposing ((∀i:ℕ||A||. P[A[i]]) and (A = B ∈ (T List)))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
prop: ℙ,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T
Lemmas referenced :
list_wf,
equal_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
select_wf,
length_wf,
int_seg_wf,
all_wf,
list-set-type2,
strong-subtype-self,
strong-subtype-set3,
strong-subtype-equal-lists
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
independent_isectElimination,
lambdaEquality,
natural_numberEquality,
cumulativity,
setElimination,
rename,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[A,B:T List]. (A = B) supposing ((\mforall{}i:\mBbbN{}||A||. P[A[i]]) and (A = B))
Date html generated:
2016_05_15-PM-04_10_32
Last ObjectModification:
2016_01_16-AM-11_05_44
Theory : general
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