hcju/isabelleps huggingface dataset:
per_supset_compare: fixes p1 :: "('a \<times> 'a) set"   and p2 :: "('a \<times> 'a) set"   and x1 :: "'a"   and x2 :: "'a" assumes "(p1, p2) \<in> SeprefUF.per_supset_rel"   and "x1 \<in> Domain p2"   and "x2 \<in> Domain p2" shows "per_compare p1 x1 x2 = per_compare p2 x1 x2"
step_z_sound: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and Z :: "('f \<Rightarrow> 'g) set"   and l' :: "'e"   and Z' :: "('f \<Rightarrow> 'g) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>" shows "\<forall>u'\<in>Z'. \<exists>u\<in>Z. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"
inf_chain_def: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" shows "inf_chain p f = (\<forall>i. p (f i) (f (Suc i)))"
ladder_\<alpha>_def: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" shows "ladder_\<alpha> a D L index = (if index = 0 then a else ladder_\<gamma> a D L (index - 1))"
isVar_asSort_Var: fixes xs :: "'varSort"   and x :: "'var" shows "isVar (asSort xs) (Var xs x)"
assms_2: shows "\<forall>\<sigma>. \<not> strategy_winning_by_Even \<sigma> p"
times_mat_def: fixes A :: "'a Matrix.mat"   and B :: "'a Matrix.mat" shows "A * B \<equiv> Matrix.mat (dim_row A) (dim_col B) (\<lambda>(i, j). Matrix.row A i \<bullet> Matrix.col B j)"
extend_state_oracle_def: fixes callee :: "'i \<Rightarrow> 'j \<Rightarrow> ('k \<times> 'i) spmf" shows "\<dagger>callee = (\<lambda>(s', s) x. map_spmf (\<lambda>(y, s). (y, s', s)) (callee s x))"
tell_state_def: fixes tell :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and w :: "'f"   and m :: "('d, 'e) stateT" shows "tell_state tell w m = StateT (\<lambda>s. tell w (run_state m s))"
content_cball: shows "content (cball c r) = unit_ball_vol (real DIM('a)) * r ^ DIM('a)"
Dijkstra_Impl_axioms: shows "Dijkstra_Impl G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin g G_\<alpha>"
ok3_empty_mrexp3: fixes r :: "'b rexp" shows "ok3 (empty_mrexp3 r)"
in_imp_not_empty: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "A \<noteq> {}"
subst_LLq: fixes t1 :: "'trm"   and t2 :: "'trm"   and s :: "'trm"   and x :: "'var" assumes "t1 \<in> atrm"   and "t2 \<in> atrm"   and "s \<in> atrm"   and "x \<in> var" shows "subst (LLq t1 t2) s x = LLq (substT t1 s x) (substT t2 s x)"
inducts: fixes xa :: "'e"   and f :: "'e \<Rightarrow> 'e"   and x :: "'e"   and P :: "'e \<Rightarrow> bool" assumes "xa \<in> orbit f x"   and "P (f x)"   and "\<And>y. \<lbrakk>y \<in> orbit f x; P y\<rbrakk> \<Longrightarrow> P (f y)" shows "P xa"
Float_cases: fixes f :: "float" obtains m :: "int" and e :: "int" where "f = Float m e"
unsigned_eq_0_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and w :: "'c word" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.unsigned one plus zero w = zero) = (w = 0)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_subst: fixes S :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list"   and \<delta> :: "'g \<Rightarrow> ('f, 'g) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>"
unat_mask: fixes n :: "nat" shows "unat (mask n) = 2 ^ min n LENGTH('b) - 1"
commutes_with_unit: fixes \<eta> :: "'b \<Rightarrow> 'c"   and M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and M' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'" shows "\<eta> unit = unit'"
dec_clocked_total: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec" shows "\<exists>res. evaluate_dec True mn env s d res"
open_imp_subset: fixes U :: "'a set set" assumes "is_zariski_open U" shows "U \<subseteq> Spec"
