hcju/isabelleps huggingface dataset:
eWlsInp_simp2: fixes einp :: "'index \<Rightarrow> 'c withERR option"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym" assumes "\<not> liftAll (\<lambda>eX. eX \<noteq> ERR) einp" shows "\<not> eWlsInp MOD delta einp"
wellformed'_replace_bound_eq: fixes n :: "nat"   and t :: "term"   and k :: "nat"   and u :: "term" assumes "wellformed' n t"   and "n \<le> k" shows "replace_bound k t u = t"
collapse_flipped_signed: fixes s :: "'a"   and b :: "bool" shows "prappend_signed_list [(s, b)] [(s, \<not> b)] = []"
shows_prec_nat_def: fixes d :: "nat"   and n :: "nat" shows "shows_prec d n = shows_nat n"
pdevs_domain_aform_err_to_aform: fixes X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "degree_aform_err X \<le> n" shows "pdevs_domain (snd (aform_err_to_aform X n)) = pdevs_domain (snd (fst X)) \<union> (if snd X = 0 then {} else {n})"
index_mat_swaprows_2: fixes k :: "nat"   and l :: "nat"   and A :: "'c mat" shows "dim_row (swaprows k l A) = dim_row A"
intro: fixes A :: "V"   and f :: "V" assumes "nop_v11 A (2\<^sub>\<nat>) f" shows "binop_v11 A f"
closure_unique: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "closed T"   and "\<And>T'. \<lbrakk>S \<subseteq> T'; closed T'\<rbrakk> \<Longrightarrow> T \<subseteq> T'" shows "closure S = T"
sep_substate_def: fixes x :: "'c"   and y :: "'c" shows "x \<preceq> y \<equiv> \<exists>z. x ## z \<and> x + z = y"
ir_While_backwards_frontier: fixes P :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<And>n. ir_hoare (\<lambda>s s'. P n s s' \<and> bval b s) c SKIP (P (Suc n))"   and "ir_hoare (\<lambda>s s'. \<exists>n. P n s s') (WHILE b DO c) c' Q" shows "ir_hoare (P 0) (WHILE b DO c) c' Q"
sum_list_ge_mono: fixes as :: "'b list"   and bs :: "'b list" assumes "length as = length bs"   and "\<And>i. i < length bs \<Longrightarrow> bs ! i \<le> as ! i" shows "sum_list bs \<le> sum_list as"
map_concat: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list list" shows "map f (concat xs) = concat (map (map f) xs)"
lu_1: shows "convergent (\<lambda>n. u n / real n)"
ide_is_iso: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C"   and "partial_magma.ide C a" shows "category.iso C a"
gL_gH_rel: fixes x :: "'a"   and y :: "'a" shows "(x \<preceq>\<^sub>L y) = (x \<preceq>\<^sub>H y)"
\<phi>_def: shows "\<phi> \<equiv> \<lambda>a. if a \<in> HRepHomSet (\<star>) W then a \<circ> induced_vector else 0"
eq2: shows "fpxs_compose_power (fpxs_of_fls (fls_of_fpxs f)) (1 / rat_of_nat q) = f"
isnpolyh_unique: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "(\<forall>bs. \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>) = (p = q)"
finite_in: fixes y :: "'a" shows "f (x, y) \<noteq> \<top>"
jw_Let_inv: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e\<^sub>1 :: "Syntax.term"   and x :: "Syntax.var"   and e\<^sub>2 :: "Syntax.term"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile>\<^sub>W Syntax.Let e\<^sub>1 x e\<^sub>2 : \<tau>\<^sub>2"   and "atom x \<sharp> (e\<^sub>1, \<Gamma>)" obtains \<tau>\<^sub>1 :: "ty" where "\<Gamma> \<turnstile>\<^sub>W e\<^sub>1 : \<tau>\<^sub>1" and "\<Gamma>(x $$:= \<tau>\<^sub>1) \<turnstile>\<^sub>W e\<^sub>2 : \<tau>\<^sub>2"
