hcju/isabelleps huggingface dataset:
lub_cong_lemma: fixes f :: "nat \<Rightarrow> ordinal"   and g :: "nat \<Rightarrow> ordinal"   and F :: "ordinal \<Rightarrow> 'a" assumes "\<forall>n. f n < oLimit f"   and "\<forall>m. g m < oLimit g"   and "oLimit f \<le> oLimit g"   and "\<forall>y<oLimit g. \<forall>x\<le>y. F x << F y" shows "lub (\<lambda>n. F (f n)) << lub (\<lambda>n. F (g n))"
integral_of_real: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f" shows "LINT x|M. of_real (f x) = of_real (integral\<^sup>L M f)"
eng: fixes h1 :: "'c"   and G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and h2 :: "'c" assumes "h1 \<in> generate G H"   and "h2 \<in> generate G H" shows "h1 \<otimes>\<^bsub>G\<^esub> h2 \<in> generate G H"
sel_8: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" shows "getId (Send x21 x22 x23 x24 x25 x26) = x21"
sup_state_refl: fixes P :: "'b prog"   and s :: "ty list \<times> ty err list" shows "P \<turnstile> s \<le>\<^sub>i s"
m_comm: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
mine: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "honest nid"   and "active nid (t (Suc n))"   and "mining (cmp nid (t (Suc n)))" shows "active nid (t n)"
eclose_def: fixes X :: "hf" shows "eclose X = \<Sqinter>\<lbrace>Y \<^bold>\<in> HPow (Vset (rank X)). Transset Y \<and> X \<le> Y\<rbrace>"
V_def: shows "V \<equiv> {v. \<forall>x\<in>\<X>. 0 \<le> v x}"
A: fixes u :: "nat \<Rightarrow> 'a"   and v :: "nat \<Rightarrow> 'a" assumes "u \<longlonglongrightarrow> x" shows "liminf (\<lambda>n. ereal (Gromov_product_at e (u n) (v n))) = liminf (\<lambda>n. ereal (Gromov_product_at e x (v n)))"
eq_\<I>_gpvD: fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and \<I> :: "('j, 'k) \<I>"   and gpv :: "('h, 'j, 'k) gpv"   and gpv' :: "('i, 'j, 'k) gpv" assumes "eq_\<I>_gpv A \<I> gpv gpv'" shows "rel_spmf (eq_\<I>_generat A \<I> (eq_\<I>_gpv A \<I>)) (the_gpv gpv) (the_gpv gpv')"
simps_4: fixes v :: "'a formula" shows "is_disj (\<^bold>\<not> v) = is_lit_plus (\<^bold>\<not> v)"
tl_def: shows "DList_Set.tl \<equiv> map_fun list_of_dlist Abs_dlist list.tl"
gen2_unique: fixes i :: "nat"   and j :: "nat"   and A :: "nat"   and m :: "nat"   and B :: "nat"   and n :: "nat" assumes "i < j"   and "j < length (gen2 A B m n)" shows "gen2 A B m n ! i \<noteq> gen2 A B m n ! j"
mono: fixes xs :: "'a llist" shows "mono_llist (\<lambda>lmap. F lmap xs)"
uint16_sshiftr_def: fixes x :: "uint16"   and n :: "integer" shows "uint16_sshiftr x n = (if n < 0 \<or> 16 \<le> n then undefined (>>>) x n else x >>> nat_of_integer n)"
nat_term_compare_inv_conv: fixes to :: "'c nat_term_order" shows "nat_term_compare_inv to = key_compare (key_order_of_nat_term_order_inv to)"
assign\<^sub>\<C>: fixes x :: "'Var"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'AExp"   and t :: "'BExp set"   and P :: "'BExp set"   and P' :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set" assumes "x \<in> \<C>"   and "\<Gamma> \<turnstile>\<^sub>a e \<in> t"   and "P \<turnstile> t"   and "\<forall>v\<in>dom \<Gamma>. x \<notin> vars_of_type (the (\<Gamma> v))"   and "P' = assign_post P x e |` {v. stable \<S> v}"   and "\<forall>v. x \<in> \<C>_vars v \<and> v \<notin> snd \<S> \<longrightarrow> P \<turnstile> to_total \<Gamma> v \<and> to_total \<Gamma> v \<le>:\<^sub>P' dma_type v" shows "\<turnstile> \<Gamma>,\<S>,P {x \<leftarrow> e} \<Gamma>,\<S>,P'"
