hcju/isabelleps huggingface dataset:
Smith_Impl_axioms: shows "Smith_Impl Smith_1x2 Smith_2x2 div_op"
SPEC_eq_is_RETURN_2: fixes y :: "'c" shows "SPEC (\<lambda>x. x = y) = RETURN y"
decisionLess_def: shows "decisionLess = {(l1, l2). isDecision l1 \<and> \<not> isDecision l2}"
gets_def: fixes f :: "'c \<Rightarrow> 'd" shows "gets f \<equiv> do s \<leftarrow> get; return (f s) od"
rel_map_1: fixes Sb :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'d \<Rightarrow> 'b"   and x :: "'d list"   and y :: "'c list" shows "list_all2 Sb (map i x) y = list_all2 (\<lambda>x. Sb (i x)) x y"
assms: fixes g :: "'a set" assumes "g \<in> G" shows "g \<noteq> {}"
rationalizable_binary_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "rationalizable_binary_on A f = (\<forall>B\<subseteq>A. f B = {y \<in> B. \<forall>x\<in>B. y \<in> f {x, y}})"
YY: shows "complex_of_real (cmod (inner_prod v (X N *\<^sub>v v)) - cmod (inner_prod v (A *\<^sub>v v))) < - inner_prod v (A *\<^sub>v v)"
assms_3: fixes s :: "'a" shows "<I s * true> bi s <\<lambda>r. I s * \<up> (r = b s)>\<^sub>t"
int_msb_and: fixes x :: "int"   and y :: "int" shows "msb (x AND y) = (msb x \<and> msb y)"
simps_4: fixes x21 :: "freeRes"   and x22 :: "pi"   and x11 :: "name"   and x12 :: "name \<Rightarrow> pi noption" shows "FreeR_Rep x21 x22 \<noteq> BoundOutputR_Rep x11 x12"
imp_map_def: fixes is_map :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'c \<Rightarrow> assn" shows "imp_map is_map \<equiv> precise is_map"
ext_p_def: fixes n :: "nat" shows "ext_p n = q @ mk_cycles_path n c @ r"
quad_form_congruence: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and z :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0" shows "quad_form (M *\<^sub>m\<^sub>v z) (congruence (mat_inv M) H) = quad_form z H"
fmpredD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and x :: "'c"   and y :: "'d" assumes "fmpred P m"   and "fmlookup m x = Some y" shows "P x y"
INF_cong: fixes A :: "'k set"   and B :: "'k set"   and C :: "'k \<Rightarrow> 'l"   and D :: "'k \<Rightarrow> 'l"   and Inf :: "'l set \<Rightarrow> 'l" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
reflect_poly_reflect_poly: fixes p :: "'b poly" assumes "coeff p 0 \<noteq> (0::'b)" shows "reflect_poly (reflect_poly p) = p"
sum_image: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite X" shows "sum f X = \<Sum> (f ` X)"
comp_converter_map_converter1: fixes f :: "'h \<Rightarrow> 'n"   and g :: "'o \<Rightarrow> 'i"   and h :: "'p \<Rightarrow> 'l"   and k :: "'m \<Rightarrow> 'q"   and conv :: "('n, 'o, 'p, 'q) converter"   and conv' :: "('l, 'm, 'j, 'k) converter" shows "map_converter f g h k conv \<odot> conv' = map_converter f g id id (conv \<odot> map_converter h k id id conv')"
sublocale_sas_plus_finite_domain_representation_ii_3: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "dom (\<Psi>\<^sub>I\<^sub>+) = set (\<Psi>\<^sub>\<V>\<^sub>+)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x xs. \<lbrakk>P (filter (\<lambda>y. y < x) xs); P (filter ((<) x) xs)\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P a0"
Zero_def: shows "quiver.Zero \<equiv> MkArr False False []"
winding_number_def: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" shows "winding_number \<gamma> z \<equiv> SOME n. \<forall>e>0. \<exists>p. winding_number_prop \<gamma> z e p n"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "Ik.satL \<xi> (Neg a) = (\<not> Ik.satA \<xi> a)"