simps_2: fixes u :: "nat"   and e :: "nat \<times> nat"   and es :: "(nat \<times> nat) list" shows "pathVertices u (e # es) = fst e # pathVertices (snd e) es"
prop_in_f_3_b: fixes v :: "i" shows "[\<^bold>\<not>Indiscriminate (E!\<^sup>-) in v]"
QuoteP_Mem_imp_QMem: fixes i :: "name"   and j :: "name"   and j' :: "name"   and i' :: "name" assumes "atom i \<sharp> (j, j', i')"   and "atom i' \<sharp> (j, j')"   and "atom j \<sharp> j'" shows "{QuoteP (Var i) (Var i'), QuoteP (Var j) (Var j'), Var i IN Var j} \<turnstile> PfP (Q_Mem (Var i') (Var j'))"
fresh_fact: fixes a :: "name"   and t1 :: "lam"   and t2 :: "lam"   and b :: "name" assumes "a \<sharp> t1"   and "a \<sharp> t2" shows "a \<sharp> t1[b::=t2]"
inext_nth_card_append_eq1: fixes A :: "nat set"   and B :: "nat set"   and n :: "nat" assumes "Max A < iMin B"   and "n < card A" shows "(A \<union> B) \<rightarrow> n = A \<rightarrow> n"
simps_2: fixes a :: "'b"   and t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.remove1Prio a (t # bq) = (if BinomialHeapStruc.prio t = a then bq else t # BinomialHeapStruc.remove1Prio a bq)"
is_subfield: shows "subfield (indexed_const ` carrier R) (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>)"
unique_halloc: fixes G :: "prog"   and s :: "state"   and oi :: "obj_tag"   and a :: "loc"   and s' :: "state"   and a' :: "loc"   and s'' :: "state" assumes "G\<turnstile>s \<midarrow>halloc oi\<succ>a\<rightarrow> s'"   and "G\<turnstile>s \<midarrow>halloc oi\<succ>a'\<rightarrow> s''" shows "a' = a \<and> s'' = s'"
Cons_less_Cons: fixes a :: "'b"   and x :: "'b list"   and b :: "'b"   and y :: "'b list" shows "list_less (a # x) (b # y) = (a < b \<or> a = b \<and> list_less x y)"
columns_Gram_Schmidt_upt_k_rw: fixes k :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "Suc k < ncols A" shows "{column i (Gram_Schmidt_upt_k A (Suc k)) |i. i < mod_type_class.from_nat (Suc k)} = {column i (Gram_Schmidt_upt_k A k) |i. i < mod_type_class.from_nat (Suc k)}"
safe_regex_induct: fixes P :: "modality \<Rightarrow> safety \<Rightarrow> 'a Regex.regex \<Rightarrow> bool"   and a0 :: "modality"   and a1 :: "safety"   and a2 :: "'a Regex.regex" assumes "\<And>m uu_ n. P m uu_ (regex.Skip n)"   and "\<And>m g \<phi>. P m g (regex.Test \<phi>)"   and "\<And>m g r s. \<lbrakk>P m g r; P m g s\<rbrakk> \<Longrightarrow> P m g (regex.Plus r s)"   and "\<And>g r s. \<lbrakk>P Futu g s; P Futu Lax r\<rbrakk> \<Longrightarrow> P Futu g (regex.Times r s)"   and "\<And>g r s. \<lbrakk>P Past g r; P Past Lax s\<rbrakk> \<Longrightarrow> P Past g (regex.Times r s)"   and "\<And>m g r. P m g r \<Longrightarrow> P m g (regex.Star r)" shows "P a0 a1 a2"
ivl_disj_un_singleton_3: fixes l :: "'a"   and u :: "'a" assumes "l < u" shows "{l} \<union> {l<..<u} = {l..<u}"
his_determ_ext: fixes n' :: "nat"   and n :: "nat"   and t :: "nat \<Rightarrow> cnf"   and nid :: "'nid" assumes "n' \<le> n"   and "\<exists>nid'. (n', nid') \<in> his t n nid" shows "(\<exists>!nid'. (n', nid') \<in> his t n nid) \<and> ((\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> his t n nid) \<longrightarrow> (\<exists>x. (\<exists>n. latestAct_cond (THE nid'. (n', nid') \<in> his t n nid) t n' n) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>THE nid'. (n', nid') \<in> his t n nid \<leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')))) \<and> (hisPred t n nid n', SOME nid'. (hisPred t n nid n', nid') \<in> his t n nid) = (SOME x. (\<exists>n. latestAct_cond (THE nid'. (n', nid') \<in> his t n nid) t n' n) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>THE nid'. (n', nid') \<in> his t n nid \<leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>THE nid'. (n', nid') \<in> his t n nid\<^esub>t n')))))"