annil: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and x :: "'b" assumes "class.ab_near_semiring_one_zerol plus times one zero" shows "times zero x = zero"
H_parts_1: shows "verts H = HU \<union> HV"
integral_cos_Z': fixes n :: "real" assumes "n \<in> \<int>" shows "LINT x|lebesgue_on {- pi..pi}. cos (n * x) = (if n = 0 then 2 * pi else 0)"
IH: fixes xs :: "real list" shows "eval (QE_dnf opt step \<phi>) xs = eval \<phi> xs"
simps: fixes prog :: "cmd"   and a1 :: "w_node"   and a2 :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and a3 :: "w_node" shows "While_CFG.path prog a1 a2 a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> While_CFG.valid_node prog n) \<or> (\<exists>n'' as n' a n. a1 = n \<and> a2 = a # as \<and> a3 = n' \<and> While_CFG.path prog n'' as n' \<and> valid_edge prog a \<and> sourcenode a = n \<and> targetnode a = n''))"
greaterThanAtMost_eq_atLeastAtMost_diff: fixes a :: "'a"   and b :: "'a" shows "{a<..b} = {a..b} - {a}"
tLength_0_conv: fixes t :: "'a Trace" shows "(tLength t = 0) = (\<exists>s. t = tInit s)"
assms: shows "\<forall>vs'. vs' \<in> svs \<longrightarrow> \<not> dep_var_set PROB vs' vs"
rank_Gauss_Jordan: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A = rank (Gauss_Jordan A)"
integer_of_uint64_signed_def: fixes n :: "uint64" shows "integer_of_uint64_signed n = (if bit n 63 then undefined integer_of_uint64 n else integer_of_uint64 n)"
line_equation: fixes z1 :: "complex"   and z2 :: "complex"   and \<mu> :: "complex" assumes "z1 \<noteq> z2"   and "\<mu> = rot90 (z2 - z1)" shows "line z1 z2 = {z. cnj_mix \<mu> z - cnj_mix \<mu> z1 = 0}"
wf_measure: fixes f :: "'d \<Rightarrow> nat" shows "wf (measure f)"
predecessor_is_node: fixes n :: "'node"   and g :: "'g"   and n' :: "'node" assumes "n \<in> set (predecessors g n')"   and "invar g" shows "n \<in> set (\<alpha>n g)"
model_def: fixes e :: "nat \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b list \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'b list \<Rightarrow> bool"   and ps :: "('c, 'd) form list"   and p :: "('c, 'd) form" shows "(e,f,g,ps \<Turnstile> p) = (list_all (eval e f g) ps \<longrightarrow> eval e f g p)"
simps_7: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm" shows "usubstappt \<sigma> U (Differential \<theta>) = Differentialo (usubstappt \<sigma> Denotational_Semantics.allvars \<theta>)"
continuous_on_no_limpt: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. \<not> x islimpt S" shows "continuous_on S f"
assms_1: shows "x = A *\<^sub>m\<^sub>v y"
deleteMin_def: fixes bq :: "('c, 'd) SkewBinomialTree list" shows "deleteMin bq = (let min = getMinTree bq in insertList (filter (\<lambda>t. rank t = 0) (children min)) (meld (rev (filter (\<lambda>t. 0 < rank t) (children min))) (remove1Prio (prio min) bq)))"
