hcju/isabelleps huggingface dataset:
ext_inject: fixes carrier :: "'a set"   and more :: "'b"   and carrier' :: "'a set"   and more' :: "'b" shows "(\<lparr>carrier = carrier, \<dots> = more\<rparr> = \<lparr>carrier = carrier', \<dots> = more'\<rparr>) = (carrier = carrier' \<and> more = more')"
cmod_square_real: fixes n :: "real" shows "(cmod (complex_of_real n))\<^sup>2 = n\<^sup>2"
8: shows "irreflexive ((f \<sqinter> q)\<^sup>T\<^sup>+ * (f \<sqinter> - q)\<^sup>+)"
simps_46: fixes \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula" shows "mmonitorable_exec (formula.Since \<phi> I \<psi>) = (fv \<phi> \<subseteq> fv \<psi> \<and> (mmonitorable_exec \<phi> \<or> (case \<phi> of formula.Neg x \<Rightarrow> mmonitorable_exec x | _ \<Rightarrow> False)) \<and> mmonitorable_exec \<psi>)"
constant_transformation_axioms: shows "constant_transformation (\<cdot>\<^sub>J) (\<cdot>) f"
rep_eq: fixes x :: "uint8" shows "Rep_uint16 (uint16_of_uint8 x) = ucast (Rep_uint8 x)"
ide_char: fixes a :: "'a1" shows "A1.ide a = (A1.arr a \<and> A1.dom a = a \<and> A1.cod a = a)"
interpolation_poly: fixes xs_ys :: "('a \<times> 'a) list"   and p :: "'a poly"   and alg :: "interpolation_algorithm"   and x :: "'a"   and y :: "'a" assumes "distinct (map fst xs_ys)"   and "p = interpolation_poly alg xs_ys"   and "(x, y) \<in> set xs_ys" shows "poly p x = y"
path_join_def: fixes p :: "nat list"   and q :: "nat list" shows "path_join p q = p @ tl q"
sup_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd err" shows "Listn.sup f = (\<lambda>xs ys. if length xs = length ys then coalesce (xs [\<squnion>\<^bsub>f\<^esub>] ys) else Err)"
AKcryptSK_analz_insert: fixes K :: "nat"   and K' :: "nat"   and evs :: "event list" assumes "AKcryptSK K K' evs"   and "K \<in> symKeys"   and "evs \<in> kerbIV" shows "Key K' \<in> analz (insert (Key K) (knows Spy evs))"
GcdI: fixes A :: "'a set"   and b :: "'a" assumes "\<And>a. a \<in> A \<Longrightarrow> b dvd a"   and "\<And>c. (\<And>a. a \<in> A \<Longrightarrow> c dvd a) \<Longrightarrow> c dvd b"   and "normalize b = b" shows "b = Gcd A"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and atom :: "'b atom" shows "P (case atom of Leq x xa \<Rightarrow> f1 x xa | Geq x xa \<Rightarrow> f2 x xa) = ((\<forall>x11 x12. atom = Leq x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x21 x22. atom = Geq x21 x22 \<longrightarrow> P (f2 x21 x22)))"
axioms_2: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and basis\<^sub>2 :: "'f set" assumes "finite_dimensional_vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2" shows "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 basis\<^sub>2"
simps: fixes a1 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list"   and a2 :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" shows "a1 \<hookrightarrow> a2 = ((\<exists>\<Gamma>\<^sub>1 n\<^sub>1 \<Psi>\<^sub>1 \<Phi>\<^sub>1 \<Gamma>\<^sub>2 n\<^sub>2 \<Psi>\<^sub>2 \<Phi>\<^sub>2. a1 = \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<and> a2 = \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2 \<and> \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>i \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2) \<or> (\<exists>\<Gamma>\<^sub>1 n\<^sub>1 \<Psi>\<^sub>1 \<Phi>\<^sub>1 \<Gamma>\<^sub>2 n\<^sub>2 \<Psi>\<^sub>2 \<Phi>\<^sub>2. a1 = \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<and> a2 = \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2 \<and> \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>e \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2))"
atLeastatMost_subset_iff: fixes 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.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b \<subseteq> ord.atLeastAtMost less_eq c d) = (\<not> less_eq a b \<or> less_eq c a \<and> less_eq b d)"
