Nuprl Lemma : cons_sublist_cons
∀[T:Type]. ∀x1,x2:T. ∀L1,L2:T List. ([x1 / L1] ⊆ [x2 / L2] ⇐⇒ ((x1 = x2 ∈ T) ∧ L1 ⊆ L2) ∨ [x1 / L1] ⊆ L2)
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
sublist: L1 ⊆ L2,
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
or: P ∨ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
decidable: Dec(P),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
guard: {T},
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
sq_type: SQType(T),
true: True,
increasing: increasing(f;k),
nequal: a ≠ b ∈ T ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
select: L[n],
cons: [a / b]
Lemmas referenced :
length_of_cons_lemma,
sublist_wf,
cons_wf,
list_wf,
istype-universe,
istype-false,
add_nat_plus,
length_wf_nat,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
nat_plus_properties,
add-is-int-iff,
intformand_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
istype-le,
length_wf,
decidable__assert,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
int_seg_wf,
increasing_wf,
add_nat_wf,
istype-void,
nat_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
select_wf,
int_seg_properties,
non_neg_length,
subtype_base_sq,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
select_cons_hd,
subtype_rel_self,
iff_weakening_equal,
increasing_lower_bound,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtract-is-int-iff,
select_cons_tl,
le_wf,
less_than_wf,
add-subtract-cancel,
decidable__equal_int,
subtract_nat_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
add-member-int_seg2,
int_seg_subtype,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
minus-minus,
add-commutes,
le-add-cancel,
select-cons-hd,
select-cons-tl,
select_cons_tl_sq2,
int_seg_subtype_nat,
subtract-add-cancel,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
productElimination,
universeIsType,
isectElimination,
hypothesisEquality,
unionIsType,
productIsType,
equalityIstype,
inhabitedIsType,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
voidElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
int_eqEquality,
because_Cache,
addEquality,
applyEquality,
inlFormation_alt,
functionIsType,
functionExtensionality,
imageElimination,
cumulativity,
intEquality,
imageMemberEquality,
productEquality,
hyp_replacement,
inrFormation_alt,
equalityElimination,
minusEquality,
multiplyEquality
Latex:
\mforall{}[T:Type]
\mforall{}x1,x2:T. \mforall{}L1,L2:T List. ([x1 / L1] \msubseteq{} [x2 / L2] \mLeftarrow{}{}\mRightarrow{} ((x1 = x2) \mwedge{} L1 \msubseteq{} L2) \mvee{} [x1 / L1] \msubseteq{} L2)
Date html generated:
2020_05_19-PM-09_42_01
Last ObjectModification:
2019_12_31-PM-00_15_48
Theory : list_1
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