hcju/isabelleps huggingface dataset:
elims: fixes x :: "'a \<times> 'b \<times> 'a"   and xa :: "('a \<times> 'b \<times> 'a) list"   and y :: "bool \<times> 'a \<times> 'b \<times> 'a" assumes "norm_fold x xa = y" obtains q :: "'a" and \<nu> :: "'b" and p :: "'a" and xs :: "('a \<times> 'b \<times> 'a) list" where "x = (q, \<nu>, p)" and "xa = xs" and "y = foldl_break norm_rep fst (False, q, \<nu>, if q = p then q else p) xs"
Par_ZObis_discrR2: fixes c :: "('test, 'atom) com"   and d2 :: "('test, 'atom) com"   and d1 :: "('test, 'atom) com" assumes "c \<approx>01 d2"   and "discr d1" shows "c \<approx>01 d1 | d2"
FOREACHoi_refine: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "('e \<times> 'f) set"   and RR :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and RR' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<Phi>'' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "(\<sigma>0, \<sigma>0') \<in> R"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; RR x y\<rbrakk> \<Longrightarrow> RR' (\<alpha> x) (\<alpha> y)"   and "\<Phi>'' S \<sigma>0 (\<alpha> ` S) \<sigma>0'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi>'' it \<sigma> it' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> \<Phi> it \<sigma>"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>\<forall>y\<in>it - {x}. RR x y; x' = \<alpha> x; x \<in> it; x' \<in> it'; it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; \<Phi>'' it \<sigma> it' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). (\<sigma>, \<sigma>') \<in> R \<and> \<Phi>'' (it - {x}) \<sigma> (it' - {x'}) \<sigma>'} (f' x' \<sigma>')" shows "FOREACH\<^sub>O\<^bsup>RR,\<Phi>\<^esup> S f \<sigma>0 \<le> \<Down> R (FOREACH\<^sub>O\<^bsup>RR',\<Phi>'\<^esup> S' f' \<sigma>0')"
x_plus_top_ennreal: fixes x :: "ennreal" shows "x + \<top> = \<top>"
clop_lsu_lsl: shows "clop (\<nu>\<^sup>\<natural> \<circ> \<nu>)"
sel_4: fixes x21 :: "('a, 'b) tm"   and x22 :: "('a, 'b) tm" shows "fun (App x21 x22) = x21"
cod_simps_1: shows "H\<mu>H\<nu>\<pi>.cod.apex = H\<mu>H\<nu>\<pi>.apex"
strand_leaks\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and Secrets :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "\<A> leaks Secrets under \<I> \<equiv> \<exists>t\<in>Secrets - declassified\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<I>. \<exists>n. \<lbrakk>{}; {}; proj_unl n \<A> @ [send\<langle>t\<rangle>]\<rbrakk>\<^sub>s \<I>"
simps_3: fixes uu :: "t list" shows "c_types_agree Bot uu = False"
2: fixes d :: "nat" assumes "d \<in> set (nths (dims Aw) (Collect even))" shows "r \<le> d"
erfc_holomorphic_on: fixes A :: "complex set" shows "erfc holomorphic_on A"
sigma_sets_UNION: fixes B :: "'d set set"   and X :: "'d set"   and A :: "'d set set" assumes "countable B"   and "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets X A" shows "\<Union> B \<in> sigma_sets X A"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "color \<Rightarrow> ('d, 'e) rbt \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('d, 'e) rbt \<Rightarrow> 'c"   and rbt :: "('d, 'e) rbt" shows "P (case rbt of rbt.Empty \<Rightarrow> f1 | Branch x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = ((rbt = rbt.Empty \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23 x24 x25. rbt = Branch x21 x22 x23 x24 x25 \<longrightarrow> P (f2 x21 x22 x23 x24 x25)))"
PivotEq_axioms: shows "PivotEq TYPE('a) pivot_eq"
span_breakdown_eq: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b"   and S :: "'b set" assumes "Modules.module scale" shows "(x \<in> module.span scale (insert a S)) = (\<exists>k. x - scale k a \<in> module.span scale S)"
inj_image_mem_iff: fixes f :: "'e \<Rightarrow> 'f"   and a :: "'e"   and A :: "'e set" assumes "inj f" shows "(f a \<in> f ` A) = (a \<in> A)"
simps_5: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "extractFrame (P \<parallel> Q) = extractFrame P \<otimes>\<^sub>F extractFrame Q"
