Nuprl Lemma : permutation-strong-subtype
∀[A,B:Type].
∀L1:B List. ∀L2:A List. (permutation(A;L1;L2) ⇒ {(L2 ∈ B List) ∧ permutation(B;L1;L2)})
supposing strong-subtype(B;A)
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
list: T List,
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
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],
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
guard: {T},
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
permutation: permutation(T;L1;L2)
Lemmas referenced :
strong-subtype_witness,
member-permutation,
subtype_rel_list,
permutation_wf,
list_wf,
strong-subtype_wf,
strong-subtype-implies,
list-subtype,
l_member_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,
inject_wf,
int_seg_wf,
length_wf,
equal_wf,
permute_list_wf,
strong-subtype-list
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
lambdaFormation,
because_Cache,
dependent_functionElimination,
applyEquality,
independent_isectElimination,
productElimination,
sqequalRule,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
lambdaEquality,
setElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
productEquality,
functionExtensionality
Latex:
\mforall{}[A,B:Type].
\mforall{}L1:B List. \mforall{}L2:A List. (permutation(A;L1;L2) {}\mRightarrow{} \{(L2 \mmember{} B List) \mwedge{} permutation(B;L1;L2)\})
supposing strong-subtype(B;A)
Date html generated:
2017_04_17-AM-08_13_34
Last ObjectModification:
2017_02_27-PM-04_39_13
Theory : list_1
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