hcju/isabelleps huggingface dataset:
diff_chain_within: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'b \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'c" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at (f x) within f ` s)" shows "(g \<circ> f has_derivative g' \<circ> f') (at x within s)"
h: fixes a :: "complex"   and f1 :: "complex poly" shows "norm2 ([:- a, 1:] * f1) = norm2 ((if 1 < cmod a then [:- 1, cnj a:] else [:- a, 1:]) * f1)"
nn: fixes t :: "real" assumes "t \<in> T" shows "0 \<le> t"
runit_char_2: fixes f :: "'a" assumes "S.ide f" shows "R \<r>[f] = (f \<star> \<iota>) \<cdot>\<^sub>S \<a>[f, a, a]"
Smith_normal_form_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Smith_normal_form A = ((\<forall>a b. to_nat a = to_nat b \<and> to_nat a + 1 < nrows A \<and> to_nat b + 1 < ncols A \<longrightarrow> A $h a $h b dvd A $h (a + (1::'d)) $h (b + (1::'c))) \<and> isDiagonal A)"
complete_lattice_ccpo: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and 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 top :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "class.ccpo Sup less_eq less"
homeomorphic_maps_map: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" shows "homeomorphic_maps X Y f g = (homeomorphic_map X Y f \<and> homeomorphic_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y))"
spec_ops_distinct_fst: fixes ops :: "('b \<times> 'c) list"   and deps :: "'c \<Rightarrow> 'b set" assumes "spec_ops ops deps" shows "distinct (map fst ops)"
eq_iff: fixes x :: "real poly"   and y :: "real poly" shows "(map_poly complex_of_real x = map_poly complex_of_real y) = (x = y)"
set_of_pred_def: shows "set_of_pred = Collect \<circ> eval"
multiplicity_lessI: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one"   and "\<not> dvd.dvd times (power.power one times p n) x" shows "factorial_semiring.multiplicity times one p x < n"
lookup_single_eq: fixes k :: "'c"   and v :: "'b" shows "lookup (single k v) k = v"
conversep_restrict_relp: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "(R \<upharpoonleft> P \<otimes> Q)\<inverse>\<inverse> = R\<inverse>\<inverse> \<upharpoonleft> Q \<otimes> P"
adm: shows "r.adm_wo (\<lambda>g a. s.suc (g ` r.underS a))"
Red1AAccBounds: fixes s :: "'heap \<times> 'addr val list"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id" assumes "typeof_addr (hp s) a = \<lfloor>Array_type T n\<rfloor>"   and "i <s 0 \<or> int n \<le> sint i" shows "uf,P,t \<turnstile>1 \<langle>addr a\<lfloor>Val (Intg i)\<rceil>,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt ArrayIndexOutOfBounds),s\<rangle>"
operators_of_def: shows "operators_of \<equiv> id \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst sas_plus_problem_ext_Tuple_Iso"
induct: fixes P :: "('a state \<Rightarrow> exp0 \<Rightarrow> 'a state \<times> (v, v) result) \<Rightarrow> 'a state \<Rightarrow> exp0 list \<Rightarrow> bool"   and a0 :: "'a state \<Rightarrow> exp0 \<Rightarrow> 'a state \<times> (v, v) result"   and a1 :: "'a state"   and a2 :: "exp0 list" assumes "\<And>eval s. P eval s []"   and "\<And>eval s e es. (\<And>x y x1. \<lbrakk>(x, y) = fix_clock s (eval s e); y = Rval x1\<rbrakk> \<Longrightarrow> P eval x es) \<Longrightarrow> P eval s (e # es)" shows "P a0 a1 a2"
nlistToTree_def: fixes xs :: "('a, 'b) Node list" shows "FingerTreeStruc.nlistToTree xs = FingerTreeStruc.lconsNlist xs Empty"
isOK_check: fixes b :: "bool"   and e :: "'a" shows "isOK (check b e) = b"
