hcju/isabelleps huggingface dataset:
top_eq: fixes L :: "('a, 'b) gorder_scheme"   and t :: "'a" assumes "bounded_lattice L"   and "t \<in> carrier L"   and "\<And>x. x \<in> carrier L \<Longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> t" shows "t = \<top>\<^bsub>L\<^esub>"
Keys_def: fixes F :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "Keys F = \<Union> (keys ` F)"
fold_fun_chain_2: fixes m :: "nat"   and g :: "'a \<Rightarrow> 'a"   and n :: "nat"   and x :: "'a" shows "(g ^^ m) ((g ^^ n) x) = (g ^^ (m + n)) x"
simps: fixes n :: "nat"   and x :: "'a" shows "v n x = (if n = 0 then max (u 0 x) 0 else if n = 1 then u 1 x else min (u n x) (MIN k\<in>{0<..<n}. v k x + v (n - k) ((T ^^ k) x)))"
assms: shows "\<psi> \<sim>[at_top] (\<lambda>x. x)"
ncp: fixes D' :: "char list" assumes "P \<turnstile> C\<^sub>0 \<preceq>\<^sup>* D'" shows "D' \<noteq> C"
analz_image_pan: fixes evs :: "event list"   and KK :: "nat set"   and P :: "nat" assumes "evs \<in> set_pur"   and "\<forall>K\<in>KK. K \<notin> range priEK" shows "(Pan P \<in> analz (Key ` KK \<union> knows Spy evs)) = (Pan P \<in> analz (knows Spy evs))"
span_superset: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set" assumes "module scale" shows "S \<subseteq> module.span scale S"
key_sets_pwd_empty: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and X :: "msg" assumes "s\<^sub>0 \<Turnstile> s" shows "key_sets (Hash (Pwd n)) (crypts (Log -` spied s)) = {} \<and> key_sets \<lbrace>Pwd n, X\<rbrace> (crypts (Log -` spied s)) = {} \<and> key_sets \<lbrace>X, Pwd n\<rbrace> (crypts (Log -` spied s)) = {}"
Diamond_for_One: fixes t :: "lam"   and t1 :: "lam"   and t2 :: "lam" assumes " t \<longrightarrow>\<^sub>1 t1"   and " t \<longrightarrow>\<^sub>1 t2" shows "\<exists>t3.  t2 \<longrightarrow>\<^sub>1 t3 \<and>  t1 \<longrightarrow>\<^sub>1 t3"
heap_wellformed_induct: fixes h :: "(_) heap"   and P :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "\<And>parent. (\<And>children child. \<lbrakk>h \<turnstile> Shadow_DOM.get_child_nodes parent \<rightarrow>\<^sub>r children; child \<in> set children\<rbrakk> \<Longrightarrow> P (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)) \<Longrightarrow> P parent" shows "P ptr"
col_carrier_vec: fixes j :: "nat"   and nc :: "nat"   and A :: "'c mat"   and nr :: "nat" assumes "j < nc"   and "A \<in> carrier_mat nr nc" shows "col A j \<in> carrier_vec nr"
Riemann_lebesgue_square_integrable: fixes S :: "real set"   and w :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "orthonormal_system S w"   and "\<And>i. w i square_integrable S"   and "f square_integrable S" shows "orthonormal_coeff S w f \<longlonglongrightarrow> 0"
differentiable_subset: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and t :: "'a set" assumes "f differentiable at x within s"   and "t \<subseteq> s" shows "f differentiable at x within t"
simple_match_and_NoneD: fixes m1 :: "'a simple_match"   and m2 :: "'a simple_match"   and p :: "('a, 'b) simple_packet_scheme" assumes "simple_match_and m1 m2 = None" shows "\<not> (simple_matches m1 p \<and> simple_matches m2 p)"
executions_def: fixes ioa :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set" shows "executions ioa \<equiv> {e. snd e 0 \<in> starts_of ioa \<and> is_execution_fragment ioa e}"
