Nuprl Lemma : strong-subtype-equal-lists
∀[A,B:Type]. ∀[L1:A List]. ∀[L2:B List]. L1 = L2 ∈ (A List) supposing L1 = L2 ∈ (B List) supposing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
list: T List,
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
squash: ↓T,
true: True,
all: ∀x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
cand: A c∧ B,
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
so_lambda: λ2x.t[x],
so_apply: x[s],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
subtract: n - m,
le: A ≤ B
Lemmas referenced :
strong-subtype-implies,
list_extensionality,
length_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,
and_wf,
equal_wf,
list_wf,
length_wf_nat,
nat_wf,
less_than_wf,
subtype_rel_list,
strong-subtype_wf,
exists_wf,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
equal-wf-T-base,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
nil_wf,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
cons_wf,
hd_wf,
length_of_nil_lemma,
cons_neq_nil,
length_of_cons_lemma,
false_wf,
not-ge-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
reduce_hd_cons_lemma,
squash_wf,
tl_wf,
reduce_tl_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalitySymmetry,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
lambdaFormation,
dependent_functionElimination,
setElimination,
rename,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addLevel,
hyp_replacement,
dependent_set_memberEquality,
equalityTransitivity,
applyLambdaEquality,
productElimination,
levelHypothesis,
cumulativity,
universeEquality,
setEquality,
intWeakElimination,
axiomEquality,
promote_hyp,
hypothesis_subsumption,
addEquality,
instantiate,
minusEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[L1:A List]. \mforall{}[L2:B List]. L1 = L2 supposing L1 = L2 supposing strong-subtype(A;B)
Date html generated:
2017_04_14-AM-09_27_31
Last ObjectModification:
2017_02_27-PM-04_02_23
Theory : list_1
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