equivalence_comp_right_complement: fixes x :: "'a" assumes "(top \<le> x \<and> x \<sqinter> x \<le> x) \<and> local.comp_inf.symmetric x" shows "- x \<sqinter> x = - x"
dvd_diff_commute: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a dvd c - b) = (a dvd b - c)"
list_all2_induct: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and R :: "'b list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "R [] []"   and "\<And>x xs y ys. \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)" shows "R xs ys"
pOp_assocTr43: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> (c \<plusminus>\<^bsub>A\<^esub> d) = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c) \<plusminus>\<^bsub>A\<^esub> d"
assms_1: shows "wf\<^sub>s\<^sub>t {} A"
cobounded2: fixes b :: "'b"   and a :: "'b" shows "b \<le> sup a b"
eq2: shows "punit.lt (rep_list (monom_mult c t f)) = t + punit.lt (rep_list f)"
post_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and post :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.post_mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top post" shows "times (times (post x) x) top = top"
GB_image_monic: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_Groebner_basis (monic ` G) = is_Groebner_basis G"
locals_lupd: fixes vn :: "lname"   and v :: "val"   and s :: "st" shows "locals (lupd(vn\<mapsto>v) s) = locals s(vn \<mapsto> v)"
prod_atLeastAtMost_code: fixes f :: "nat \<Rightarrow> 'a"   and a :: "nat"   and b :: "nat" shows "prod f {a..b} = fold_atLeastAtMost_nat (\<lambda>a. (*) (f a)) a b (1::'a)"
scaleR: fixes r :: "real"   and b :: "'b" shows "frechet_derivative f (at (g x)) (r *\<^sub>R b) = r *\<^sub>R frechet_derivative f (at (g x)) b"
simps_1: fixes x :: "quat" shows "quat_proj x 0 = quat.Re x"
power2_abs: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "power.power one times (abs a) 2 = power.power one times a 2"
eval_prefs_from_table_aux_3: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "('a \<times> 'b set list) list"   and agents :: "'a set"   and alts :: "'b set" assumes "R \<equiv> prefs_from_table xs"   and "prefs_from_table_wf agents alts xs" shows "anonymous_profile R = mset (map snd xs)"
prod_emb_iff: fixes f :: "'b \<Rightarrow> 'c"   and I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and K :: "'b set"   and X :: "('b \<Rightarrow> 'c) set" shows "(f \<in> prod_emb I M K X) = (f \<in> extensional I \<and> restrict f K \<in> X \<and> (\<forall>i\<in>I. f i \<in> space (M i)))"
set_subset_Univ: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "local.set a \<subseteq> Univ"
list_all2_nthD2: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length ys" shows "P (xs ! p) (ys ! p)"
fmpred_filter: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and Q :: "'c \<Rightarrow> bool" assumes "fmpred P m" shows "fmpred P (fmfilter Q m)"
abs_eq: fixes xa :: "complex \<times> complex \<times> complex \<times> complex"   and x :: "complex \<times> complex \<times> complex \<times> complex" assumes "eq_onp (\<lambda>M. mat_det M \<noteq> 0) xa xa"   and "eq_onp (\<lambda>M. mat_det M \<noteq> 0) x x" shows "moebius_mat_eq (Abs_moebius_mat xa) (Abs_moebius_mat x) = moebius_cmat_eq xa x"
GREATEST_FP_def: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" shows "GFP\<^bsub>L\<^esub> f = Lattice.sup L {u. u \<in> carrier L \<and> le L u (f u)}"
scale_zero_left: fixes x :: "'b" shows "(0::'b) * x = (0::'b)"
bc: shows "finite {a. y a \<noteq> (b, c)}"
HaddP_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm"   and v :: "tm"   and v' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'"   and "H \<turnstile> v EQ v'" shows "H \<turnstile> HaddP t u v IFF HaddP t' u' v'"
rewrite_sum_of_powers: fixes p :: "nat"   and n :: "nat" assumes "1 < p" shows "\<Sum> ((^) p ` {0..n}) = sum ((^) p) {0..n}"
assms: shows "\<exists>e. f e = top \<and> (\<forall>d. e \<noteq> d \<longrightarrow> f d = bot)"
app_iso_inv_subgraph: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and H :: "('c, 'd) pre_digraph" assumes "pre_digraph.digraph_isomorphism G hom"   and "subgraph H G" shows "app_iso (pre_digraph.inv_iso G hom) (app_iso hom H) = H"
