hcju/isabelleps huggingface dataset:
f2: shows "\<forall>e n. e \<notin> set (history n) \<or> pre n e @ [e] prefix of n"
assoc: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word" shows "signed.max (signed.max a b) c = signed.max a (signed.max b c)"
monoid_def: fixes M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b" shows "Group_Theory.monoid M composition unit \<equiv> ((\<forall>a b. a \<in> M \<longrightarrow> b \<in> M \<longrightarrow> composition a b \<in> M) \<and> unit \<in> M) \<and> (\<forall>a b c. a \<in> M \<longrightarrow> b \<in> M \<longrightarrow> c \<in> M \<longrightarrow> composition (composition a b) c = composition a (composition b c)) \<and> (\<forall>a. a \<in> M \<longrightarrow> composition unit a = a) \<and> (\<forall>a. a \<in> M \<longrightarrow> composition a unit = a)"
Neg_I: fixes A :: "fm"   and H :: "fm set" assumes "insert A H \<turnstile> Fls" shows "H \<turnstile> Neg A"
del_vert_simps_4: fixes u :: "'a" shows "head (del_vert u) = head G"
iTILL_mult: fixes n :: "nat"   and k :: "nat" shows "[\<dots>n] \<otimes> k = [ 0, mod k, n ]"
set_lebesgue_integral_eq_integral_1: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "f absolutely_integrable_on S" shows "f integrable_on S"
simps_7: fixes n :: "'n"   and k :: "'k"   and \<phi> :: "('a, 'k) aformula" shows "local.wf n (FAll k \<phi>) = local.wf (SUC k n) \<phi>"
all: shows "\<forall>x. v \<noteq> x \<longrightarrow> \<alpha> x = \<beta> x"
ba_ab: fixes b :: "'b" assumes "b \<in> B" shows "a_to_b (b_to_a b) = b"
perfect_hiding_ind_cpa_def: fixes key_gen :: "('a \<times> 'b) spmf"   and commit :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf"   and valid_msg :: "'c \<Rightarrow> bool"   and \<A> :: "('b \<Rightarrow> (('c \<times> 'c) \<times> 'f) spmf) \<times> ('d \<Rightarrow> 'f \<Rightarrow> bool spmf)" shows "abstract_commitment.perfect_hiding_ind_cpa key_gen commit valid_msg \<A> \<equiv> abstract_commitment.hiding_advantage_ind_cpa key_gen commit valid_msg \<A> = 0"
lf_1: shows "Polynomial.smult (lead_coeff u) u = f * g"
lset_induct': fixes x :: "'c"   and xs :: "'c llist"   and P :: "'c llist \<Rightarrow> bool" assumes "x \<in> lset xs"   and "\<And>xs. P (LCons x xs)"   and "\<And>x' xs. \<lbrakk>x \<in> lset xs; P xs\<rbrakk> \<Longrightarrow> P (LCons x' xs)" shows "P xs"
List_def: fixes h :: "nat \<Rightarrow> nat option"   and x :: "nat"   and as :: "nat list" shows "List h x as = Path h x as 0"
tendsto_Lim: fixes net :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "net \<noteq> \<bottom>"   and "(f \<longlongrightarrow> l) net" shows "Lim net f = l"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, s\<rangle> \<Rightarrow> t"
fib_final_def: shows "fib_final \<equiv> Abs_cfun ((!!) fib_work_final)"
finite_Int: fixes F :: "'e set"   and G :: "'e set" assumes "finite F \<or> finite G" shows "finite (F \<inter> G)"
splitFace_new_f12_norm: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF newVs"   and "minGraphProps g"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs" shows "normFace f12 \<notin> set (normFaces (faces g))"
steps_append: fixes A :: "'a \<times> ('b \<Rightarrow> 'a \<Rightarrow> 'a set) \<times> ('a \<Rightarrow> bool)"   and v :: "'b list"   and w :: "'b list" shows "steps A (v @ w) = steps A v O steps A w"
const_def: fixes c :: "nat" shows "const c = (\<forall>x\<in>{i. i < 2 ^ n}. f x = c)"
GmultD: fixes g :: "'g"   and v :: "'v" shows "g *\<cdot> v = ((1::'f) \<delta>\<delta> g) \<cdot> v"
assms_1: shows "chain_on (LEX P) C (SEQ A)"
e_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.e p q = card (QR.E p q)"
finite_dimension_simple_extension: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R" shows "finite_dimension K (simple_extension K x) = (algebraic over K) x"