scale_minus_left: fixes a :: "'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" shows "- a \<odot> x = - (a \<odot> x)"
S2TTP_sender: fixes S :: "agent"   and AO :: "nat"   and K :: "nat"   and R :: "agent"   and hs :: "msg"   and evs :: "event list" assumes "Crypt (pubK TTP) \<lbrace>Agent S, Number AO, Key K, Agent R, hs\<rbrace> \<in> used evs"   and "Key K \<notin> analz (knows Spy evs)"   and "evs \<in> certified_mail" shows "\<exists>m ctxt q. hs = Hash \<lbrace>Number ctxt, Nonce q, response S R q, Crypt K (Number m)\<rbrace> \<and> Says S R \<lbrace>Agent S, Agent TTP, Crypt K (Number m), Number AO, Number ctxt, Nonce q, Crypt (pubK TTP) \<lbrace>Agent S, Number AO, Key K, Agent R, hs\<rbrace>\<rbrace> \<in> set evs"
shiftl_word_eq: fixes w :: "'b word"   and n :: "nat" shows "w << n = push_bit n w"
are_all_operators_applicable_in_def: fixes s :: "'a \<Rightarrow> 'b option"   and ops :: "('a, 'b) sas_plus_operator list" shows "are_all_operators_applicable_in s ops \<equiv> list_all (is_operator_applicable_in s) ops"
bal_eq: fixes n :: "nat"   and xs :: "'b list" shows "bal n xs = Balance.bal n xs"
intros_1: fixes \<sigma> :: "'tp"   and c :: "('fsym, 'psym) lit list" assumes "infTp \<sigma>" shows "\<sigma> \<turnstile>2 c"
I: shows "((\<lambda>t. sum_upto \<chi> t * complex_of_real (f' t)) has_integral sum_upto \<chi> b * complex_of_real (f b) - sum_upto \<chi> a * complex_of_real (f a) - (\<Sum>n\<in>real -` {a<..b}. \<chi> n * complex_of_real (f (real n)))) {a..b}"
convex_prod: fixes P :: "'b \<Rightarrow> real \<Rightarrow> bool" assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}" shows "convex {x. \<forall>i\<in>Basis. P i (x \<bullet> i)}"
le_minus_one_simps_2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus one) one"
of_nat_le_iff: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m \<le> n)"
card_of_empty1: fixes r :: "('n \<times> 'n) set" assumes "Well_order r \<or> Card_order r" shows "|{}| \<le>o r"
compile: shows "rec_ci (Mn n f) = (ap, arity, fp)"
single_use: fixes rs :: "(proc \<times> val) list"   and r :: "proc \<times> val" shows "\<bottom> \<star> append rs [r] = Some (snd r)"
assms_3: fixes x :: "'a" assumes "P x" shows "\<exists>x'. (x, x') \<in> R \<and> P' x'"
ok_iff_Allowed: fixes F :: "'c program"   and G :: "'c program" shows "F ok G = (F \<in> Allowed G \<and> G \<in> Allowed F)"
fds_scale_0: fixes f :: "'b fds" shows "fds_scale 0 f = 0"
retract_of_empty_2: fixes S :: "'a set" shows "(S retract_of {}) = (S = {})"
cop_2: shows "comm_monoid_mult_class.coprime (lead_coeff g) p"
find_Some: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list"   and p :: "'a" assumes "List.find P l = Some p" shows "p \<in> set l \<and> P p"
append_butlast_last_id: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "butlast xs @ [List.last xs] = xs"
get_dom_component_get_child_nodes: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "(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 \<in> set c) = (ptr' \<in> set c)"
matrix_to_iarray_nrows: fixes A :: "(('c, 'd) vec, 'e) vec" shows "nrows A = nrows_iarray (matrix_to_iarray A)"
no_Emptys_hps: fixes h :: "'b heap" assumes "no_Empty h" shows "no_Emptys (hps h)"
linear_scale: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and r :: "'a"   and b :: "('a, 'b) vec" assumes "linear (*s) (*s) f" shows "f (r *s b) = r *s f b"
