Nuprl Lemma : list-set-type3
∀[T:Type]. ∀[L:T List]. ∀[P:T ⟶ ℙ]. L ∈ {x:T| P[x]} List supposing ∃L':{x:T| P[x]} List. (L = L' ∈ (T List))
Proof
Definitions occuring in Statement :
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
guard: {T},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
uiff: uiff(P;Q)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
exists_wf,
list_wf,
equal_wf,
less_than_transitivity1,
less_than_irreflexivity,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
nil_wf,
equal-wf-base-T,
subtype_rel_list,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
set_wf,
cons_wf,
null_nil_lemma,
btrue_wf,
and_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
cons_one_one,
reduce_hd_cons_lemma,
hd_wf,
squash_wf,
length_wf,
length_cons_ge_one,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
cumulativity,
applyEquality,
functionExtensionality,
because_Cache,
functionEquality,
universeEquality,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
instantiate,
imageElimination,
hyp_replacement,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. L \mmember{} \{x:T| P[x]\} List supposing \mexists{}L':\{x:T| P[x]\} List. (L = L'\000C)
Date html generated:
2017_04_17-AM-07_25_12
Last ObjectModification:
2017_02_27-PM-04_03_52
Theory : list_1
Home
Index