decseq_def: fixes X :: "nat \<Rightarrow> 'b" shows "decseq X = (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
foldli_map: fixes f1 :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and c :: "'d \<Rightarrow> bool"   and f2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldli (map f1 xs) c f2 \<sigma> = foldli xs c (f2 \<circ> f1) \<sigma>"
chine_in_hom_1: shows "\<guillemotleft>chine : src r\<^sub>0 \<rightarrow> src s\<^sub>0\<guillemotright>"
set_sorted_domain_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "set (sorted_domain_raw ko xs) = fst ` set (fst xs)"
polyhedron_eq_finite_faces: fixes S :: "'a set" shows "polyhedron S = (closed S \<and> convex S \<and> finite {F. F face_of S})"
is_proT2: fixes s :: "'c event list"   and X :: "'c event set"   and P :: "'c process" assumes "(s, X) \<in> F P"   and "s \<noteq> []" shows "tick \<notin> set (tl (rev s))"
mono_neutral_cong: fixes T :: "'b set"   and S :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite T"   and "finite S"   and "\<And>i. i \<in> T - S \<Longrightarrow> h i = (0::'c)"   and "\<And>i. i \<in> S - T \<Longrightarrow> g i = (0::'c)"   and "\<And>x. x \<in> S \<inter> T \<Longrightarrow> g x = h x" shows "sum g S = sum h T"
lincomb_zero: fixes U :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "U \<subseteq> carrier_vec n"   and "a \<in> U \<rightarrow> {0::'a}" shows "lincomb a U = 0\<^sub>v n"
set_of_variables_is_finite_lit: fixes l :: "'b literal" shows "finite (vars_of_lit l)"
swapidseq_ext_imp_swapidseq: fixes S :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a" assumes "swapidseq_ext S n p" shows "swapidseq n p"
prems_2: fixes i :: "nat" assumes "i < Suc n" shows "(nderiv i f {0<..} \<longlongrightarrow> f0 i) (at_right 0)"
supp_trans: fixes a :: "'a"   and e :: "exp" shows "supp (trans a e) \<subseteq> supp e"
choice_axiom_def: shows "choice_axiom \<equiv> \<langle> Game hgid1 \<union>\<union> Game hgid2 \<rangle> pusall \<leftrightarrow> (\<langle> Game hgid1 \<rangle> pusall || \<langle> Game hgid2 \<rangle> pusall)"
code_free_basis_hull: fixes C :: "'b list set" assumes "code C" shows "C = \<BB>\<^sub>F \<langle>C\<rangle>"
mat_map: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and f :: "'c \<Rightarrow> 'c" assumes "mat nr nc m" shows "mat nr nc (mat_map f m)"
fdisjnt_subset_right: fixes N' :: "'c fset"   and N :: "'c fset"   and M :: "'c fset" assumes "N' |\<subseteq>| N"   and "fdisjnt M N" shows "fdisjnt M N'"
simps_3: fixes x :: "int"   and y :: "int" shows "Num_arc x + Num_arc y = Num_arc (max x y)"
lspasl_eq: fixes Gamma :: "bool"   and h2 :: "'a"   and h1 :: "'a"   and Delta :: "bool" assumes "Gamma \<and> (0::'a,h2\<triangleright>h2) \<and> h1 = h2 \<longrightarrow> Delta" shows "Gamma \<and> (0::'a,h1\<triangleright>h2) \<longrightarrow> Delta"
dom_def: shows "dom = flow0 x ` {0..} \<union> {p}"
Keys_not_empty: fixes a :: "'c \<Rightarrow>\<^sub>0 'd"   and A :: "('c \<Rightarrow>\<^sub>0 'd) set" assumes "a \<in> A"   and "a \<noteq> 0" shows "Keys A \<noteq> {}"
insert_inorder: fixes t :: "'a btree"   and k :: "nat"   and x :: "'a" assumes "sorted_less (inorder t)" shows "inorder (abs_split.insert k x t) = ins_list x (inorder t)"
B_2: shows "S \<subseteq> ball (0::'a) B"
fresherI1: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and r' :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "nsqn\<^sub>r r < nsqn\<^sub>r r'" shows "r \<sqsubseteq> r'"
OK_Env: fixes e :: "char list \<Rightarrow> 'b option" shows "ok (Env e) = finite (dom e)"
v2_in_type: shows "(left (ext v) + x, left (ext v) + (x + y)) \<in> {r. fst r \<le> snd r}"
simps_1: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "Hermite_reduce_above A 0 i j res = A"
