hcju/isabelleps huggingface dataset:
"case": fixes f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and x1 :: "'l"   and x2 :: "'m" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
mem_Ch_CH: fixes x :: "'a"   and h :: "'c"   and X :: "'a set" assumes "x \<in> Ch h X" shows "x \<in> CH X"
that_2: fixes m :: "int"   and e :: "int" assumes "real_of_float f = real_of_int m * 2 powr real_of_int e"   and "odd m"   and "f \<noteq> 0" shows "thesis"
dehomogenize_monom_mult: fixes x :: "'b"   and c :: "'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "dehomogenize x (MPoly_Type_Class.punit.monom_mult c t p) = MPoly_Type_Class.punit.monom_mult c (except t {x}) (dehomogenize x p)"
wT_3: shows "X = w +o \<langle>T\<rangle>"
sel_2: fixes x1 :: "'b list"   and x2 :: "'b" shows "concl_of (Infer x1 x2) = x2"
CondReds2F: fixes e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr"   and e' :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>false,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>"
4: shows "LBINT x=0..1. - ((1 - (1 - q powr log q (1 - x)) ^ n) / (ln q * (1 - x))) = - (harm n / ln q)"
add_indices_def: fixes ns :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h"   and data :: "nat \<times> 'h" shows "add_indices ns data = (map_idx (\<lambda>h i. (fst h, i, snd h)) (fst ns) (fst data), fst data + length (fst ns), snd ns)"
const3_def: fixes x :: "'h" shows "const3 x \<equiv> \<lambda>_ _ _. x"
CondTrue_reds: fixes b :: "expr"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and s :: "char list \<Rightarrow> val option"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>if (b) c\<^sub>1 else c\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>"   and "\<lbrakk>b\<rbrakk>s = Some true" shows "\<langle>c\<^sub>1,s\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>"
eq1: shows "\<forall>Z\<in>Sols f U S t\<^sub>0 s. \<forall>t\<in>U s. Z t = Y t"
dependent_insertD: fixes a :: "'b"   and S :: "'b set" assumes "a \<notin> vs1.span S"   and "vs1.dependent (insert a S)" shows "vs1.dependent S"
fvi_less_nfv: fixes \<phi> :: "Formula.formula" shows "\<forall>i\<in>fv \<phi>. i < Formula.nfv \<phi>"
IH: fixes y :: "nat"   and \<alpha> :: "fm" assumes "y < n"   and "ground_fm \<alpha>"   and "size \<alpha> < y"   and "se_fm \<alpha>"   and "eval_fm e \<alpha>" shows "{} \<turnstile> \<alpha>"
intros_3: fixes u :: "'d \<Rightarrow> 'c"   and r :: "'d"   and d :: "'c" assumes "u r < d" shows "dbm_entry_val u (Some r) None (Lt d)"
max_2: fixes y :: "real" assumes "y \<in> connected_component_set ((\<lambda>t. x t - y t) -` {0::'a} \<inter> ({t0--t1} \<inter> {t0--t2})) t0" shows "dist t_ne t_max \<le> dist t_ne y"
initMsg_def: fixes ind :: "nat"   and m :: "('p, 'v) message" shows "initMsg ind m = (if \<exists>p. m = <p, inM \<exists>i<ind. procList ! i = p> then 1 else 0)"
linorder_axioms_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "class.linorder_axioms less_eq \<equiv> \<forall>x y. less_eq x y \<or> less_eq y x"
val_ifex: fixes b :: "'a bexp"   and s :: "'a \<Rightarrow> bool" shows "val_ifex (Boolean_Expression_Example.ifex_of b) s = bval b s"
bisims1_Invoke_stkD: fixes P :: "'addr J1_prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and esxs :: "'addr expr1 list \<times> 'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and M :: "String.literal"   and n' :: "nat" assumes "P,es,h \<turnstile> esxs [\<leftrightarrow>] (stk, loc, pc, None)"   and "pc < length (compEs2 es)"   and "compEs2 es ! pc = Invoke M n'" shows "\<exists>vs v stk'. stk = vs @ v # stk' \<and> length vs = n'"