monad_fail_writerT: fixes return :: "'d \<times> 'e list \<Rightarrow> 'f"   and bind :: "'f \<Rightarrow> ('d \<times> 'e list \<Rightarrow> 'f) \<Rightarrow> 'f"   and fail :: "'f" assumes "monad_fail return bind fail" shows "monad_fail (return_writer return) (bind_writer return bind) (fail_writer fail)"
alt_char: shows "mat_of_rows (dim_col A) (map ((!) (rows A)) (filter (\<lambda>y. y < dim_col A) [0..<length (pivot_positions A)])) = mat_of_rows (dim_col A) (map ((!) (rows A)) [0..<length (pivot_positions A)])"
lappend_simps_2: fixes xs :: "'c llist"   and ys :: "'c llist" shows "ltl (lappend xs ys) = (if lnull xs then ltl ys else lappend (ltl xs) ys)"
code_is_basis: fixes \<C> :: "'b list set" assumes "code \<C>" shows "\<BB> \<C> = \<C>"
assms_1: shows "a \<or> b \<or> c \<or> d"
digraph_axioms': fixes \<alpha>' :: "V" assumes "\<alpha>' = \<alpha>" shows "digraph \<alpha>' \<BB>"
ltl_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "ltl (lappend xs ys) = (if lnull xs then ltl ys else lappend (ltl xs) ys)"
inv_character_inv_character: fixes \<chi> :: "'c \<Rightarrow> complex" shows "inv_character (inv_character \<chi>) = \<chi>"
B: shows "(\<lambda>x. (g1 * h1) x * (g2 * h2) x) = g1 * g2 * (h1 * h2)"
extended_Gromov_product_at_def: fixes e :: "'b"   and x :: "'b Gromov_completion"   and y :: "'b Gromov_completion" shows "extended_Gromov_product_at e x y = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e (u n) (v n))) |u v. abs_Gromov_completion u = x \<and> abs_Gromov_completion v = y \<and> Gromov_completion_rel u u \<and> Gromov_completion_rel v v}"
IH: fixes Y :: "('fun, 'var) Term.term list" assumes "f \<in> \<Union> (range \<Sigma>n) - \<C>"   and "g \<in> \<Union> (range \<Sigma>n) - \<C>"   and "\<Gamma> (Fun f (X @ x # X')) = \<Gamma> (Fun g (Y @ y # Y'))"   and "length X = length Y" shows "\<Gamma> x = \<Gamma> y"
sum_set_simps_2: fixes x :: "'d" shows "setl (Inr x) = {}"
unadjoin_adjoined: shows "H.unadjoin x = (\<one>, 1)"
inner_zero_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner zero x = 0"
v_3: shows "x2 = (1 - v) *\<^sub>R x1 + v *\<^sub>R b"
represents_def: fixes p :: "int poly"   and x :: "'b" shows "p represents x = (ipoly p x = (0::'b) \<and> p \<noteq> 0)"
Fin: fixes x :: "'a" assumes "x \<in> S" shows "F x \<in> \<C>"
getX_def: fixes r :: "'a point_scheme" shows "getX r = xpos r"
C2r: fixes x :: "'a"   and y :: "'a" assumes "x = x \<cdot> y" shows "x = x \<cdot> y\<^sup>\<star>"
get_get: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'm" shows "get (\<lambda>s. get (f s)) = get (\<lambda>s. f s s)"
map_S_parametric: fixes L1 :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and L1' :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and L2 :: "'g \<Rightarrow> 'r \<Rightarrow> bool"   and L2' :: "'h \<Rightarrow> 's \<Rightarrow> bool"   and Co1 :: "'i \<Rightarrow> 't \<Rightarrow> bool"   and Co1' :: "'j \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'k \<Rightarrow> 'v \<Rightarrow> bool"   and Co2' :: "'l \<Rightarrow> 'w \<Rightarrow> bool"   and Contra1' :: "'m \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'n \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2' :: "'o \<Rightarrow> 'z \<Rightarrow> bool"   and Contra2 :: "'p \<Rightarrow> 'aa \<Rightarrow> bool" shows "rel_fun (rel_fun L1 L1') (rel_fun (rel_fun L2 L2') (rel_fun (rel_fun Co1 Co1') (rel_fun (rel_fun Co2 Co2') (rel_fun (rel_fun Contra1' Contra1) (rel_fun (rel_fun Contra2' Contra2) (rel_fun (rel_S L1 L2 Co1 Co2 Contra1 Contra2) (rel_S L1' L2' Co1' Co2' Contra1' Contra2'))))))) map_S map_S"