lm030: fixes P :: "'c set set"   and X :: "'c set" assumes "\<Union> P = X" shows "P \<subseteq> Pow X"
finite_dimensional_vector_space_axioms: shows "finite_dimensional_vector_space (*a) B1"
order_le_less_subst1: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and b :: "'c"   and c :: "'c" assumes "a \<le> f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
zeta_real: fixes x :: "real" shows "zeta (complex_of_real x) \<in> \<real>"
assms: fixes x :: "'a" shows "mono (\<lambda>s. f s x)"
typed: shows "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"
prems: shows "invar \<langle>l, (xy, n), r\<rangle>"
ProblemIkTpart_axioms: shows "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw"
contains_def: fixes A :: "'a set"   and x :: "'a" shows "Predicate_Compile.contains A x = (x \<in> A)"
remdups_gen_subset: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "set (remdups_gen f xs) \<subseteq> set xs"
k_def: shows "k = nat \<lfloor>(D - C / 2 - 15 / 2 * delta) / (5 * delta)\<rfloor>"
im_jointfun: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and A :: "'e set"   and g :: "nat \<Rightarrow> 'e"   and m :: "nat"   and B :: "'e set" assumes "f \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> B" shows "jointfun n f m g ` {j. j \<le> Suc (n + m)} = f ` {j. j \<le> n} \<union> g ` {j. j \<le> m}"
Cons_5: shows "lookup_pair (map f xs) (fst (f (k, v))) = snd (f (k, lookup_pair xs k))"
map_transform_append: fixes t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "map_transform t ae (\<Delta> @ \<Gamma>) = map_transform t ae \<Delta> @ map_transform t ae \<Gamma>"
load_tmps_write_tmps_distinct: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "load_tmps is \<inter> \<Union> (fst ` write_sops sb) = {}"
overlaps_universe: fixes x :: "i" assumes "CEMU" shows "O x u"
up3_tm_def: fixes x :: "'a"   and sib :: "'a tree"   and twist :: "bool"   and u :: "'a up2" shows "up3_tm x sib twist u =1 (case u of Same2 \<Rightarrow> return Same2 | Bal2 t \<Rightarrow> return (Bal2 (node twist t x sib)) | Unbal2 t n1 h1 \<Rightarrow> size_tree_tm sib \<bind> (\<lambda>n2. let t' = node twist t x sib; n' = n1 + n2 + 1; h' = h1 + 1 in if bal_i n' h' then return (Unbal2 t' n' h') else bal_tree_tm n' t' \<bind> (\<lambda>t''. return (Bal2 t''))))"
mult_oner: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b" assumes "class.ab_near_semiring_one plus times one" shows "times x one = x"
fin_steps_plusI: fixes A :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and q :: "bool list"   and u :: "'b list"   and p :: "bool list" assumes "(start A, q) \<in> steps A u"   and "u \<noteq> []"   and "fin A p" shows "(p, q) \<in> steps (RegExp2NA.plus A) u"