spec_ops_distinct_fst: fixes ops :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" assumes "spec_ops ops deps" shows "distinct (map fst ops)"
chain_map_add: fixes p :: "nat"   and g :: "'d \<Rightarrow> 'c"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int" shows "chain_map p g (a + b) = chain_map p g a + chain_map p g b"
less_imp_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.preorder less_eq less"   and "less x y" shows "less_eq x y"
simps: fixes s1 :: "('a, 'b, 'c) SLin_state"   and s2 :: "('a, 'b, 'c) SLin_state" shows "P22 (s1, s2) = (initialized s2 \<longrightarrow> dstate (f (s1, s2)) = dstate s2)"
inca_axioms: shows "inca Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"
vertical_composite_axioms: shows "vertical_composite (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G H \<sigma> \<tau>"
measurable_count_space_extend: fixes A :: "'c set"   and B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure" assumes "A \<subseteq> B"   and "f \<in> space M \<rightarrow> A"   and "f \<in> M \<rightarrow>\<^sub>M count_space B" shows "f \<in> M \<rightarrow>\<^sub>M count_space A"
minus_mult_distrib_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and C :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat nr n"   and "C \<in> carrier_mat n nc" shows "(A - B) * C = A * C - B * C"
entailment_subset: fixes S2 :: "'at Literal set set"   and S1 :: "'at Literal set set" assumes "S2 \<subseteq> S1" shows "entails_formula S1 S2"
assert_comp_choice: fixes p :: "'b"   and S :: "'a \<Rightarrow> 'b"   and T :: "'a \<Rightarrow> 'b" shows "{. p .} \<circ> S \<sqinter> T = ({. p .} \<circ> S) \<sqinter> ({. p .} \<circ> T)"
frontier_def: fixes S :: "'c set" shows "frontier S = closure S - interior S"
rep_eq: fixes x :: "'a Executable_Relation.rel" shows "Image x = (``) (set_of_rel x)"
normalize_alt_simps_15: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and ys :: "'a expr list" shows "(\<Gamma> \<turnstile>\<^sub>L [] \<Rrightarrow> ys) = (ys = [])"
eq2: shows "mat 1 (dim_vec v) (\<lambda>(i, y). conjugate v $ y) * B = mat 1 (dim_vec (adjoint B *\<^sub>v v)) (\<lambda>(i, y). conjugate (adjoint B *\<^sub>v v) $ y)"
\<sigma>_def: fixes n :: "nat" shows "\<sigma> n = List.n_lists n [True, False]"
well_analyzed_singleton_2: fixes t :: "('fun, 'var) Term.term" shows "well_analyzed [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)]"
split_ivl_at_halfspace_def: fixes sctn1 :: "'b sctn"   and x :: "'b set" shows "split_ivl_at_halfspace sctn1 x = ivl_rep x \<bind> (\<lambda>(i, s). let sctn2 = Sctn (- normal sctn1) (- pstn sctn1) in restrict_to_halfspace sctn1 x \<bind> (\<lambda>x1. restrict_to_halfspace sctn2 x \<bind> (\<lambda>x2. RETURN (x1, x2))))"
assms_2: shows "(xs, S) \<in> \<langle>\<langle>A\<rangle>list_set_rel\<rangle>list_set_rel"
m2eq: shows "m2 = m1(x := ExprEval e m1)"
coprime_dvd_mult_left_iff: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "coprime a c" shows "(a dvd b * c) = (a dvd b)"
assms_1: shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t A Sec"
pdigraphI: fixes \<alpha> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "pdigraph_base \<alpha> I \<AA>"   and "\<And>J. J \<subseteq>\<^sub>\<circ> I \<Longrightarrow> dg_prod J \<AA>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "\<And>J A B. \<lbrakk>J \<subseteq>\<^sub>\<circ> I; A \<subseteq>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Obj\<rparr>; B \<subseteq>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Obj\<rparr>; A \<in>\<^sub>\<circ> Vset \<alpha>; B \<in>\<^sub>\<circ> Vset \<alpha>\<rbrakk> \<Longrightarrow> \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom (dg_prod J \<AA>) a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>" shows "pdigraph \<alpha> I \<AA>"
