hcju/isabelleps huggingface dataset:
valid_program_history_axioms: shows "valid_program_history ts\<^sub>s\<^sub>b"
intros_8: fixes \<C> :: "'a"   and \<D> :: "'a" assumes "\<C> < \<D>" shows "\<langle>\<C>\<rangle>\<^sub>\<T> \<sqsubset>\<^sub>B \<langle>\<D>\<rangle>\<^sub>\<T>"
not_in_set_insert: fixes x :: "'g"   and xs :: "'g list" assumes "x \<notin> set xs" shows "List.insert x xs = x # xs"
split_C: fixes \<omega> :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat"   and l :: "nat" assumes "\<omega> \<in> space (Pi\<^sub>M {0..<n} (\<lambda>_. M))" shows "C n m \<omega> \<bind> C (n + m) l = C n (m + l) \<omega>"
bounded_linear_left: fixes b :: "'b" shows "bounded_linear (\<lambda>a. h a b)"
assms: shows "\<forall>Y. upclosed_set Y \<longrightarrow> upclosed_set (f -` Y)"
set_of_remove: fixes t :: "tree"   and x :: "nat" assumes "BinaryTree_TacticStyle.sorted t" shows "set_of (remove x t) = set_of t - {x}"
rel_G_witness2: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and tytok :: "('o \<times> ('o \<times> 'p) \<times> 'p \<times> 'q \<times> ('q \<times> 'r) \<times> 'r \<times> 's) itself"   and L1 :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and L2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and x :: "('o, 'q, 'b, 'f, 'i, 'l, 's) G"   and y :: "('p, 'r, 'e, 'h, 'k, 'n, 's) G" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok"   and "rel_G L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_G (\<lambda>(x, y') y. y' = y \<and> L1 x y) (\<lambda>(x, y') y. y' = y \<and> L2 x y) Co1' Co2' Contra1' Contra2' (rel_G_witness L1 L2 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y)) y"
refl: fixes a :: "'x \<Rightarrow>\<^sub>0 nat" shows "a \<preceq> a"
get_scdom_component_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_scdom_component ptr"
rot_x_ivl: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> 1"   and "0 \<le> y"   and "y \<le> 1" shows "0 \<le> x_coord_inv ((2 * y - 1) * circle_y (x_coord x)) \<and> x_coord_inv ((2 * y - 1) * circle_y (x_coord x)) \<le> 1"
fib_asymptotics: shows "(\<lambda>n. real (fib n) / (((1 + sqrt 5) / 2) ^ n / sqrt 5)) \<longlonglongrightarrow> 1"
minus_Mp_2: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.Mp m (f - poly_mod.Mp m g) = poly_mod.Mp m (f - g)"
trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_subst: fixes a :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) list"   and \<theta> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (a \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) = trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
expectation_return_pmf: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "measure_pmf.expectation (return_pmf x) f = f x"
nR2: shows "\<forall>D\<in>MajoritySet. \<forall>qa. \<exists>d\<in>D. bal (dblock s' q) < mbal (disk s' d qa) \<longrightarrow> hasRead s' q d qa"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data"   and T :: "'b span_data" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "span_in_category C S"   and "span_in_category C T"   and "composite_span_axioms C S T" shows "composite_span C prj0 prj1 S T"
bij_betw_the_inv_into: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw (the_inv_into A f) B A"
dagger_plus_one: fixes x :: "'a ppath set" shows "pp_one \<union> pp_star x = pp_star x"
take_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list" shows "take n (drop m xs) = drop m (take (n + m) xs)"