injective_down_closed: fixes x :: "'a"   and y :: "'a" assumes "is_inj x"   and "y \<le> x" shows "is_inj y"
H: shows "\<forall>x\<in>U. \<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> y \<in> U"
eq_rat_2: fixes a :: "int" shows "Fract a 0 = Fract 0 1"
is_ext_def: fixes y' :: "'c"   and g :: "'d"   and f :: "'c" shows "is_ext y' g f \<equiv> \<guillemotleft>f : y \<rightarrow>\<^sub>C y'\<guillemotright> \<and> g = G f \<cdot>\<^sub>D e"
finite_arcs: fixes a :: "'a \<times> nat \<times> 'a" shows "finite (arcs (add_arc a))"
rank_iarray_def: fixes A :: "'d iarray iarray" shows "rank_iarray A = (let A' = Gauss_Jordan_iarrays A; nrows = IArray.length A' in card {i. i < nrows \<and> \<not> is_zero_iarray (A' !! i)})"
StepActEvent_HAEvents: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<in> HPT ST" shows "StepActEvent TS \<subseteq> HAEvents (HA ST)"
divide_neg_pos: fixes x :: "'a"   and y :: "'a" assumes "x < (0::'a)"   and "(0::'a) < y" shows "x / y < (0::'a)"
succ_in_Limit_iff: fixes \<xi> :: "V"   and \<alpha> :: "V" assumes "Limit \<xi>" shows "(ZFC_in_HOL.succ \<alpha> \<in> elts \<xi>) = (\<alpha> \<in> elts \<xi>)"
iteratei_rule_insert_P: assumes "set_iterator_genord it S0 (\<lambda>_ _. True)"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>True; x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S0 - S - {x}. True; \<forall>y\<in>S. True\<rbrakk> \<Longrightarrow> I (insert x S) (f x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> True; I S \<sigma>; \<forall>x\<in>S0 - S. \<forall>y\<in>S. True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
Exit_not_standard_control_dependent: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g"   and n :: "'g" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit"   and "Postdomination.standard_control_dependence sourcenode targetnode valid_edge Exit n Exit" shows "False"
gso_int_tail': fixes acc :: "int vec"   and i :: "nat"   and l :: "nat" assumes "acc = gso_int i l"   and "0 < i"   and "0 < l"   and "l \<le> i" shows "gso_int_tail' i l acc = gso_int i i"
abs_eq: fixes x :: "'b list" shows "vec_of_list x = Abs_vec (length x, mk_vec (length x) ((!) x))"
linorder_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.linorder less_eq less \<equiv> class.order less_eq less \<and> class.linorder_axioms less_eq"
ide_dom: fixes f :: "'a" assumes "B.arr f" shows "B.ide (B.dom f)"
oghoare_Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and p1 :: "('e, 'd, 'f) com"   and p2 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Seq p1 p2) Q, A" shows "\<exists>P\<^sub>1 P\<^sub>2 P' Q' A'. P = AnnComp P\<^sub>1 P\<^sub>2 \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>1 p1 P', A' \<and> P' \<subseteq> pre P\<^sub>2 \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>2 p2 Q', A' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A"