assms: shows "s \<in> set (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
le_imp_less_or_eq: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" assumes "ko.le ko x y" shows "ko.lt ko x y \<or> x = y"
real_inner_inner: fixes x :: "'b"   and y :: "'b" shows "real_of (inner x x * inner y y) = real_of (inner x x) * real_of (inner y y)"
represented_generic_objects_nonnull: fixes \<tau> :: "'d state \<times> 'd state"   and pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>'e\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> OclAllInstances_generic pre_post H->includes\<^sub>S\<^sub>e\<^sub>t(x)" shows "\<tau> \<Turnstile> not (x \<triangleq> null)"
sup_ty_opt_refl: fixes P :: "'a prog"   and T :: "ty err" shows "P \<turnstile> T \<le>\<^sub>\<top> T"
l_insert_before_wf2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_insert_before_wf2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs set_child_nodes set_child_nodes_locs get_shadow_root get_shadow_root_locs set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs heap_is_wellformed parent_child_rel get_ancestors_si get_ancestors_si_locs get_parent get_parent_locs adopt_node adopt_node_locs get_owner_document insert_before insert_before_locs known_ptrs get_host get_host_locs get_root_node_si get_root_node_si_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs"
T_get_min_estimate: fixes ts :: "'b tree list" assumes "ts \<noteq> []" shows "T_get_min ts = length ts"
lset_lnull: fixes xs :: "'c llist" assumes "lnull xs" shows "lset xs = {}"
mset_subset_eqD: fixes A :: "'a multiset"   and B :: "'a multiset"   and x :: "'a" assumes "A \<subseteq># B"   and "x \<in># A" shows "x \<in># B"
lunionM_def: fixes m1 :: "'m"   and m2 :: "'m" shows "lunionM m1 m2 \<equiv> bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A @ B)))"
imageI: fixes x :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> A" shows "f x \<in> f ` A"
charts_submanifold_def: fixes S :: "'c set" shows "manifold_eucl.dest.charts_submanifold S = restrict_chart S ` charts_eucl"
disj_commute: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "disj x y = disj y x"
cases: fixes len :: "nat"   and a1 :: "nat \<times> 'a option"   and a2 :: "nat \<times> 'a option" assumes "sim21_size_aux len a1 a2" obtains pc1 :: "nat" and pc2 :: "nat" and xcp1 :: "'a option" and xcp2 :: "'a option" where "a1 = (pc1, xcp1)" and "a2 = (pc2, xcp2)" and "pc1 \<le> len" and "pc2 \<le> len" and "xcp1 \<noteq> None \<and> xcp2 = None \<and> pc1 = pc2 \<or> xcp1 = None \<and> pc2 < pc1"
I_notcarr: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "primeideal I R" shows "carrier R \<noteq> I"
below_antisym: fixes x :: "'b"   and y :: "'b" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
FtorB: shows "Ftor (Hom\<^bsub>C\<^esub>[\<midarrow>,B]) : Op C \<longrightarrow> SET"
s_mul_ext_union_compat: fixes A :: "'a multiset"   and B :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and C :: "'a multiset" assumes "(A, B) \<in> s_mul_ext ns s"   and "locally_refl ns C" shows "(A + C, B + C) \<in> s_mul_ext ns s"
inf_bot_right: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and x :: "'a" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf x bot = bot"
