hcju/isabelleps huggingface dataset:
term_ok'_subst_bv2: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and s :: "Core.term"   and u :: "Core.term"   and lev :: "nat" assumes "term_ok' \<Sigma> s"   and "term_ok' \<Sigma> u" shows "term_ok' \<Sigma> (subst_bv2 s lev u)"
pmf_map_spmf_None: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf" shows "pmf (map_spmf f p) None = pmf p None"
pmf_K: fixes n :: "nat"   and s :: "'s"   and t :: "'s" assumes "0 < prob {\<omega> \<in> space M. X n \<omega> = s}" shows "pmf (K s) t = cond_prob M (\<lambda>\<omega>. X (Suc n) \<omega> = t) (\<lambda>\<omega>. X n \<omega> = s)"
Tau: fixes P :: "pi" shows "\<tau>.P \<Longrightarrow>\<^sup>^\<tau> \<prec> P"
show_law_nat: fixes n :: "nat" shows "show_law showsp_nat n"
abs_conv_abscissa_diff_leI: fixes f :: "'a fds"   and d :: "ereal"   and g :: "'a fds" assumes "abs_conv_abscissa f \<le> d"   and "abs_conv_abscissa g \<le> d" shows "abs_conv_abscissa (f - g) \<le> d"
closedin_path_components_of_locally_path_connected_space: fixes X :: "'c topology"   and c :: "'c set" assumes "locally_path_connected_space X"   and "c \<in> path_components_of X" shows "closedin X c"
countable_cycles: fixes G :: "('a, 'b) graph_scheme" assumes "countable \<^bold>V\<^bsub>G\<^esub>" shows "countable (cycles G)"
eventually_ball_finite: fixes A :: "'a set"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and net :: "'b filter" assumes "finite A"   and "\<forall>y\<in>A. \<forall>\<^sub>F x in net. P x y" shows "\<forall>\<^sub>F x in net. \<forall>y\<in>A. P x y"
ccTTree_cc_restr: fixes S :: "Vars.var set"   and G :: "CoCalls" shows "ccTTree S G = ccTTree S (G G|` S)"
cod_cod: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.cod (VV.cod f) = VV.cod f"
assms_1: shows "step s1 a1 = (ou1, s1')"
iter_step_mono: fixes c :: "'a::refinement_lattice" shows "mono (\<lambda>x::'a::refinement_lattice. nil \<sqinter> (c::'a::refinement_lattice) ; x)"
res_invarm_def: fixes res :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option" shows "res_invarm res \<equiv> \<forall>v. case res v of None \<Rightarrow> True | Some (p, w) \<Rightarrow> w = path_weight (rev p)"
dickson_grading_varnum_pp: shows "dickson_grading varnum_pp"
commute: fixes a :: "'c \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'd" shows "a \<star> b = b \<star> a"
p1: shows "ket x = ket (enum_class.enum ! enum_idx x)"
iT_Div_not_empty: fixes I :: "nat set"   and k :: "nat" assumes "I \<noteq> {}" shows "I \<oslash> k \<noteq> {}"
cmp_fun: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "cmp g f \<in> A \<rightarrow> C"
float_induct: fixes P :: "('a, 'b) IEEE.float \<Rightarrow> bool"   and a :: "('a, 'b) IEEE.float" assumes "\<And>x. \<lbrakk>is_normal x; valof x = real_of_int (normal_mantissa x) * 2 powr real_of_int (normal_exponent x)\<rbrakk> \<Longrightarrow> P x"   and "\<And>x. \<lbrakk>is_denormal x; valof x = real_of_int (denormal_mantissa x) * 2 powr real_of_int (denormal_exponent TYPE(('a, 'b) IEEE.float))\<rbrakk> \<Longrightarrow> P x"   and "P 0"   and "P (- 0)"   and "P \<infinity>"   and "P (- \<infinity>)"   and "\<And>x. is_nan x \<Longrightarrow> P x" shows "P a"
dg_Dom_app_in_Obj: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<AA>\<lparr>Dom\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
p_bgrid: shows "p \<in> grid (base {0..<d} p) {0..<d}"