consis': shows "sharing_consistent \<S> \<O> (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
cong: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "f = g"   and "x = y" shows "f x = g y"
step_invariantD: fixes A :: "('d, 'e) automaton"   and I :: "'e \<Rightarrow> bool"   and P :: "'d \<times> 'e \<times> 'd \<Rightarrow> bool"   and s :: "'d"   and a :: "'e"   and s' :: "'d" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "P (s, a, s')"
Nmul_def: shows "(*\<^sub>N) = (\<lambda>(a, b) (a', b'). let g = gcd (a * a') (b * b') in (a * a' div g, b * b' div g))"
intros_2: fixes \<Gamma> :: "(name \<times> ty) list"   and t1 :: "llam"   and T1 :: "ty"   and T2 :: "ty"   and t2 :: "llam" assumes " \<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2 "   and " \<Gamma> \<turnstile> t2 : T1 " shows " \<Gamma> \<turnstile> lApp t1 t2 : T2 "
list_of_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lfinite xs"   and "lfinite ys" shows "list_of (lappend xs ys) = list_of xs @ list_of ys"
leg: fixes n :: "nat" shows "g n \<le>\<^sub>L M"
qfree_list_disj: fixes fs :: "'a fm list" assumes "\<forall>\<phi>\<in>set fs. qfree \<phi>" shows "qfree (list_disj fs)"
iv2: shows "2 ^ r \<le> size (tree_to_multiset (Node e2 a2 r ts2))"
M'_eq: shows "M' = distr M (Pi\<^sub>M UNIV (\<lambda>i. borel)) (\<lambda>x. \<lambda>i\<in>UNIV. X i x)"
dom_c: shows "\<D>\<^sub>\<circ> c = K"
assms: fixes sctn :: "'a sctn" assumes "sctn \<in> sctns" shows "normal sctn \<noteq> (0::'a)"
chain: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph"   and i :: "nat"   and j :: "nat" assumes "chain S"   and "i \<le> j" shows "subgraph (S i) (S j)"
hyps_4: fixes b :: "(W.tvar \<times> ty) list" shows " b<\<Gamma>> \<turnstile> t1 : b<T1> "
fold_equality: fixes z :: "'a\<^sub>m option"   and A :: "'a\<^sub>m set"   and y :: "'a\<^sub>m option" assumes "fold_graph (\<lambda>x yo. yo \<bind> m x) z A y" shows "Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z A = y"
dg_Rel_is_arrD_3: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> B" shows "S\<lparr>ArrCod\<rparr> = B"
ccard_freq: fixes n :: "nat"   and n' :: "nat"   and Q :: "nat \<Rightarrow> bool"   and cnf :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "n \<le> n'"   and "card {i. n < i \<and> i \<le> n' \<and> Q i} + cnf < card {i. n < i \<and> i \<le> n' \<and> P i}" shows "\<exists>n' n''. cnf < card {i. n' < i \<and> i \<le> n'' \<and> P i} \<and> card {i. n' < i \<and> i \<le> n'' \<and> Q i} \<le> cnf"
openin_path_component_locally_path_connected: fixes S :: "'b set"   and x :: "'b" assumes "locally path_connected S" shows "openin (top_of_set S) (path_component_set S x)"
simps: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "app (Fn f) x = f x"
cases: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "a \<in> nnfis f M" obtains u :: "nat \<Rightarrow> 'b \<Rightarrow> real" and x :: "nat \<Rightarrow> real" and y :: "real" where "a = y" and "u\<up>f" and "\<And>n. x n \<in> sfis (u n) M" and "x\<up>y"
supp_plus_perm: fixes p :: "perm"   and q :: "perm" shows "supp (p + q) \<subseteq> supp p \<union> supp q"
normalize_def'_2: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R" shows "ring.normalize R p = drop (length p - length (ring.normalize R p)) p"
