hcju/isabelleps huggingface dataset:
add_strict_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.strict_ordered_ab_semigroup_add plus less_eq less"   and "less a b"   and "less c d" shows "less (plus a c) (plus b d)"
t2: assumes "Can t2" shows "D.ide \<lbrace>t2\<rbrace>"
list_strip_comb: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app (fst (term.strip_comb unapp t)) (snd (term.strip_comb unapp t)) = t"
seqI': fixes f :: "'s"   and a :: "'s"   and b :: "'s"   and g :: "'s"   and c :: "'s" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "seq g f"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "x \<noteq> y" shows "g x \<inter> g y = {}"
that: fixes t :: "('fun, 'var) Term.term" assumes "\<Gamma> t = \<Gamma> s"   and "{} \<turnstile>\<^sub>c t"   and "\<forall>s\<in>T. \<forall>u\<in>subterms s. u \<notin> subterms t" shows "thesis"
mod_add_right_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (modulo b c)) c = modulo (plus a b) c"
nn_integral_indicator_finite: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> sets M" shows "set_nn_integral M A f = (\<Sum>x\<in>A. f x * emeasure M {x})"
RUV_precise: fixes u :: "'a"   and Ml :: "'d set"   and Me :: "'d set"   and U :: "'a set"   and V :: "'a set" assumes "(u, Ml, Me) \<in> RUV_cs fg U V" shows "\<exists>w s' c'. (([u], {#}), w, s', c') \<in> trcl (ntrp fg) \<and> atUV U V ({#s'#} + c') \<and> mon_loc fg w = Ml \<and> mon_env fg w = Me"
inter_aform_plane_ortho_overappr: fixes p :: "nat"   and Z :: "'a \<times> 'a pdevs"   and n :: "'a"   and g :: "real"   and X :: "'a \<times> 'a pdevs" assumes "inter_aform_plane_ortho p Z n g = Some X" shows "{x. \<forall>i\<in>Basis. x \<bullet> i \<in> {y. (g, y) \<in> (\<lambda>x. (x \<bullet> n, x \<bullet> i)) ` Affine Z}} \<subseteq> Affine X"
order_less_imp_le: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "x \<le> y"
continuous_map_closedin: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f = ((\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>C. closedin Y C \<longrightarrow> closedin X {x \<in> topspace X. f x \<in> C}))"
apply_zero_matrix_def: fixes A :: "'c matrix" shows "apply_matrix (\<lambda>x. 0::'b) A = 0"
assms_5: shows "ptr \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 |h \<turnstile> create_document|\<^sub>r"
that: fixes g :: "real \<Rightarrow> 'a" assumes "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative f x) (at x within {a..b})" shows "thesis"
y_onAxis: shows "xval y = (0::'a) \<and> yval y = (0::'a) \<and> zval y = (0::'a)"
alpha_A: fixes A :: "type_scheme list"   and n :: "nat" shows "$ (\<lambda>x. TVar (if x \<in> free_tv A then x else n + x)) A = A"
divides_primepow_weak: fixes p :: "'a"   and a :: "'a"   and n :: "nat" assumes "prime p"   and "a dvd p ^ n" obtains m :: "nat" where "m \<le> n" and "normalize a = normalize (p ^ m)"
distinct_45: fixes x3 :: "('a \<times> 'a) set"   and x7 :: "'b" shows "Spec x3 \<noteq> Call x7"
elements_four_block_mat_id: fixes A :: "'b mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat"   and C :: "'b mat"   and nr2 :: "nat"   and D :: "'b mat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "elements_mat (four_block_mat A B C D) = elements_mat A \<union> elements_mat B \<union> elements_mat C \<union> elements_mat D"
bidirected_digraph_axioms: shows "bidirected_digraph (with_proj G) (edge_rev M)"
