hcju/isabelleps huggingface dataset:
select_convs_3: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and more :: "'d" shows "finished \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = finished"
maddux_20: fixes x :: "'a" shows "x \<le> x ; 1"
eventually_nonzero_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "eventually_nonzero F f = (\<forall>\<^sub>F x in F. f x \<noteq> (0::'c))"
nrename_NotL': fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = NotL <c>.N u"   and "c \<sharp> R"   and "x \<noteq> y" shows "(\<exists>N'. R = NotL <c>.N' u \<and> N'[x\<turnstile>n>y] = N) \<or> (\<exists>N'. R = NotL <c>.N' x \<and> y = u \<and> N'[x\<turnstile>n>y] = N)"
interaction_bounded_by_obsf_distinguisher: fixes "consider" :: "'m \<Rightarrow> bool"   and \<A> :: "('m, 'n) distinguisher"   and bound :: "enat" assumes "interaction_bounded_by consider \<A> bound" shows "interaction_bounded_by consider (obsf_distinguisher \<A>) bound"
add: fixes S1 :: "'b set"   and S2 :: "'c set"   and s1 :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and b1 :: "'b"   and b2 :: "'b" assumes "module_hom_on S1 S2 s1 s2 f"   and "b1 \<in> S1"   and "b2 \<in> S1" shows "f (b1 + b2) = f b1 + f b2"
ts_c_def_2: shows "ts' = ts_c @ t2s"
pi_boundaries: fixes n :: "nat" shows "pi \<in> {real_of_float (lb_pi n)..real_of_float (ub_pi n)}"
size_12: fixes x61 :: "'b formula"   and x62 :: "'b formula" shows "size (x61 \<^bold>\<rightarrow> x62) = size x61 + size x62 + Suc 0"
fps_compose_0_right: fixes a :: "'a fps" shows "a oo 0 = fps_const (a $ 0)"
disc_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) tllist" shows "\<not> is_TNil (TCons x21 x22)"
prod_def: shows "prod \<equiv> \<lparr>g_V = igba.V \<times> sa.V, g_E = {((q, s), q', s'). local.igba.L q (local.sa.L s) \<and> (q, q') \<in> igba.E \<and> (s, s') \<in> sa.E}, g_V0 = igba.V0 \<times> sa.V0, igbg_num_acc = igba.num_acc, igbg_acc = \<lambda>(q, s). if s \<in> sa.V then igba.acc q else {}\<rparr>"
max_subgraphI: fixes P :: "('a, 'b) pre_digraph \<Rightarrow> bool"   and x :: "('a, 'b) pre_digraph" assumes "P x"   and "subgraph x G"   and "\<And>y. \<lbrakk>x \<noteq> y; subgraph x y; subgraph y G\<rbrakk> \<Longrightarrow> \<not> P y" shows "max_subgraph P x"
prv_imp_cnj_scnj: fixes F :: "'fmla set"   and \<phi> :: "'fmla" assumes "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla" shows "prv (imp (cnj \<phi> (scnj F)) (scnj (insert \<phi> F)))"
extend_indets_ideal: fixes F :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "extend_indets ` ideal F = ideal (extend_indets ` F) \<inter> P[- {None}]"
intros_2: fixes A :: "'a set"   and a :: "'a" assumes "finite A" shows "finite (insert a A)"
I_def: shows "I = {uu_. \<exists>i bi li nn la. uu_ = i \<and> i < n \<and> j_to_jb_index n_as i = (bi, li) \<and> n_as ! bi = (nn, la) \<and> cmod la = 1 \<and> nn = m \<and> li = nn - 1}"
lselect_discard_start: fixes s :: "nat set"   and k :: "nat"   and w :: "'b llist" assumes "\<And>i. i \<in> s \<Longrightarrow> k \<le> i" shows "lselect {i. k + i \<in> s} (ldropn k w) = lselect s w"
mul_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (mul_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
prj_simps_9: shows "C.cod \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0 = \<rho>.apex"
gen_dfs_pre_imp_fe: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and S :: "'a set"   and V :: "'a set"   and u0 :: "'a" assumes "gen_dfs_pre E U S V u0" shows "gen_dfs_fe_inv E U (insert u0 S) (insert u0 V) u0 (E `` {u0}) (insert u0 V) False"
