Nuprl Lemma : first-member_wf
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. ∀[x:T]. (first-member(T;x;L;P) ∈ ℙ)
Proof
Definitions occuring in Statement :
first-member: first-member(T;x;L;P),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
first-member: first-member(T;x;L;P),
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
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,
so_apply: x[s]
Lemmas referenced :
exists_wf,
int_seg_wf,
length_wf,
equal_wf,
select_wf,
int_seg_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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
assert_wf,
all_wf,
not_wf,
list_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
productEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
applyEquality,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. \mforall{}[x:T]. (first-member(T;x;L;P) \mmember{} \mBbbP{})
Date html generated:
2018_05_21-PM-06_33_23
Last ObjectModification:
2017_07_26-PM-04_52_12
Theory : general
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