is_offending_flows_min_set_minimalize_offending_overapprox: fixes G :: "'a graph"   and ff :: "('a \<times> 'a) list"   and nP :: "'a \<Rightarrow> 'b" assumes "sinvar_mono"   and "wf_graph G"   and "is_offending_flows (set ff) G nP"   and "set ff \<subseteq> edges G"   and "distinct ff" shows "is_offending_flows_min_set (set (minimalize_offending_overapprox ff [] G nP)) G nP"
product_run_project_fst: fixes \<delta>\<^sub>1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and \<delta>\<^sub>2 :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>1 :: "'a"   and q\<^sub>2 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "project_transition_fst (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w i) = run\<^sub>t \<delta>\<^sub>1 q\<^sub>1 w i"
face_cycle_succ_cyclic: fixes a :: "'b" shows "cyclic_on GM.face_cycle_succ (GM.face_cycle_set a)"
norm_minus_commute: fixes a :: "'b"   and b :: "'b" shows "norm (a - b) = norm (b - a)"
open_PInfty2: fixes A :: "ereal set" assumes "open A"   and "\<infinity> \<in> A" obtains x :: "real" where "{ereal x<..} \<subseteq> A"
cases: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "char list exp list"   and a3 :: "ty list" assumes "P,a1 \<turnstile> a2 [::] a3" obtains E :: "char list \<Rightarrow> ty option" where "a1 = E__" and "a2 = []" and "a3 = []"   | E :: "char list \<Rightarrow> ty option" and e :: "char list exp" and T :: "ty" and es :: "char list exp list" and Ts :: "ty list" where "a1 = E" and "a2 = e # es" and "a3 = T # Ts" and "P,E \<turnstile> e :: T" and "P,E \<turnstile> es [::] Ts"
of_radiant_plus_2pi: fixes \<omega> :: "real" shows "of_radiant (\<omega> + 2 * pi) = of_radiant \<omega>"
mk_minsky_seq': fixes P :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" assumes "mk_minsky P"   and "mk_minsky Q" shows "mk_minsky (\<lambda>vs vs''. \<exists>vs'. P vs vs' \<and> Q vs' vs'')"
open_cover_of_subset_axioms_def: fixes is_open :: "'c set \<Rightarrow> bool"   and I :: "real set"   and C :: "real \<Rightarrow> 'c set" shows "open_cover_of_subset_axioms is_open I C \<equiv> \<forall>i. i \<in> I \<longrightarrow> is_open (C i)"
canonical_saturated_2: fixes X :: "'b set"   and v :: "'b \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'b"   and r :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and c2 :: "'b" assumes "Beta_Regions' X v n not_in_X"   and "Le r \<le> M 0 (v c2)"   and "Le (- r) \<le> M (v c2) 0"   and "cycle_free M n"   and "canonical M n"   and "v c2 \<le> n"   and "0 < v c2"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" obtains u :: "'b \<Rightarrow> 'c" where "u \<in> [M]\<^bsub>v,n\<^esub>" and "u c2 = - r"
prems_7: shows "R \<subseteq> {v. v \<turnstile> g}"
singleton_lst_proj_1: fixes l :: "'b"   and a :: "'a" shows "proj_unl l [(ln l, a)] = [a]"
elims: fixes x :: "'c \<Rightarrow> bool"   and xa :: "'c option"   and y :: "'c option" assumes "filter_option x xa = y" obtains P :: "'c \<Rightarrow> bool" where "x = P__" and "xa = None" and "y = None"   | P :: "'c \<Rightarrow> bool" and xb :: "'c" where "x = P" and "xa = Some xb" and "y = (if P xb then Some xb else None)"
sw_def: fixes ys :: "'c"   and y1 :: "'b"   and y2 :: "'b"   and xs :: "'c"   and x :: "'b" shows "x @xs[y1 \<and> y2]_ys \<equiv> if ys = xs then if x = y1 then y2 else if x = y2 then y1 else x else x"
Block_undefined: fixes a :: "'a" assumes "a \<notin> S" shows "Block a = undefined"