distinct_13: fixes x21 :: "('a, 'b) com"   and x22 :: "('a, 'b) com"   and x51 :: "('a, 'b) com"   and x52 :: "('a, 'b) com" shows "x21 ;; x22 \<noteq> x51 | x52"
iso_between_subgroups: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd"   and S :: "'b set"   and T :: "'d set" assumes "group_hom G H h"   and "h \<in> Group.iso G H"   and "S \<subseteq> carrier G"   and "h ` S = T" shows "h \<in> Group.iso (subgroup_generated G S) (subgroup_generated H T)"
emeasure_point_measure_finite: fixes A :: "'a set"   and X :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "finite A"   and "X \<subseteq> A" shows "emeasure (point_measure A f) X = sum f X"
hyp: shows "\<not> (finite (cl_ecl C) \<and> finite C')"
CK_nf_pos_lattice_2: shows "id \<le> K"
tauChainInduct: fixes P :: "pi"   and P' :: "pi"   and F :: "pi \<Rightarrow> bool" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "F P"   and "\<And>P' P''. \<lbrakk>P \<Longrightarrow>\<^sub>\<tau> P'; P' \<longmapsto> \<tau> \<prec> P''; F P'\<rbrakk> \<Longrightarrow> F P''" shows "F P'"
b1: shows "(fps_exp a - 1) $ 1 \<noteq> (0::'a)"
lebesgue_integral_real_affine: fixes c :: "real"   and f :: "real \<Rightarrow> 'c"   and t :: "real" assumes "c \<noteq> 0" shows "integral\<^sup>L lebesgue f = \<bar>c\<bar> *\<^sub>R (LINT x|lebesgue. f (t + c * x))"
x0: shows "real_of_1 (p1, l1, r1) < 0 \<or> 0 < real_of_1 (p1, l1, r1)"
bij_betw_comp_iff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and f' :: "'c \<Rightarrow> 'd"   and A'' :: "'d set" assumes "bij_betw f A A'" shows "bij_betw f' A' A'' = bij_betw (f' \<circ> f) A A''"
insertion_add_ind: fixes L :: "'a list"   and var :: "nat"   and x :: "'a"   and p :: "'a mpoly"   and d :: "nat" shows "insertion (nth_default (0::'a) (L[var := x])) (\<Sum>i\<le>d. isolate_variable_sparse p var i * Var var ^ i) = (\<Sum>i = 0..<d + 1. insertion (nth_default (0::'a) (L[var := x])) (isolate_variable_sparse p var i * Var var ^ i))"
that: fixes x :: "V" assumes "xs = [x]\<^sub>\<circ>"   and "x \<in>\<^sub>\<circ> A" shows "thesis"
edges_are_in_V_2: fixes v :: "'c"   and w :: "'c" assumes "v \<rightarrow>\<^bsub>T\<^esub> w" shows "w \<in> V\<^bsub>T\<^esub>"
len_join_adj_div2: fixes ts :: "'a tree23s" assumes "not_T ts" shows "len (join_adj ts) \<le> len ts div 2"
lemma_3_8_i: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> filters" shows "cong_r F a b = (\<exists>x y. x \<in> F \<and> y \<in> F \<and> x * a = y * b)"
autoref_WHILEIT: fixes I :: "'d \<Rightarrow> bool"   and R :: "('c \<times> 'd) set"   and c :: "'c \<Rightarrow> bool"   and c' :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and f' :: "'d \<Rightarrow> 'd nres"   and s' :: "'d"   and s :: "'c" assumes "\<And>x x'. \<lbrakk>I x'; (x, x') \<in> R\<rbrakk> \<Longrightarrow> (c x, c' $ x') \<in> bool_rel"   and "\<And>x x'. \<lbrakk>REMOVE_INTERNAL c' x'; I x'; (x, x') \<in> R\<rbrakk> \<Longrightarrow> (f x, f' $ x') \<in> \<langle>R\<rangle>nres_rel"   and "I s' \<Longrightarrow> (s, s') \<in> R" shows "(WHILE\<^sub>T c f s, (OP WHILE\<^sub>T\<^bsup>I\<^esup> ::: (R \<rightarrow> bool_rel) \<rightarrow> (R \<rightarrow> \<langle>R\<rangle>nres_rel) \<rightarrow> R \<rightarrow> \<langle>R\<rangle>nres_rel) $ c' $ f' $ s') \<in> \<langle>R\<rangle>nres_rel"