xy_relativorder_mtf2: fixes q :: "'c"   and x :: "'c"   and y :: "'c"   and xs :: "'c list"   and n :: "nat" assumes "q \<noteq> x"   and "q \<noteq> y"   and "distinct xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "q \<in> set xs" shows "x < y in mtf2 n q xs = x < y in xs"
Cons_1: fixes p :: "real poly"   and q :: "real poly"   and a :: "real"   and b :: "real" assumes "xs = smods p q"   and "a < b"   and "poly p a \<noteq> 0"   and "poly p b \<noteq> 0" shows "cindex_poly a b q p = changes_itv_smods a b p q"
uminus_eq_vec: fixes v :: "'a vec"   and w :: "'a vec" shows "(- v = - w) = (v = w)"
hyps: assumes "xs \<noteq> []" shows "max_list xs \<in> set xs"
xless: fixes x :: "real" assumes "0 \<le> x"   and "x < r" shows "x < r * cmod c"
comp_char: fixes g :: "'a"   and f :: "'a" shows "Left_a.comp g f = (if Left_a.arr f \<and> Left_a.arr g \<and> seq g f then g \<cdot> f else null)"
inner_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
to_Gromov_completion_continuous: shows "continuous_on UNIV to_Gromov_completion"
lb_ln2: fixes prec :: "nat" shows "real_of_float (lb_ln2 prec) \<le> ln 2"
Shleg_def: fixes n :: "nat" shows "Shleg n = poly (of_int_poly (shleg_poly n))"
strip_eq_While: fixes c :: "'a acom"   and b :: "bexp"   and c1 :: "com" shows "(strip c = WHILE b DO c1) = (\<exists>I d1 P. c = {I} WHILE b DO d1 {P} \<and> strip d1 = c1)"
img: fixes i :: "nat"   and x :: "real" assumes "g (l i) \<le> x"   and "x \<le> g (u i)" shows "\<exists>c\<ge>l i. c \<le> u i \<and> x = g c"
H_def: shows "H \<equiv> homology_group (int (p - Suc 0)) (subtopology X S)"
distrib_right: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a \<squnion> b) * c = a * c \<squnion> b * c"
nn_integral_mono_AE: fixes u :: "'d \<Rightarrow> ennreal"   and v :: "'d \<Rightarrow> ennreal"   and M :: "'d measure" assumes "AE x in M. u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
assms_1: shows "factorize_rat_poly_monic p = (c, fs)"
negatex_def: fixes x :: "(real, 2) vec" shows "negatex x = vector [- x $ 1, x $ 2]"
image_mset_eq_replicate_msetD: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset"   and n :: "nat"   and y :: "'a" assumes "image_mset f A = replicate_mset n y" shows "\<forall>x\<in>#A. f x = y"
pdg_dg_prod_vunion_is_arr: fixes J :: "V"   and K :: "V"   and a :: "V"   and b :: "V"   and g :: "V"   and c :: "V"   and d :: "V"   and f :: "V" assumes "J \<inter>\<^sub>\<circ> K = []\<^sub>\<circ>"   and "J \<subseteq>\<^sub>\<circ> I"   and "K \<subseteq>\<^sub>\<circ> I"   and "g : a \<mapsto>\<^bsub>dg_prod J \<AA>\<^esub> b"   and "f : c \<mapsto>\<^bsub>dg_prod K \<AA>\<^esub> d" shows "g \<union>\<^sub>\<circ> f : a \<union>\<^sub>\<circ> c \<mapsto>\<^bsub>dg_prod (J \<union>\<^sub>\<circ> K) \<AA>\<^esub> b \<union>\<^sub>\<circ> d"
1: shows "eq_onp ord_class.filter (Filters.filter.Rep_filter (phi x)) (Filters.filter.Rep_filter (phi x)) \<and> eq_onp ord_class.filter (\<up>y) (\<up>y)"
perp_foot_incident_1: fixes p :: "proj2"   and l :: "proj2_line" shows "proj2_incident (perp_foot p l) l"
1: shows "x \<sqinter> (1::'a) \<le> (x \<sqinter> - (1::'a))\<^sup>\<star>"
timpls_transformable_to_if_term_variants: fixes a :: "'g set"   and b :: "'g set"   and s :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and TI :: "('g set \<times> 'g set) list" assumes "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) s t"   and "(a, b) \<in> set TI" shows "timpls_transformable_to TI s t"