INF_mono': fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Inf (f ` A)) (Inf (g ` A))"
disc_1: fixes x1 :: "'f" shows "is_Done (Done x1)"
Dflow_zero: fixes x :: "'a" assumes "x \<in> X" shows "Dflow x 0 = 1\<^sub>L"
induct: fixes x1 :: "('c, 'd) Expr"   and x2 :: "Dom set"   and P :: "('c, 'd) Expr \<Rightarrow> Dom set \<Rightarrow> bool" assumes "\<turnstile>\<^bsub>\<E>\<^esub> x1 : x2"   and "\<And>v d. P (Const v) {d}"   and "\<And>x. P (Var x) {DA x}"   and "\<And>arglist dl f. \<forall>i<length arglist. \<turnstile>\<^bsub>\<E>\<^esub> arglist ! i : dl ! i \<and> P (arglist ! i) (dl ! i) \<Longrightarrow> P (Op f arglist) (\<Union> {d. \<exists>i<length arglist. d = dl ! i})" shows "P x1 x2"
max_bag_card_lower_bound_bag: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set"   and t :: "'e" assumes "TreeDecomposition G T bag"   and "t \<in> V\<^bsub>T\<^esub>" shows "card (bag t) \<le> TreeDecomposition.max_bag_card T bag"
F_def: shows "F = (\<lambda>K e x. real_of_ereal (max (l x) (- ereal K)) + e)"
common_matcher_SrcDst_Inter_1: fixes X :: "'a ipt_iprange list"   and \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" shows "(\<forall>m\<in>set X. matches (common_matcher, \<alpha>) (Match (Src m)) a p) = (p_src p \<in> \<Inter> (ipt_iprange_to_set ` set X))"
psiFreshVec_7: fixes xvec :: "name list"   and P :: "('e, 'f, 'g) psi" shows "xvec \<sharp>* !P = xvec \<sharp>* P"
inv_solve_right: fixes a :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a"   and c :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier F"   and "b \<in> carrier F"   and "c \<in> carrier F" shows "(a = b \<oplus>\<^bsub>F\<^esub> \<ominus>\<^bsub>F\<^esub> c) = (b = a \<oplus>\<^bsub>F\<^esub> c)"
ws: fixes c :: "'b"   and c1 :: "'b"   and c2 :: "'b" shows "{c \<triangleright> D |D. D = {c1} \<or> D = {c2}} = {{c, c1}, {c, c2}}"
ClassCast_def: shows "ClassCast = STR ''java/lang/ClassCastException''"
cone_iff: fixes S :: "'a set" assumes "S \<noteq> {}" shows "cone S = ((0::'a) \<in> S \<and> (\<forall>c>0. (*\<^sub>R) c ` S = S))"
prv_imp_refl: fixes \<phi> :: "fm" assumes "\<phi> \<in> UNIV" shows "{} \<turnstile> \<phi> IMP \<phi>"
out_var_def: fixes x :: "'c \<Longrightarrow> 'e" shows "$x\<acute> = x ;\<^sub>L snd\<^sub>L"
star_inductr_var_eq2: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x = y" shows "y \<cdot> x\<^sup>\<star> = y"
induced_map_unique: fixes r\<^sub>0 :: "'a"   and \<omega> :: "'a"   and w :: "'a"   and w' :: "'a" assumes "tab.is_induced_by_cell w r\<^sub>0 \<omega>"   and "tab.is_induced_by_cell w' r\<^sub>0 \<omega>" shows "w \<cong> w'"
UP_a_inv_closed: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
NN_neg_unique: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (imp (NN \<langle>\<phi>\<rangle> (Var yy)) (eql \<langle>neg \<phi>\<rangle> (Var yy)))"
hcomp_obj_arr: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and b :: "'c"   and f :: "'c" assumes "normal_bicategory V H \<a> \<i> src trg"   and "horizontal_homs.obj V src b"   and "partial_magma.arr V f"   and "b = trg f" shows "H b f = f"
l9_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and A :: "'a"   and C :: "'a"   and M :: "'a"   and R :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet P Q A C"   and "Tarski_neutral_dimensionless.Col Bet M P Q"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A C"   and "Tarski_neutral_dimensionless.Col Bet R P Q"   and "Tarski_neutral_dimensionless.Out Bet R A B" shows "Tarski_neutral_dimensionless.TS Bet P Q B C"