simps_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and x' :: "'a"   and y :: "'a" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.split f (Vertex x, Edge x' y) = (if x' = x then f (x, y) else 0)"
conjugate_symmetric: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "conjugate f g" shows "conjugate g f"
heap_is_wellformed_disc_nodes_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node \<in> set disc_nodes" shows "node |\<in>| node_ptr_kinds h"
simps_2: fixes T :: "typ"   and body :: "Core.term"   and n :: "nat" shows "lift (Abs T body) n = Abs T (lift body (n + 1))"
exists_wls_imp_exists_wlsAbs: fixes us :: "'varSort"   and s :: "'sort" assumes "isInBar (us, s)"   and "\<exists>X. wls s X" shows "\<exists>A. wlsAbs (us, s) A"
fps_const_inverse: fixes a :: "'a" shows "inverse (fps_const a) = fps_const (inverse a)"
axiom_doubleneg_def: fixes p :: "fol fm" shows "axiom_doubleneg p \<equiv> Thm (Imp (Imp (Imp p Falsity) Falsity) p)"
undual_dual: fixes x :: "'c" shows "undual (dual x) = x"
singleton_weak_ranking: fixes A :: "'a set" assumes "A \<in> set (weak_ranking le)" shows "is_singleton A"
DIAndCut34Elim1_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAndCut34Elim1 vid1 pid3 pid4 = (([[$\<alpha> vid1]](Pc pid3 && Pc pid4) \<rightarrow> Pc pid3) \<rightarrow> ([[$\<alpha> vid1]]Pc pid3 && Pc pid4) \<rightarrow> [[$\<alpha> vid1]]Pc pid3)"
span_insert_zero: fixes B :: "'b set" shows "span (insert (0::'b) B) = span B"
oprod_Refl: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Refl r"   and "Refl r'" shows "Refl (r *o r')"
not_ks_1: fixes x :: "k" shows "(x \<noteq> kno) = (x = unk)"
vprop: shows "open (SOME U. open U \<and> U \<inter> A = {x}) \<and> (SOME U. open U \<and> U \<inter> A = {x}) \<inter> A = {x}"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "arr (f \<otimes> cod \<iota>)"
left_adjoint_determines_right_up_to_iso: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "Cop.adjoint_pair f g"   and "Cop.adjoint_pair f g'" shows "g \<cong> g'"
T_merge_length: fixes ts\<^sub>1 :: "'b tree list"   and ts\<^sub>2 :: "'b tree list" shows "length (merge ts\<^sub>1 ts\<^sub>2) + T_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * (length ts\<^sub>1 + length ts\<^sub>2) + 1"
A6_DAcc_level0: shows "DAcc level0 sA6 = {sA7}"
span_eq: fixes S :: "('t \<Rightarrow>\<^sub>0 'd) set"   and T :: "('t \<Rightarrow>\<^sub>0 'd) set" shows "(pmdl S = pmdl T) = (S \<subseteq> pmdl T \<and> T \<subseteq> pmdl S)"
extend: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> sub.diff_fun_space" shows "extend f \<in> diff_fun_space \<and> (\<forall>x\<in>closure B. f x = extend f x)"
triangle_def: shows "triangle = {l \<in> lists angles. length l = 3 \<and> sum_list l = 180 \<and> sorted l}"
wf_trm_TComp_subterm: fixes s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m s"   and "t \<sqsubset> s" obtains f :: "'fun" and T :: "('fun, 'atom) Term.term list" where "\<Gamma> s = Fun f T"
boundedD: fixes step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and n :: "nat"   and p :: "nat"   and q :: "nat"   and t :: "'c"   and xs :: "'c" assumes "bounded step n"   and "p < n"   and "(q, t) \<in> set (step p xs)" shows "q < n"
cases: fixes a1 :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and a2 :: "'Var\<^sub>C \<Rightarrow> 'Val"   and a3 :: "nat"   and a4 :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and a5 :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "conc.mevalp a1 a2 a3 a4 a5" obtains cms :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list" and n :: "nat" and mem :: "'Var\<^sub>C \<Rightarrow> 'Val" and cm' :: "'Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)" and mem' :: "'Var\<^sub>C \<Rightarrow> 'Val" where "a1 = cms" and "a2 = mem" and "a3 = n" and "a4 = cms[n := cm']" and "a5 = mem'" and "(cms ! n, mem) \<leadsto>\<^sub>C (cm', mem')" and "n < length cms"