simps_6: fixes r :: "'a set"   and b :: "'a set"   and c :: "'a com" shows "atomics (AnnAwait r b c) = {(r \<inter> b, c)}"
InvariantVarsM_def: fixes M :: "(Literal \<times> bool) list"   and F0 :: "Literal list list"   and Vbl :: "nat set" shows "InvariantVarsM M F0 Vbl \<equiv> vars (elements M) \<subseteq> vars F0 \<union> Vbl"
black_heap_def: shows "black_heap = (\<lambda>s. \<forall>x. obj_at (\<lambda>s. True) x s \<longrightarrow> black x s)"
right_idem: fixes a :: "'g"   and b :: "'g" shows "min (min a b) b = min a b"
FieldDecls'_def: 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 C :: "char list"   and F :: "char list"   and Cs :: "char list list"   and T :: "ty" shows "FieldDecls' P C F Cs T \<equiv> (Cs, T) \<in> FieldDecls P C F"
Edka_Impl_axioms: shows "Edka_Impl c s t N"
hyps: fixes y :: "nat"   and c :: "int" assumes "y < l"   and "c \<noteq> 0"   and "\<bar>c\<bar> < p ^ y"   and "poly_mod.dvdm (p ^ y) u (smult c f)" shows "u dvdm f"
l0: fixes p :: "int"   and F :: "int poly"   and l :: "nat" assumes "prime p"   and "0 < degree F"   and "2 ^ (degree F)\<^sup>2 * B2_LLL F ^ (2 * degree F) \<le> (p ^ l)\<^sup>2" shows "l \<noteq> 0"
int_nem: shows "cbox (- 1) 1 \<noteq> {}"
ideD_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.cod C a = a"
surj_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" shows "surj_on f A B \<equiv> \<forall>b\<in>B. \<exists>a\<in>A. b = f a"
fst_map_prod: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and x :: "'c \<times> 'd" shows "fst (map_prod f g x) = f (fst x)"
iptables_bigstep_to_undecided: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> Undecided" shows "s = Undecided"
f'_le: fixes x :: "real" assumes "x \<in> {real_of_float l..real_of_float u}" shows "min (f' (real_of_float l)) (f' (real_of_float u)) \<le> f' x"
is_full: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and a :: "'b"   and a' :: "'b"   and g :: "'c" assumes "full_functor A B F"   and "partial_magma.ide A a"   and "partial_magma.ide A a'"   and "partial_magma.in_hom B g (F a') (F a)" shows "\<exists>f. partial_magma.in_hom A f a' a \<and> F f = g"
sum_fact: shows "int (\<Sum>b2\<in>#remove1_mset b1 \<B>. nat (b1 |\<inter>| b2) choose 2) = int (nat \<k> choose 2) * (\<Lambda> - 1)"
additive_wp_Seq: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "Transformers.additive (wp a)"   and "Transformers.additive (wp b)"   and "well_def b" shows "Transformers.additive (wp (a ;; b))"
simps_2: fixes x :: "nat"   and m :: "nat" shows "exp x (Suc m) = x * exp x m"
abs_eq: fixes xa :: "'a ltl"   and x :: "'a set" shows "af_G_abs_opt.f_abs (Abs xa) x = Abs (af_G_letter_opt xa x)"
tensor_of_H_values: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)))"   and "j < dim_col (Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)))" shows "Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)) $$ (i, j) = complex_of_real ((- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)"
plus_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>+ = (x\<^sup>\<star> * y)\<^sup>\<star> * x\<^sup>+ \<squnion> (x\<^sup>\<star> * y)\<^sup>+"
eval: shows "P \<turnstile> \<langle>body,(h, [this # pns [\<mapsto>] vs], sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>"
