hcju/isabelleps huggingface dataset:
u2: fixes x :: "'a" shows "P *\<^sub>V x \<in> space_as_set M"
two_mod_two: shows "2 mod Suc (Suc 0) = 0"
simps_18: fixes Vs :: "String.literal list"   and U :: "unit"   and a :: "'a"   and e :: "'a expr" shows "compE1 Vs (insync\<^bsub>U\<^esub> (a) e) = insync\<^bsub>length Vs\<^esub> (a) compE1 (Vs @ [fresh_var Vs]) e"
map: shows "map_of FDTs (F, C) = \<lfloor>(b, T)\<rfloor>"
writes: fixes v :: "'var"   and n :: "'n" assumes "\<exists>\<sigma>. \<sigma> v \<noteq> sem n \<sigma> v" shows "v \<in> local.writes n"
horizontal_composition_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c" shows "horizontal_composition V H src trg \<equiv> horizontal_homs V src trg \<and> functor (subcategory.comp (product_category.comp V V) (\<lambda>\<mu>\<nu>. partial_magma.arr V (fst \<mu>\<nu>) \<and> partial_magma.arr V (snd \<mu>\<nu>) \<and> src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>))) V (\<lambda>\<mu>\<nu>. H (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<and> horizontal_composition_axioms V H src trg"
perm_append_Cons: fixes a :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "a # xs @ ys <~~> xs @ a # ys"
dim_eq: fixes X :: "(('a \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a" assumes "X \<subseteq> tangent_space p" shows "tangent_space.dim p X = dim X"
assms_2: shows "(rel_interior S = {}) = (rel_interior T = {})"
chine_simps_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "partial_magma.dom C (Chn \<mu>) = span_in_category.apex C (Dom \<mu>)"
span_base: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S" shows "a \<in> More_Modules.ideal S"
dg_Cod_vdomain: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
cases: fixes x :: "tm" obtains t :: "name" and u :: "name" and y :: "tm" where "atom t \<sharp> (u, y)" and "atom u \<sharp> y" and "x = y"
sliceable: fixes v :: "'a list"   and S :: "'a list set"   and \<sigma> :: "'b trace"   and i :: "nat" assumes "v \<in> S" shows "sat (slice S \<sigma>) v i = sat \<sigma> v i"
seq: shows "C.arr (fst gf) \<and> C.arr (snd gf) \<and> C.dom (snd gf') = C.cod (snd gf) \<and> C.arr (fst gf') \<and> C.arr (snd gf') \<and> C.dom (fst gf) = C.cod (fst gf')"
preserves_modes_mem_mds\<^sub>A_simp: fixes mds\<^sub>A :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set" assumes "\<forall>m. var\<^sub>C_of ` mds\<^sub>A m = range var\<^sub>C_of \<inter> mds\<^sub>C m" shows "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"
restrict_extensional_sub: fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "restrict f A \<in> extensional B"
gap_\<alpha>_def: shows "gap_\<alpha> \<equiv> \<lambda>(l, r, buf). BUF (take l buf) (drop r buf)"
shortestPath_ex: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)"   and "invar g" obtains ns :: "'node list" where "g \<turnstile> Entry g-ns\<rightarrow>n" and "distinct ns" and "length ns = shortestPath g n"
intro: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and default_node_properties :: "'d" assumes "\<And>G f nP. \<lbrakk>wf_graph G; f \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP\<rbrakk> \<Longrightarrow> \<forall>i\<in>snd ` f. \<not> sinvar G (nP(i := default_node_properties))"   and "\<And>otherbot. \<forall>G f nP i. wf_graph G \<and> f \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<and> i \<in> snd ` f \<longrightarrow> \<not> sinvar G (nP(i := otherbot)) \<Longrightarrow> otherbot = default_node_properties" shows "SecurityInvariant_IFS_axioms sinvar default_node_properties"
prime_factorization_empty_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "(factorial_semiring.prime_factorization zero times one normalize x = {#}) = (x = zero \<or> dvd.dvd times x one)"
adapt_stoch_proc_borel_measurable: fixes X :: "'b \<Rightarrow> 'a \<Rightarrow> 'd"   and N :: "'d measure" assumes "adapt_stoch_proc F X N" shows "\<forall>n. random_variable N (X n)"
