Nuprl Lemma : strong-subtype-l_member
∀[A,B:Type]. ∀L:A List. ∀x:B. ((x ∈ L) ⇒ (x ∈ L)) supposing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
l_member: (x ∈ l),
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
guard: {T},
label: ...$L... t
Lemmas referenced :
strong-subtype_witness,
l_member_wf,
subtype_rel_list,
list_wf,
strong-subtype_wf,
exists_wf,
equal_wf,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
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,
strong-subtype-implies,
less_than_wf,
length_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
lambdaFormation,
productElimination,
cumulativity,
applyEquality,
independent_isectElimination,
sqequalRule,
because_Cache,
universeEquality,
dependent_set_memberEquality,
lambdaEquality,
dependent_pairFormation,
setElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
productEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}L:A List. \mforall{}x:B. ((x \mmember{} L) {}\mRightarrow{} (x \mmember{} L)) supposing strong-subtype(A;B)
Date html generated:
2019_06_20-PM-01_20_20
Last ObjectModification:
2018_09_17-PM-05_54_59
Theory : list_1
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