Nuprl Lemma : l_all-set
∀[T:Type]. ∀[P:T ⟶ ℙ]. ((∀x:T. SqStable(P[x])) ⇒ (∀L:{x:T| P[x]} List. (∀x∈L.P[x])))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
list: T List,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
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 :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
l_all: (∀x∈L.P[x]),
member: t ∈ T,
so_apply: x[s],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
uimplies: b supposing a,
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,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
sq_stable: SqStable(P)
Lemmas referenced :
sq_stable_wf,
all_wf,
list_wf,
int_seg_wf,
equal_wf,
set_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,
length_wf,
int_seg_properties,
select_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
introduction,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
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\mmember{}L.P[x])))
Date html generated:
2016_05_14-AM-07_49_55
Last ObjectModification:
2016_01_15-AM-08_32_06
Theory : list_1
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