genideal_one: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "Idl\<^bsub>R\<^esub> {\<one>\<^bsub>R\<^esub>} = carrier R"
dual_inv_surj: shows "surj \<partial>\<^sup>-"
hyps: assumes "\<forall>c. c \<in> set w \<longrightarrow> c = a" shows "\<exists>k. w = [a] \<^sup>@ k"
inf_sup_aci_5: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
cnf_\<Phi>\<^sub>F_is_A_union_B: shows "cnf (encode_all_frame_axioms \<Pi> (length \<pi>)) = \<Union> (\<Union>(k, v)\<in>{0..<length \<pi>} \<times> set (\<Pi>\<^sub>\<V>). {{{(State k (index (\<Pi>\<^sub>\<V>) v))\<^sup>+, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<inverse>} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (add_effects_of op)}}}) \<union> \<Union> (\<Union>(k, v)\<in>{0..<length \<pi>} \<times> set (\<Pi>\<^sub>\<V>). {{{(State k (index (\<Pi>\<^sub>\<V>) v))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (delete_effects_of op)}}})"
tok_prik_rev: shows "Rev_PriK \<notin> range Tok_PriK"
\<phi>_def: shows "\<phi> \<equiv> inv_into D (ordermap D VWF)"
Lim_PInfty: fixes f :: "nat \<Rightarrow> ereal" shows "f \<longlonglongrightarrow> \<infinity> = (\<forall>B. \<exists>N. \<forall>n\<ge>N. ereal B \<le> f n)"
Gp_mult_induced2: fixes H :: "'a set"   and K :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "a \<in> H \<inter> K"   and "b \<in> H \<inter> K" shows "a \<cdot>\<^bsub>\<natural>H \<inter> K\<^esub> b = a \<cdot>\<^bsub>\<natural>K\<^esub> b"
heap_sift_down: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, a, r\<rangle>"   and "heap l"   and "heap r" shows "heap (sift_down l a r)"
eq: shows "f ` A' \<inter> X = f ` (A' \<inter> (f -` X \<inter> A))"
transpose_mat_def: fixes A :: "'a mat" shows "A\<^sup>T \<equiv> mat (dim_col A) (dim_row A) (\<lambda>(i, j). A $$ (j, i))"
chine_assoc_inverse: shows "C.inverse_arrows fgf.chine_assoc fgf.chine_assoc'"
rep_eq: fixes S :: "'b set" assumes "S \<noteq> {}"   and "finite S" shows "pmf (pmf_of_set S) = (\<lambda>x. indicat_real S x / real (card S))"
seq_vs_steps: fixes a :: "'a"   and b :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set" shows "((a, b) \<in> (unlabel ars)\<^sup>*) = (\<exists>ss. fst ss = a \<and> lst ss = b \<and> ss \<in> seq ars)"
iso_iff_mon_epi: fixes f :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "(f \<in> Group.iso G H) = (f \<in> mon G H \<and> f \<in> epi G H)"
GB_replace_lt_adds_stable_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis G"   and "q \<noteq> 0"   and "q \<in> pmdl G"   and "lt q adds\<^sub>t lt p" shows "is_Groebner_basis (insert q (G - {p}))"
comp_wiring_simps: fixes f :: "'h \<Rightarrow> 'l"   and g :: "'m \<Rightarrow> 'k"   and f' :: "'l \<Rightarrow> 'i"   and g' :: "'j \<Rightarrow> 'm" shows "(f, g) \<circ>\<^sub>w (f', g') = (f' \<circ> f, g \<circ> g')"
one_le_inverse: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less_eq a one" shows "less_eq one (inverse a)"
ack2_le_ack1: fixes i :: "nat"   and j :: "nat" shows "ack i (Suc j) \<le> ack (Suc i) j"
path_component_of_subset: fixes S :: "'b set"   and T :: "'b set"   and x :: "'b"   and y :: "'b" assumes "S \<subseteq> T"   and "path_component S x y" shows "path_component T x y"
opt: shows "real (T_splay a t) + \<Phi> (splay a t) - \<Phi> t \<le> 3 * \<phi> t + 1"
commut_normal_subgroup: fixes H :: "'a set"   and N :: "'a set"   and K :: "'a set" assumes "subgroup H G"   and "N \<lhd> G\<lparr>carrier := H\<rparr>"   and "subgroup K (G\<lparr>carrier := H\<rparr>)" shows "K <#> N = N <#> K"
dbm_entry_dbm_min3': fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "dbm_entry_val u (Some c) None (min a b)" shows "dbm_entry_val u (Some c) None a"
inj: fixes Q :: "process set" shows "inj_on fst (vote_set (votes s) Q)"
add_cancel_right_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus b a) = (b = zero)"
Ln'_i: shows "Ln' \<i> = \<i> * complex_of_real pi / 2"
imp_wf_dfa: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and N :: "nat bdd list \<times> bool list" assumes "wf_dfa M n"   and "wf_dfa N n" shows "wf_dfa (imp_dfa M N) n"
