Nuprl Lemma : list-set-type-member
∀[T:Type]. ∀[P:T ⟶ ℙ]. ((∀x:T. SqStable(P[x])) ⇒ (∀L:{x:T| P[x]} List. ∀x:T. {(x ∈ L) ⇒ P[x]}))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
set_wf,
equal_wf,
l_member_wf,
subtype_rel_list,
list_wf,
all_wf,
sq_stable_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
universeEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:T. SqStable(P[x])) {}\mRightarrow{} (\mforall{}L:\{x:T| P[x]\} List. \mforall{}x:T. \{(x \mmember{} L) {}\mRightarrow{} P[x]\})\000C)
Date html generated:
2016_10_21-AM-10_03_31
Last ObjectModification:
2016_07_12-AM-05_24_27
Theory : list_1
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