notin_fset: fixes x :: "'c"   and S :: "'c fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
map_id: fixes t :: "('a, 'b, 'c, 'd) converter" shows "map_converter' id id t = t"
diamond_cube_boundary_explicit: shows "boundary diamond_cube = {diamond_top_edges, diamond_bot_edges, (- 1, \<lambda>y. (diamond_x 0, (2 * y - 1) * diamond_y (diamond_x 0))), (1, \<lambda>y. (diamond_x 1, (2 * y - 1) * diamond_y (diamond_x 1)))}"
wf_digraphI_induced: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "induced_subgraph H G" shows "wf_digraph H"
Some: fixes x' :: "'a" assumes "x = Some x'"   and "x' \<in> results'_gpv gpv" shows "thesis"
strategy_attracts_viaE: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts_via p \<sigma> v A W" shows "visits_via P A W"
in_hom: shows "\<guillemotleft>\<pi> : s \<Rightarrow> t\<guillemotright>"
indep_in_imp_subset_basis_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> insert x X"   and "X_insert.indep_in \<E> X" shows "\<exists>B. X_insert.basis_in \<E> B \<and> X \<subseteq> B"
drinks_0_rejects: fixes a :: "String.literal \<times> value list"   and r :: "nat \<Rightarrow>f value option" assumes "\<not> (fst a = STR ''select'' \<and> length (snd a) = 1)" shows "possible_steps drinks 0 r (fst a) (snd a) = {||}"
op_set_npick_remove_def: fixes X :: "'a set" shows "op_set_npick_remove X = SPEC (\<lambda>(x, X'). x \<in> X \<and> (X' = X - {x} \<or> X' = X))"
reachable_def: fixes ioa :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and s :: "'f" shows "reachable ioa s \<equiv> \<exists>ex\<in>executions ioa. \<exists>n. snd ex n = s"
s_mono: fixes u :: "real"   and t :: "real" assumes "u \<le> t"   and "0 \<le> u" shows "ego.s u \<le> ego.s t"
consensus_def: fixes vals :: "'f \<Rightarrow> 'g"   and dec :: "'h \<Rightarrow> 'g option"   and rho :: "nat \<Rightarrow> 'f \<Rightarrow> 'h" shows "consensus vals dec rho \<equiv> (\<forall>n p v. dec (rho n p) = Some v \<longrightarrow> v \<in> range vals) \<and> (\<forall>m n p q v w. dec (rho m p) = Some v \<and> dec (rho n q) = Some w \<longrightarrow> v = w) \<and> (\<forall>p. \<exists>n. dec (rho n p) \<noteq> None)"
refine_IdI: fixes m :: "'c nres"   and m' :: "'c nres" assumes "m \<le> m'" shows "m \<le> \<Down> Id m'"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "semiring_hom hom" shows "Ring_Hom.ring_hom hom"
llist_gen_induct: fixes P :: "'c llist \<Rightarrow> bool"   and xs :: "'c llist" assumes "ccpo.admissible lSup (\<sqsubseteq>) P"   and "\<exists>zs. zs \<sqsubseteq> xs \<and> lfinite zs \<and> (\<forall>ys. zs \<sqsubseteq> ys \<longrightarrow> ys \<sqsubseteq> xs \<longrightarrow> lfinite ys \<longrightarrow> P ys)" shows "P xs"
assms_2: shows "\<forall>\<^sup>\<infinity>n. 0 \<le> f n"
transitive_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "transitive B (\<sim>)"
prod_carr_def: fixes M :: "('e, 'g, 'h) Module_scheme"   and N :: "('f, 'g, 'h) Module_scheme" shows "M \<times>\<^sub>c N = carrier M \<times> carrier N"
irs_SDG_path_Append: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n irs-ns\<rightarrow>\<^sub>d* n''"   and "n'' irs-ns'\<rightarrow>\<^sub>d* n'" shows "n irs-ns @ ns'\<rightarrow>\<^sub>d* n'"
bind_assoc_pmf: fixes A :: "'c pmf"   and B :: "'c \<Rightarrow> 'b pmf"   and C :: "'b \<Rightarrow> 'a pmf" shows "bind_pmf (bind_pmf A B) C = bind_pmf A (\<lambda>x. bind_pmf (B x) C)"
disj: shows "pairwise (\<lambda>x y. disjnt (C x) (C y)) A"
mset_union_subset: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "A + B \<subseteq># C" shows "A \<subseteq># C \<and> B \<subseteq># C"