fls_regpart_as_fls_nth: fixes f :: "'a fls"   and n :: "int" shows "fls_nth (fls_regpart_as_fls f) n = (if n < 0 then 0::'a else fls_nth f n)"
w_simps_5: shows "local.dom \<chi>.chine = \<chi>.chine"
proj_nil_2: fixes n :: "'e" shows "proj_unl n [] = []"
enat_cosplit: fixes P :: "'a \<Rightarrow> bool"   and c :: "'a"   and d :: "enat \<Rightarrow> 'a"   and n :: "enat" shows "P (case n of 0 \<Rightarrow> c | eSuc x \<Rightarrow> d x) = ((n = 0 \<longrightarrow> P c) \<and> (\<forall>m. n = eSuc m \<longrightarrow> P (d m)))"
prems_9: shows "is_generic p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3)"
X_def: fixes j :: "nat" shows "local.X j = [:- local.x j, 1::'a:]"
generate_valid_stateful_policy_IFSACS_all_security_requirements_fulfilled_IFS: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and edgesList :: "('b \<times> 'b) list" assumes "valid_reqs M"   and "wf_graph G"   and "all_security_requirements_fulfilled M G"   and "set edgesList \<subseteq> edges G" shows "all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph (generate_valid_stateful_policy_IFSACS G M edgesList))"
I_hd_coeffs1: fixes as :: "atom list"   and xs :: "int list" assumes "\<forall>a\<in>set as. hd_coeff a \<noteq> 0" shows "(\<exists>x. \<forall>a\<in>set (hd_coeffs1 as). I\<^sub>Z a (x # xs)) = (\<exists>x. \<forall>a\<in>set as. I\<^sub>Z a (x # xs))"
disjoint_notin1: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<inter> B = {}"   and "x \<in> B" shows "x \<notin> A"
cons_perm_eq: fixes z :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "(z # xs <~~> z # ys) = (xs <~~> ys)"
tensor_mkarr: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u" shows "\<F>C.mkarr t \<otimes> \<F>C.mkarr u = \<F>C.mkarr (t \<^bold>\<otimes> u)"
num_syms_1: fixes t :: "('d, 'e) Term.term" shows "1 \<le> num_syms t"
lr_of_tran_s1_split: fixes a :: "('b, 'c) routing_rule_scheme"   and rt :: "('b, 'c) routing_rule_scheme list" shows "lr_of_tran_s1 (a # rt) = (route2match a, routing_oiface a) # lr_of_tran_s1 rt"
irreducible\<^sub>dD_1: fixes p :: "'d poly" assumes "irreducible\<^sub>d p" shows "0 < degree p"
pHom_mem: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and n :: "nat"   and c :: "nat \<Rightarrow> 'a" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "pol_coeff S (n, c)" shows "f (polyn_expr R X n (n, c)) = polyn_expr A Y n (n, cmp f c)"
comm_add_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A + B = B + A"
2: shows "d \<sqinter> H * a * top \<le> a * top"
card_bad_times: fixes x :: "'a"   and B :: "nat" assumes "x \<in> Og" shows "real (card (bad_times x \<inter> {..<B})) \<le> d2 * real B"
measurable_density_eq2: fixes h :: "'b \<Rightarrow> 'c"   and Mh :: "'b measure"   and Mh' :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "(h \<in> Mh \<rightarrow>\<^sub>M density Mh' f) = (h \<in> Mh \<rightarrow>\<^sub>M Mh')"
lookup_eq_None_iff: fixes vo :: "'d option"   and kvs :: "('c \<times> ('c, 'd) trie) list"   and ks :: "'c list" assumes "invar_trie (Trie vo kvs)" shows "(lookup_trie (Trie vo kvs) ks = None) = (ks = [] \<and> vo = None \<or> (\<exists>k ks'. ks = k # ks' \<and> (\<forall>t. (k, t) \<in> set kvs \<longrightarrow> lookup_trie t ks' = None)))"
P1: shows "Bet B A A' \<and> Cong A A' E D"
br_rel_prod: fixes a :: "'c \<Rightarrow> 'e"   and I :: "'c \<Rightarrow> bool"   and b :: "'d \<Rightarrow> 'f"   and J :: "'d \<Rightarrow> bool" shows "br a I \<times>\<^sub>r br b J = br (\<lambda>(x, y). (a x, b y)) (\<lambda>(x, y). I x \<and> J y)"