braun_indices_if_braun: fixes t :: "'b tree" assumes "braun t" shows "braun_indices t = {1..size t}"
steps_preserves_types: 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 es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P,E,hp s \<turnstile> es [:] Ts" shows "P,E,hp s' \<turnstile> es' [:]\<^bsub>NT\<^esub> Ts"
sop_distr1Tr: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i (s_set R s A))"   and "assoc_bpp (aug_pm_set z i (s_set R s A)) f"   and "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr1 R s f i (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_inv R s i (s_set R s A)"   and "sop_one R s (aug_pm_set z i (s_set R s A))"   and "sop_z R s z" shows "\<forall>a\<in>carrier R. \<forall>b\<in>carrier R. \<forall>x\<in>add_set f (aug_pm_set z i (s_set R s A)) n. (a \<plusminus> b) \<^sub>s\<cdot> x = a \<^sub>s\<cdot> x \<^sub>f+ b \<^sub>s\<cdot> x"
f_im: fixes v :: "'c" assumes "v \<in> carrier M" shows "f v \<in> carrier N"
cbox_2D: fixes x\<^sub>0 :: "real"   and y\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and y\<^sub>1 :: "real" shows "cbox (x\<^sub>0, y\<^sub>0) (x\<^sub>1, y\<^sub>1) = {(x, y). x\<^sub>0 \<le> x \<and> x \<le> x\<^sub>1 \<and> y\<^sub>0 \<le> y \<and> y \<le> y\<^sub>1}"
min_satisfying_def: fixes P :: "'b \<Rightarrow> bool"   and l :: "'b list" shows "min_satisfying P l \<equiv> let xs = filter P l in if xs = [] then None else Some (list_min xs)"
T_del_min_def: fixes t :: "'b tree" shows "T_del_min t = int (case t of \<langle>\<rangle> \<Rightarrow> 1 | \<langle>t1, a, t2\<rangle> \<Rightarrow> T_merge t1 t2 + 1)"
kcomp_isol: fixes f :: "'a \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b set"   and h :: "'c \<Rightarrow> 'a set" assumes "f \<le> g" shows "h \<circ>\<^sub>K f \<le> h \<circ>\<^sub>K g"
unat_sub_if_size: fixes x :: "'b word"   and y :: "'b word" shows "unat (x - y) = (if unat y \<le> unat x then unat x - unat y else unat x + 2 ^ size x - unat y)"
simps_47: fixes x4 :: "val"   and x21 :: "char list"   and x22 :: "expr" shows "Val x4 \<noteq> Cast x21 x22"
length_list_of_conv_the_enat: fixes xs :: "'a llist" assumes "lfinite xs" shows "length (list_of xs) = the_enat (llength xs)"
respond_imp_responses: fixes PA :: "msg"   and RB :: "msg"   and KAB :: "nat"   and evs :: "event list" assumes "(PA, RB, KAB) \<in> respond evs" shows "RB \<in> responses evs"
scale_left_commute: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "a * (b * x) = b * (a * x)"
subdiv_step_def: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b"   and u :: "'a"   and v :: "'a"   and w :: "'a"   and uv :: "'b"   and uw :: "'b"   and vw :: "'b" shows "subdiv_step G rev_G H rev_H u v w uv uw vw \<equiv> subdivision_step G rev_G H rev_H (u, v, w) (uv, uw, vw)"
endpoint_in_S_unique: fixes a :: "proj2"   and b :: "proj2"   and p :: "proj2" assumes "a \<noteq> b"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S"   and "p \<in> S"   and "B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt p" shows "p = endpoint_in_S a b"