d_1: shows "D \<in> Q_of_state (lnth Sts i)"
finprod_cong': fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a vec"   and f :: "'b \<Rightarrow> 'a vec" assumes "A = B"   and "g \<in> B \<rightarrow> carrier_vec n"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum V f A = finsum V g B"
map_poly_map_poly: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and p :: "'d poly" assumes "f (0::'c) = (0::'b)" shows "map_poly f (map_poly g p) = map_poly (f \<circ> g) p"
Abel_lemma: fixes r :: "real"   and r0 :: "real"   and a :: "nat \<Rightarrow> 'a"   and M :: "real" assumes "0 \<le> r"   and "r < r0"   and "\<And>n. norm (a n) * r0 ^ n \<le> M" shows "summable (\<lambda>n. norm (a n) * r ^ n)"
transp_on_map': fixes Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and h :: "'c \<Rightarrow> 'b" assumes "transp_on Q B"   and "g ` A \<subseteq> B"   and "h ` A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> Q\<^sup>=\<^sup>= (h x) (g x)" shows "transp_on (\<lambda>x y. Q (g x) (h y)) A"
d_interr_U: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> U \<parallel> (d y \<cdot> U) = d (x \<parallel> y) \<cdot> U"
simps: shows "local.tighten_poly_bounds_epsilon l r sr = (l', r', sr')"
parts_sign: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "Sign n A \<in> parts (used s)" shows "A = Auth_PriK n"
cmp_def: fixes \<mu>\<nu> :: "'b \<times> 'b" shows "HoG.cmp \<mu>\<nu> = (if B.VV.arr \<mu>\<nu> then H (G (fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>)) \<cdot>\<^sub>D H (\<Phi>\<^sub>G (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D \<Phi>\<^sub>H (G (B.dom (fst \<mu>\<nu>)), G (B.dom (snd \<mu>\<nu>))) else D.null)"
Abs_status_inverse: fixes y :: "('a, 'b, 'c) hierauto \<times> 'a set \<times> 'b set \<times> 'c data" assumes "y \<in> status" shows "Rep_status (Abs_status y) = y"
30: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (x \<squnion> y) = x \<squnion> y"
isMinimalBackjumpLevelEnsuresIsNotUnitBeforePrefix: fixes M :: "(Literal \<times> bool) list"   and c :: "Literal list"   and level :: "nat"   and l :: "Literal"   and level' :: "nat" assumes "consistent (elements M)"   and "uniq (elements M)"   and "clauseFalse c (elements M)"   and "isMinimalBackjumpLevel level l c M"   and "level' < level" shows "\<nexists>l'. isUnitClause c l' (elements (prefixToLevel level' M))"
linear_compose_add: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "linear (*a) (*b) f"   and "linear (*a) (*b) g" shows "linear (*a) (*b) (\<lambda>x. f x + g x)"
type_less_left_simps_5: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "(Set \<tau> < \<sigma>) = (\<exists>\<phi>. \<sigma> = OclSuper \<or> \<sigma> = Collection \<phi> \<and> \<tau> \<le> \<phi> \<or> \<sigma> = Set \<phi> \<and> \<tau> < \<phi>)"
map_converter_id: fixes conv :: "('a, 'b, 'c, 'd) converter" shows "map_converter id id id id conv = conv"
simps_37: fixes x41 :: "stmt"   and x42 :: "stmt"   and x2 :: "expr" shows "x41;; x42 \<noteq> Expr x2"
infinite: fixes A :: "'d set" assumes "infinite A" shows "mset_set A = {#}"
assms_3: shows "length ms\<^sub>1 = length s\<^sub>1"
HFun_Sigma_single: fixes H :: "fm set"   and d :: "tm"   and d' :: "tm" assumes "H \<turnstile> OrdP d" shows "H \<turnstile> HFun_Sigma (Eats Zero (HPair d d'))"
exits_Cons: fixes x :: "instr"   and xs :: "instr list" shows "exits (x # xs) = isuccs x 0 - {0} \<union> (+) 1 ` exits xs - {0..<1 + size xs}"