\<Delta>'''_def: shows "\<Delta>'''.\<Delta>''' = \<lparr>edge = \<lambda>x y. edge \<Delta>''' x y \<and> 0 < capacity \<Delta>''' (x, y) \<and> y \<noteq> source \<Delta>''' \<and> x \<noteq> sink \<Delta>''', capacity = \<lambda>(x, y). if x = sink \<Delta>''' \<or> y = source \<Delta>''' then 0 else capacity \<Delta>''' (x, y), source = source \<Delta>''', sink = sink \<Delta>'''\<rparr>"
fmfilter_comp: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and m :: "('a, 'b) fmap" shows "fmfilter P (fmfilter Q m) = fmfilter (\<lambda>x. P x \<and> Q x) m"
fst_sym_preproc_addnew_less: fixes vs :: "'t list"   and v :: "'t"   and u :: "'t"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "\<And>u. u \<in> set vs \<Longrightarrow> u \<prec>\<^sub>t v"   and "u \<in> set (fst (sym_preproc_addnew gs vs fs v))" shows "u \<prec>\<^sub>t v"
order_iff: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (a = ord_class.min a b)"
execute_serial_plan_sas_plus_is_execute_parallel_plan_sas_plus: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi>" shows "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> (List_Supplement.embed \<psi>)"
integrable_const_bound: fixes f :: "'a \<Rightarrow> 'c"   and B :: "real" assumes "AE x in M. norm (f x) \<le> B"   and "f \<in> borel_measurable M" shows "integrable M f"
subgame_without_deadends: fixes V' :: "'a set"   and p :: "Player"   and v :: "'a" assumes "V' = V - attractor p (deadends p**) - attractor p** (deadends p****)"   and "v \<in> V\<^bsub>subgame V'\<^esub>" shows "\<not> \<not> (\<exists>w\<in>V\<^bsub>subgame V'\<^esub>. v \<rightarrow>\<^bsub>subgame V'\<^esub> w)"
assms_1: fixes n :: "nat" shows "coeff p n \<in> \<int>"
real_distribution_interval_measure: fixes F :: "real \<Rightarrow> real" assumes "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"   and "\<And>a. continuous (at_right a) F"   and "(F \<longlongrightarrow> 0) at_bot"   and "(F \<longlongrightarrow> 1) at_top" shows "real_distribution (interval_measure F)"
subsumption1: fixes P :: "'a upred"   and Q :: "'a upred" assumes "`P \<Rightarrow> Q`" shows "(P \<or> Q) = Q"
sat_push': fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "Height_Bounded_Labeling c s t f l"   and "sat_push_precond f l e" shows "((f, l), SAT_PUSH' e, push_effect f e, l) \<in> pr_algo_lts'"
17: shows "(w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f \<le> top * selected_edge h j g\<^sup>T * (forest_components f \<sqinter> (w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f))\<^sup>T\<^sup>\<star>"
card_of_Un_singl_ordLess_infinite: fixes B :: "'a set"   and A :: "'b set"   and a :: "'b" assumes "infinite B" shows "(|A| <o |B|) = (|{a} \<union> A| <o |B|)"
mat_times_vec_leqI: fixes A :: "'a mat"   and b :: "'a vec"   and x :: "'a vec" assumes "dim_row A = dim_vec b"   and "dim_col A = dim_vec x"   and "\<And>i. i < dim_vec b \<Longrightarrow> (A *\<^sub>v x) $ i \<le> b $ i" shows "[A*\<^sub>vx]\<le>b"
authKeysI: fixes A :: "agent"   and K :: "nat"   and Ta :: "nat"   and evs :: "event list" assumes "Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key K, Agent Tgs, Number Ta\<rbrace>, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key K, Number Ta\<rbrace>\<rbrace> \<in> set evs" shows "K \<in> authKeys evs"
comp_right_dist_sum: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'a" shows "comm_monoid_add.sum (\<squnion>) bot (\<lambda>k. f k \<sqinter> x) {k. True} = comm_monoid_add.sum (\<squnion>) bot f {k. True} \<sqinter> x"