min_int_poly_def: fixes x :: "'b" shows "min_int_poly x = (if algebraic x then THE p. p represents x \<and> irreducible p \<and> 0 < Polynomial.lead_coeff p else [:0, 1:])"
parallel_fix_ind: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'a \<rightarrow> 'a"   and G :: "'b \<rightarrow> 'b" assumes "adm (\<lambda>x. P (prod.fst x) (prod.snd x))"   and "P \<bottom> \<bottom>"   and "\<And>x y. P x y \<Longrightarrow> P (F\<cdot>x) (G\<cdot>y)" shows "P (fix\<cdot>F) (fix\<cdot>G)"
subid_par_idem: fixes R :: "('a \<times> 'a set) set" assumes "R \<subseteq> 1\<^sub>\<sigma>" shows "R \<parallel> R = R"
hbt_delete_auto_1: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "hbt t" shows "hbt (delete x t)"
error_free_call_access: fixes G :: "prog"   and s1 :: "state"   and args :: "expr list"   and vs :: "val list"   and s2 :: "state"   and accC :: "qtname"   and L :: "lenv"   and e :: "expr"   and statT :: "ref_ty"   and mn :: "mname"   and pTs :: "ty list"   and statDeclT :: "ref_ty"   and statM :: "mhead"   and pTs' :: "ty list"   and a :: "val"   and s3 :: "state"   and invC :: "qtname"   and invDeclC :: "qtname" assumes "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<rightarrow> s2"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-RefT statT"   and "max_spec G accC statT \<lparr>name = mn, parTs = pTs\<rparr> = {((statDeclT, statM), pTs')}"   and "s2\<Colon>\<preceq>(G, L)"   and "normal s1 \<Longrightarrow> G,snd s1\<turnstile>a\<Colon>\<preceq>RefT statT"   and "normal s3 \<Longrightarrow> G\<turnstile>invmode statM e\<rightarrow>invC\<preceq>statT"   and "s3 = init_lvars G invDeclC \<lparr>name = mn, parTs = pTs'\<rparr> (invmode statM e) a vs s2"   and "invC = invocation_class (invmode statM e) (snd s2) a statT"   and "invDeclC = invocation_declclass G (invmode statM e) (snd s2) a statT \<lparr>name = mn, parTs = pTs'\<rparr>"   and "wf_prog G" shows "check_method_access G accC statT (invmode statM e) \<lparr>name = mn, parTs = pTs'\<rparr> a s3 = s3"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(S_action.Act x1 = S_action.Act y1) = (x1 = y1)"
hfref_def: fixes P :: "'h \<Rightarrow> bool"   and RS :: "('h \<Rightarrow> 'f \<Rightarrow> assn) \<times> ('h \<Rightarrow> 'f \<Rightarrow> assn)"   and T :: "'i \<Rightarrow> 'g \<Rightarrow> assn" shows "[P]\<^sub>a RS \<rightarrow> T \<equiv> {(f, g). \<forall>c a. P a \<longrightarrow> hn_refine (fst RS a c) (f c) (snd RS a c) T (g a)}"
infdist_def: fixes x :: "'a"   and A :: "'a set" shows "infdist x A = (if A = {} then 0 else \<Sqinter> (dist x ` A))"
map_sq_matrix_cong: fixes M :: "('b, 'c) sq_matrix"   and N :: "('b, 'c) sq_matrix"   and f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" assumes "M = N"   and "\<And>i j. f (to_fun N i j) = g (to_fun N i j)" shows "map_sq_matrix f M = map_sq_matrix g N"
aux26: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set"   and p :: "protocol \<times> int" assumes "twoNetsDistinct a b c d" shows "dom (Cp (AllowPortFromTo a b p)) \<inter> dom (Cp (DenyAllFromTo c d)) = {}"
Cons: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and x :: "'a"   and xs :: "'a list" assumes "monoid_list f z" shows "monoid_list.F f z (x # xs) = f x (monoid_list.F f z xs)"
med12s_def: fixes t :: "l2_state" shows "med12s t \<equiv> \<lparr>ik = ik t, secret = secret t, progress = progress t, signals = l1_state.signals t\<rparr>"