UNpart1_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list"   and P :: "nat \<Rightarrow> nat set set" shows "UNpart1 cl dl P \<equiv> \<Union>n\<in>theNFTBoth cl dl. local.shift cl n ` P n"
MIN': shows "min_dist s t = length p1 + 1 + length p2"
unlabel_mono: fixes A :: "('e \<times> 'f) list"   and B :: "('e \<times> 'f) list" assumes "set A \<subseteq> set B" shows "set (unlabel A) \<subseteq> set (unlabel B)"
divisor_sigma_def: fixes x :: "'a"   and n :: "nat" shows "divisor_sigma x n = (\<Sum>d | d dvd n. nat_power d x)"
inv2Inv_1: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "Category C"   and "cinv\<^bsub>C\<^esub> f g" shows "ciso\<^bsub>C\<^esub> f"
eq: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq A B"
lm055: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd"   and X :: "'c set" shows "(x \<in> argmax f X) = (x \<in> X \<and> f x = Max (f ` X))"
lookup_env_delete: fixes x' :: "'a"   and x :: "'a"   and m :: "'a \<Rightarrow> 'b" assumes "x' \<noteq> x" shows "env_delete x m x' = m x'"
inject_4: fixes x5 :: "'d"   and y5 :: "'d" shows "(Inca.instr.IOp x5 = Inca.instr.IOp y5) = (x5 = y5)"
RedFAcc: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and v :: "'addr val"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "heap_read h a (CField D F) v" shows "extTA,P,t \<turnstile> \<langle>addr a\<bullet>F{D},(h, l)\<rangle> -\<lbrace>ReadMem a (CField D F) v\<rbrace>\<rightarrow> \<langle>Val v,(h, l)\<rangle>"
Copied_6: shows "((ps, a) # branch) !. w = Some (rs, b)"
le_transpose_matrix: fixes A :: "'b matrix"   and B :: "'b matrix" shows "(A \<le> B) = (transpose_matrix A \<le> transpose_matrix B)"
nataction_add_eq0_flip: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.group_add minus plus zero uminus"   and "power.power zero plus (plus a b) n = zero" shows "power.power zero plus (plus b a) n = zero"
invert_one_side: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OS Bet A B P Q" shows "Tarski_neutral_dimensionless.OS Bet B A P Q"
length_disj_filter_le: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> (P x \<and> Q x)" shows "|filter P xs| + |filter Q xs| \<le> |xs|"
abs_eq: fixes x :: "('b, 'c, 'd, 'e) Tree" shows "Not\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tNot x)"
shows_prec_xml_append: fixes d :: "nat"   and xml :: "xml"   and r :: "char list"   and s :: "char list" shows "shows_prec d xml (r @ s) = shows_prec d xml r @ s"
OclInt1_def: shows "\<one> = (\<lambda>_. \<lfloor>\<lfloor>1\<rfloor>\<rfloor>)"
assms: shows "(ftv T - ftv \<Gamma>) \<sharp>* \<theta>"
wf: shows "is_density_expr ([], vs @ vs', \<Gamma>, CReal 1) BOOL f"
tabulate_alt: fixes f :: "nat \<Rightarrow> 'a"   and x :: "nat"   and n :: "nat" shows "tabulate f x n = map f [x..<x + n]"
distrib_right: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
IH_1: shows "subst_dnf' (list_dnf \<phi>1) m = list_dnf (subst \<phi>1 m)"
class_leq_antisym: fixes c1 :: "String.literal"   and c2 :: "String.literal" assumes "class_leq cs c1 c2"   and "class_leq cs c2 c1" shows "c1 = c2"
content_cball_conv_ball: fixes c :: "'c"   and r :: "real" shows "content (cball c r) = content (ball c r)"
rep_eq: fixes x :: "nat"   and xa :: "'d pdevs"   and xb :: "'d pdevs" shows "pdevs_apply (msum_pdevs x xa xb) = msum_pdevs_raw x (pdevs_apply xa) (pdevs_apply xb)"
assms: shows "(f, h) \<in> [comp_PRE S1 Q (\<lambda>x _. P x) X]\<^sub>f R \<rightarrow> S"