a: shows "\<lbrakk>Final i1\<rbrakk> < \<lbrakk>Final i2\<rbrakk>"
nat_neq_4k1: fixes m :: "nat"   and k :: "nat"   and n :: "nat" shows "of_nat m \<noteq> ((4::'b) * of_nat k + (1::'b)) / ((2::'b) * (2::'b) ^ n)"
subsetD: fixes U :: "'b set"   and V :: "'b set"   and x :: "'b" assumes "Subspace.subspace U V"   and "x \<in> U" shows "x \<in> V"
sylow_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" shows "sylow_axioms G p a m \<equiv> prime p \<and> order G = p ^ a * m \<and> finite (carrier G)"
filter_mono': fixes F1 :: "'b filter"   and F2 :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "F1 \<le> F2" shows "\<Omega>[F2](f) \<subseteq> \<Omega>[F1](f)"
fls_inverse_delta_nonzero: fixes b :: "'a"   and a :: "int" assumes "b \<noteq> (0::'a)" shows "inverse (Abs_fls (\<lambda>n. if n = a then b else (0::'a))) = Abs_fls (\<lambda>n. if n = - a then inverse b else (0::'a))"
filter_min_append_def: fixes xs :: "'a list"   and ys :: "'a list" shows "local.filter_min_append xs ys = (let P = \<lambda>zs x. \<not> (\<exists>z\<in>set zs. rel z x); ys1 = filter (P xs) ys in filter (P ys1) xs @ ys1)"
ex_Loop_stream_enc: fixes I :: "(nat + nat set) list"   and w :: "'a list" assumes "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P" shows "\<exists>n. sdrop n (stream_enc (w, I)) = sconst (any, replicate (length I) False)"
approx_add_right_cancel: fixes b :: "'a star"   and d :: "'a star"   and c :: "'a star" assumes "b + d \<approx> c + d" shows "b \<approx> c"
Vd_succ: fixes u :: "nat"   and d :: "nat"   and v :: "nat" assumes "u \<in> Vd d"   and "(u, v) \<in> E"   and "\<forall>i\<le>d. v \<notin> Vd i" shows "v \<in> Vd (Suc d)"
sum_permutations_inverse: fixes f :: "('d \<Rightarrow> 'd) \<Rightarrow> 'c"   and S :: "'d set" shows "sum f {p. p permutes S} = (\<Sum>p | p permutes S. f (inv p))"
5: shows "\<forall>x y z. z \<sqinter> y \<le> x \<sqinter> y \<or> (z \<squnion> x) \<sqinter> y \<noteq> x \<sqinter> y"
eq_onp_live_step: fixes x :: "bool"   and y :: "bool"   and P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "x = y" shows "(eq_onp P a a \<and> x) = (P a \<and> y)"
eq_vecI: fixes w :: "'a Matrix.vec"   and v :: "'a Matrix.vec" assumes "\<And>i. i < dim_vec w \<Longrightarrow> v $v i = w $v i"   and "dim_vec v = dim_vec w" shows "v = w"
inversion_infty: shows "inversion \<infinity>\<^sub>h = 0\<^sub>h"
mem_cball_leI: fixes x :: "'b"   and y :: "'b"   and e :: "real"   and f :: "real" assumes "x \<in> cball y e"   and "e \<le> f" shows "x \<in> cball y f"
ComplexLikePoset_axioms: shows "ComplexLikePoset (\<^bold>\<le>) (\<^bold><) P"
rep_eq: fixes x :: "'d"   and xa :: "'d set!['e]" shows "set_nebset (nebinsert x xa) = insert x (set_nebset xa)"
surjI: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'e" assumes "\<And>x. g (f x) = x" shows "surj g"
Class_unit_normal_subgroup: shows "contained.Class \<one> = L"
lb_arctan: fixes x :: "float"   and y :: "real"   and prec :: "nat" assumes "arctan (real_of_float x) \<le> y" shows "real_of_float (lb_arctan prec x) \<le> y"
indicator_single_Some: fixes x :: "'b"   and y :: "'b" shows "indicator {Some x} (Some y) = indicator {x} y"
symKey_compromise: fixes evs :: "event list"   and SK :: "nat"   and KK :: "nat set" assumes "evs \<in> set_cr"   and "SK \<in> symKeys"   and "\<forall>K\<in>KK. \<not> KeyCryptKey K SK evs" shows "(Key SK \<in> analz (Key ` KK \<union> knows Spy evs)) = (SK \<in> KK \<or> Key SK \<in> analz (knows Spy evs))"