spawn: fixes \<iota> :: "nat"   and V :: "('exp, 'id) MWLsCom list"   and m :: "'id \<Rightarrow> 'val" shows "\<langle>spawn\<^bsub>\<iota>\<^esub> V,m\<rangle> \<rightarrow>\<lhd>V\<rhd> \<langle>None,m\<rangle>"
assms_1: shows "finite {v. f1 v \<noteq> 0}"
lookup_tabulate: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "distinct xs" shows "Mapping.lookup (Mapping.tabulate xs f) x = (if x \<in> set xs then Some (f x) else None)"
store_word_mem_def: fixes state :: "'b sparc_state"   and va :: "word32"   and wd :: "word32"   and byte_mask :: "word4"   and asi :: "word8" shows "store_word_mem state va wd byte_mask asi \<equiv> let pair = virt_to_phys va (mmu state) (mem state) in case pair of None \<Rightarrow> None | Some pair \<Rightarrow> if mmu_writable (get_acc_flag (snd pair)) asi then Some (mem_mod_w32 asi (fst pair) byte_mask wd state) else None"
axioms_1: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" assumes "discrete_diagram J C D" shows "category J"
poly_f_nonzero: fixes qs :: "real poly list" shows "poly_f qs \<noteq> 0"
elims: fixes x :: "'b list \<times> 'b list"   and y :: "'b list \<times> 'b list" assumes "norm x = y" obtains fs :: "'b list" and rs :: "'b list" where "x = (fs, rs)" and "y = (if fs = [] then (itrev rs [], []) else (fs, rs))"
VE_prop: shows "\<lparr>nodes = V, edges = E\<rparr> = generate_valid_topology M (fully_connected G)"
subtopology_Times: fixes X :: "'d topology"   and Y :: "'e topology"   and S :: "'d set"   and T :: "'e set" shows "subtopology (prod_topology X Y) (S \<times> T) = prod_topology (subtopology X S) (subtopology Y T)"
A_dim_not0: shows "\<not> (dim_row A = 0 \<or> dim_col A = 0)"
inj_permute: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "perm" assumes "class.pt permute" shows "inj (permute p)"
bridgingInjection: fixes N :: "'c set"   and G :: "'d list" assumes "0 < card N"   and "distinct G" shows "\<forall>l\<in>set (all_partitions_list G). set (injections_alg l N) = injections (set l) N"
AbruptIs_1: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and x :: "expr" shows "G\<turnstile>(Some xc, s) \<midarrow>x-\<succ>undefined\<rightarrow> (Some xc, s)"
convex_hull_finite: fixes S :: "'a set" assumes "finite S" shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y}"
surj_to_def: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set" shows "surj_to f A B = (f ` A = B)"
itrm_brackets_all_unlift_vars: fixes x :: "nat itrm"   and vs :: "nat list"   and x' :: "nat itrm" assumes "set (opaque x) \<subseteq> set vs"   and "itrm_brackets vs x = Some x'" shows "x' \<simeq>\<^sup>+ Pure (unlift_vars 0 x')"
PPs_closed_tpp: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'x set" assumes "p \<in> P[X]" shows "tpp p \<in> .[X]"
g_eq: fixes z :: "complex" assumes "- 1 < Re z"   and "Re z < 2" shows "g (z / 2) * g ((z + 1) / 2) = (Gamma (1 / 2))\<^sup>2 * g z"
h4: shows "m * (gcd a b)\<^sup>2 = n"
map_id: fixes t :: "('d, 'e, 'f) expr" shows "\<R>\<^sub>E id id t = t"
lemFromToTo: fixes p :: "'a Point"   and q :: "'a Point"   and r :: "'a Point" shows "from p to q \<oplus> from q to r = from p to r"
that: fixes a :: "'a" assumes "p = pCons a 0" shows "thesis"
less_max_iff_disj: fixes z :: "'a word"   and x :: "'a word"   and y :: "'a word" shows "(z <s signed.max x y) = (z <s x \<or> z <s y)"