inf_not_le_Lt: fixes x :: "'b" shows "(\<infinity> \<le> Lt x) = False"
schur_decomposition_exists: fixes A :: "'b mat"   and n :: "nat"   and es :: "'b list" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>es. [:- a, 1::'b:])" shows "\<exists>B\<in>carrier_mat n n. upper_triangular B \<and> similar_mat A B"
split_vec_dims_2: fixes i :: "nat"   and j :: "nat"   and x :: "(nat, 'b) mapping"   and a :: "'b vec"   and b :: "'b vec" assumes "split_i_j_x i j x = (a, b)" shows "dim_vec b = j - i"
adjoint_stays_mon: fixes I :: "interp"   and a :: "char"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)" shows "mono (case SGames \<sigma> a of None \<Rightarrow> Games I a | Some r \<Rightarrow> game_sem I r)"
numer_cf6_def: shows "numer_cf6 \<equiv> \<lambda>x. x ^ 6 + 42 * x ^ 5 + 840 * x ^ 4 + 10080 * x ^ 3 + 75600 * x\<^sup>2 + 332640 * x + 665280"
integrable_on_open_interval: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c" shows "(f integrable_on box a b) = (f integrable_on cbox a b)"
swapBinp_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and binp :: "'b \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qAbs set option" shows "binp %%[x \<and> y]_xs \<equiv> lift (swapAbs xs x y) binp"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and negation_type :: "'b negation_type" shows "P (case negation_type of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. negation_type = Pos x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. negation_type = Neg x2 \<and> \<not> P (f2 x2))))"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vreal_inverse"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vreal_inverse" shows "b = c"
bounded_linear: fixes v :: "(real, 'sz) vec" shows "bounded_linear (\<lambda>v'. sterm_sem I \<theta>\<^sub>1 v * frechet I \<theta>\<^sub>2 v v' + frechet I \<theta>\<^sub>1 v v' * sterm_sem I \<theta>\<^sub>2 v)"
assms_2: shows "- \<infinity> < subcocycle_avg_ereal u"
mset_subseteq_add_iff2: fixes i :: "nat"   and j :: "nat"   and u :: "'k multiset"   and m :: "'k multiset"   and n :: "'k multiset" assumes "i \<le> j" shows "(repeat_mset i u + m \<subseteq># repeat_mset j u + n) = (m \<subseteq># repeat_mset (j - i) u + n)"
insertion_code: fixes i :: "nat \<Rightarrow> 'b"   and mp :: "'b mpoly" shows "insertion i mp = (\<Sum>m\<in>monomials mp. MPoly_Type.coeff mp m * (\<Prod>k\<in>keys m. i k ^ lookup m k))"
e: shows "stp \<le> stpa \<and> crsp ly (as', am') (steps0 (Suc 0, l, r) tp stpa) ires"
Quotient3I: fixes Abs :: "'f \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'f"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>a. Abs (Rep a) = a"   and "\<And>a. R (Rep a) (Rep a)"   and "\<And>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" shows "Quotient3 R Abs Rep"
enc_equiv: shows "[\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,F\<rbrace> in v]"
monotone_times_ennreal1: fixes y :: "ennreal" shows "monotone (\<le>) (\<le>) (\<lambda>x. x * y)"
elem_in_original_in_repeat: fixes n :: "nat"   and a :: "'b"   and A :: "'b multiset" assumes "0 < n"   and "a \<in># A" shows "a \<in># repeat_mset n A"
assms: fixes f :: "nat \<Rightarrow> 'a" assumes "\<forall>n. f n \<in> s"   and "Cauchy f" shows "\<exists>l\<in>s. f \<longlonglongrightarrow> l"
sub_q64_array: fixes i :: "nat" assumes "i \<in> {..<64}" shows "q64_array !! i = (i \<in> q64)"
lm077: fixes g :: "('c \<times> 'd) set"   and A :: "'c set"   and B :: "'d set"   and C :: "'c set" assumes "g \<in> injections A B" shows "Outside' C g \<in> injections (A - C) B"
sym_def: fixes r :: "('g \<times> 'g) set" shows "sym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
r_lt_eq_b: shows "\<Lambda> * (\<v> - 1) div (\<k> - 1) \<le> \<b>"
SeqRed: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e;; e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e';; e\<^sub>2,s'\<rangle>"
sconjunctive_assert: fixes p :: "'d" shows "sconjunctive {. p .}"