connected_root_iff2: fixes r :: "'a"   and x :: "'a" assumes "point r" shows "connected_root r x = (x\<^sup>T ; 1 \<le> x\<^sup>T\<^sup>+ ; r)"
Some: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "ord x y" shows "ord_option ord (Some x) (Some y)"
leg1_is_map: shows "is_left_adjoint t\<^sub>1"
Class_right_inverse: fixes a :: "'a" assumes "a \<in> G" shows "Class a [\<cdot>] Class (inverse a) = Class \<one>"
starfun_star_of: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(*f* f) (star_of x) = star_of (f x)"
that: fixes d :: "real"   and u :: "'a \<Rightarrow> real" assumes "0 < d"   and "u x = 0"   and "\<And>y. y \<in> cball x d \<Longrightarrow> s (flow0 y (u y)) = 0"   and "\<And>y. y \<in> cball x d \<Longrightarrow> \<bar>u y\<bar> < e"   and "\<And>y. y \<in> cball x d \<Longrightarrow> u y \<in> existence_ivl0 y"   and "continuous_on (cball x d) u"   and "(u has_derivative blinfun_apply (- Ds x /\<^sub>R blinfun_apply (Ds x) (f x))) (at x)" shows "thesis"
hcomp_def: fixes g :: "'a"   and f :: "'a" shows "g \<star> f = (if arr f \<and> arr g \<and> trg\<^sub>B f = src\<^sub>B g then g \<star>\<^sub>B f else null)"
wp_def: fixes pr :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "wp pr \<equiv> pr True"
prv_eqv_scnj_insert: fixes F :: "'fmla set"   and \<phi> :: "'fmla" assumes "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla" shows "prv (eqv (scnj (insert \<phi> F)) (cnj \<phi> (scnj F)))"
cong_def: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "cong a c b = (\<exists>k. a - c = b * k)"
connectedin_eq_not_separated: fixes X :: "'a topology"   and S :: "'a set" shows "connectedin X S = (S \<subseteq> topspace X \<and> (\<nexists>C1 C2. C1 \<union> C2 = S \<and> C1 \<noteq> {} \<and> C2 \<noteq> {} \<and> separatedin X C1 C2))"
inter_add_right2: fixes x :: "'k"   and N :: "'k multiset"   and M :: "'k multiset" assumes "x \<in># N" shows "N \<inter># add_mset x M = add_mset x ((N - {#x#}) \<inter># M)"
numeral_3_eq_3: shows "ny = Suc (Suc (Suc 0))"
wtC_fax: fixes f :: "'fsym" assumes "wtFsym f"   and "prot (resOf f) \<or> protFw (resOf f)" shows "GE.wtC (fax f)"
nat_2: shows "nat 2 = Suc (Suc 0)"
nn_integral_erlang_ith_moment: fixes l :: "real"   and k :: "nat"   and i :: "nat" assumes "0 < l" shows "\<integral>\<^sup>+ x. ennreal (erlang_density k l x * x ^ i) \<partial>lborel = ennreal (fact (k + i) / (fact k * l ^ i))"
union_disjoint: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum g (A \<union> B) = sum g A + sum g B"
le_less: fixes x :: "'c word"   and y :: "'c word" shows "(x \<le>s y) = (x <s y \<or> x = y)"
lin_vectorspace: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "vectorspace (lin x)"
atLeastLessThan_empty_iff2: fixes a :: "'c"   and b :: "'c" shows "({} = {a..<b}) = (\<not> a < b)"
notstutstep_nextsuffix1: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<not> stutstep s n" shows "nextsuffix (s |\<^sub>s n) = s |\<^sub>s Suc n"
closed_open: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open" shows "topological_space.closed open S = open (- S)"
assms_2: fixes gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('d, 'b, 'c) gpv" assumes "X gpv gpv'" shows "rel_spmf (eq_\<I>_generat A \<I> (\<lambda>gpv gpv'. X gpv gpv' \<or> eq_\<I>_gpv A \<I> gpv gpv')) (the_gpv gpv) (the_gpv gpv')"
icard_cartesian_product_singleton: fixes x :: "'a"   and A :: "'b set" shows "icard ({x} \<times> A) = icard A"
g'_def: shows "g' = (\<lambda>s t. - (complex_of_real t powr - s) * complex_of_real (ln t) * complex_of_real (exp (- (a * t))) / complex_of_real (1 - exp (- t)))"