assms_2: shows "are_operators_interfering op\<^sub>1 op\<^sub>2"
step_3: shows "h \<turnstile> get_root_node_si c \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child"
iterate_Suc2: fixes n :: "nat"   and F :: "'d \<rightarrow> 'd"   and x :: "'d" shows "iterate (Suc n)\<cdot>F\<cdot>x = iterate n\<cdot>F\<cdot>(F\<cdot>x)"
card_order_csum: fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set" assumes "card_order r1"   and "card_order r2" shows "card_order (r1 +c r2)"
flip_const: fixes c :: "'j" shows "flip (\<lambda>a b. c) = (\<lambda>a b. c)"
lessThan_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b" shows "ord.lessThan less u \<equiv> {x. less x u}"
has_integral_spike_eq: fixes S :: "'b set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "(f has_integral y) T = (g has_integral y) T"
multi_psub_of_add_self: fixes A :: "'a multiset"   and x :: "'a" shows "A \<subset># add_mset x A"
minus_bit1_def: fixes x :: "'a bit1"   and y :: "'a bit1" shows "x - y = Abs_bit1' (Rep_bit1 x - Rep_bit1 y)"
assms_1: shows "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"
snd_splitAt_rev: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "snd (splitAt x (rev xs)) = rev (fst (splitAt x xs))"
WT_resourceD: fixes \<I> :: "('c, 'd) \<I>"   and res :: "('c, 'd) resource"   and q :: "'c"   and r :: "'d"   and res' :: "('c, 'd) resource" assumes "\<I> \<turnstile>res res \<surd>"   and "q \<in> outs_\<I> \<I>"   and "(r, res') \<in> set_spmf (run_resource res q)" shows "r \<in> responses_\<I> \<I> q \<and> \<I> \<turnstile>res res' \<surd>"
size_jump2: fixes l :: "nat"   and cs :: "instr list" shows "length (jump l cs) < length cs \<or> jump l cs = cs"
spec_ops_rem_last: fixes xs :: "('c \<times> 'd) list"   and x :: "'c \<times> 'd"   and deps :: "'d \<Rightarrow> 'c set" assumes "spec_ops (xs @ [x]) deps" shows "spec_ops xs deps"
c'_n: shows "c' n = (u_ts, u_m, u_shared)"
normalize_polynomial: fixes K :: "'a list set"   and p :: "'a list list" assumes "polynomial\<^bsub>K [X]\<^esub> K p" shows "UP.normalize p = p"
boundedInsert: fixes a :: "nat \<times> 'b"   and B :: "(nat \<times> 'b) set" shows "bounded (insert a B) = bounded B"
assms: fixes x :: "'a" shows "\<lbrace>P x\<rbrace> c \<lbrace>Q x\<rbrace>"
rep_eq: fixes x :: "(('d, 'c) vec, 'c) vec" shows "to_fun (from_vec x) = (\<lambda>i. ($) (x $ i))"
pseudominimal_ne_bottom: fixes P :: "'a set"   and x :: "'a" assumes "domain.pseudominimal_in P x" shows "x \<noteq> domain.bottom P"
assms_2: shows "(s, t) \<notin> {\<rhd>}"
mat_eq_iff: fixes x :: "'c mat"   and y :: "'c mat" shows "(x = y) = (dim_row x = dim_row y \<and> dim_col x = dim_col y \<and> (\<forall>i j. i < dim_row y \<longrightarrow> j < dim_col y \<longrightarrow> x $$ (i, j) = y $$ (i, j)))"
impl_test: fixes b1 :: "'d bool_expr"   and b2 :: "'d bool_expr" shows "impl_test ifex_of b1 b2 = (\<forall>s. val_bool_expr b1 s \<longrightarrow> val_bool_expr b2 s)"
case_prodI2': fixes p :: "'d \<times> 'e"   and c :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f" assumes "\<And>a b. (a, b) = p \<Longrightarrow> c a b x" shows "(case p of (a, b) \<Rightarrow> c a b) x"
monad_writer_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad return bind" shows "monad_writer (return_writer return) (bind_writer return bind) (tell_writer return bind)"
Cons_eq_append_conv: fixes x :: "'k"   and xs :: "'k list"   and ys :: "'k list"   and zs :: "'k list" shows "(x # xs = ys @ zs) = (ys = [] \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' @ zs))"
sup_top_right: fixes x :: "'a" shows "x \<squnion> top = top"
diff_right_commute: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a c) b = minus (minus a b) c"
lookup_ev_None: fixes ev :: "'b"   and j :: "nat"   and A :: "'b mat"   and i :: "nat" assumes "??.Jordan_Normal_Form_Existence.lookup_ev ev j A = None"   and "i < j" shows "A $$ (i, i) \<noteq> ev"