UN_I: fixes a :: "'g"   and A :: "'g set"   and b :: "'h"   and B :: "'g \<Rightarrow> 'h set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
h: shows "h \<in> Group.iso (relative_homology_group p (nsphere n) {x. x k \<le> 0}) (relative_homology_group p (nsphere n) {x. 0 \<le> x k})"
mat_of_rows_row: fixes i :: "nat"   and vs :: "'c vec list"   and n :: "nat" assumes "i < length vs"   and "vs ! i \<in> carrier_vec n" shows "row (mat_of_rows n vs) i = vs ! i"
P_mod_mult: fixes I :: "'a set"   and J :: "'a set"   and n :: "nat"   and c :: "nat \<Rightarrow> 'a"   and m :: "nat"   and d :: "nat \<Rightarrow> 'a" assumes "ideal S I"   and "ideal S J"   and "pol_coeff S (n, c)"   and "pol_coeff S (m, d)"   and "P_mod R S X I (polyn_expr R X n (n, c))"   and "P_mod R S X J (polyn_expr R X m (m, d))" shows "P_mod R S X (I \<diamondsuit>\<^sub>r\<^bsub>S\<^esub> J) (polyn_expr R X n (n, c) \<cdot>\<^sub>r polyn_expr R X m (m, d))"
proj_def: shows "proj \<equiv> \<lambda>x. x /\<^sub>R norm x"
pol_of_deg0: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "(deg_n R S X p = 0) = (p \<in> carrier S)"
assms_1: shows "S \<subseteq> 1\<^sub>\<sigma>"
isomorphic_to_local_is_local: fixes B :: "'d set"   and addB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multB :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroB :: "'d"   and oneB :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and addA :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multA :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zeroA :: "'e"   and oneA :: "'e" assumes "local_ring B addB multB zeroB oneB"   and "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "local_ring A addA multA zeroA oneA"
min_cost_awalk: fixes u :: "'a"   and v :: "'a"   and c :: "'b \<Rightarrow> real" assumes "u \<rightarrow>\<^sup>* v"   and "\<And>e. e \<in> arcs G \<Longrightarrow> 0 \<le> c e" shows "\<exists>p. apath u p v \<and> \<mu> c u v = ereal (awalk_cost c p)"
case_5: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and f4 :: "'c"   and f5 :: "char list \<Rightarrow> 'c"   and f6 :: "'c"   and f7 :: "char list \<Rightarrow> 'c"   and f8 :: "'c"   and f9 :: "'c"   and x5 :: "char list" shows "(case Call x5 of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2 | Log \<Rightarrow> f3 | Reject \<Rightarrow> f4 | Call x \<Rightarrow> f5 x | Return \<Rightarrow> f6 | Goto x \<Rightarrow> f7 x | Empty \<Rightarrow> f8 | Unknown \<Rightarrow> f9) = f5 x5"
simps_1: fixes lformula0 :: "'a \<Rightarrow> bool"   and uu :: "bool" shows "Formula_Operations.lformula lformula0 (FBool uu) = True"
beukers_nn_integral1_series: shows "beukers_nn_integral1 r s = (\<Sum>k. ennreal (1 / real ((k + r + 1)\<^sup>2 * (k + s + 1)) + 1 / real ((k + r + 1) * (k + s + 1)\<^sup>2)))"
projr_def: fixes sum :: "'d + 'c" shows "projr sum = (case sum of Inr x2 \<Rightarrow> x2)"
finI2: fixes K :: "'c \<Rightarrow> 'c DBMEntry"   and g :: "'c \<Rightarrow> 'd"   and f :: "(nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry) \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "theLe \<circ> K = id" shows "finite {g d |d. f M (v c) = K d}"
distinct_list_update: fixes xs :: "'g list"   and a :: "'g"   and i :: "nat" assumes "distinct xs"   and "a \<notin> set xs - {xs ! i}" shows "distinct (xs[i := a])"
negative_y_axis_def: shows "negative_y_axis = {z \<in> circline_set y_axis. z \<noteq> \<infinity>\<^sub>h \<and> Im (to_complex z) < 0}"
base_in_hom_2: shows "\<guillemotleft>u : u \<Rightarrow> u\<guillemotright>"
sylvester_mat_dim_1: fixes p :: "'b poly"   and q :: "'b poly" shows "dim_row (sylvester_mat p q) = degree p + degree q"
Axiom: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "t \<in> M" shows "M \<turnstile> t"
filter_insort_triv: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and P :: "'h \<Rightarrow> bool"   and x :: "'h"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" assumes "class.linorder less_eq less"   and "\<not> P x" shows "filter P (linorder.insort_key less_eq f x xs) = filter P xs"