sumhr_split_diff: fixes n :: "nat star"   and p :: "nat star"   and f :: "nat \<Rightarrow> real" assumes "n < p" shows "sumhr (0, p, f) - sumhr (0, n, f) = sumhr (n, p, f)"
mem_interior: fixes x :: "'a"   and S :: "'a set" shows "(x \<in> interior S) = (\<exists>e>0. ball x e \<subseteq> S)"
big_small_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Omega>[F](g)"   and "g \<in> \<omega>[F](h)" shows "f \<in> \<omega>[F](h)"
mcont2mcont: fixes lub' :: "'f set \<Rightarrow> 'f"   and ord' :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and lub :: "'h set \<Rightarrow> 'h"   and ord :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and t :: "'h \<Rightarrow> 'f" assumes "mcont lub' ord' Sup (\<le>) f"   and "mcont lub ord lub' ord' t" shows "mcont lub ord Sup (\<le>) (\<lambda>x. f (t x))"
simps_2: fixes div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and i :: "nat"   and xs :: "nat list"   and H :: "'a mat"   and P :: "'a mat"   and K :: "'a mat" shows "reduce_column_aux div_op (i # xs) H (P, K) = (let k = div_op (H $$ (i, 0)) (H $$ (0, 0)); P' = addrow_mat (dim_row H) (- k) i 0; K' = addrow (- k) i 0 K in reduce_column_aux div_op xs H (P' * P, K'))"
B'_altdef: shows "B' = (*\<^sub>R) (1 / 2) ` B"
odd_nodes_no_edge: fixes g :: "('a, 'b) graph" assumes "finite (nodes g)" shows "num_of_odd_nodes (g\<lparr>edges := {}\<rparr>) = 0"
wf_Term_quot_dbtm: fixes u :: "dbtm"   and e :: "(name, hf) finfun" assumes "wf_dbtm u" shows "Term \<lbrakk>quot_dbtm u\<rbrakk>e"
fls_X_intpow_diff_conv_times: fixes i :: "int"   and j :: "int" shows "fls_X_intpow (i - j) = fls_X_intpow i * fls_X_intpow (- j)"
segment_open_subset_closed: fixes a :: "'a"   and b :: "'a" shows "open_segment a b \<subseteq> closed_segment a b"
simps_2: fixes \<mu> :: "'a" shows "Src \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>src \<mu>\<^bold>\<rangle>\<^sub>0"
left_adjoint_preserved_by_iso': fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.is_left_adjoint f"   and "Span.isomorphic f f'" shows "Span.is_left_adjoint f'"
plus: fixes x :: "V"   and y :: "V" shows "x + y = x \<squnion> set ((+) x ` elts y)"
mult_2_ereal: fixes x :: "ereal" shows "ereal 2 * x = x + x"
cod_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VVV.cod f = (if VVV.arr f then VxVxV.cod f else VxVxV.null)"
check_instr_stk_offer: shows "ci_stk_offer check_instr"
igFreshIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igFreshIGOp MOD \<equiv> \<forall>ys y delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igFreshInp MOD ys y inp \<and> igFreshBinp MOD ys y binp \<longrightarrow> igFresh MOD ys y (igOp MOD delta inp binp)"
Fvars_fmla_simps_2: fixes A :: "fmla"   and B :: "fmla" shows "Fvars (A OR B) = Fvars A \<union> Fvars B"
type_pgwt_inhabited: fixes t :: "('fun, 'var) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t" shows "\<exists>t'. \<Gamma> t = \<Gamma> t' \<and> public_ground_wf_term t'"
payer_def: fixes dc :: "'a \<times> 'b" shows "payer dc = fst dc"
cases: fixes x :: "('a\<^sub>m + 'b\<^sub>m) \<times> ('a\<^sub>m + 'b\<^sub>m)" obtains xa :: "'a\<^sub>m" and y :: "'a\<^sub>m" where "x = (Inl xa__, Inl y__)"   | xa :: "'b\<^sub>m" and y :: "'b\<^sub>m" where "x = (Inr xa, Inr y)"   | v :: "'b\<^sub>m" and va :: "'a\<^sub>m" where "x = (Inr v__, Inl va__)"   | va :: "'a\<^sub>m" and v :: "'b\<^sub>m" where "x = (Inl va, Inr v)"