inv_is_inverse: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.iso f" shows "S.inverse_arrows f (S.inv f)"
IH: fixes t :: "pdf_type" assumes "\<Gamma> \<turnstile> e : t"   and "density_context V V' \<Gamma> \<delta>"   and "free_vars e \<subseteq> V \<union> V'" shows "has_parametrized_subprob_density (state_measure V' \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) (stock_measure t) f"
sign_times: fixes x :: "'a"   and y :: "'a" shows "sign (x * y) = sign x * sign y"
head_isnpolyh_Suc': fixes n :: "nat"   and p :: "poly" assumes "0 < n"   and "isnpolyh p n" shows "head p = p"
fold_insert_idem: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b set"   and z :: "'c"   and x :: "'b" assumes "comp_fun_idem f"   and "finite A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)"
minus_left_distributivity: fixes p :: "int poly fract"   and q :: "int poly fract"   and r :: "int poly fract" shows "rat_poly_times p (q - r) = rat_poly_times p q - rat_poly_times p r"
M: shows "(map (inv p) (filter (\<lambda>x. x \<in> range p) (xs @ zs)), inv p ` R) \<in> failures P"
split_spec_params_def: fixes d :: "nat"   and n :: "'b"   and X :: "'a list set" shows "split_spec_params d n X = SPEC (\<lambda>(A, B). env_len A d \<and> env_len B d \<and> X \<subseteq> A \<union> B)"
one_closed: fixes H :: "'f set"   and G :: "('f, 'g) monoid_scheme" assumes "subgroup H G" shows "\<one>\<^bsub>G\<^esub> \<in> H"
Q_1: shows "DisjointPaths H_x v0 v1 Q"
H: shows "T --` A \<Delta> A \<in> null_sets M"
supp_def: fixes f :: "'c \<Rightarrow> 'd" shows "supp f = {x. f x \<noteq> (0::'d)}"
partition_on_P': shows "partition_on (B - B' \<union> C) P'"
ring_hom_ring_axioms_def: fixes R :: "('d, 'e) ring_scheme"   and S :: "('f, 'g) ring_scheme"   and h :: "'d \<Rightarrow> 'f" shows "ring_hom_ring_axioms R S h \<equiv> h \<in> ring_hom R S"
ucast_ucast_eq: fixes x :: "'c word"   and y :: "'d word" assumes "UCAST('c \<rightarrow> 'b) x = UCAST('c \<rightarrow> 'b) (UCAST('d \<rightarrow> 'c) y)"   and "LENGTH('c) \<le> LENGTH('d)"   and "LENGTH('d) \<le> LENGTH('b)" shows "x = UCAST('d \<rightarrow> 'c) y"
height_balance_tree: fixes t :: "'b tree" shows "height (balance_tree t) = nat \<lceil>log 2 (real (size t + 1))\<rceil>"
simps_2: fixes I :: "('b, 'c, 'a) interp"   and x :: "'a" shows "dterm_sem I ($' x) = (\<lambda>v. snd v $ x)"
oo_def: shows "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))"
tts_AA_A_transfer: fixes \<UU> :: "'d set"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> \<UU>; b \<in> \<UU>\<rbrakk> \<Longrightarrow> f a b \<in> \<UU>"   and "Domainp A = (\<lambda>x. x \<in> \<UU>)"   and "bi_unique A"   and "right_total A" shows "\<exists>rcdt. rel_fun A (rel_fun A A) f rcdt"
eq: shows "ideal (set (removeAll 0 fs)) = ideal (set fs)"
\<D>_def: shows "\<D> \<equiv> {S \<in> \<B>. \<exists>U. U \<in> \<F> \<and> S \<subseteq> U}"