\<R>_mds: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds', mem\<^sub>2\<rangle>" shows "mds = mds'"
functor_axioms: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) ((\<otimes>\<^sub>C) (C.cod \<iota>\<^sub>C))"
heq: fixes x :: "real"   and x' :: "real" assumes "open_segment x x' \<inter> T = {}" shows "h x = h x'"
continuous_at: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "isCont f x = f \<midarrow>x\<rightarrow> f x"
set_less_aux_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubset>' B) = (finite A \<and> finite B \<and> (\<exists>y\<in>B - A. \<forall>z\<in>A - B \<union> (B - A). y \<le> z \<and> (z \<le> y \<longrightarrow> y = z)))"
elims: fixes x :: "nat"   and y :: "'a bro set" assumes "B x = y" obtains "x = 0" and "y = {N0}"   | h :: "nat" where "x = Suc h" and "y = {N2 t1 a t2 |t1 a t2. t1 \<in> T h \<and> t2 \<in> B h \<or> t1 \<in> B h \<and> t2 \<in> T h}"
retraction_is_epi: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b" assumes "category C"   and "category.retraction C g" shows "category.epi C g"
schroeder_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "(inf (times x y) z = bot) = (inf (times (conv x) z) y = bot)"
subword_def: fixes w :: "'c list"   and i :: "nat"   and j :: "nat" shows "subword w i j = take j (drop i w)"
MVT_ivl: fixes D :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and J :: "'c \<Rightarrow> 'c \<Rightarrow> 'd"   and J0 :: "'d"   and u :: "'c"   and a :: "'c" assumes "\<And>x. x \<in> D \<Longrightarrow> (f has_derivative J x) (at x within D)"   and "\<And>x. x \<in> D \<Longrightarrow> J0 \<le> J x u"   and "\<And>x. x \<in> {0..1} \<Longrightarrow> a + x *\<^sub>R u \<in> D" shows "J0 \<le> f (a + u) - f a"
simps_13: shows "extN.Top + extN.Bot = extN.N 1"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and H :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "category A"   and "category B"   and "functor A B F"   and "functor A B G"   and "functor A B H"   and "natural_transformation A B F G \<sigma>"   and "natural_transformation A B G H \<tau>" shows "vertical_composite A B F G H \<sigma> \<tau>"
inv_joinL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invc l"   and "invc r"   and "invh l"   and "invh r"   and "bheight l \<le> bheight r" shows "invc2 (joinL l x r) \<and> (bheight l \<noteq> bheight r \<and> color r = Black \<longrightarrow> invc (joinL l x r)) \<and> invh (joinL l x r) \<and> bheight (joinL l x r) = bheight r"
wiring_parallel_converter2: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>1' :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and fg :: "('i \<Rightarrow> 'k) \<times> ('l \<Rightarrow> 'j)"   and \<I>2 :: "('m, 'n) \<I>"   and \<I>2' :: "('o, 'p) \<I>"   and conv2 :: "('m, 'n, 'o, 'p) converter"   and fg' :: "('m \<Rightarrow> 'o) \<times> ('p \<Rightarrow> 'n)" assumes "wiring \<I>1 \<I>1' conv1 fg"   and "wiring \<I>2 \<I>2' conv2 fg'" shows "wiring (\<I>1 \<oplus>\<^sub>\<I> \<I>2) (\<I>1' \<oplus>\<^sub>\<I> \<I>2') (conv1 |\<^sub>= conv2) (fg |\<^sub>w fg')"
rtrancl_start: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "(x, y) \<in> r\<^sup>*"   and "P y"   and "\<not> P x" shows "\<exists>u v. (x, u) \<in> r\<^sup>* \<and> (u, v) \<in> r \<and> (v, y) \<in> r\<^sup>* \<and> \<not> P u \<and> P v"
integer_hull_def: fixes n :: "nat"   and P :: "'b vec set" shows "gram_schmidt.integer_hull n P = gram_schmidt.convex_hull n (P \<inter> \<int>\<^sub>v)"