simps_2: fixes f :: "'d \<Rightarrow> 'c"   and i :: "'d" shows "sub f (Nom i) = Nom (f i)"
closed_segment_def: fixes a :: "'a"   and b :: "'a" shows "{a--b} = {(1 - u) *\<^sub>R a + u *\<^sub>R b |u. 0 \<le> u \<and> u \<le> 1}"
simps: fixes l :: "'a"   and u :: "'a" shows "is_interval (l, u) = (l < u)"
nfoldli_to_monadic: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres" shows "nfoldli l c f = monadic_nfoldli l (\<lambda>x. RETURN (c x)) f"
residue_a_inv: fixes k :: "nat"   and y :: "int" shows "\<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> y = \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> (y mod p ^ k)"
L: fixes K :: "ennreal" assumes "K < \<infinity>" shows "{x \<in> A. (\<Sum>n. (nn_transfer_operator ^^ n) f x) \<le> K} \<in> null_sets M"
lists_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "lists (f ` A) = map f ` lists A"
product_complement_character: fixes x :: "i"   and y :: "i" assumes "\<exists>w. P w x \<and> \<not> O w y" shows "\<forall>w. P w (x \<otimes> \<midarrow> y) = (P w x \<and> \<not> O w y)"
run_rel_consistent: fixes ra :: "nat \<Rightarrow> 'a"   and rb :: "nat \<Rightarrow> 'b" assumes "(ra, rb) \<in> run_rel R" shows "a.L \<circ> ra = b.L \<circ> rb"
is_group: shows "Group.group (G\<lparr>carrier := H\<rparr>)"
shift_append: fixes xs :: "'d list"   and ys :: "'d list"   and s :: "'d stream" shows "(xs @ ys) @- s = xs @- ys @- s"
select_convs_2: fixes NTDom :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and NTCod :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and NatTransMap :: "'d \<Rightarrow> 'g"   and more :: "'j" shows "NTCod \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NTCod"
borel_measurableI_greater: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" assumes "\<And>y. {x \<in> space M. y < f x} \<in> sets M" shows "f \<in> borel_measurable M"
typ_of1_subst_bv: fixes Ts :: "typ list"   and uty :: "typ"   and f :: "Core.term"   and fty :: "typ"   and u :: "Core.term" assumes "typ_of1 (Ts @ [uty]) f = Some fty"   and "typ_of u = Some uty" shows "typ_of1 Ts (subst_bv1 f (length Ts) u) = Some fty"
assms: fixes u :: "'a"   and v :: "'a" assumes "dg \<Longrightarrow> deg_pp (fst (rep_nat_term u)) = deg_pp (fst (rep_nat_term v))"   and "ps \<Longrightarrow> snd (rep_nat_term u) = snd (rep_nat_term v)" shows "nat_term_compare to1 u v = nat_term_compare to2 u v"
cases: fixes a :: "'s1" assumes "a -\<tau>1\<rightarrow> \<infinity>" obtains s :: "'s1" and s' :: "'s1" where "a = s" and "s -\<tau>1\<rightarrow> s'" and "s' -\<tau>1\<rightarrow> \<infinity>"
obj_char: fixes a :: "('a E.term, 'a) arr" shows "obj a = (endo a \<and> E.Obj (Dom a) \<and> Map a = EVAL (Dom a))"
set_update_with_aux: fixes xs :: "('c \<times> 'd) list"   and v :: "'d"   and k :: "'c"   and f :: "'d \<Rightarrow> 'd" assumes "distinct (map fst xs)" shows "set (AList.update_with_aux v k f xs) = set xs - {k} \<times> UNIV \<union> {(k, f (case map_of xs k of None \<Rightarrow> v | Some v \<Rightarrow> v))}"
raw_induct: fixes P :: "'s \<Rightarrow> nat \<Rightarrow> bool"   and s :: "'s"   and y :: "nat" assumes "\<And>terminates_within s sa. \<lbrakk>\<And>x y. terminates_within x = Some y \<Longrightarrow> P x y; (let (has_next, next) = g in if has_next s then map_option (\<lambda>n. n + 1) (terminates_within (snd (next s))) else Some 0) = Some sa\<rbrakk> \<Longrightarrow> P s sa"   and "local.terminates_within s = Some y" shows "P s y"
simps_1: fixes i :: "nat" shows "[]\<langle>i\<rangle> = \<bottom>"
UPFDefs_8: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option" shows "m2 \<Oplus> m1 = (\<lambda>x. case m2 x of \<bottom> \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"
