Nuprl Lemma : cons_cancel_wrt_permutation
∀[A:Type]. ∀a:A. ∀bs,cs:A List. (permutation(A;[a / bs];[a / cs]) ⇒ permutation(A;bs;cs))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
permutation: permutation(T;L1;L2),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
squash: ↓T,
less_than: a < b,
inject: Inj(A;B;f),
compose: f o g,
flip: (i, j),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
cons: [a / b],
select: L[n],
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
subtract: n - m
Lemmas referenced :
permutation-length,
cons_wf,
length_of_cons_lemma,
non_neg_length,
decidable__equal_int,
length_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
permutation_wf,
list_wf,
istype-universe,
decidable__lt,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
compose_wf,
int_seg_wf,
flip_wf,
decidable__le,
istype-le,
istype-less_than,
inject_wf,
permute_list_wf,
istype-false,
set_subtype_base,
lelt_wf,
int_subtype_base,
istype-void,
istype-assert,
not_wf,
bnot_wf,
int_seg_properties,
equal-wf-base,
assert_wf,
bool_wf,
equal-wf-T-base,
eq_int_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
le_weakening2,
length_wf_nat,
nat_properties,
list_extensionality,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
permute_list_length,
istype-nat,
permute_list_select,
select_wf,
less_than_wf,
le_wf,
subtype_base_sq,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
add-member-int_seg2,
false_wf,
subtract_nat_wf,
zero-add,
add-commutes,
add-swap,
add-associates,
select_cons_tl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
Error :memTop,
because_Cache,
unionElimination,
productElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
voidElimination,
inhabitedIsType,
instantiate,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
productIsType,
applyEquality,
equalityIstype,
applyLambdaEquality,
addEquality,
intEquality,
baseClosed,
sqequalBase,
functionIsType,
imageElimination,
closedConclusion,
rename,
setElimination,
equalityElimination,
imageMemberEquality,
productEquality,
cumulativity,
promote_hyp,
pointwiseFunctionality,
baseApply
Latex:
\mforall{}[A:Type]. \mforall{}a:A. \mforall{}bs,cs:A List. (permutation(A;[a / bs];[a / cs]) {}\mRightarrow{} permutation(A;bs;cs))
Date html generated:
2020_05_19-PM-09_44_24
Last ObjectModification:
2019_12_31-PM-00_14_35
Theory : list_1
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