IH: fixes F :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" shows "(F, F, \<Gamma> \<Rightarrow> \<Delta> \<down> n \<longrightarrow> F, \<Gamma> \<Rightarrow> \<Delta> \<down> n) \<and> (\<Gamma> \<Rightarrow> F, F, \<Delta> \<down> n \<longrightarrow> \<Gamma> \<Rightarrow> F, \<Delta> \<down> n)"
image_cong_simp: fixes M :: "'a set"   and N :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = N"   and "\<And>x. x \<in> N =simp=> f x = g x" shows "f ` M = g ` N"
add_monom: fixes a :: "'b"   and n :: "nat"   and b :: "'b" shows "monom a n + monom b n = monom (a + b) n"
fst_comp_map_prod: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "fst \<circ> map_prod f g = f \<circ> fst"
that: fixes x :: "'a" assumes "x \<in> src.carrier" shows "f x = g x"
MinkowskiBetweenness_axioms: shows "MinkowskiBetweenness \<E> \<P> betw"
a_set_tag_name_locs_def: fixes element_ptr :: "(_) element_ptr" shows "a_set_tag_name_locs element_ptr \<equiv> all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name_update)"
plus_mat0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and nr :: "nat"   and nc :: "nat"   and m :: "'c list list" assumes "class.monoid_add plus zero"   and "mat nr nc m" shows "mat_plusI plus m (mat0I zero nr nc) = m"
suffixes_fin: fixes x :: "'c list" shows "finite {t. \<exists>t1. x = t1 @ t}"
poly_to_univar: fixes p :: "real mpoly"   and var :: "nat"   and A :: "real mpoly"   and B :: "real mpoly"   and C :: "real mpoly"   and a :: "real"   and xs' :: "real list"   and y :: "real"   and xs :: "real list"   and b :: "real"   and c :: "real"   and x :: "real" assumes "MPoly_Type.degree p var < 3"   and "get_coeffs var p = (A, B, C)"   and "a = insertion (nth_default 0 (xs' @ y # xs)) A"   and "b = insertion (nth_default 0 (xs' @ y # xs)) B"   and "c = insertion (nth_default 0 (xs' @ y # xs)) C"   and "length xs' = var" shows "insertion (nth_default 0 (xs' @ x # xs)) p = a * x\<^sup>2 + b * x + c"
Torder_adjunction: fixes X :: "'a set"   and a :: "'a" assumes "X \<subseteq> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>X. x \<preceq> a"   and "Torder (Iod D X)" shows "Torder (Iod D (X \<union> {a}))"
chain_reparam_weak_path_pathimg_subset: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "chain_reparam_weak_path \<gamma> subdiv" shows "\<forall>(k', \<gamma>'')\<in>subdiv. path_image \<gamma>'' \<subseteq> path_image \<gamma>"
of_nat_nat_def: fixes n :: "nat" shows "of_nat_class.of_nat n = n"
E_def: shows "E = C \<inter> {(int p - 1) div 2<..}"
leg0_commutes: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "C (Leg0 (Cod \<mu>)) (Chn \<mu>) = Leg0 (Dom \<mu>)"
almost_full_onI: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>f. \<forall>i. f i \<in> A \<Longrightarrow> good P f" shows "almost_full_on P A"
x: shows "r|\<tau> @ \<upsilon>| + r|\<sigma>''| -s r \<down>l (\<tau> @ \<upsilon>) = r|\<tau> @ \<upsilon>| + r|\<sigma>''| -s r \<down>l \<upsilon> -s r \<down>l \<tau>"
mod_mod_trivial: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (modulo a b) b = modulo a b"
assms_4: fixes a :: "act"   and P' :: "ccs"   and P'' :: "ccs"   and \<C> :: "'a" assumes "P \<longmapsto>a \<prec> P'"   and "!P \<longmapsto>coAction a \<prec> P''"   and "\<And>\<C>. Prop \<C> !P (coAction a) P''"   and "a \<noteq> \<tau>" shows "Prop \<C> (P \<parallel> !P) (\<tau>) (P' \<parallel> P'')"