ext_special_cf_hi: fixes n :: "nat" shows "snd (ext_cf S n (C\<^sub>0 S)) n = 1\<^sub>r\<^bsub>S\<^esub>"
disc_eq_case_2: fixes option :: "'f option" shows "(option \<noteq> None) = (case option of None \<Rightarrow> False | Some uu_ \<Rightarrow> True)"
fps_sin_cos_sum_of_squares: fixes c :: "'b" shows "(fps_cos c)\<^sup>2 + (fps_sin c)\<^sup>2 = 1"
next_snapshot: fixes c :: "('p, 's, 'm) configuration"   and p :: "'p"   and c' :: "('p, 's, 'm) configuration" shows "c \<turnstile> Snapshot p \<mapsto> c' = (can_occur (Snapshot p) c \<and> ps c' p = Some (states c p) \<and> (\<forall>p'. states c p' = states c' p') \<and> (\<forall>p'. p' \<noteq> p \<longrightarrow> ps c' p' = ps c p') \<and> (\<forall>i. (\<exists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i @ [Marker]) \<and> (\<forall>i. (\<exists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = (fst (cs c i), Recording)) \<and> (\<forall>i. (\<nexists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i) \<and> (\<forall>i. (\<nexists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = cs c i))"
monic_normalize: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "monic (normalize p)"
eint_1_iff_2: fixes x :: "int" shows "(1 = eint x) = (x = 1)"
cases: fixes x :: "'a set list" obtains "x = []"   | xa :: "'a set" where "x = [xa__]"   | xa :: "'a set" and xs :: "'a set list" and y :: "'a set" where "x = xa # xs @ [y]"
prems_1: shows "gram_schmidt_sub n us (w # ws) = us'"
all_info: shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"
get_env_smaller: fixes x :: "char list"   and e :: "'b environment"   and a :: "char list"   and X :: "'b" assumes "x \<in> env_dom e"   and "a \<notin> env_dom e" shows "e\<lparr>a:X\<rparr>!x = e!x"
crel_vs_execute_None: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "['d]"   and heap :: "heap" assumes "crel_vs R a b"   and "execute b heap = None"   and "P heap"   and "Q heap"   and "state_dp_consistency.cmem heap" shows "False"
foundation18'': fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<upsilon> X) = (\<tau> |\<noteq> X \<triangleq> invalid)"
floor_numeral: fixes v :: "num" shows "\<lfloor>numeral v\<rfloor> = numeral v"
comp: fixes r :: "Vertex \<Rightarrow> Vertex"   and s :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations"   and "complex_rotationsp s" shows "complex_rotationsp (r \<circ> s)"
surj_hom_induced_retraction_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and p :: "int" assumes "retraction_map X Y f" shows "carrier (homology_group p Y) = hom_induced p X {} Y {} f ` carrier (homology_group p X)"
traces_contain_events_def: fixes ES :: "'b ES_rec" shows "traces_contain_events ES \<equiv> \<forall>l\<in>Tr\<^bsub>ES\<^esub>. \<forall>e\<in>set l. e \<in> E\<^bsub>ES\<^esub>"
m_o1: fixes h :: "nat"   and m :: "'a \<Rightarrow> nat"   and X :: "char list set"   and o1 :: "(char list \<Rightarrow> 'a) option"   and o2 :: "(char list \<Rightarrow> 'a) option" assumes "Measure_fun h m"   and "finite X"   and "Measure_fun.top_on_opt o1 (- X)"   and "Measure_fun.top_on_opt o2 (- X)"   and "o1 \<le> o2" shows "Measure1_fun.m_o m h o2 X \<le> Measure1_fun.m_o m h o1 X"
count_mset_set': fixes A :: "'c set"   and x :: "'c" shows "count (mset_set A) x = (if finite A \<and> x \<in> A then 1 else 0)"
character_data_ptr_kinds_eq_h2: shows "character_data_ptr_kinds h3 = character_data_ptr_kinds h2"
MakeCatComp: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "f \<approx>>\<^bsub>C\<^esub> g" shows "f ;;\<^bsub>MakeCat C\<^esub> g = f ;;\<^bsub>C\<^esub> g"
omax_isOmax: fixes R :: "('c \<times> 'c) set set" assumes "finite R"   and "R \<noteq> {}"   and "\<forall>r\<in>R. Well_order r" shows "isOmax R (omax R)"