mkArr_eqI: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b"   and A' :: "'b set"   and B' :: "'b set"   and F' :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)"   and "A = A'"   and "B = B'"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "set_category_given_img.mkArr S img \<AA> A B F = set_category_given_img.mkArr S img \<AA> A' B' F'"
assms: shows "\<forall>n. \<exists>m>n. P (s m)"
assms_2: fixes x :: "nat"   and y :: "nat" shows "eval r [x, y] = f x y"
cases: fixes \<Gamma> :: "('a, 'b) \<I>"   and a :: "('c, 'a, 'b) gpv" assumes "\<Gamma> \<turnstile>g a \<surd>" obtains gpv :: "('c, 'a, ('c, 'a, 'b) rpv) generat spmf" where "a = GPV gpv" and "\<And>out c. IO out c \<in> set_spmf gpv \<Longrightarrow> out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. \<Gamma> \<turnstile>g c input \<surd>)"
modular_var_3: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'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 composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "less_eq (composition x y) (composition (inf x (composition top (converse y))) y)"
wfp_on_iff_minimal: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "wfp_on P A = (\<forall>Q x. x \<in> Q \<and> Q \<subseteq> A \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. P y z \<longrightarrow> y \<notin> Q))"
word_shiftr_lt: fixes w :: "'a word"   and n :: "nat" shows "unat (w >> n) < 2 ^ (LENGTH('a) - n)"
reachable_nodes_step: fixes g :: "'b ugraph"   and r :: "'b" shows "edges g `` reachable_nodes g r \<subseteq> reachable_nodes g r"
bounded_def: fixes P :: "'c \<Rightarrow> real" shows "Expectations.bounded P \<equiv> \<exists>b. bounded_by b P"
int_pow_neg: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier \<G>" shows "x [^] - i = inv (x [^] i)"
sign_changes_decompose: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "x \<noteq> (0::'a)" shows "sign_changes (xs @ x # ys) = sign_changes (xs @ [x]) + sign_changes (x # ys)"
carrier_polynomial: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "polynomial\<^bsub>K [X]\<^esub> K p" shows "polynomial\<^bsub>K [X]\<^esub> (carrier (K [X])) p"
rep_eq: fixes x :: "'a regular" shows "Rep_regular (x\<^sup>T) = Rep_regular x\<^sup>T"
m_1: fixes n :: "nat" assumes "m \<le> n" shows "0 < f n"
remdups_fwd_acc_append: fixes Acc :: "'b set"   and xs :: "'b list"   and ys :: "'b list" shows "remdups_fwd_acc Acc (xs @ ys) = remdups_fwd_acc Acc xs @ remdups_fwd_acc (Acc \<union> set xs) ys"
card_of_ordLeq_finite: fixes A :: "'d set"   and B :: "'e set" assumes "ordLeq3 (card_of A) (card_of B)"   and "finite B" shows "finite A"
cases: fixes x :: "'b trm" obtains xa :: "'b" where "x = Var xa__"   | xa :: "'b" where "x = Const xaa__"   | xa :: "'b trm" and y :: "'b trm" where "x = xa \<cdot> y"
h_2: shows "rep (stabilizer m #> inv g) = h \<otimes> inv g"
rel_spmf_bindI1: fixes p :: "'b spmf"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c spmf"   and q :: "'d spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf R (f x) q"   and "lossless_spmf p" shows "rel_spmf R (p \<bind> f) q"
split_paired_all: fixes P :: "'a \<times> 'b \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
dfree_Fun_simps: fixes i :: "'a"   and args :: "'b \<Rightarrow> ('a, 'b) trm" shows "dfree ($f i args) = (\<forall>x. dfree (args x))"
\<gamma>1'_eq: shows "lst \<gamma>1' = lst \<gamma>1''"
a': shows "a' \<in>\<^sub>A frontier (c_pts c loc)"
proj_ord_def: fixes cmp :: "(('c, 'd) pp \<times> nat) nat_term_order" shows "proj_ord cmp = Abs_nat_term_order (proj_comp cmp)"
gen_dfs_simps_1: fixes succs :: "'d \<Rightarrow> 'd list"   and ins :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and memb :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c" shows "gen_dfs succs ins memb S [] = S"