IdFtor'PreFunctor: fixes C :: "('a, 'b, 'c) Category_scheme" assumes "Category C" shows "PreFunctor FId' C"
less_minus_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (uminus b) = less b (uminus a)"
infinite: fixes A :: "'a set" assumes "infinite A" shows "card A = 0"
G_3: shows "\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y"
eval_init_seq: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>INIT C (Cs,b) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>(INIT C (Cs,b) \<leftarrow> unit);; e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
map_project_def: fixes f :: "'c \<Rightarrow> 'b option"   and A :: "'c set" shows "List.map_project f A = {b. \<exists>a\<in>A. f a = Some b}"
simps: fixes xs :: "('a \<times> 'd) list"   and ko :: "'o" shows "lookup_raw (xs, ko) = lookup_pair ko xs"
simps_1: fixes x :: "'b" shows "init\<cdot>[x] = []"
assms_2: fixes i :: "'a" assumes "i \<le> least A"   and "i \<le> least B"   and "i \<in> B" shows "i \<in> A"
ide_char: fixes Obj :: "'b set"   and Null :: "'b"   and f :: "'b" assumes "discrete_category Obj Null" shows "partial_magma.ide (discrete_category.comp Obj Null) f = (f \<in> Obj)"
intro: fixes type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf known_ptr set_child_nodes set_child_nodes_locs" shows "l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
proots_card_pcompose_bij_eq: fixes q :: "'a poly"   and A :: "'a set"   and B :: "'a set"   and p :: "'a poly" assumes "bij_betw (poly q) A B"   and "p \<noteq> 0"   and "degree q = 1" shows "card (proots_within p B) = card (proots_within (p \<circ>\<^sub>p q) A)"
polytope_imp_convex: fixes S :: "'a set" assumes "polytope S" shows "convex S"
top_eq_inf_iff: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "(top = inf x y) = (x = top \<and> y = top)"
that: fixes T :: "'a set" assumes "T \<subseteq> S"   and "int (card T) = n"   and "finite T" shows "thesis"
one_MonoTran_def: shows "1 \<equiv> Abs_MonoTran id"
strict_prefix_simps_1: fixes xs :: "'c list" shows "strict_prefix xs [] = False"
pmf_of_set_union_split: fixes A :: "'a set"   and B :: "'a set"   and p :: "real" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "A \<union> B \<noteq> {}"   and "p = real (card A) / real (card A + card B)" shows "bernoulli_pmf p \<bind> (\<lambda>b. if b then pmf_of_set A else pmf_of_set B) = pmf_of_set (A \<union> B)"
var_def: fixes R :: "('c, 'd) ring_scheme" shows "X\<^bsub>R\<^esub> = [\<one>\<^bsub>R\<^esub>, \<zero>\<^bsub>R\<^esub>]"
inject: fixes x2 :: "'x"   and y2 :: "'x" shows "(Some x2 = Some y2) = (x2 = y2)"
assms_1: shows "safe_distance_2r < s\<^sub>o - s\<^sub>e"
differentiable_add: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f differentiable F"   and "g differentiable F" shows "(\<lambda>x. f x + g x) differentiable F"
simps_5: fixes edges' :: "'a edge set \<Rightarrow> 'a edge set"   and root :: "'a"   and edges :: "'a edge set"   and more :: "'b" shows "edges_update edges' \<lparr>root = root, edges = edges, \<dots> = more\<rparr> = \<lparr>root = root, edges = edges' edges, \<dots> = more\<rparr>"
smallomegaD_filterlim_at_infinity: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<omega>[F](g)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)" shows "LIM x F. f x / g x :> at_infinity"
contD: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and Y :: "'f set" assumes "cont luba orda lubb ordb f"   and "Complete_Partial_Order.chain orda Y"   and "Y \<noteq> {}" shows "f (luba Y) = lubb (f ` Y)"
substitute_simps_1: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree" shows "([] \<in> paths (substitute f T t)) = True"