a1: shows "P (upd (map_hd (\<lambda>_. rhs \<sigma>)) (\<sigma>\<lparr>return_status := True\<rparr>))"
pseudo_mod_def: fixes f :: "'a poly"   and g :: "'a poly" shows "pseudo_mod f g = snd (pseudo_divmod f g)"
str_del_point_blocks_def: fixes p :: "'a" shows "str_del_point_blocks p \<equiv> filter_mset ((\<notin>) p) \<B>"
mult_zero_ow: shows "class.mult_zero = mult_zero_ow UNIV"
reflexive: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme"   and p :: "'f" shows "transition_system.reachablep (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p) p p"
aboveS_pred: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "isSucc r i" shows "aboveS r (pred r i) \<noteq> {}"
map_poly_normalize: fixes p :: "'a poly" shows "map_poly hom (normalize p) = normalize (map_poly hom p)"
rcm_B: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b * a r\<rightarrow> c = a r\<rightarrow> b r\<rightarrow> c"
matroid_subset: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set" assumes "matroid carrier indep"   and "\<E> \<subseteq> carrier" shows "matroid \<E> (indep_system.indep_in indep \<E>)"
read_tmps_mono: fixes xs :: "'b memref list"   and ys :: "'b memref list" assumes "set xs \<subseteq> set ys" shows "read_tmps xs \<subseteq> read_tmps ys"
is_unique: fixes I :: "'a set"   and J :: "'a set" assumes "max_lideal I A addA multA zeroA oneA"   and "max_lideal J A addA multA zeroA oneA" shows "I = J"
connected_imp_connected_closure: fixes S :: "'a set" assumes "connected S" shows "connected (closure S)"
in_set_simps_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and ys :: "'a list" shows "(x \<in> set (y # z # ys)) = (x = y \<or> x \<in> set (z # ys))"
Liminf_le_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd" assumes "F \<noteq> bot" shows "Liminf F f \<le> Limsup F f"
higher_differentiable_on_real_SucI: fixes S :: "real set"   and n :: "nat"   and f :: "real \<Rightarrow> 'b" assumes "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>x. nth_derivative n f x 1) differentiable at x"   and "continuous_on S (\<lambda>x. nth_derivative (Suc n) f x 1)"   and "higher_differentiable_on S f n"   and "open S" shows "higher_differentiable_on S f (Suc n)"
rep_eq: fixes x :: "'b fract" shows "quot_of_fract x = normalize_quot (rep_fract x)"
hsize_chop_lt: fixes t :: "('a, 'b) tm" assumes "is_App t" shows "hsize (chop t) < hsize t"
type_less_eq_x_ObjectType_intro: fixes \<tau> :: "'a basic_type"   and \<C> :: "'a"   and \<D> :: "'a" assumes "\<tau> = \<langle>\<C>\<rangle>\<^sub>\<T>"   and "\<C> \<le> \<D>" shows "\<tau> \<le> \<langle>\<D>\<rangle>\<^sub>\<T>"
uniqAppendIff: fixes l1 :: "'a list"   and l2 :: "'a list" shows "uniq (l1 @ l2) = (uniq l1 \<and> uniq l2 \<and> set l1 \<inter> set l2 = {})"
perm_commute: fixes a :: "atom"   and p :: "perm"   and a' :: "atom" assumes "a \<sharp> p"   and "a' \<sharp> p" shows "(a \<rightleftharpoons> a') + p = p + (a \<rightleftharpoons> a')"
monic_degree_0: fixes p :: "'b poly" assumes "monic p" shows "(degree p = 0) = (p = 1)"
r_cancel_one': fixes x :: "'a"   and a :: "'a" shows "(x = a + x) = (a = (0::'a))"
inverse_arrowsE: fixes f :: "'s"   and g :: "'s" assumes "S.inverse_arrows f g" obtains "S.ide (g \<cdot>\<^sub>S f)" and "S.ide (f \<cdot>\<^sub>S g)"
lub_upper: fixes A :: "'c llist set"   and x :: "'c llist" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) A"   and "x \<in> A" shows "x \<sqsubseteq> lSup A"
is_cadjoint_unique: fixes F1 :: "'b \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'b"   and F2 :: "'b \<Rightarrow> 'c" assumes "is_cadjoint F1 G"   and "is_cadjoint F2 G" shows "F1 = F2"
rel_gpv_colossless_gpvD1: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'j) gpv"   and \<I> :: "('h, 'j) \<I>"   and \<I>' :: "('i, 'j) \<I>" assumes "rel_gpv A C gpv gpv'"   and "colossless_gpv \<I> gpv"   and "rel_\<I> C (=) \<I> \<I>'" shows "colossless_gpv \<I>' gpv'"