simps_1: fixes prog :: "cmd"   and n :: "w_node"   and c :: "cmd" shows "labels_LDCFG_nodes prog (LDCFG_node.Node n) c = labels_nodes prog n c"
opp_def: fixes d :: "dir" shows "opp d = (if d = dir.Right then dir.Left else dir.Right)"
eval_poly_monom: fixes a :: "'a"   and n :: "nat"   and x :: "'a poly" shows "eval_poly (\<lambda>a. [:a:]) (Polynomial.monom a n) x = [:a:] * x ^ n"
Irr_minimal: fixes K :: "'a set"   and x :: "'a"   and p :: "'a list" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x"   and "p \<in> carrier (K [X])"   and "local.eval p x = \<zero>" shows "Irr K x pdivides p"
S_def: fixes h :: "'a"   and e :: "nat" shows "Schnorr_\<Sigma>.S h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S2 h e)"
topological_basis_def: fixes B :: "'c set set" shows "topological_basis B = ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union> B' = x)))"
get_nth_im1_def: fixes ba :: "'b list \<times> 'b list" shows "get_nth_im1 ba = hd (fst ba)"
HEndPhase2_HInv4c_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv4c s p" shows "HInv4c s' p"
idem: fixes a :: "'c" shows "a \<sqinter> a = a"
unique_factorization_m_smult: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset"   and di :: "int"   and d :: "int" assumes "unique_factorization_m f (c, fs)"   and "M (di * d) = 1" shows "unique_factorization_m (smult d f) (c * d, fs)"
mo_def: shows "mo = find_map_filter (\<lambda>x. (x, poly p x)) (\<lambda>(uu_, res). res = 0) tests"
sorted_lems_2: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "sorted (x # xs @ y # ys) = (sorted (x # xs) \<and> x < y \<and> sorted (xs @ [y]) \<and> sorted (y # ys))"
permutation_delete_def: fixes p :: "nat \<Rightarrow> nat"   and i :: "nat" shows "permutation_delete p i = delete_ran (delete_dom p i) (p i)"
assms_2: shows "X \<subseteq>\<^sub>L' Y"
igFreshIGAbs1_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igFreshIGAbs1 MOD \<equiv> \<forall>ys y s X. isInBar (ys, s) \<and> igWls MOD s X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD ys y X)"
"4_1": fixes x2 :: "nat \<times> tid_or_exn"   and x :: "nat"   and y :: "tid_or_exn"   and xa :: "pat"   and ya :: "v"   and init :: "(char list \<times> v) list match_result"   and x3 :: "(char list \<times> v) list" assumes "nsLookup envC_ n_ = Some x2"   and "(x, y) = x2"   and "same_tid y t'_ \<and> length ps_ = x"   and "same_ctor (id_to_n n_, y) (n'_, t'_)"   and "xa \<in> set ps_"   and "ya \<in> set vs_"   and "init = Match x3"   and "is_cupcake_pat xa" shows "pmatch_single envC_ s_ xa ya x3 = cupcake_pmatch envC_ xa ya x3"
cont_eff: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and b :: "nat" shows "cont_eff cf b = (local.cont (fst cf) (snd cf) b, eff (fst cf) (snd cf) b)"
U_def: shows "U \<equiv> {z \<in> S. \<exists>B Z. 0 < B \<and> open Z \<and> z \<in> Z \<and> Z \<subseteq> S \<and> (\<forall>h z'. h \<in> Y \<and> z' \<in> Z \<longrightarrow> cmod (h z') \<le> B)}"
orderI: fixes a :: "'c"   and b :: "'c" assumes "a = max a b" shows "b \<le> a"
mono: shows "AE x in M. incseq (\<lambda>n. f n x)"
Rep_nd_fun_inverse: fixes x :: "'b nd_fun" shows "Abs_nd_fun (Rep_nd_fun x) = x"
assms_2: fixes i :: "'i" assumes "i \<in> I" shows "(0::'a) \<le> f i"
nth_default_geq: fixes i :: "nat" shows "nth_default 0 (Bernstein_coeffs p c d P) i \<le> nth_default 0 (Bernstein_coeffs p c d [:b, a:]) i"
in_scope: fixes v' :: "'form \<times> nat list"   and p' :: "('form, 'var) antecedent"   and v :: "'form \<times> nat list" assumes "valid_in_port (v', p')" shows "(v \<in> scope (v', p')) = scope' v' p' v"
inv_right: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bijection f" shows "f (inv f a) = a"
vfun_le_inv: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" assumes "VFun t1 \<sqsubseteq> VFun t2" obtains "t1 \<lesssim> t2"
f'_def: shows "f' \<equiv> \<lambda>i x. if integrable M (f x) then simple_bochner_integral M (\<lambda>y. s i (x, y)) else (0::'f)"
