Nuprl Lemma : l_member_non_nil
∀[T:Type]. ∀[x:T]. ∀[L:T List]. ¬(L = [] ∈ (T List)) supposing (x ∈ L)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
and: P ∧ Q,
so_apply: x[s]
Lemmas referenced :
equal-wf-T-base,
list_wf,
exists_wf,
nat_wf,
less_than_wf,
length_wf,
equal_wf,
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,
length_of_nil_lemma,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
productElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
baseClosed,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
because_Cache,
productEquality,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
universeEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[x:T]. \mforall{}[L:T List]. \mneg{}(L = []) supposing (x \mmember{} L)
Date html generated:
2017_04_14-AM-09_26_52
Last ObjectModification:
2017_02_27-PM-04_00_37
Theory : list_1
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