new_element_tag_name: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr" shows "h' \<turnstile> get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t new_element_ptr tag_name \<rightarrow>\<^sub>r []"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f' g ((g \<star> \<phi>) \<cdot> \<eta>) (\<epsilon> \<cdot> (local.inv \<phi> \<star> g))"
longestImpWitnessPath_def: fixes c :: "('loc, 't, 'b) configuration_scheme"   and loc1 :: "'loc"   and loc2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and t :: "'t" shows "longestImpWitnessPath c loc1 loc2 xs t = (impWitnessPath c loc1 loc2 xs t \<and> (\<forall>loc' xs'. impWitnessPath c loc' loc2 xs' t \<longrightarrow> length xs' \<le> length xs))"
upd_cs_intra_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and as :: "'a list"   and cs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "\<forall>a\<in>set as. intra_kind (kind a)" shows "CFG.upd_cs kind cs as = cs"
ex1E: fixes P :: "'g \<Rightarrow> bool" assumes "\<exists>!x. P x" obtains x :: "'g" where "P x" and "\<forall>y. P y \<longrightarrow> y = x"
h: shows "\<forall>a\<in>A. R `` {a} \<subseteq> A"
Simple_WPC_Chain: fixes t :: "'e itself"   and Rs :: "(('f, 'g) labeled_graph \<times> ('f, 'g) labeled_graph) set"   and S :: "nat \<Rightarrow> ('f, 'h) labeled_graph" assumes "Simple_WPC t Rs S" shows "chain S"
lambda: shows "size {#b \<in># \<B>. x \<in> b \<and> y \<in> b#} = nat \<Lambda>"
l2_skr_def: fixes R :: "fid_t"   and K :: "msg" shows "l2_skr R K \<equiv> {(s, s'). R \<noteq> test \<and> R \<notin> partners \<and> in_progress (progress s R) xsk \<and> guessed_frame R xsk = Some K \<and> s' = s\<lparr>ik := {K} \<union> ik s\<rparr>}"
rel_gpv_conv_rel_gpv'': fixes A :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_gpv A C = rel_gpv'' A C (=)"
map_append_inv: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and ys :: "'c list"   and zs :: "'c list" assumes "map f xs = ys @ zs" shows "\<exists>vs ws. xs = vs @ ws \<and> map f vs = ys \<and> map f ws = zs"
abs_summable_on_restrict': fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "(f abs_summable_on A) = ((\<lambda>x. if x \<in> A then f x else (0::'e)) abs_summable_on B)"
Affine_aform_of_point_eq: fixes p :: "'a" shows "Affine (aform_of_point p) = {p}"
ord_mod: fixes n :: "'a"   and k :: "'a" shows "ord n (k mod n) = ord n k"
prime_elem_power_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and p :: "'b"   and n :: "nat" assumes "class.algebraic_semidom divide plus minus zero times one" shows "comm_semiring_1.prime_elem times one zero (power.power one times p n) = (comm_semiring_1.prime_elem times one zero p \<and> n = 1)"
that: fixes i :: "nat" assumes "i < card N"   and "x = enum N i" shows "thesis"
vsv_vdomain_vrange_vsingleton: fixes a :: "V"   and b :: "V" assumes "\<D>\<^sub>\<circ> (\<BB>\<lparr>Comp\<rparr>) = ZFC_in_HOL.set {a}"   and "\<R>\<^sub>\<circ> (\<BB>\<lparr>Comp\<rparr>) = ZFC_in_HOL.set {b}" shows "\<BB>\<lparr>Comp\<rparr> = ZFC_in_HOL.set {\<langle>a, b\<rangle>}"