K_def: shows "K = 2 ^ (5 * (degree f - 1) * (degree f - 1)) * \<parallel>f\<parallel>\<^sup>2 ^ (2 * (degree f - 1))"
okamoto_base_def: fixes \<G> :: "'a cyclic_group" shows "okamoto_base \<G> \<equiv> prime (order \<G>)"
cases: fixes a :: "chmsg"   and T :: "atom set"   and U :: "fid_t set"   and H :: "chmsg set" assumes "a \<in> fake T U H" obtains M :: "chmsg" where "a = M__" and "M__ \<in> H"   | M :: "atom list" and c :: "secprop set" and A :: "agent" and B :: "agent" where "a = StatCh c__ A B (Msg Ma__)" and "set Ma__ \<subseteq> extr T H" and "auth \<notin> c__ \<or> A \<in> bad \<or> B \<in> bad"   | M :: "atom list" and c :: "secprop set" and K :: "key" where "a = DynCh c K (Msg M)" and "set M \<subseteq> extr T H" and "auth \<notin> c \<and> K \<in> rkeys U \<or> aKey K \<in> extr T H"
dg_is_arrD_5: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "\<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
mangoldt_even_def: fixes d :: "nat" shows "mangoldt_even d = (if primepow_even d then ln (real (aprimedivisor d)) else 0)"
Aexp_fresh_bot: fixes v :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "atom v \<sharp> e" shows "Aexp_syn' a e v = \<bottom>"
noFault_start: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate"   and f :: "'c" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "t \<noteq> Fault f" shows "s \<noteq> Fault f"
CCC_eq_VVV: shows "CCC.comp = VVV.comp"
Abs_finfun_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> finfun" shows "finfun_apply (Abs_finfun y) = y"
test_bit_rcat: fixes sw :: "nat"   and wl :: "'c word list"   and rc :: "'d word"   and n :: "nat" assumes "sw = size (hd wl)"   and "rc = word_rcat wl" shows "rc !! n = (n < size rc \<and> n div sw < length wl \<and> rev wl ! (n div sw) !! (n mod sw))"
invar_update: fixes m :: "'m"   and a :: "'v"   and b :: "('v \<times> nat) list" assumes "M_invar m" shows "M_invar (M_update a b m)"
translate_plus': fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" shows "ccw (x + a) (x + b) (x + c) = ccw a b c"
Pair_def: fixes a :: "'b"   and b :: "'c" shows "(a, b) = Abs_prod (Pair_Rep a b)"
swapsxy: fixes A :: "'a list"   and x :: "'a"   and y :: "'a"   and sws :: "nat list" assumes "A \<in> {[x, y], [y, x]}" shows "swaps sws A \<in> {[x, y], [y, x]}"
lformula_decr: fixes \<phi> :: "('a, 'k) aformula"   and k :: "'k"   and l :: "nat" assumes "lformula \<phi>" shows "lformula (decr k l \<phi>)"
bounded_clinear_mult_right: fixes x :: "'a" shows "bounded_clinear ((*) x)"
strt'_lemma: fixes xs :: "nat list"   and vl :: "nat"   and n :: "nat" assumes "length xs = vl"   and "n \<le> vl" shows "rec_exec (rec_strt' vl n) xs = strt' xs n"
pCons_0_as_mult: shows "pCons (0::'a) = (*) (pCons (0::'a) 1)"
ladj_radj_dual: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'d" shows "ladj f x = \<partial> (radj (\<partial>\<^sub>F f) (\<partial> x))"
simps_1: fixes xss :: "'a list list" shows "radix_sort 0 xss = xss"
validated_part_def: fixes I :: "'at set"   and C :: "'at Literal set" shows "validated_part I C = {L \<in> C. I \<Turnstile> L}"
unif: shows "unifier\<^sub>l\<^sub>s \<sigma>' L"
assms_1: shows "constructive_security_obsf middle ideal sim_inner \<I>_middle \<I>_inner \<I>_common (absorb \<A> (obsf_converter (sim_outer |\<^sub>= 1\<^sub>C))) adv1"
add_ac_2: fixes a :: "'c"   and b :: "'c" shows "a + b = b + a"
proj_stoch_proc_component_2: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and w :: "'c" assumes "n \<le> k" shows "proj_stoch_proc f n w !! k = f n w"
closed_segment_eq_open: fixes a :: "'b"   and b :: "'b" shows "closed_segment a b = open_segment a b \<union> {a, b}"
red_single_ord: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "red_single p q f t" shows "q \<prec>\<^sub>p p"
lcm_neg1: 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 uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "lcm (uminus a) b = lcm a b"