not_empty_irreducible: fixes C :: "'s option set" assumes "C \<in> UNIV // K2.communicating" shows "C \<noteq> {}"
continuous_at_imp_continuous_at_within: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "isCont f x" shows "continuous (at x within s) f"
in_set_tlD: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set (tl xs)" shows "x \<in> set xs"
set_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o \<AA>" shows "local.set (mkIde A) = A"
is_Prim0_Src: fixes t :: "'c local.term" shows "is_Prim\<^sub>0 (Src t)"
res: shows "factorize_rat_poly_monic f = (c, fs)"
less_floor_iff: fixes z :: "int"   and x :: "'b" shows "(z < \<lfloor>x\<rfloor>) = (of_int z + (1::'b) \<le> x)"
linear_add: fixes f :: "'a \<Rightarrow> 'b"   and b1 :: "'a"   and b2 :: "'a" assumes "\<forall>x. x \<in> S \<longrightarrow> f x \<in> T"   and "b1 \<in> S"   and "b2 \<in> S"   and "linear_on S T (*\<^sub>R) (*\<^sub>R) f" shows "f (b1 + b2) = f b1 + f b2"
assms: fixes i :: "'a" assumes "i \<in> Basis" shows "convex {x. P i x}"
R2: shows "RETURN l' \<le> set_to_sorted_list R s'"
unit_in_vhom: shows "\<guillemotleft>\<zeta> : src f \<Rightarrow> g \<star> f\<guillemotright>"
Al: fixes y :: "'a list set"   and x :: "'a list set"   and z :: "'a list set" assumes "\<not> l_ewp y"   and "x = y \<cdot> x + z" shows "x = y\<^sup>\<star> \<cdot> z"
even_numeral: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "num" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (numeral.numeral one plus (num.Bit0 n))"
visible_methods_exist: fixes P :: "'a prog"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option"   and M :: "char list"   and m :: "staticb \<times> ty list \<times> ty \<times> 'a"   and D :: "char list" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "\<exists>D' fs ms. class P D = \<lfloor>(D', fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>m\<rfloor>"
initial_state_2: shows "fs_state (initial_state m fs_init) = fs_init"
arc_in_partition: fixes x :: "'a"   and y :: "'a" assumes "arc x" shows "x \<le> - y \<or> x \<le> - - y"
assms_1: shows "(_Exit_) \<in> set (sourcenodes as)"
intensional_simps_17: fixes y :: "'b \<Rightarrow> bool" shows "#True \<and> y \<equiv> y"
prv_allE: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "bprv (all x \<phi>)" shows "bprv (subst \<phi> t x)"
diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and s :: "'a"   and S :: "'a set"   and t :: "'a" assumes "class.ab_group_add plus zero minus uminus"   and "s \<in> ab_group_add.group_closure zero minus S"   and "t \<in> ab_group_add.group_closure zero minus S" shows "minus s t \<in> ab_group_add.group_closure zero minus S"
sum_cls_val_process_Suc: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market" assumes "portfolio pf1"   and "portfolio pf2" shows "\<forall>n w. cls_val_process Mkt (qty_sum pf1 pf2) (Suc n) w = cls_val_process Mkt pf1 (Suc n) w + cls_val_process Mkt pf2 (Suc n) w"
start_len_gt_0: fixes G :: "nat graph_rec" assumes "dom_sl G" shows "0 < length ([] # replicate (length (g_V G) - 1) (rev [0..<length (cfg_doms.nodes G)]))"
norm_matrix_le_op_norm: fixes x :: "('b, 'c) vec"   and A :: "(('b, 'c) vec, 'd) vec" assumes "\<parallel>x\<parallel> = 1" shows "\<parallel>A *v x\<parallel> \<le> \<parallel>A\<parallel>\<^sub>o\<^sub>p"