order2_hd_in_lconjseq_deletion: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b"   and ss :: "'b list" assumes "class.group_add minus plus zero uminus"   and "plus s s = zero"   and "s \<in> set (group_add.lconjseq minus plus ss)" shows "\<exists>as b bs. ss = as @ [b] @ bs \<and> monoid_add.sum_list plus zero (s # ss) = monoid_add.sum_list plus zero (as @ bs)"
simps_3: fixes c :: "int"   and e :: "Cooper.num" shows "\<beta> (Eq (CN 0 c e)) = [Sub (C (- 1)) e]"
"case": fixes f :: "'a" shows "(case () of () \<Rightarrow> f) = f"
derangements_inverse_in_image: fixes d :: "nat \<Rightarrow> nat"   and A :: "nat set"   and x :: "nat" assumes "d \<in> derangements A"   and "x \<in> A" shows "inv d x \<in> A"
simps_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and x :: "Preliminaries.var" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.gT (Var x) = Var x"
comm_add_var: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and x :: "'a"   and y :: "'a" assumes "test p"   and "test q"   and "test r"   and "p \<cdot> x = x \<cdot> p"   and "p \<cdot> y = y \<cdot> p" shows "p \<cdot> (q \<cdot> x + r \<cdot> y) = (q \<cdot> x + r \<cdot> y) \<cdot> p"
take_bit_minus_one_eq_mask: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "take_bit n (uminus one) = mask n"
op_vertices_def: fixes vs :: "nat list" shows "vs\<^bsup>op\<^esup> \<equiv> rev vs"
content_pos_lt_eq: fixes a :: "'b"   and b :: "'b" shows "(0 < content (cbox a b)) = (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
repr_independence2: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and U :: "'c set"   and g :: "'c" assumes "subgroup H G"   and "Group.group G"   and "U \<in> rcosets\<^bsub>G\<^esub> H"   and "g \<in> U" shows "U = H #>\<^bsub>G\<^esub> g"
disc_iff_2: fixes xs :: "('a, 'b) tllist" shows "(\<not> lnull (llist_of_tllist xs)) = (\<not> is_TNil xs)"
code: shows "octo_e0 = Octo 1 0 0 0 0 0 0 0"
subset_segment_2: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a--b} \<subseteq> {c<--<d}) = (a \<in> {c<--<d} \<and> b \<in> {c<--<d})"
intruder_synth_timpl_closure_set: fixes M :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term set"   and t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and s :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term"   and TI :: "('sets set \<times> 'sets set) set" assumes "M \<turnstile>\<^sub>c t"   and "s \<in> timpl_closure t TI" shows "timpl_closure_set M TI \<turnstile>\<^sub>c s"
sl3: shows "- abscissa A * (ordinate A - ordinate B) + ordinate A * (abscissa A - abscissa B) \<in> radical_sqrt"
simps_1: fixes S :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" shows "release [] S \<R> = \<R>"
list_down_induct: fixes ys :: "'b list"   and Q :: "'b \<Rightarrow> bool"   and P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "\<forall>y\<in>set ys. Q y"   and "P (ys @ xs)"   and "\<And>y xs. \<lbrakk>Q y; P (y # xs)\<rbrakk> \<Longrightarrow> P xs" shows "P xs"
simps_2: fixes b :: "real"   and c :: "real"   and p :: "real \<times> real \<times> real" shows "substInfinitesimalLinearUni b c (LessUni p) = map_atomUni (linearSubstitutionUni b c) (convertDerivativeUni p)"
intros_1: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state" shows "evaluate_decs ck mn env s [] (s, Rval (make_sem_env nsEmpty nsEmpty))"
target: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelRTPO Enc TRel"