induct: fixes P :: "int \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int" assumes "\<And>i j. \<lbrakk>\<And>x. \<lbrakk>\<not> j < i; x \<in> {i..j}\<rbrakk> \<Longrightarrow> P i (x - 1); \<And>x. \<lbrakk>\<not> j < i; x \<in> {i..j}\<rbrakk> \<Longrightarrow> P (x + 1) j\<rbrakk> \<Longrightarrow> P i j" shows "P a0 a1"
oreceiveT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and u\<^sub>m\<^sub>s\<^sub>g :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and msg :: "'f"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = u\<^sub>m\<^sub>s\<^sub>g msg (\<sigma> i)" shows "((\<sigma>, {l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p), receive msg, \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
det_interchange_rows: fixes A :: "(('d, 'e) vec, 'e) vec"   and i :: "'e"   and j :: "'e" shows "det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A"
cobounded2: fixes b :: "'d"   and a :: "'d" shows "b \<le> sup a b"
index_transpose_mat_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat" assumes "i < dim_col A"   and "j < dim_row A" shows "A\<^sup>T $$ (i, j) = A $$ (j, i)"
ARR_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.ARR (\<F>C.mkarr t)"
Suc_3: shows "KD_Tree.size_kdt kdt = 2 ^ height kdt"
simps: fixes a :: "('a, 'b, 'c) exp" shows "final a = ((\<exists>v. a = Val v) \<or> (\<exists>aa. a = Throw aa))"
take_i_f_join_eq2: fixes I :: "nat set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "finite I"   and "card I \<le> n" shows "f \<Join>\<^sub> I \<down> n = f \<Join>\<^sub> I"
Word_def: shows "Word.Word \<equiv> quot_type.abs (\<lambda>k l. take_bit LENGTH('k) k = take_bit LENGTH('k) l) Word.word.Abs_word"
Entry_path: fixes n :: "'b" assumes "valid_node n" shows "\<exists>as. Entry -as\<rightarrow>* n"
int_set_bit_False_conv_NAND: fixes i :: "int"   and n :: "nat" shows "set_bit_class.set_bit i n False = i AND NOT (1 << n)"
assms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A))"
Exit_no_SDG_edge_source: fixes Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and n' :: "'node SDG_node" assumes "SDG_edge (CFG_node (_Exit_)) Vopt popt n'" shows "False"
eq_Min_iff: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and m :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "A \<noteq> {}" shows "(m = ordered_powerprod_lin.Min A) = (m \<in> A \<and> (\<forall>a\<in>A. m \<preceq> a))"
distF: shows "dist_over_inf (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
assms_3: shows "pseudominimal_in (P .\<^bold>\<le> x) y"
rep_eq: fixes x :: "'b" shows "Formula.formula.Rep_formula (Formula.Pred x) = Formula.Pred\<^sub>\<alpha> x"
rep_eq: fixes x :: "integer" shows "Rep_uint8 (Uint8 x) = word_of_int (int_of_integer x)"
mplus_mzero_left: fixes m :: "'a\<cdot>'b" shows "mplus\<cdot>mzero\<cdot>m = m"
Ks'_1: shows "set Ks' \<subseteq> K"
kh: fixes x :: "'a" assumes "x \<in> rel_frontier S" shows "k (h x) = x"
image_chamber_system: shows "ChamberComplex.chamber_system (g \<turnstile> X) = g \<turnstile> folding_g.\<C>"
edges_are_in_V_2: fixes G :: "('e, 'f) Graph_scheme"   and v :: "'e"   and w :: "'e" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "w \<in> V\<^bsub>G\<^esub>"
assms: shows "F = {f \<in> {0..<n} \<rightarrow> {0..<m}. \<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i}"
Gmult_closed: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and g :: "'a"   and v :: "'c" assumes "FGModule G smult V"   and "g \<in> G"   and "v \<in> V" shows "aezfun_scalar_mult.Gmult smult g v \<in> V"