BuildSeq_succ_E: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and y :: "hf" assumes "BuildSeq B C s k y" obtains "B y"   | m :: "hf" and n :: "hf" where "m \<^bold>\<in> k" and "n \<^bold>\<in> k" and "C y (app s m) (app s n)"
abc_bcd_abd: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c"   and "betw b c d" shows "betw a b d"
abs_eq: fixes xa :: "nat \<Rightarrow> rat"   and x :: "nat \<Rightarrow> rat" assumes "realrel xa xa"   and "realrel x x" shows "Real xa * Real x = Real (\<lambda>n. xa n * x n)"
ce_set_lm_7: fixes A :: "nat set"   and p :: "nat \<Rightarrow> nat" assumes "A \<in> ce_sets"   and "p \<in> PrimRec1" shows "{p x |x. x \<in> A} \<in> ce_sets"
is_run_alt: fixes s :: "nat \<Rightarrow> 'a" shows "is_run s = (s 0 = s\<^sub>0 \<and> (\<forall>i. step (s i) (s (Suc i))))"
Berlekamp_subspace_eq_dim_vec: shows "Berlekamp_subspace.dim = RV.dim"
impl_l_d1: fixes a :: "'a"   and b :: "'a" shows "a l\<rightarrow> b = d1 a (a \<sqinter> b)"
h3: shows "\<turnstile> \<box>P \<and> \<box>[N]_v \<and> \<box>F \<longrightarrow> \<diamond>Enabled \<langle>A\<rangle>_v"
SDG_call_edge_CFG_node: fixes n :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "n -p\<rightarrow>\<^bsub>call\<^esub> n'" shows "\<exists>m. n = CFG_node m"
add_left_imp_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a + b = a + c" shows "b = c"
t_mult_closed: fixes x :: "'a"   and y :: "'a" shows "t(tx \<cdot> ty) = tx \<cdot> ty"
induct: fixes x1 :: "'a list"   and x2 :: "'b list"   and R :: "('a \<times> 'b) set"   and P :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "(x1, x2) \<in> seq_lift R"   and "P [] []"   and "\<And>xs ys x y. \<lbrakk>(xs, ys) \<in> seq_lift R; P xs ys; (x, y) \<in> R\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
PE_base_lf_rep_def: shows "PE_base_lf_rep \<equiv> \<lambda>(mR, pR). {\<bottom>} \<union> {\<lambda>i. ValTT} \<union> {\<lambda>i. ValFF} \<union> (\<Union>n. {\<lambda>i. ValN\<cdot>n}) \<union> {f. f 1 = \<bottom> \<or> f 2 = \<bottom>}"
matrix_inv_def: fixes A :: "(('b, 'd) vec, 'c) vec" shows "matrix_inv A = (SOME A'. A ** A' = mat (1::'b) \<and> A' ** A = mat (1::'b))"
red_rtrancl_subset: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "(punit.red G)\<^sup>*\<^sup>* p q"   and "G \<subseteq> F" shows "(punit.red F)\<^sup>*\<^sup>* p q"
slice_take': fixes n :: "nat"   and w :: "'c word" shows "slice n w = of_bl (take (size w - n) (to_bl w))"
intro: fixes charts :: "('c, 'd) chart set"   and k :: "enat" assumes "c_manifold charts k" shows "c_manifold' charts k"
case_llist_Lazy_llist: fixes n :: "'a"   and c :: "'b \<Rightarrow> 'b llist \<Rightarrow> 'a"   and xs :: "unit \<Rightarrow> ('b \<times> 'b llist) option" shows "(case Lazy_llist xs of LNil \<Rightarrow> n | LCons x xa \<Rightarrow> c x xa) = (case xs () of None \<Rightarrow> n | Some (x, ys) \<Rightarrow> c x ys)"
p: shows "p \<in> (\<lambda>p a. if a \<in> A then b_to_a (p (a_to_b a)) else a) ` {p. p permutes B}"
cong: fixes a :: "nat"   and b :: "nat" assumes "[a = b] (mod n)" shows "principal_dchar n a = principal_dchar n b"
singlPart_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set" shows "inj_on ((`) f) (singlPart I) = inj_on f I"
is_glb_bin: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.po below"   and "below x y" shows "po.is_glb below {x, y} x"