split_all_prime: fixes ms :: "nat list"   and ns :: "nat list" assumes "all_prime ms"   and "all_prime ns" shows "\<exists>qs. all_prime qs \<and> (\<Prod>m\<in>#mset qs. m) = (\<Prod>m\<in>#mset ms. m) * (\<Prod>m\<in>#mset ns. m)"
fail_converter_parametric: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_converter A B C R fail_converter fail_converter"
length_take: fixes n :: "nat"   and xs :: "'a list" shows "length (take n xs) = min (length xs) n"
Res: fixes y :: "name"   and R :: "pi"   and S :: "pi" assumes "y \<sharp> R" shows "(R \<parallel> <\<nu>y>S, <\<nu>y>(R \<parallel> S)) \<in> Rel"
that: fixes k :: "'a"   and s :: "nat \<Rightarrow> real" assumes "k \<in> X"   and "k \<noteq> l"   and "(f \<circ> s) \<longlonglongrightarrow> k"   and "strict_mono s"   and "\<forall>n. real n \<le> s n" shows "thesis"
ide_in_hom: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and a :: "'ag" assumes "partial_magma C" shows "partial_magma.ide C a = partial_magma.in_hom C a a a"
distinct_1: fixes name :: "name"   and name' :: "name" shows "InputS name \<noteq> BoundOutputS name'"
inverse_eq2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)" shows "((\<lambda>x. inverse (f x)) \<in> \<omega>[F](g)) = ((\<lambda>x. 1::'b) \<in> \<omega>[F](\<lambda>x. f x * g x))"
bintr_ge0: fixes n :: "nat"   and w :: "int" shows "0 \<le> take_bit n w"
unlabel_Cons_2: fixes b :: "'f \<times> 'e"   and A :: "('f \<times> 'e) list" shows "unlabel (b # A) = snd b # unlabel A"
conv_sum_norm: shows "infsetsum'_converges (\<lambda>x. norm (f x)) A"
generate_topology_Union: fixes I :: "'a set"   and S :: "'b set set"   and K :: "'a \<Rightarrow> 'b set" assumes "\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)" shows "generate_topology S (\<Union> (K ` I))"
cobounded1: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "linordered_monoid.min a b \<preceq> a"
m_def: shows "m = nat \<lceil>norm z * e\<rceil>"
strict_trans2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ordering less_eq less"   and "less a b"   and "less_eq b c" shows "less a c"
and_mask_dvd_nat: fixes n :: "nat"   and w :: "'b word" shows "(2 ^ n dvd unat w) = (w AND mask n = 0)"
farkas_coefficients_atoms_tableau_def: fixes as :: "'a atom set"   and t :: "(nat \<times> linear_poly) list"   and C :: "(rat \<times> 'a atom) list" shows "farkas_coefficients_atoms_tableau as t C = (\<exists>p c. (\<forall>(r, a)\<in>set C. a \<in> as \<and> is_leq_ns (r *R nsc_of_atom a) \<and> r \<noteq> 0) \<and> (\<Sum>(r, a)\<leftarrow>C. lec_of_nsc (r *R nsc_of_atom a)) = Le_Constraint Leq_Rel p c \<and> c < (0::'a) \<and> (\<forall>v. v \<Turnstile>\<^sub>t t \<longrightarrow> p \<lbrace> v \<rbrace> = (0::'a)))"
insert_is_Un: fixes a :: "'b"   and A :: "'b set" shows "a \<triangleright> A = {a} \<union> A"
preserves_reflects_arr: fixes f :: "'a" shows "arr (\<tau> f) = arr f"
effective_tensor_compose_distribution2: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "matrix_match A1 A2 B1 B2" shows "mat (row_length A1 * row_length B1) (length A2 * length B2) ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2))"
retract_of_imp_subset: fixes S :: "'a set"   and T :: "'a set" assumes "S retract_of T" shows "S \<subseteq> T"
ordered_insert_set: fixes x :: "'a"   and ys :: "'a list" shows "set (ordered_insert x ys) = {x} \<union> set ys"