relabel_to_front_def: shows "relabel_to_front \<equiv> let f = pp_init_f; l = pp_init_l; n = rtf_init_n; L_left = [] in (spec l. distinct l \<and> set l = V - {s, t}) \<bind> (\<lambda>L_right. while\<^sub>T (\<lambda>(f, l, n, L_left, L_right). L_right \<noteq> []) (\<lambda>(f, l, n, L_left, L_right). let u = hd L_right in assert (u \<in> V) \<bind> (\<lambda>_. let old_lu = l u in discharge f l n u \<bind> (\<lambda>(f, l, n). if l u \<noteq> old_lu then let (L_left, L_right) = ([u], L_left @ tl L_right) in return (f, l, n, L_left, L_right) else let (L_left, L_right) = (L_left @ [u], tl L_right) in return (f, l, n, L_left, L_right)))) (f, l, n, L_left, L_right) \<bind> (\<lambda>(f, l, n, L_left, L_right). assert (neighbor_invar c s t f l n) \<bind> (\<lambda>_. return f)))"
tail_face_cycle_succ: fixes a :: "'b" assumes "a \<in> arcs G" shows "tail G (face_cycle_succ a) = head G a"
eq_0_height: fixes t :: "'b tree" shows "(0 = height t) = (t = \<langle>\<rangle>)"
RID\<^sub>GE: fixes r :: "'d"   and s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" assumes "r \<in> RID\<^sub>G s" obtains "r \<in> dom s"   | r' :: "'d" and ls :: "('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr" where "s r' = Some ls" and "r \<in> RID\<^sub>L ls"
ds: shows "matrix_sum d2 f n \<in> carrier_mat d2 d2"
assms_3: shows "fst ` set bs \<subseteq> dgrad_p_set d m"
le_mult: fixes \<beta> :: "V"   and \<alpha> :: "V" assumes "Ord \<beta>"   and "\<beta> \<noteq> 0" shows "\<alpha> \<le> \<alpha> * \<beta>"
L_def: shows "L = inv_into {0..dist x y} g ` H"
recurrent_acc_4: fixes x :: "'s"   and y :: "'s" assumes "recurrent x"   and "(x, y) \<in> acc" shows "(x, y) \<in> communicating"
assms: shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<longmapsto> Rs"
less_not_refl3: fixes s :: "nat"   and t :: "nat" assumes "order.greater t s" shows "s \<noteq> t"
conv_invol': fixes x :: "'a" shows "(1::'a) \<lhd> ((1::'a) \<lhd> x) = x"
f: shows "\<forall>x y. HMA_I x y \<longrightarrow> f x = fT y"
S_edge_reachable: fixes u :: "'v"   and v :: "'v" assumes "u \<in> S (A Q \<pi>)"   and "(u, v) \<in> edges g" shows "(u, v) \<in> edges rg"
if_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'b"   and u :: "'b"   and y :: "'b"   and v :: "'b" assumes "b = c"   and "c \<Longrightarrow> x = u"   and "\<not> c \<Longrightarrow> y = v" shows "(if b then x else y) = (if c then u else v)"
line_integral_exists_point_path: fixes basis :: "'a set"   and \<gamma> :: "real \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a" assumes "finite basis"   and "point_path \<gamma>" shows "line_integral_exists F basis \<gamma>"
is_swap_values_2: assumes "is_swap" shows "f 1 = 0"
joinpaths_def: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" shows "g1 +++ g2 = (\<lambda>x. if x \<le> 1 / 2 then g1 (2 * x) else g2 (2 * x - 1))"
order_eq_0_iff: fixes p :: "'b poly"   and x :: "'b" assumes "p \<noteq> 0" shows "(order x p = 0) = (poly p x \<noteq> (0::'b))"
ipoly_complex_interval_def: fixes p :: "int poly"   and x :: "complex_interval" shows "ipoly_complex_interval p x = fold_coeffs (\<lambda>a b. of_int_complex_interval a + x * b) p 0"
set_ConsD: fixes y :: "'d"   and x :: "'d"   and xs :: "'d list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
carrier_subgroup_generated_subgroup: shows "carrier (subgroup_generated G H') = H'"
list_less_iff_prefix: fixes a :: "'b list"   and b :: "'b list" shows "(a < b) = strict_prefix a b"
atLeast_Suc_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..Suc n} = sum (g \<circ> Suc) {m..n}"
singleton_transp: fixes p :: "'a" assumes "singleton p" shows "singleton (p\<^sup>T)"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e rose_tree\<^sub>h" shows "map_rose_tree\<^sub>h g (map_rose_tree\<^sub>h f v) = map_rose_tree\<^sub>h (g \<circ> f) v"
span_base: fixes a :: "'b"   and S :: "'b set" assumes "a \<in> S" shows "a \<in> local.span S"