filterlim_times_pos: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'b"   and F1 :: "'a filter"   and c :: "'b"   and l :: "'b" assumes "filterlim f (at_right p) F1"   and "(0::'b) < c"   and "l = c * p" shows "LIM x F1. c * f x :> at_right l"
chine_simps_3: shows "C.cod assoc\<nu>\<pi>\<rho>.chine = assoc\<nu>\<pi>\<rho>.cod.apex"
rel_with_target_impl_reflC_transC_TRel_is_weak_reduction_simulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_simulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>*" shows "weak_reduction_simulation (TRel\<^sup>*) Target"
E_1: assumes "QuadRes (int p) a" shows "[a ^ ((p - 1) div 2) = 1] (mod int p)"
insert_remove: fixes A :: "'k set"   and x :: "'k" assumes "finite A" shows "card (insert x A) = Suc (card (A - {x}))"
le_plus_1: fixes s :: "'lbl"   and s' :: "'lbl" shows "s \<le> s + s'"
starfun_eq: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d" shows "(*f* f) (star_of a) = star_of (f a)"
valid_distinct: fixes \<Gamma> :: "(name \<times> ty) list" shows "valid \<Gamma> = distinct (map fst \<Gamma>)"
terminates_to_terminates_merge_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<Gamma>\<turnstile>merge_guards c \<down> s"
cancels_to_1_def: fixes l1 :: "(bool \<times> 'c) list"   and l2 :: "(bool \<times> 'c) list" shows "cancels_to_1 l1 l2 = (\<exists>i. cancels_to_1_at i l1 l2)"
mset_preorder: fixes t :: "'b tree" shows "mset (preorder t) = mset_tree t"
extremum_uniqueI: fixes a :: "'c" assumes "top \<le> a" shows "a = top"
Hausdorff_Euclidean_space: fixes n :: "nat" shows "Hausdorff_space (Euclidean_space n)"
bigo_powr_iff: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "0 < p"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "\<forall>\<^sub>F x in F. 0 \<le> g x" shows "((\<lambda>x. f x powr p) \<in> O[F](\<lambda>x. g x powr p)) = (f \<in> O[F](g))"
Gamma_real_pos: fixes x :: "real" assumes "0 < x" shows "0 < Gamma x"
mem_val_alt_8_mem_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 (pa AND 68719476732) s1 = mem_val_alt 8 (pa AND 68719476732) s2 \<and> mem_val_alt 8 ((pa AND 68719476732) + 1) s1 = mem_val_alt 8 ((pa AND 68719476732) + 1) s2 \<and> mem_val_alt 8 ((pa AND 68719476732) + 2) s1 = mem_val_alt 8 ((pa AND 68719476732) + 2) s2 \<and> mem_val_alt 8 ((pa AND 68719476732) + 3) s1 = mem_val_alt 8 ((pa AND 68719476732) + 3) s2"
\<beta>_int: fixes p :: "fm"   and bs :: "real list" assumes "iszlfm p bs" shows "\<forall>b\<in>set (\<beta> p). isint b bs"
hom2_2: shows "homotopic_with (\<lambda>x. True) U U (f2 \<circ> g2) id"
Infinitesimal_square_iff: fixes x :: "'a star" shows "(x \<in> Infinitesimal) = (x * x \<in> Infinitesimal)"
rbt_lookup_rbt_insert: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt"   and k :: "'a"   and v :: "'b" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.rbt_lookup less (ord.rbt_insert less k v t) = ord.rbt_lookup less t(k \<mapsto> v)"
sg6a: shows "ack (t + 2 + k) = [connection_ok]"
compliant_stateful_ACS_only_state_violations': shows "\<forall>F\<in>get_offending_flows M (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (flows_state \<T>)"
abs_eq: fixes x :: "('c \<times> 'd) list" shows "fmap_of_list x = Abs_fmap (map_of x)"
list_e_eq_lel_2: fixes l1 :: "'a list"   and e' :: "'a"   and l2 :: "'a list"   and e :: "'a" shows "(l1 @ e' # l2 = [e]) = (l1 = [] \<and> e' = e \<and> l2 = [])"
permutation: fixes p :: "'b \<Rightarrow> 'b" shows "permutation p = (bij p \<and> finite {x. p x \<noteq> x})"