sum_list_map_filter: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "class.monoid_add plus zero"   and "\<And>x. \<lbrakk>x \<in> set xs; \<not> P x\<rbrakk> \<Longrightarrow> f x = zero" shows "monoid_add.sum_list plus zero (map f (filter P xs)) = monoid_add.sum_list plus zero (map f xs)"
valid_data_dependency_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "valid_data_dependency ts \<equiv> (\<forall>\<R> i p is \<O> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> data_dependency_consistent_instrs (dom \<theta>) is) \<and> (\<forall>\<R> i p is \<O> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> load_tmps is \<inter> \<Union> (fst ` write_sops sb) = {})"
eval_NUM: fixes n :: "nat" shows "NUM n \<Down> NUM n"
nth_replicate: fixes i :: "nat"   and n :: "nat"   and x :: "'c" assumes "i < n" shows "replicate n x ! i = x"
simps_7: fixes f1 :: "'g"   and f2 :: "nat \<Rightarrow> 'g \<Rightarrow> 'g"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
simps_5: fixes F :: "'f set"   and b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "strip_guards F (Cond b c\<^sub>1 c\<^sub>2) = Cond b (strip_guards F c\<^sub>1) (strip_guards F c\<^sub>2)"
unit_closed: shows "stx.class_of X (zero_str\<^sub>X X) \<in> stx.carrier_direct_lim"
ass: shows "coprime a b \<and> odd (int (\<Prod>i\<in>#ps + {#p#}. i)) \<and> a\<^sup>2 + 3 * b\<^sup>2 = int (\<Prod>i\<in>#ps + {#p#}. i) ^ 3 \<and> (\<forall>a\<in>#ps. prime a) \<and> prime (int p)"
primes_coprime: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "p \<noteq> q" shows "algebraic_semidom.coprime times one p q"
max_bot: fixes x :: "'c list" shows "prefix_order.max [] x = x"
P_LCons': shows "P = LCons v (LCons comp.w0 (ltl (ltl P)))"
0: shows "prv (imp \<phi> (imp (neg \<phi>) fls))"
subcls1I: 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 Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "D \<in> baseClasses Bs" shows "subcls1p P C D"
has_density_emeasure: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and X :: "'b set" assumes "has_density M N f"   and "X \<in> sets M" shows "emeasure M X = set_nn_integral N X f"
extNTA2J_def: fixes P :: "'e J_prog" shows "extNTA2J P = (\<lambda>(C, M, a). let (D, Ts, T, meth) = method P C M; (pns, body) = the meth in ({this:Class D=\<lfloor>Addr a\<rfloor>; body}, Map.empty))"
comp_cod_arr: fixes f :: "'b"   and b :: "'b" assumes "B.arr f"   and "B.cod f = b" shows "b \<cdot>\<^sub>B f = f"
ipresSubst_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'c, 'h, 'i) Iteration.model_scheme" shows "ipresSubst h MOD \<equiv> \<forall>ys Y y s X. wls (asSort ys) Y \<and> wls s X \<longrightarrow> h (X #[Y / y]_ys) = igSubst MOD ys (h Y) y (h X)"
arr_char: fixes Obj :: "'a set"   and f :: "('a, 'a) concrete_category.arr" shows "partial_magma.arr (concrete_category.COMP Obj (\<lambda>x y. if x = y then {x} else {}) (\<lambda>_ _ x _ _. x)) f = (concrete_category.Dom f \<in> Obj \<and> f = MkArr (concrete_category.Dom f) (concrete_category.Dom f) (concrete_category.Dom f))"
open_cballE: fixes S :: "'b set"   and x :: "'b" assumes "open S"   and "x \<in> S" obtains e :: "real" where "0 < e" and "cball x e \<subseteq> S"
Map_by_Ordered_axioms: shows "Map_by_Ordered empty update delete lookup inorder inv"
exhaust: fixes y :: "hmultiset" obtains x :: "hmultiset multiset" where "y = HMSet x"
SucFbd_Card_order: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "Card_order (cardSuc (r1 +c r2))"
assms_3: shows "Broadcast m2 \<in> set (history j)"
eln_special_values_1: fixes x :: "ennreal" shows "(eln x = - \<infinity>) = (x = 0)"
preserves_reflects_arr: fixes f :: "'b" shows "D.arr (D.L f) = D.arr f"
stop_no_eval': fixes mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val"   and P :: "bool" assumes "(Stop, mem) \<leadsto>\<^sub>s (c', mem')" shows "P"
