Nuprl Lemma : l-all-and_wf
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. ∀[B:ℙ]. (B ∧ ∀x∈L.P[x] ∈ ℙ)
Proof
Definitions occuring in Statement :
l-all-and: B ∧ ∀x∈L.P[x]
,
l_member: (x ∈ l)
,
list: T List
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
so_apply: x[s]
,
member: t ∈ T
,
set: {x:A| B[x]}
,
function: x:A ⟶ B[x]
,
universe: Type
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
,
subtype_rel: A ⊆r B
,
guard: {T}
,
or: P ∨ Q
,
l-all-and: B ∧ ∀x∈L.P[x]
,
cons: [a / b]
,
colength: colength(L)
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
,
decidable: Dec(P)
,
nil: []
,
it: ⋅
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
sq_type: SQType(T)
,
less_than: a < b
,
squash: ↓T
,
less_than': less_than'(a;b)
,
iff: P
⇐⇒ Q
,
rev_implies: 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,
l_member_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
reduce_nil_lemma,
nil_wf,
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,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
reduce_cons_lemma,
cons_wf,
cons_member,
list_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,
universeEquality,
functionEquality,
cumulativity,
setEquality,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
productEquality,
functionExtensionality,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]. \mforall{}[B:\mBbbP{}]. (B \mwedge{} \mforall{}x\mmember{}L.P[x] \mmember{} \mBbbP{})
Date html generated:
2018_05_21-PM-06_45_07
Last ObjectModification:
2017_07_26-PM-04_55_32
Theory : general
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