t_star_contl: fixes X :: "('a \<times> ('b \<times> 'a) list) set"   and Y :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod X (t_star Y) = (\<Union>n. t_prod X (trace_dioid.power Y n))"
halfchsys_decomp_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplex.chamber_system X = f \<turnstile> ChamberComplex.chamber_system X \<union> g \<turnstile> ChamberComplex.chamber_system X"
assms: shows "homotopic_with (\<lambda>x. True) (nsphere p) (nsphere p) f g"
mono_neutral_cong_right: fixes T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = \<^bold>1"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g T = F h S"
subsumes_refl: fixes C :: "'a literal multiset" shows "subsumes C C"
matrix_add_vect_distrib: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "('a, 'c) Finite_Cartesian_Product.vec" shows "(A + B) *v v = A *v v + B *v v"
hd_take_nths: fixes i :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "i < length xs" shows "hd (take_nths i k xs) = xs ! i"
mdecl: shows "P \<turnstile> C sees M: Ts\<rightarrow>T = (pns, body) in D"
minus_plus: fixes s :: "'a"   and t :: "'a"   and u :: "'a" assumes "s adds t" shows "t - s + u = t + u - s"
eq_refl: fixes x :: "'a"   and y :: "'a"   and ko :: "'a key_order" assumes "x = y" shows "ko.le ko x y"
bidirected_digraph_del_vert: fixes u :: "'a" shows "bidirected_digraph (H.del_vert u) (perm_restrict (edge_rev M) (arcs (H.del_vert u)))"
of_bl_rev_eq: shows "of_bl (rev (rev bs)) = horner_sum of_bool 2 (rev bs)"
final_Power_dfa: fixes M :: "'b nfa" assumes "nfa M" shows "dfa.final (nfa.Power_dfa M) = {HF (nfa.epsclo M Q) |Q. Q \<subseteq> nfa.states M \<and> Q \<inter> nfa.final M \<noteq> {}}"
z_2: shows "0 \<le> f z \<bullet> rot (x - y)"
suntil_induct_strong: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and x :: "'a stream"   and P :: "'a stream \<Rightarrow> bool" assumes "(\<phi> suntil \<psi>) x"   and "\<And>\<omega>. \<psi> \<omega> \<Longrightarrow> P \<omega>"   and "\<And>\<omega>. \<lbrakk>\<phi> \<omega>; \<not> \<psi> \<omega>; (\<phi> suntil \<psi>) (stl \<omega>); P (stl \<omega>)\<rbrakk> \<Longrightarrow> P \<omega>" shows "P x"
simps_9: fixes f :: "'f"   and f' :: "'g"   and g :: "'d set"   and c :: "('d, 'e, 'g) com" shows "mark_guards f (Guard f' g c) = Guard f g (mark_guards f c)"
distinct_89: fixes x91 :: "nat"   and x92 :: "'a fm"   and x101 :: "nat"   and x102 :: "'a fm" shows "ExN x91 x92 \<noteq> AllN x101 x102"
IH: fixes p :: "real mpoly" assumes "p \<noteq> 0"   and "vars p \<subseteq> {..<n}" shows "{f. f \<in> space (lborel_f n) \<and> insertion f p = 0} \<in> null_sets (lborel_f n)"
is_none_def: fixes x :: "'c option" shows "Option.is_none x = (x = None)"
remove_vertex_def: fixes G :: "nat set \<times> nat set set"   and u :: "nat" shows "G -- u \<equiv> (uverts G - {u}, uedges G - {A \<in> uedges G. u \<in> A})"
Heap_cases: fixes f :: "'c Heap"   and h :: "heap" obtains x :: "'c" and h' :: "heap" where "execute f h = Some (x, h')"   | "execute f h = None"
eucl_ivals_5: fixes a :: "'c"   and b :: "'c" shows "{a..b} \<in> sets borel"
isFDERIV_lengthD_2: fixes n :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and vs :: "real list" assumes "isFDERIV n xs fas vs" shows "length xs = n"
freeIn_substNegInfinity: fixes var :: "nat"   and A :: "atom" shows "freeIn var (substNegInfinity var A)"
image_mono: fixes A :: "'g set"   and B :: "'g set"   and f :: "'g \<Rightarrow> 'h" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
sum_msets_chop: fixes n :: "nat"   and xs :: "'a list" assumes "0 < n" shows "sum_list (map mset (chop n xs)) = mset xs"