hyps: fixes y :: "nat" assumes "y < n"   and "\<exists>a. inf_chain (\<lambda>ys xs. length ys \<le> y \<and> length xs \<le> y \<and> ext gt ys xs) a" shows "False"
doesnt_modify_annos: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var"   and annos :: "'Var ModeUpd list" assumes "doesnt_modify c x" shows "doesnt_modify (c \<otimes> annos) x"
length_greaters_le_sorted: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list"   and i :: "nat" assumes "Linorder_Relations.sorted_wrt R xs"   and "i < length xs" shows "length (greaters R (xs ! i) xs) \<le> length xs - i - 1"
BCh'_def: shows "BCh' \<equiv> BCh(Xh d1h2 := \<lambda>X. set_option (find (\<lambda>x. x \<in> X) [d1h2, d2h2, d3h2]))"
flush_unchanged_addresses: fixes a :: "nat"   and sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat" assumes "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b sb" shows "flush sb m a = m a"
transfer: shows "rel_fun cr_uint32 (rel_fun cr_uint32 cr_uint32) (+) (+)"
uminus_perm_def: fixes p :: "'a perm" shows "- p = Abs_perm (inv (Rep_perm p))"
decode_state_at_goal_state: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "\<Pi>\<^sub>G \<subseteq>\<^sub>m \<Phi>\<^sub>S\<inverse> \<Pi> \<A> t"
ab_group_add_ow_axioms_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" shows "ab_group_add_ow_axioms U plus zero minus uminus \<equiv> (\<forall>a. a \<in> U \<longrightarrow> plus (uminus a) a = zero) \<and> (\<forall>a b. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> minus a b = plus a (uminus b))"
A: shows "Esigma (a * u * q) = Esigma (a * r * c)"
p2: shows "{OrdP (Var k), VarP v} \<turnstile> (SyntaxN.All j' (SyntaxN.All k' (SeqStTermP v a t (Var j) s (Var k) IMP SeqStTermP v a t (Var j') s' (Var k') IMP Var j' EQ Var j)))(j::=u)"
adopt_node_locs_impl: shows "adopt_node_locs = local.a_adopt_node_locs"
llist_all2_trans: fixes P :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'e llist"   and ys :: "'e llist"   and zs :: "'e llist" assumes "llist_all2 P xs ys"   and "llist_all2 P ys zs"   and "transp P" shows "llist_all2 P xs zs"
eval\<^sub>t_bij: fixes b_of_a :: "'c \<Rightarrow> 'd"   and E :: "char list \<Rightarrow> 'c"   and F :: "char list \<Rightarrow> 'c list \<Rightarrow> 'c"   and t :: "fterm" assumes "bij b_of_a" shows "eval\<^sub>t (E_conv b_of_a E) (F_conv b_of_a F) t = b_of_a (eval\<^sub>t E F t)"
simps_1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sl :: "'a list" shows "quicksort_by_rel R sl [] = sl"
simps_3: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and y :: "'a"   and x :: "'a" assumes "efficient_power f"   and "n \<noteq> 0"   and "even n" shows "efficient_power.efficient_power f y x n = efficient_power.efficient_power f y (f x x) (n div 2)"
count_digit_def: shows "count_digit \<equiv> rec_Digit count_node (\<lambda>a b. count_node a + count_node b) (\<lambda>a b c. count_node a + count_node b + count_node c) (\<lambda>a b c d. count_node a + count_node b + count_node c + count_node d)"
HK_1: shows "H \<lhd> homology_group p (subtopology X S)"
abs_eq: fixes xa :: "ereal"   and x :: "ereal" assumes "eq_onp ((\<le>) 0) xa xa"   and "eq_onp ((\<le>) 0) x x" shows "(e2ennreal xa \<le> e2ennreal x) = (xa \<le> x)"