alpha_Tree_L_transform_Tree: fixes t1 :: "('a, 'b, 'c, 'd) FL_Formula.Tree"   and t2 :: "('a, 'b, 'c, 'd) FL_Formula.Tree" assumes "t1 =\<^sub>\<alpha> t2" shows "L_transform_Tree t1 = L_transform_Tree t2"
strong_del_point_finite: fixes p :: "'a" shows "finite_incidence_system (fin.del_point p) (fin.str_del_point_blocks p)"
inj_on_pminusx_E: shows "inj_on ((-) (int p)) E"
induct: fixes x1 :: "TP"   and x2 :: "OBJ"   and P :: "TP \<Rightarrow> OBJ \<Rightarrow> bool" assumes "(x1, x2) \<in> VS_com"   and "\<And>y f x. \<lbrakk>CONTEXT y = low; GAMMA f = low\<rbrakk> \<Longrightarrow> P low (Get x y f)"   and "\<And>x y f. CONTEXT x = high \<Longrightarrow> P high (Get x y f)"   and "\<And>x f e. \<lbrakk>CONTEXT x = low; GAMMA f = low; (e, low) \<in> VS_expr\<rbrakk> \<Longrightarrow> P low (Put x f e)"   and "\<And>f e x. \<lbrakk>GAMMA f = high; \<forall>s. noLowDPs s \<longrightarrow> Expr_good e s\<rbrakk> \<Longrightarrow> P high (Put x f e)"   and "\<And>x c. CONTEXT x = low \<Longrightarrow> P low (New x c)"   and "\<And>x C. CONTEXT x = high \<Longrightarrow> P high (New x C)"   and "\<And>x e. \<lbrakk>CONTEXT x = low; (e, low) \<in> VS_expr\<rbrakk> \<Longrightarrow> P low (Assign x e)"   and "\<And>x e. \<lbrakk>CONTEXT x = high; \<forall>s. noLowDPs s \<longrightarrow> Expr_good e s\<rbrakk> \<Longrightarrow> P high (Assign x e)"   and "\<And>pc. P pc Skip"   and "\<And>pc c d. \<lbrakk>(pc, c) \<in> VS_com; P pc c; (pc, d) \<in> VS_com; P pc d\<rbrakk> \<Longrightarrow> P pc (Comp c d)"   and "\<And>b pc c d. \<lbrakk>(b, pc) \<in> VS_Bexpr; (pc, c) \<in> VS_com; P pc c; (pc, d) \<in> VS_com; P pc d\<rbrakk> \<Longrightarrow> P pc (Iff b c d)"   and "\<And>b pc c. \<lbrakk>(b, pc) \<in> VS_Bexpr; (pc, c) \<in> VS_com; P pc c\<rbrakk> \<Longrightarrow> P pc (While b c)"   and "\<And>c. \<lbrakk>(high, c) \<in> VS_com; P high c\<rbrakk> \<Longrightarrow> P low c" shows "P x1 x2"
f_def: shows "f = Cn 2 r_singleton_encode [recf.Id 2 0]"
prems_1: shows "weak_sharing_consistent \<O> (x # sb)"
sub_inf_def: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y = - (- - x \<squnion> - - y)"
lunit'_in_hom_1: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<l>\<^sup>-\<^sup>1[f] : src f \<rightarrow> trg f\<guillemotright>"
ln_exp_cf3_upper_bound_nonneg: fixes x :: "real" assumes "0 \<le> x"   and "0 < numer_cf3 (- x)" shows "x \<le> ln (exp_cf3 x)"
sublist_valid_plan__alt: fixes as1 :: "'c list"   and PROB :: "'c set"   and as2 :: "'c list" assumes "as1 \<in> valid_plans PROB"   and "subseq as2 as1" shows "as2 \<in> valid_plans PROB"
sigma_finite_state_measure: fixes V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" assumes "finite V" shows "sigma_finite_measure (state_measure V \<Gamma>)"
tranclp_into_rtranclp: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "r\<^sup>+\<^sup>+ a b" shows "r\<^sup>*\<^sup>* a b"
absorp2: fixes x :: "'a"   and y :: "'a" shows "x + x \<sqinter> y = x"
three_composable_arrows_of_spans_axioms: shows "three_composable_arrows_of_spans (\<cdot>) prj0 prj1 f g h"
init_segment_iff: fixes S :: "nat set"   and T :: "nat set" shows "init_segment S T = (S = T \<or> (\<exists>m\<in>T. S = {n \<in> T. n < m}))"
perfect_sec_P1_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool \<times> bool" shows "xor_sim_det.perfect_sec_P1 m1 m2 \<equiv> R1_xor m1 m2 = xor_funct m1 m2 \<bind> (\<lambda>(s1, s2). S1_xor m1 s1)"
circ_square: fixes x :: "'a" shows "(x * x)\<^sup>\<circ> \<le> x\<^sup>\<circ>"