aux_2_3: fixes m :: "'node"   and v :: "'var" shows "(if m \<in> phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty)))) (\<alpha>n g)) n \<and> v \<in> uses g n then Some (map (\<lambda>m. lookupDef g m v) (predecessors g m)) else Mapping.lookup (snd (foldr (aux_2 g) ns (Mapping.empty, Mapping.empty))) (m, v)) = Mapping.lookup p' (m, v)"
lfilter_lfind_1: fixes P :: "'b \<Rightarrow> bool"   and lst :: "'b llist" assumes "lfilter P lst \<noteq> LNil" shows "P (lst $ lfind P lst)"
c_def: fixes i :: "nat" shows "c i = insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var i)"
norm_increases_online: fixes d :: "'a"   and a :: "'a" assumes "d \<noteq> (0::'a)" shows "norm a < norm (a + d) \<or> norm a < norm (a - d)"
contractible_space_subtopology_singleton: fixes X :: "'b topology"   and a :: "'b" shows "contractible_space (subtopology X {a})"
prv_subst: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "bprv \<phi>" shows "bprv (subst \<phi> t x)"
1: shows "Inf (PLS s as) \<le> problem_plan_bound PROB"
listsE: fixes x :: "'b"   and l :: "'b list"   and A :: "'b set" assumes "x # l \<in> lists A" obtains "x \<in> A" and "l \<in> lists A"
intros_1: fixes x :: "name"   and a :: "coname" shows "fic (Ax x a) a"
countable_image: fixes A :: "'k set"   and f :: "'k \<Rightarrow> 'l" assumes "countable A" shows "countable (f ` A)"
perp_up_non_degenerate_in_S_at_end_1: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident b l"   and "\<not> proj2_incident a l" shows "perp_up a l \<in> S"
simps_2: fixes f :: "'a vec"   and g :: "'a vec"   and ng :: "'a"   and n_gs :: "('a vec \<times> 'a) list"   and mus :: "'a list"   and g' :: "'a vec" shows "mus_adjuster f ((g, ng) # n_gs) mus g' = (let a = f \<bullet> g / ng in mus_adjuster f n_gs (a # mus) (- a \<cdot>\<^sub>v g + g'))"
generate_sincl: fixes A :: "int set" shows "A \<subseteq> generate (Z 1) A"
PSP_Stable2: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto>w A'"   and "F \<in> Stable B" shows "F \<in> B \<inter> A \<longmapsto>w B \<inter> A'"
f_1: fixes G :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "G \<in> F" shows "flowsto (f G - trap) T CX G"
comp_cone_functor: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b"   and J' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'a" assumes "Limit.cone J C D a \<chi>"   and "functor J' J F" shows "Limit.cone J' C (D \<circ> F) a (\<chi> \<circ> F)"
match_fvi_trm_Some: fixes f :: "nat \<Rightarrow> 'a option"   and ts :: "'a MFOTL.trm list"   and xs :: "'a list"   and t :: "'a MFOTL.trm"   and x :: "nat" assumes "Some f = match ts xs"   and "t \<in> set ts"   and "x \<in> fv_trm t" shows "f x \<noteq> None"
abs_prts: fixes a :: "'a" shows "\<bar>a\<bar> = pprt a - nprt a"
strict_prefixI': fixes ys :: "'b list"   and xs :: "'b list"   and z :: "'b"   and zs :: "'b list" assumes "ys = xs @ z # zs" shows "strict_prefix xs ys"
0: fixes y :: "int" assumes "y dvd x" shows "y \<noteq> 0"
c1: fixes x :: "('a \<times> 'a set) set" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> x = x"
rel_frontier_frontier: fixes S :: "'a set" assumes "affine hull S = UNIV" shows "rel_frontier S = frontier S"
D_cyc_free_preservation: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m' :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'b) \<le> len m i i xs"   and "\<forall>i\<le>n. \<forall>j\<le>n. D m i j n = m' i j" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'b) \<le> len m' i i xs"
prj_emb_prj: fixes x :: "udom" assumes "DEFL('c) \<sqsubseteq> DEFL('d)" shows "PRJ('c)\<cdot>(EMB('d)\<cdot>(PRJ('d)\<cdot>x)) = PRJ('c)\<cdot>x"