gpv_pred_mono_strong: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and x :: "('d, 'e, 'f) gpv"   and P' :: "'d \<Rightarrow> bool"   and Q' :: "'e \<Rightarrow> bool" assumes "pred_gpv P Q x"   and "\<And>a. \<lbrakk>a \<in> results'_gpv x; P a\<rbrakk> \<Longrightarrow> P' a"   and "\<And>b. \<lbrakk>b \<in> outs'_gpv x; Q b\<rbrakk> \<Longrightarrow> Q' b" shows "pred_gpv P' Q' x"
ln_Gamma_series'_aux: fixes z :: "complex" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(\<lambda>k. z / of_nat (Suc k) - Ln (1 + z / of_nat (Suc k))) sums (ln_Gamma z + euler_mascheroni * z + Ln z)"
rcons_correct_1: fixes t :: "('a, 'b) FingerTreeStruc"   and a :: "'a \<times> 'b" assumes "FingerTreeStruc.ft_invar t" shows "FingerTreeStruc.ft_invar (FingerTreeStruc.rcons t a)"
kuratowski_def: shows "kuratowski \<equiv> \<lambda>x. K\<^bsub>3,3\<^esub> (with_proj x) \<or> K\<^bsub>5\<^esub> (with_proj x)"
in_set_conv_nth: fixes x :: "'e"   and xs :: "'e list" shows "(x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)"
r_parallel_2: fixes i :: "nat"   and f :: "recf"   and j :: "nat"   and g :: "recf"   and x :: "nat" assumes "\<And>x. eval r_phi [i, x] = eval f [x]"   and "\<And>x. eval r_phi [j, x] = eval g [x]"   and "eval f [x] \<down> \<and> eval g [x] \<up>" shows "eval r_parallel [i, j, x] \<down>= prod_encode (0, the (eval f [x]))"
precise_def: fixes P :: "'a \<Rightarrow> bool" shows "precise P = (\<forall>h hp hp'. hp \<preceq> h \<and> P hp \<and> hp' \<preceq> h \<and> P hp' \<longrightarrow> hp = hp')"
intros_1: fixes A :: "('b, unit) Old_Datatype.node set set" shows "NIL \<in> list A"
convolution_zero: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f = 0 \<or> g = 0" shows "convolution f g = 0"
strip_while_snoc: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" shows "strip_while P (xs @ [x]) = (if P x then strip_while P xs else xs @ [x])"
o0: shows "Abs_fps (\<lambda>n. 1::'a) $ 0 \<noteq> (0::'a)"
name_exists_fresh: fixes x :: "'a" obtains c :: "name" where "c \<sharp> x"
LeftDerivation_implies_leftderives: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "LeftDerivation a D b" shows "leftderives a b"
assms_5: shows "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'"
in_ccApprox: fixes x :: "Vars.var"   and y :: "Vars.var"   and t :: "Vars.var ttree" shows "x--y\<in>ccApprox t = (\<exists>xs\<in>paths t. x--y\<in>ccFromList xs)"
rel_refl_strong: fixes x :: "'c llist"   and Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z. z \<in> lset x \<Longrightarrow> Ra z z" shows "llist_all2 Ra x x"
Y_elem_eq: shows "(\<lambda>X. X - {elem}) ` {Y} = {Y - {elem}}"
alist_and': fixes \<gamma> :: "'c \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'b \<Rightarrow> bool"   and as :: "'c negation_type list" shows "matches (\<gamma>, \<alpha>) (alist_and' as) = matches (\<gamma>, \<alpha>) (alist_and as)"
preciseD: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a :: "'a"   and p :: "'b"   and F :: "assn"   and a' :: "'a"   and F' :: "assn"   and h :: "heap \<times> nat set" assumes "precise R"   and "h \<Turnstile> R a p * F \<and>\<^sub>A R a' p * F'" shows "a = a'"
not_neutral_obtains_not_neutral: fixes g :: "'c \<Rightarrow> 'a" assumes "G g \<noteq> \<^bold>1" obtains a :: "'c" where "g a \<noteq> \<^bold>1"
axioms_1: fixes G :: "('c, 'd) pre_digraph" assumes "sym_digraph G" shows "wf_digraph G"
gcd_mult_left_right_cancel: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one b c" shows "gcd (times a c) b = gcd a b"
w_simps_3: shows "src HHfgh_HfHgh.chine = src (tab\<^sub>0 (local.dom (\<alpha> (f, g, h))))"