simps_13: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case T_i32 of T_i32 \<Rightarrow> f1 | T_i64 \<Rightarrow> f2 | T_f32 \<Rightarrow> f3 | T_f64 \<Rightarrow> f4) = f1"
cos_mono_le_eq: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> pi"   and "0 \<le> y"   and "y \<le> pi" shows "(cos x \<le> cos y) = (y \<le> x)"
atLeastatMost_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "class.order less_eq less"   and "less b a" shows "ord.atLeastAtMost less_eq a b = {}"
simps_2: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and uw :: "char list"   and rs :: "'c rule list" shows "no_matching_Goto \<gamma> p (Rule m (Goto uw) # rs) = (\<not> matches \<gamma> m p \<and> no_matching_Goto \<gamma> p rs)"
0: fixes xs :: "'a list" shows "real (time (mergesort_tm f xs)) \<le> (mergesort_recurrence \<circ> length) xs"
raw_has_prod_nonzero: fixes f :: "nat \<Rightarrow> 'e"   and M :: "nat" shows "\<not> raw_has_prod f M (0::'e)"
inorder_split_maxD: fixes t :: "('b \<times> nat) tree"   and t' :: "('b \<times> nat) tree"   and a :: "'b" assumes "split_max t = (t', a)"   and "t \<noteq> \<langle>\<rangle>" shows "Tree2.inorder t' @ [a] = Tree2.inorder t"
last_index_less: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat" assumes "x \<in> set xs"   and "length xs \<le> n" shows "last_index xs x < n"
twoCisTypeII_5: shows "g1 piecewise_C1_differentiable_on {a..b}"
indep_sets_mono: fixes F :: "'e \<Rightarrow> 'a set set"   and I :: "'e set"   and J :: "'e set"   and G :: "'e \<Rightarrow> 'a set set" assumes "indep_sets F I"   and "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> G i \<subseteq> F i" shows "indep_sets G J"
msubstle: fixes c :: "poly"   and d :: "poly"   and a :: "poly"   and r :: "tm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and t :: "tm"   and s :: "tm" assumes "isnpoly c"   and "isnpoly d"   and "islin (Le (CNP 0 a r))" shows "Ifm vs (x # bs) (msubstle c t d s a r) = Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'b) # bs) (Le (CNP 0 a r))"
mu_G_spec: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)" shows "\<mu>G (emb I J X) = emeasure (P J) X"
basic_type_less_left_simps_1: fixes \<sigma> :: "'a basic_type" shows "(OclAny < \<sigma>) = False"
assms: fixes a :: "'a"   and b\<^sub>1 :: "'b"   and b\<^sub>2 :: "'b" assumes "(a, b\<^sub>1) |\<in>| M"   and "(a, b\<^sub>2) |\<in>| M" shows "b\<^sub>1 = b\<^sub>2"
inext_nth_mono: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a \<le> b" shows "I \<rightarrow> a \<le> I \<rightarrow> b"
successively_conv_sorted_wrt: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "transp P" shows "successively P xs = sorted_wrt P xs"
seq_char: fixes Obj :: "'a set"   and g :: "('a, 'a) concrete_category.arr"   and f :: "('a, 'a) concrete_category.arr" shows "partial_magma.arr (concrete_category.COMP Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>_ _ x _ _. x)) (concrete_category.COMP Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>_ _ x _ _. x) g f) = (partial_magma.arr (concrete_category.COMP Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>_ _ x _ _. x)) f \<and> f = g)"
all_same_arities_def: fixes F1 :: "'a \<Rightarrow> 'b fundef option"   and F2 :: "'a \<Rightarrow> 'c fundef option" shows "all_same_arities F1 F2 \<equiv> \<forall>f. rel_option (\<lambda>fd gd. arity fd = arity gd) (F1 f) (F2 f)"
punit_decr_grading_term: shows "punit.decr_grading_term = decr_grading"
dimM_2: shows "card {i. i < 2 ^ n \<and> f i} < 2 ^ n"
Suc_2: shows "higher_differentiable_on S g (Suc n)"
gauss_cnj_cnj: fixes z :: "gauss_int" shows "gauss_cnj (gauss_cnj z) = z"