xor_ac_2: fixes x :: "'e"   and y :: "'e" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<oplus> y = y \<oplus> x"
leg1_commutes: shows "f.cod.leg1 \<cdot> f.chine = f.leg1"
id_comp: fixes g :: "'e \<Rightarrow> 'f" shows "id \<circ> g = g"
g: shows "\<forall>\<^sub>F x in F. 0 \<le> g x"
A1: shows "\<bar>spmf (R1_14 M (c0, c1) \<bind> D) True - spmf (R1_14_interm1 M (c0, c1) \<bind> D) True\<bar> = \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c0 \<bind> (\<lambda>view. A1 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A1 view (m0, m1)))) True\<bar>"
ipresIGSwapAll_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'i, 'j, 'k, 'l, 'c, 'e, 'm) model_scheme"   and MOD' :: "('n, 'o, 'i, 'p, 'q, 'l, 'd, 'f, 'r) model_scheme" shows "ipresIGSwapAll h hA MOD MOD' \<equiv> ipresIGSwap h MOD MOD' \<and> ipresIGSwapAbs hA MOD MOD'"
lm033: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" assumes "Domain P \<subseteq> Domain Q" shows "P +< Q = Q"
expectation_greater: fixes X :: "'a \<Rightarrow> real"   and a :: "real" assumes "integrable M X"   and "AE x in M. a < X x" shows "a < expectation X"
assms: shows "poly p1 x \<noteq> 0 \<or> poly p2 x \<noteq> 0"
card_Inv_sym: fixes xs :: "'a list"   and ys :: "'a list" shows "card (Inv xs ys) = card (Inv ys xs)"
unat_lt2p: fixes x :: "'b word" shows "unat x < 2 ^ LENGTH('b)"
path_defs_3: fixes g :: "real \<Rightarrow> 'c" shows "pathfinish g = g 1"
span_Un: fixes S :: "'c set"   and T :: "'c set" shows "ideal (S \<union> T) = {x + y |x y. x \<in> ideal S \<and> y \<in> ideal T}"
execn_Normal_elim_cases_9: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Cond b c1 c2,Normal s\<rangle> =n\<Rightarrow> t" obtains "s \<in> b" and "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> =n\<Rightarrow> t"   | "s \<notin> b" and "\<Gamma>\<turnstile> \<langle>c2,Normal s\<rangle> =n\<Rightarrow> t"
assms: shows "inverse_transformations A B F G \<sigma> \<sigma>'"
\<epsilon>_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'c \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'c"   and \<phi> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and \<psi> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b" assumes "meta_adjunction C D F G \<phi> \<psi>" shows "meta_adjunction.\<epsilon> C F G \<psi> \<equiv> transformation_by_components.map C C (F \<circ> G) (meta_adjunction.\<epsilon>o G \<psi>)"
g_def: shows "g = (\<lambda>x. poly (q mod p) x / poly p x)"
get_disconnected_nodes_ok: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr" assumes "type_wf h"   and "document_ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok get_disconnected_nodes document_ptr"
sel_2: fixes x1 :: "'b"   and x2 :: "'b tree"   and x3 :: "'b tree" shows "left (Node x1 x2 x3) = x2"
either_ccTTree: fixes S :: "Vars.var set"   and G :: "CoCalls"   and S' :: "Vars.var set"   and G' :: "CoCalls" shows "ccTTree S G \<oplus>\<oplus> ccTTree S' G' \<sqsubseteq> ccTTree (S \<union> S') (G \<squnion> G')"
upred_top: shows "\<top>\<^bsub>\<P>\<^esub> = false"
powhr_one_eq_one: fixes a :: "real star" shows "1 powhr a = 1"
Comp_in_comp_ARR: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set"   and t :: "'b local.term"   and u :: "'b local.term" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f"   and "free_monoidal_category.ARR C g"   and "monoidal_language.Dom C (free_monoidal_category.rep C f) = monoidal_language.Cod C (free_monoidal_category.rep C g)"   and "t \<in> f"   and "u \<in> g" shows "t \<^bold>\<cdot> u \<in> free_monoidal_category.comp C f g"