sets_PiM_eq_proj: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" assumes "I \<noteq> {}" shows "sets (Pi\<^sub>M I M) = sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))"
det_minors_diagonal2: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and I :: "nat set"   and k :: "nat"   and J :: "nat set" assumes "diagonal_mat A"   and "A \<in> carrier_mat n m"   and "card I = k"   and "card J = k"   and "I \<subseteq> {0..<n}"   and "J \<subseteq> {0..<m}"   and "0 < k" shows "Determinant.det (submatrix A I J) = (0::'b) \<or> (\<exists>S\<subseteq>{0..<min n m}. card S = k \<and> S = I \<and> (Determinant.det (submatrix A I J) = (\<Prod>i\<in>S. A $$ (i, i)) \<or> Determinant.det (submatrix A I J) = - (\<Prod>i\<in>S. A $$ (i, i))))"
\<theta>_def: shows "\<theta> = prime_sum_upto (\<lambda>p. ln (real p))"
signed_Hash_imp_used: fixes C :: "agent"   and H :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Crypt (priSK C) (Hash \<lbrace>H, Hash X\<rbrace>) \<in> parts (knows Spy evs)"   and "C \<notin> bad"   and "evs \<in> set_pur" shows "parts {X} \<subseteq> used evs"
trans_Atm: fixes atm :: "'atom"   and s :: "'state" shows "(Atm atm, s) \<rightarrow>t aval atm s"
execl_next: fixes P :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list"   and k :: "nat" assumes "P \<Turnstile> cfs"   and "k < length cfs"   and "k \<noteq> length cfs - 1" shows "(P !! fst (cfs ! k), cfs ! k) \<mapsto> cfs ! Suc k"
emeasure_insert: fixes x :: "'b"   and M :: "'b measure"   and A :: "'b set" assumes "{x} \<in> sets M"   and "A \<in> sets M"   and "x \<notin> A" shows "emeasure M (insert x A) = emeasure M {x} + emeasure M A"
solve_consistent_rref_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "solve_consistent_rref A b = (\<chi>j. if \<exists>i. A $ i $ j = (1::'d) \<and> j = (LEAST n. A $ i $ n \<noteq> (0::'d)) then b $ (THE i. A $ i $ j = (1::'d)) else (0::'d))"
set_intros_1: fixes ya :: "'c \<times> 'c list_R1.list_R1"   and x :: "unit + 'c \<times> 'c list_R1.list_R1"   and yb :: "'c" assumes "ya \<in> setr x"   and "yb \<in> fsts ya" shows "yb \<in> list_R1.set_list_R1 (list_R1.list_R1 x)"
smult_conv_mult_Const: fixes c :: "'b"   and p :: "'b mpoly" shows "MPoly_Type.smult c p = Const c * p"
mono_sup: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d"   and B :: "'d" assumes "mono f" shows "f A \<sqinter> f B \<le> f (A \<sqinter> B)"
update_def: fixes destr :: "'b \<Rightarrow> 'c"   and constr :: "'d \<Rightarrow> 'b"   and n :: "'a"   and f :: "'c \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'b" shows "update destr constr n f s = s(n := constr (f (destr (s n))))"
lvl_0_iff: fixes t :: "('c \<times> nat) tree" assumes "invar t" shows "(lvl t = 0) = (t = \<langle>\<rangle>)"
connected_space_product_topology: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set" shows "connected_space (product_topology X I) = ((\<Pi>\<^sub>E i\<in>I. topspace (X i)) = {} \<or> (\<forall>i\<in>I. connected_space (X i)))"
matrix_mul_lid: fixes A :: "(('d, 'e) vec, 'f) vec" shows "mat (1::'d) ** A = A"