divergences_def: fixes P :: "'c process" shows "divergences P \<equiv> snd (Rep_process P)"
sortedP_Nil: fixes x :: "real \<times> real" assumes "ccw'.sortedP x []" obtains
r_parallel''_1: fixes i :: "nat"   and x :: "nat"   and j :: "nat" assumes "eval r_phi [i, x] \<up> \<and> eval r_phi [j, x] \<up>" shows "eval r_parallel [i, j, x] \<up>"
argmin_in: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "argmin f xs \<in> set xs"
awalk_verts_arc1: fixes e :: "'b"   and p :: "'b list"   and u :: "'a" assumes "e \<in> set p" shows "tail G e \<in> set (G.awalk_verts u p)"
assms_5: shows "aux = f k (lookup_raw xs k) (lookup_raw ys k)"
hoare_diagram2: fixes D :: "'b \<times> 'b \<Rightarrow> 'f \<Rightarrow> 'f"   and X :: "'a \<Rightarrow> 'b \<Rightarrow> 'f" assumes "dmono D"   and "\<forall>u i j. \<Turnstile> X u i{| D (i, j) |}SUP_L_P X (pair u i) j" shows "\<Turnstile> \<Squnion>range X{| pt D |}(\<Squnion>range X \<sqinter> - grd (step D))"
fs_int: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < length fs" shows "fs ! j $ i \<in> \<int>"
Src_HcompNml: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Src (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Src u"
notempty_exist_nonstut: fixes s :: "nat \<Rightarrow> 'b"   and m :: "nat" assumes "notemptyseq (s |\<^sub>s m)" shows "\<exists>i. s i \<noteq> s m \<and> m < i"
x_coord_def: shows "x_coord \<equiv> \<lambda>t. t - 1 / 2"
cosh_ln_complex: fixes x :: "complex" assumes "x \<noteq> 0" shows "cosh (Ln x) = (x + inverse x) / 2"
cases: fixes a1 :: "'b Heap"   and a2 :: "heap option"   and a3 :: "heap option"   and a4 :: "'b" assumes "run a1 a2 a3 a4" obtains \<sigma> :: "heap option" and c :: "'b Heap" and r :: "'b" where "a1 = c__" and "a2 = \<sigma>''__" and "a3 = \<sigma>''__" and "a4 = ra__" and "is_exn \<sigma>''__"   | \<sigma> :: "heap option" and c :: "'b Heap" and r :: "'b" where "a1 = ca__" and "a2 = \<sigma>'''__" and "a3 = None" and "a4 = rb__" and "\<not> is_exn \<sigma>'''__" and "execute ca__ (the_state \<sigma>'''__) = None"   | \<sigma> :: "heap option" and c :: "'b Heap" and r :: "'b" and h' :: "heap" where "a1 = c" and "a2 = \<sigma>" and "a3 = Some h'" and "a4 = r" and "\<not> is_exn \<sigma>" and "execute c (the_state \<sigma>) = Some (r, h')"
o_def: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e" shows "f \<circ> g = (\<lambda>x. f (g x))"
Compl_atLeast: fixes k :: "'a" shows "- {k..} = {..<k}"
HMA_det: shows "rel_fun Mod_Type_Connect.HMA_M (=) Determinant.det Determinants.det"
aezdeltafun: fixes b :: "'b"   and a :: "'a" shows "aezfun (b \<delta>\<delta> a) = b \<delta> a"
l11_21_b: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and A :: "'a"   and C :: "'a"   and B' :: "'a"   and A' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet B A C"   and "Tarski_neutral_dimensionless.Out Bet B' A' C'" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"
cont_eqI: fixes lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "lub_singleton lub" shows "cont the_Sup (=) lub ord f"
rep_eq: fixes x :: "'b regular"   and xa :: "'b regular" shows "Rep_regular (x \<squnion> xa) = Rep_regular x \<squnion> Rep_regular xa"
mult_1s_ring_1_1: fixes b :: "'a" shows "- Numeral1 * b = - b"
some_trg_def: fixes \<mu> :: "'a" shows "some_trg \<mu> \<equiv> if arr \<mu> then rep (SOME b. b \<in> targets \<mu>) else null"
pred_cof_def: fixes L :: "char list set"   and t :: "sterm" shows "pred_cof L t = (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t'. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t' \<and> t = \<sigma>[s,p] t'))"