bij_swap_iff: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "bij (Fun.swap a b f) = bij f"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list"   and a2 :: "char list list" assumes "Subobjs P a1 a2" obtains C :: "char list" and Cs :: "char list list" where "a1 = C__" and "a2 = Cs__" and "Subobjs\<^sub>R P C__ Cs__"   | C :: "char list" and C' :: "char list" and D :: "char list" and Cs :: "char list list" where "a1 = C" and "a2 = Cs" and "P \<turnstile> C \<preceq>\<^sup>* C'" and "P \<turnstile> C' \<prec>\<^sub>S D" and "Subobjs\<^sub>R P D Cs"
run_asset_ii_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "case run_asset_ii n s of (s', S, A, B, x) \<Rightarrow> insert (Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) ({Asset n} \<times> {Crypt (Auth_ShaKey n) (PriKey S), \<lbrace>Num 2, PubKey B\<rbrace>}) \<subseteq> s' \<and> (Asset n, PubKey B) \<notin> s'"
conj_assoc: fixes P :: "'a upred"   and Q :: "'a upred"   and S :: "'a upred" shows "((P \<and> Q) \<and> S) = (P \<and> Q \<and> S)"
b: shows "limsup (v \<circ> r \<circ> s) < \<infinity>"
is_prefix_unsplit: fixes u :: "'b list"   and a :: "'b list" assumes "is_prefix u a" shows "u @ drop (length u) a = a"
last_message_hold_last: fixes xs :: "'b message_af list" assumes "xs \<noteq> []" shows "last (last_message_hold xs) = last_message xs"
REF_R_DONE: fixes \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a" assumes "\<Phi> {} \<sigma>"   and "\<Phi>' {} \<sigma>'"   and "(({}, \<sigma>), {}, \<sigma>') \<in> R" shows "(\<sigma>, \<sigma>') \<in> R'"
simps_3: fixes b :: "bool"   and t :: "ty"   and e :: "expr" shows "assigns_if b (Cast t e) = assigns_if b e"
rec_newleft0_def: shows "rec_newleft0 = recf.id 2 0"
rbt_greater_nit_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and t :: "('b, 'c) RBT_Impl.rbt" shows "ord.rbt_greater less k t = (\<forall>x\<in>set (RBT_Impl.keys t). less k x)"
inject_2: fixes x2 :: "'f \<Rightarrow> bool"   and y2 :: "'f \<Rightarrow> bool" shows "((x2)\<^sub>\<surd> = (y2)\<^sub>\<surd>) = (x2 = y2)"
axioms: fixes A :: "V"   and f :: "V" assumes "unop_onto A f" shows "nop_onto A (1\<^sub>\<nat>) f"
rcond_rassume_expand: fixes b :: "'a upred"   and P :: "('a, 'b) urel"   and Q :: "('a, 'b) urel" shows "P \<triangleleft> b \<triangleright>\<^sub>r Q = (?[b] ;; P) \<sqinter> (?[\<not> b] ;; Q)"
span_insert_idI: fixes r :: "'a"   and B :: "'a set" assumes "r \<in> ideal B" shows "ideal (insert r B) = ideal B"
IH_1: assumes "wbt l"   and "l \<noteq> \<langle>\<rangle>" shows "2 \<le> (1 - \<alpha>) ^ (height l - 1) * real (size1 l)"
assms_1: shows "sub_set_system V B \<V> \<B>"
j: shows "prime (9 * 2 ^ (2 * 7 - 1) - 1)"
span_eq_dim: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "vec.span S = vec.span T" shows "vec.dim S = vec.dim T"
f_join_length: fixes xs :: "'b list"   and I :: "nat set" shows "length (xs \<Join>\<^sub> I) = card (I \<down>< length xs)"
elims_3: fixes x :: "'a atom set"   and xa :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)" assumes "\<not> x \<doteq> xa" obtains as :: "'a atom set" and lb :: "nat \<Rightarrow> 'a option" and ub :: "nat \<Rightarrow> 'a option" where "x = as" and "xa = (lb, ub)" and "\<not> (\<forall>v. v \<Turnstile>\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>b (lb, ub))"
sign_changes_sturm_triple: fixes p :: "real poly"   and x :: "real"   and r :: "real poly"   and q :: "real poly" assumes "poly p x \<noteq> 0"   and "sgn (poly r x) = - sgn (poly p x)" shows "sign_changes [p, q, r] x = 1"