conga2_sams__sams: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Tarski_neutral_dimensionless.CongA Bet Cong D E F D' E' F'"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.SAMS Bet Cong A' B' C' D' E' F'"
nat_pow_distrib: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d"   and n :: "nat" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> y) [^]\<^bsub>G\<^esub> n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> y [^]\<^bsub>G\<^esub> n"
real_mon_conv_dom: fixes x :: "nat \<Rightarrow> real"   and y :: "real"   and c :: "nat \<Rightarrow> real" assumes "x\<up>y"   and "\<And>n. c n \<le> c (Suc n)"   and "c \<le> x" shows "\<exists>l. c\<up>l \<and> l \<le> y"
norm_diff_triangle_le: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and e1 :: "real"   and z :: "'b"   and e2 :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm (minus x y) \<le> e1"   and "norm (minus y z) \<le> e2" shows "norm (minus x z) \<le> e1 + e2"
assms_3: shows "(a, b\<^sub>2) |\<in>| M"
LeftDerivationFix_grow_suffix: fixes b1 :: "symbol list"   and X :: "symbol"   and b2 :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and c :: "symbol list"   and suffix :: "symbol list"   and e :: "nat"   and r :: "symbol \<times> symbol list" assumes "LeftDerivationFix (b1 @ [X] @ b2) (length b1) D j c"   and "LeftDerives1 suffix e r b2"   and "is_word (b1 @ [X])" shows "LeftDerivationFix (b1 @ [X] @ suffix) (length b1) ((e + length (b1 @ [X]), r) # D) j c"
length_abs_ast_plan: fixes \<pi>s :: "char list list" shows "length \<pi>s = length (abs_ast_plan \<pi>s)"
T_slow_select_le: fixes k :: "nat"   and xs :: "'a list" shows "T_slow_select k xs \<le> (length xs)\<^sup>2 + 3 * length xs + 3"
lessI: fixes n :: "nat" shows "order.greater (Suc n) n"
pred_rel: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a list" shows "list_all P x = list_all2 (eq_onp P) x x"
complex_div_gt_0: fixes a :: "complex"   and b :: "complex" shows "(0 < Re (a / b)) = (0 < Re (a * cnj b)) \<and> (0 < Im (a / b)) = (0 < Im (a * cnj b))"
simps_2: fixes lens_get :: "'b \<Rightarrow> 'c"   and lens_put :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and more :: "'d" shows "put\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_put"
adjust_zero: fixes us :: "'b vec list"   and n :: "nat"   and w :: "'b vec"   and i :: "nat" assumes "set us \<subseteq> carrier_vec n"   and "corthogonal us"   and "w \<in> carrier_vec n"   and "i < length us" shows "(adjuster n w us + w) \<bullet>c us ! i = (0::'b)"
integral_lebesgue_on_empty: fixes f :: "'b \<Rightarrow> 'a" shows "integral\<^sup>L (lebesgue_on {}) f = (0::'a)"
wr_sdia_is_wr_com_2: fixes \<gamma> :: "cell" shows "wr_cell \<gamma> = wr_com (com_cell \<gamma>)"
rep_eq: fixes x :: "'c multiset"   and xa :: "'c multiset" shows "count (x - xa) = (\<lambda>a. count x a - count xa a)"
C: shows "\<not> (\<exists>v\<in>sinks I D u ys. \<exists>w\<in>U. (v, w) \<in> I)"
syzygy_module_closed_map_scale: fixes s :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and B :: "('a \<Rightarrow>\<^sub>0 'c) set"   and c :: "'a \<Rightarrow>\<^sub>0 'c" assumes "s \<in> local.punit.pmdl.syzygy_module B" shows "c \<cdot> s \<in> local.punit.pmdl.syzygy_module B"
simps_12: fixes n :: "nat"   and p :: "fm" shows "boundslt n (E p) = boundslt (Suc n) p"
RPC_action_defs_5: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCReply send rcv rst p \<equiv> \<not> $Calling rcv p \<and> $(rst!p) = #rpcB \<and> Return send p res<rcv!p> \<and> (rst!p)$ = #rpcA \<and> unchanged (caller rcv!p)"
min_worst_chain_0: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "wf {(x, y). gt y x}"   and "bad p x" shows "\<not> gt (worst_chain p gt 0) x"
InvariantVarsFAfterLearn: fixes F :: "Literal list list"   and F0 :: "Literal list list"   and Vbl :: "nat set"   and C :: "Literal list"   and F' :: "Literal list list" assumes "InvariantVarsF F F0 Vbl"   and "vars C \<subseteq> vars F0 \<union> Vbl"   and "F' = F @ [C]" shows "InvariantVarsF F' F0 Vbl"
enclen_length: fixes xs :: "nat list" shows "enclen (lenc xs) = length xs"