empty_neutral_1: fixes x :: "'b multiset" shows "{#} + x = x"
natural_hom: shows "GroupHom (FreeGroup S) (\<lambda>a. if a \<in> FreeGroup S then BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {a}) else 0)"
assms: fixes h :: "'h"   and X'' :: "'x set" assumes "blocking_on ds X h X''" shows "False"
positive_ints: fixes x :: "int" assumes "x \<in> \<K>" shows "0 < x"
lookup_combine: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and m1 :: "('d, 'c) mapping"   and m2 :: "('d, 'c) mapping"   and x :: "'d" shows "Mapping.lookup (Mapping.combine f m1 m2) x = combine_options f (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
minus_zero: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd" assumes "class.group_add minus plus zero uminus" shows "uminus zero = zero"
nop_bij_n: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_bij A n f" shows "n \<in>\<^sub>\<circ> \<omega>"
1: shows "\<guillemotleft>\<l>[f] : trg f \<star> f \<Rightarrow> f\<guillemotright>"
H: shows "\<not> eq_nodes M v (dfa_trans M p bs) (dfa_trans M q bs)"
SimOut: shows "Schema FIG.Out \<preceq>Sim Schema DFIG.Out"
T_get_min_rest_estimate: fixes ts :: "'b tree list" assumes "ts \<noteq> []" shows "T_get_min_rest ts = length ts"
is_normal_normal_of_Float: fixes x :: "Float.float" assumes "is_normal_Float TYPE(('a, 'b) IEEE.float) x" shows "is_normal (normal_of_Float x)"
assms_2: shows "P' \<noteq> P \<parallel> !P"
GDD_axioms: shows "GDD (\<V> \<times> I) (\<Sum>\<^sub># (image_mset f \<B>)) {G \<times> I |G. G \<in> \<G>} 1"
llength_take: fixes t :: "'b llist"   and A :: "'b set"   and i :: "nat" assumes "t \<in> A\<^sup>\<omega>" shows "LList2.llength (t \<down> i) = i"
continuous_map_const: fixes X :: "'a topology"   and Y :: "'b topology"   and C :: "'b" shows "continuous_map X Y (\<lambda>x. C) = (topspace X = {} \<or> C \<in> topspace Y)"
cmp_simps_2: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "src\<^sub>C (\<Phi>\<^sub>H (f, g)) = src\<^sub>C (H g)"
simps_14: shows "cospan_shape.TT \<noteq> cospan_shape.AA"
echelon_form_of_column_k_def: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A' :: "(('b, 'c) vec, 'd) vec \<times> nat"   and k :: "nat" shows "echelon_form_of_column_k bezout A' k = (let (A, i) = A' in if (\<forall>m\<ge>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'b)) \<or> i = nrows A then (A, i) else if \<forall>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'b) then (A, i + 1) else let n = LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'b) \<and> mod_type_class.from_nat i \<le> n; interchange_A = interchange_rows A (mod_type_class.from_nat i) n in (bezout_iterate interchange_A (nrows A - 1) (mod_type_class.from_nat i) (mod_type_class.from_nat k) bezout, i + 1))"
assms_2: fixes s :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "s \<in> set ss" shows "homogeneous_set s"
eq_fin_concat_start: fixes w :: "'item list"   and w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" shows "(w @ w\<^sub>1 =\<^sub>F w @ w\<^sub>2) = (w\<^sub>1 =\<^sub>F w\<^sub>2)"
simps_2: fixes c :: "'d" shows "compile (Constant c) = [Const c]"
B: shows "trg f \<star> \<l>[f] = (\<i>[trg f] \<star> f) \<cdot> local.inv \<a>[trg f, trg f, f]"
qtable_unit_empty_table: fixes n :: "nat"   and R :: "'a option list \<Rightarrow> bool"   and P :: "'a option list \<Rightarrow> bool" assumes "qtable n {} R P (unit_table n)" shows "qtable n {} R (\<lambda>v. \<not> P v) empty_table"
permutation_permutes: fixes p :: "'a \<Rightarrow> 'a" shows "permutation p = (\<exists>S. finite S \<and> p permutes S)"
hyps_1: shows "Infer CC' D E \<in> \<Gamma>"
hpair_neq_fst: fixes a :: "hf"   and b :: "hf" shows "\<langle>a, b\<rangle> \<noteq> a"
rep_eq: fixes x :: "perm"   and xa :: "('a, 'b) fmap" shows "($$) (x \<bullet> xa) = x \<bullet> ($$) xa"