discI_1: fixes resumption :: "('d, 'e, 'f) resumption"   and x1 :: "'f" assumes "resumption = Done x1" shows "is_Done resumption"
takeWhile_tail: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and l :: "'b list" assumes "\<not> P x" shows "takeWhile P (xs @ x # l) = takeWhile P xs"
scalar_product_def: fixes v :: "('d, 'e) vec"   and w :: "('d, 'e) vec" shows "scalar_product v w = (\<Sum>i\<in>UNIV. v $ i * w $ i)"
cases: fixes x :: "('b, 'c) Combinators list" obtains xs :: "('b, 'c) Combinators list" where "x = DenyAll # xs"   | "x = []"   | vb :: "'b" and vc :: "'b" and va :: "('b, 'c) Combinators list" where "x = DenyAllFromTo vb__ vc__ # va__"   | vb :: "'b" and vc :: "'b" and vd :: "'c" and va :: "('b, 'c) Combinators list" where "x = AllowPortFromTo vba__ vca__ vd # vaa__"   | vb :: "('b, 'c) Combinators" and vc :: "('b, 'c) Combinators" and va :: "('b, 'c) Combinators list" where "x = vb \<oplus> vc # va"
le_measureD1: fixes A :: "'b measure"   and B :: "'b measure" assumes "A \<le> B" shows "space A \<subseteq> space B"
self_codomain_iff_ide: fixes a :: "'a1" shows "(a \<in> C1.codomains a) = C1.ide a"
real_root_pos_pos: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 < x" shows "0 \<le> root n x"
A_def: shows "A = {x \<in> space M. e \<le> f x}"
insort_wrt_def: fixes R :: "('c \<times> 'c) set"   and xs :: "'c list" shows "insort_wrt R xs = foldr (insert_wrt R) xs []"
le_zero_to_equal_zero: fixes n :: "nat" assumes "n \<le> 0" shows "n = 0"
Least1_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and z :: "'a" assumes "class.order less_eq less"   and "\<exists>!x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y)"   and "P z" shows "less_eq (ord.Least less_eq P) z"
assert_all_state_def: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" shows "assert_all_state t ats \<equiv> chk (assert_bound_loop ats (init t))"
eq_onp_eqD: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "eq_onp P = Q" shows "P x = Q x x"
comp_inv_arr: fixes f :: "'a"   and g :: "'a" assumes "inverse_arrows f g" shows "g \<cdot> f = local.dom f"
iterate_add_diff_add_eq2: fixes i :: "nat"   and j :: "nat"   and u :: "'k"   and m :: "'k"   and n :: "'k" assumes "i \<le> j" shows "iterate_add i u + m - (iterate_add j u + n) = m - (iterate_add (j - i) u + n)"
lborel_integrable_real_affine: fixes f :: "real \<Rightarrow> 'a"   and c :: "real"   and t :: "real" assumes "integrable lborel f"   and "c \<noteq> 0" shows "integrable lborel (\<lambda>x. f (t + c * x))"
po_on_strict_suffix: fixes A :: "'b list set" shows "po_on strict_suffix A"
natural_transformation_axioms: shows "natural_transformation J (\<cdot>) \<chi>.A.map D \<chi>"
IH_5: assumes "delete\<langle>t,s\<rangle> \<in> set A" shows "delete\<langle>t \<cdot> \<theta>,s \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
\<theta>\<^sub>h: shows "\<guillemotleft>\<theta>\<^sub>h : tab\<^sub>0 h \<star> w\<^sub>h \<rightarrow> u\<^sub>h\<guillemotright>"
reachable_rename_eq: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program" assumes "bij h" shows "reachable (rename h F) = h ` reachable F"
ltrm_closed: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "monom (UP R) (f (deg R f)) (deg R f) \<in> carrier (UP R)"
strict_trans: fixes b :: "'t"   and a :: "'t"   and c :: "'t" assumes "b \<prec>\<^sub>t a"   and "c \<prec>\<^sub>t b" shows "c \<prec>\<^sub>t a"
subclseq_D_C: shows "G\<turnstile>D\<preceq>\<^sub>C C"