add_uminus_minus_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A + - B = A - B"
dube_eq_3: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and j :: "nat" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F"   and "j < card X" shows "1 = (- 1) ^ (card X - Suc j) * (int (poly_deg f) - 1 gchoose (card X - Suc j)) + (- 1) ^ (card X - j) * (int (poly_deg f) - 1 gchoose (card X - j)) - 1 - (\<Sum>i = Suc j..card X. (- 1) ^ (i - j) * (int (pm_powerprod.aa X f F i) gchoose (i - j) + (int (pm_powerprod.bb ord X F i) gchoose (i - j))))"
block_set_nempty_imp_block_ex: assumes "\<B>\<^sup>C \<noteq> {#}" shows "\<exists>bl. bl \<in># \<B>\<^sup>C"
compact_imp_closed: fixes s :: "'b set" assumes "compact s" shows "closed s"
g: fixes x :: "complex" assumes "x \<in> sphere 0 1" shows "\<bar>(f \<circ> Im \<circ> Ln) x - g x\<bar> < e"
exposed_point_of_Int_supporting_hyperplane_le: fixes S :: "'b set"   and a :: "'b"   and b :: "real"   and c :: "'b" assumes "S \<inter> {x. a \<bullet> x = b} = {c}"   and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b" shows "{c} exposed_face_of S"
vunion_fun_mono: fixes f :: "V \<Rightarrow> V"   and A :: "V"   and B :: "V" assumes "mono f" shows "f A \<union>\<^sub>\<circ> f B \<subseteq>\<^sub>\<circ> f (A \<union>\<^sub>\<circ> B)"
termMOD_igWlsDisj: shows "igWlsDisj termMOD"
mem_zmset_frontier: fixes x :: "'a"   and M :: "'a zmultiset" shows "(x \<in>#\<^sub>z zmset_of (mset_set (set_antichain (frontier M)))) = (x \<in>\<^sub>A frontier M)"
for_in_RI: fixes x :: "'b"   and R :: "('b \<times> 'd) set" assumes "x \<in> Domain R" shows "(x, fun_of_rel R x) \<in> R"
apex_unique_up_to_equivalence: fixes \<rho>' :: "'a"   and f' :: "'a"   and g' :: "'a" assumes "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> w) \<rho>' f' g'" shows "\<exists>wa w' \<phi> \<psi> \<theta> \<nu> \<theta>' \<nu>'. internal_equivalence w' wa \<psi> \<phi> \<and> \<guillemotleft>wa : src m \<rightarrow> src f'\<guillemotright> \<and> \<guillemotleft>w' : src f' \<rightarrow> src m\<guillemotright> \<and> \<guillemotleft>\<theta> : f' \<star> wa \<Rightarrow> m\<guillemotright> \<and> \<guillemotleft>\<nu> : g \<star> n \<Rightarrow> g' \<star> wa\<guillemotright> \<and> local.iso \<nu> \<and> \<sigma> = ((g \<star> w) \<star> \<theta>) \<cdot> \<a>[g \<star> w, f', wa] \<cdot> (\<rho>' \<star> wa) \<cdot> \<nu> \<and> \<guillemotleft>\<theta>' : m \<star> w' \<Rightarrow> f'\<guillemotright> \<and> \<guillemotleft>\<nu>' : g' \<Rightarrow> (g \<star> n) \<star> w'\<guillemotright> \<and> local.iso \<nu>' \<and> \<rho>' = ((g \<star> w) \<star> \<theta>') \<cdot> \<a>[g \<star> w, m, w'] \<cdot> (\<sigma> \<star> w') \<cdot> \<nu>'"
Says_Tgs_message_form: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "B \<noteq> Tgs \<and> authK \<notin> range shrK \<and> authK \<in> authKeys evs \<and> authK \<in> symKeys \<and> servK \<notin> range shrK \<and> servK \<notin> authKeys evs \<and> servK \<in> symKeys \<and> servTicket = Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>"
U0_in_CP: shows "U\<^sub>0 \<in> CP"