WT_gpv_absorb: fixes \<I>' :: "('y, 'z) \<I>"   and gpv :: "('y, 'z) distinguisher"   and \<I> :: "('aa, 'ab) \<I>"   and conv :: "('y, 'z, 'aa, 'ab) converter" assumes "\<I>' \<turnstile>g gpv \<surd>"   and "\<I>', \<I> \<turnstile>\<^sub>C conv \<surd>" shows "\<I> \<turnstile>g absorb gpv conv \<surd>"
assms_4: shows "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"
less_imp_Suc_add: fixes m :: "nat"   and n :: "nat" assumes "order.greater n m" shows "\<exists>k. n = Suc (m + k)"
Ch_unilateral_substitutes: shows "\<forall>h. Pdds'.unilateral_substitutes_on UNIV (Ch h)"
distributed_joint_indep': fixes S :: "'c measure"   and T :: "'d measure"   and X :: "'a \<Rightarrow> 'c"   and Px :: "'c \<Rightarrow> ennreal"   and Y :: "'a \<Rightarrow> 'd"   and Py :: "'d \<Rightarrow> ennreal" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M S X Px"   and "distributed M T Y Py"   and "distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))" shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>(x, y). Px x * Py y)"
Suc: fixes \<rho> :: "complex mat" shows "denote_while_n M0 M1 DS n \<rho> \<in> carrier_mat d d"
ins_inv1_inv2_4: fixes t :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and x :: "'c" assumes "inv1 t"   and "inv2 t" shows "inv1l (rbt_ins f k x t)"
Cons_5: shows "set Ks' \<subseteq> carrier R"
cases: fixes a1 :: "'c"   and a2 :: "'d node_action"   and a3 :: "'c"   and S :: "('c \<times> 'd node_action \<times> 'c) set" assumes "(a1, a2, a3) \<in> cnet_sos S" obtains s :: "'c" and i :: "nat" and i' :: "nat" and s' :: "'c" where "a1 = sa__" and "a2 = connect(i__, i'__)" and "a3 = s'a__" and "(sa__, connect(i__, i'__), s'a__) \<in> S"   | s :: "'c" and i :: "nat" and i' :: "nat" and s' :: "'c" where "a1 = sb__" and "a2 = disconnect(ia__, i')" and "a3 = s'b__" and "(sb__, disconnect(ia__, i'), s'b__) \<in> S"   | s :: "'c" and R :: "nat set" and m :: "'d" and s' :: "'c" where "a1 = sc__" and "a2 = \<tau>" and "a3 = s'c__" and "(sc__, R:*cast(m), s'c__) \<in> S"   | s :: "'c" and s' :: "'c" where "a1 = sd__" and "a2 = \<tau>" and "a3 = s'd__" and "(sd__, \<tau>, s'd__) \<in> S"   | s :: "'c" and i :: "nat" and d :: "nat" and s' :: "'c" where "a1 = se__" and "a2 = ib__:deliver(d__)" and "a3 = s'e__" and "(se__, ib__:deliver(d__), s'e__) \<in> S"   | s :: "'c" and i :: "nat" and K :: "nat set" and d :: "nat" and di :: "nat" and s' :: "'c" where "a1 = s" and "a2 = i:newpkt(d, di)" and "a3 = s'" and "(s, {i}\<not>K:arrive(newpkt (d, di)), s') \<in> S"
poly_pinfty_ge: fixes p :: "real poly"   and b :: "real" assumes "0 < lead_coeff p"   and "degree p \<noteq> 0" shows "\<exists>n. \<forall>x\<ge>n. b \<le> poly p x"
gcd_primitive_part: fixes f :: "'a poly"   and g :: "'a poly" shows "gcd (primitive_part f) (primitive_part g) = normalize (primitive_part (gcd f g))"
fixp_iterates_above: fixes a :: "'a" shows "fixp_above a \<in> iterates_above a"
adjointFO_free: fixes \<sigma> :: "'a \<Rightarrow> ('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "\<And>i. dfree (\<sigma> i)" shows "adjointFO I \<sigma> \<nu> = \<lparr>Functions = \<lambda>f. case f of Inl f' \<Rightarrow> Functions I f' | Inr f' \<Rightarrow> \<lambda>_. sterm_sem I (\<sigma> f') (fst \<nu>), Predicates = Predicates I, Contexts = Contexts I, Programs = Programs I, ODEs = ODEs I, ODEBV = ODEBV I\<rparr>"
reachable: shows "\<forall>v\<in>set (g_V G). v \<in> (g_E G)\<^sup>* `` {0}"