value_Pair_inv: fixes e\<^sub>1 :: "Syntax.term"   and e\<^sub>2 :: "Syntax.term" assumes "value (Syntax.Pair e\<^sub>1 e\<^sub>2)" obtains "value e\<^sub>1" and "value e\<^sub>2"
assms_2: shows "(ref, n) \<in> succ_rel (ref # list)"
addition_set_mono: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "addition_set f A \<subseteq> addition_set f B"
power_mult: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m * n) = power.power one times (power.power one times a m) n"
not_list_all_elim: fixes P :: "'c \<Rightarrow> bool"   and A :: "'c list" assumes "\<not> list_all P A" shows "\<exists>B x C. A = B @ x # C \<and> \<not> P x \<and> list_all P B"
mbuf2_take_add'_2: fixes f :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> 'a"   and xs :: "event_data option list set list"   and ys :: "event_data option list set list"   and buf :: "event_data option list set list \<times> event_data option list set list"   and zs :: "'a list"   and buf' :: "event_data option list set list \<times> event_data option list set list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and j :: "nat"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and P' :: "String.literal \<Rightarrow> nat option"   and j' :: "nat" assumes "mbuf2_take f (mbuf2_add xs ys buf) = (zs, buf')"   and "wf_mbuf2' \<sigma> P V j n R \<phi> \<psi> buf"   and "rel_mapping (\<le>) P P'"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) [Monitor.progress \<sigma> P \<phi> j..<Monitor.progress \<sigma> P' \<phi> j'] xs"   and "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>)) [Monitor.progress \<sigma> P \<psi> j..<Monitor.progress \<sigma> P' \<psi> j'] ys"   and "j \<le> j'" shows "list_all2 (\<lambda>i Z. \<exists>X Y. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>) X \<and> qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>) Y \<and> Z = f X Y) [min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)..<min (Monitor.progress \<sigma> P' \<phi> j') (Monitor.progress \<sigma> P' \<psi> j')] zs"
le_ccINF_iff: fixes A :: "'c set"   and u :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "countable A" shows "(u \<le> Inf (f ` A)) = (\<forall>i\<in>A. u \<le> f i)"
vector_differentiable_bound_linearization: fixes S :: "real set"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and x0 :: "real"   and B :: "real" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_vector_derivative f' x) (at x within S)"   and "{a--b} \<subseteq> S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f' x - f' x0) \<le> B"   and "x0 \<in> S" shows "norm (f b - f a - (b - a) *\<^sub>R f' x0) \<le> norm (b - a) * B"
show_law_real_alg: fixes r :: "real_alg" shows "show_law showsp_real_alg r"
ennreal_divide_self: fixes a :: "ennreal" assumes "a \<noteq> 0"   and "a < \<top>" shows "a / a = 1"
BAD_iff: fixes f :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(f \<in> BAD P) = ((\<forall>i. f i \<in> trees) \<and> bad P f)"
index_smult_mat_3: fixes a :: "'a"   and A :: "'a mat" shows "dim_col (a \<cdot>\<^sub>m A) = dim_col A"
HFinite_hypreal_sqrt_imp_HFinite: fixes x :: "real star" assumes "0 \<le> x"   and "(*f* sqrt) x \<in> HFinite" shows "x \<in> HFinite"
that: fixes m :: "nat" assumes "\<sigma> m = \<tau> n"   and "\<forall>j<m. \<exists>k<n. \<sigma> j = \<tau> k" shows "thesis"
MTF_Off_def: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list" shows "MTF_Off as rs s0 \<equiv> distinct s0 \<and> length as = length rs"