p2r_neg_hom: fixes P :: "'b \<Rightarrow> bool" shows "p2r (- P) = rel_n (p2r P)"
simps_15: fixes p :: "fm"   and q :: "fm" shows "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
body_def: fixes fundef :: "'b fundef" shows "body fundef = (case fundef of Fundef x1 x2 \<Rightarrow> x1)"
ResFresh: fixes x :: "name"   and P :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>P \<equiv>\<^sub>e P"
l_get_mode_axioms: shows "l_get_mode ShadowRootClass.type_wf get_mode get_mode_locs"
is_forward_sim_def: fixes f :: "'b \<Rightarrow> 'c set"   and B :: "('b, 'd) ioa"   and A :: "('c, 'd) ioa" shows "is_forward_sim f B A \<equiv> (\<forall>s\<in>start B. f s \<inter> start A \<noteq> {}) \<and> (\<forall>s s' t a. s' \<in> f s \<and> s \<midarrow>a\<midarrow>B\<longrightarrow> t \<and> reachable B s \<longrightarrow> (\<exists>e. fst e = s' \<and> last_state e \<in> f t \<and> is_exec_frag_of A e \<and> (let tr = trace (ioa.asig A) e in if a \<in> externals (ioa.asig A) then tr = [a] else tr = [])))"
less_cSupE: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and X :: "'b set" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "less y (Sup X)"   and "X \<noteq> {}" obtains x :: "'b" where "x \<in> X" and "less y x"
max: shows "max_lideal \<ww> carrier_local_ring_at add_local_ring_at mult_local_ring_at zero_local_ring_at one_local_ring_at"
transp_ord_option: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp ord" shows "transp (ord_option ord)"
elem_0_val_if: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x = 0" shows "x \<in> carrier (Vr K v) \<and> x\<^bsup>\<hyphen> K\<^esup> \<in> carrier (Vr K v)"
topD: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and x :: "'c" assumes "Semilat.top r T" shows "x \<sqsubseteq>\<^bsub>r\<^esub> T"
sqrt_rat_def: fixes x :: "rat" shows "sqrt_rat x = root_rat 2 x"
encode_problem_has_model_then_also_partial_encodings_1: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "\<A> \<Turnstile> \<Phi>\<^sub>I \<Pi>"
hoare_assume: fixes P :: "'a"   and R :: "'a"   and Q :: "'a" shows "\<Turnstile> P{| [. R .] |}Q = (P \<sqinter> R \<sqsubseteq> Q)"
outside_compact_in_open: fixes S :: "'a set"   and T :: "'a set" assumes "compact S"   and "open T"   and "S \<subseteq> T"   and "T \<noteq> {}" shows "outside S \<inter> T \<noteq> {}"
Bseq_mult: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "Bseq f"   and "Bseq g" shows "Bseq (\<lambda>x. f x * g x)"
ladder_prev_n_def: fixes L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" shows "ladder_prev_n L index = (if index = 0 then 0 else ladder_n L (index - 1))"
HoVH_eq_ToCT: shows "HoVH = H.ToCT"
UN_constant: fixes c :: "'d set"   and A :: "'e set" shows "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
finite_dimensional_vector_space_ow_axioms: shows "finite_dimensional_vector_space_ow U\<^sub>M\<^sub>_\<^sub>2 (+) (0::'c) (-) uminus (*s\<^sub>2) basis2"
snd_snd_Gauss_Jordan_column_k_iarrays_PA_eq: fixes P :: "'b iarray iarray"   and i :: "nat"   and A :: "'b iarray iarray"   and k :: "nat" shows "snd (snd (Gauss_Jordan_column_k_iarrays_PA (P, i, A) k)) = snd (Gauss_Jordan_column_k_iarrays (i, A) k)"
flow_collect: fixes f :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal" assumes "flow \<Delta>'' f" shows "flow \<Delta> (\<lambda>(x, y). f (Edge x y, Vertex y))"
inf_continuous_mono: fixes F :: "'b \<Rightarrow> 'c" assumes "inf_continuous F" shows "mono F"
commute: fixes a :: "'h"   and b :: "'h" shows "sup a b = sup b a"
order_def: fixes S :: "('a, 'b) monoid_scheme" shows "order S = card (carrier S)"