ereal_divide_same: fixes x :: "ereal" shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
assms: shows "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))"
qr_valid_frac: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and \<pp> :: "'a set" assumes "s \<in> \<O> U"   and "\<pp> \<in> U" shows "quotient_ring.valid_frac (R\<setminus>\<pp>) R (+) (\<cdot>) \<zero> (s \<pp>)"
good_subst: fixes i :: "'sf"   and f' :: "('sf + 'sz, 'sz) trm" assumes "SFunctions \<sigma> i = Some f'" shows "dfree f'"
rel_interior_convex_shrink: fixes S :: "'a set"   and c :: "'a"   and x :: "'a"   and e :: "real" assumes "convex S"   and "c \<in> rel_interior S"   and "x \<in> S"   and "0 < e"   and "e \<le> 1" shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
IH: fixes ys :: "'a list" assumes "xs \<noteq> ys"   and "length xs = length ys" shows "\<exists>pre x xs' y ys'. x \<noteq> y \<and> xs = pre @ [x] @ xs' \<and> ys = pre @ [y] @ ys'"
distinct_remove1: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs" shows "distinct (remove1 x xs)"
star_absorb: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "x \<otimes> y = mbot" shows "matrix_stone_algebra.comp_inf.vector x"
rank_invar_ins: fixes bq :: "('c, 'd) BinomialTree list"   and t :: "('c, 'd) BinomialTree" assumes "rank_invar bq" shows "rank_invar (ins t bq)"
ring_simprules_12: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
sf: shows "square_free_factorization_int' g = (c, hs)"
primes_coprime_nat: fixes p :: "nat"   and q :: "nat" assumes "prime p"   and "prime q"   and "p \<noteq> q" shows "coprime p q"
subgraph_isomorphic_pre_iso_closed: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set"   and G\<^sub>3 :: "nat set \<times> nat set set" assumes "G\<^sub>1 \<simeq> G\<^sub>2"   and "G\<^sub>2 \<sqsubseteq> G\<^sub>3" shows "G\<^sub>1 \<sqsubseteq> G\<^sub>3"
assms: fixes i :: "'a" assumes "i \<in> I" shows "closed (X i)"
uniform_size_incomp: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and bl :: "'b set" assumes "block_design point_set block_collection u_block_size"   and "u_block_size \<le> int (card point_set) - 1"   and "bl \<in># block_collection" shows "card bl < card point_set \<and> bl \<in># block_collection"
prems_2: shows "(l # ls) ! i = (p, Fence # is, \<theta>, [], \<D>, \<O>, \<R>)"
assms_2: fixes C :: "'a set" assumes "C \<in> components S" shows "Borsukian C"
simps_2: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list" shows "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
map_map_rexp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and r :: "'c rexp" shows "map_rexp f (map_rexp g r) = map_rexp (\<lambda>r. f (g r)) r"
Vfin_imp_Efin: assumes "finite V" shows "finite E"
abs_eq: fixes xb :: "'d"   and xa :: "'e"   and x :: "'d \<Rightarrow> 'e option" shows "Mapping.update xb xa (Mapping x) = Mapping (x(xb \<mapsto> xa))"
iRelease_iUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<R> t I. Q t) = ((\<box> t I. Q t) \<or> (Q t'. t' \<U> t I. Q t \<and> P t))"
minima_def: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a" shows "minima t a b = (a \<in> alphabet t \<and> b \<in> alphabet t \<and> a \<noteq> b \<and> (\<forall>c\<in>alphabet t. c \<noteq> a \<longrightarrow> c \<noteq> b \<longrightarrow> freq t a \<le> freq t c \<and> freq t b \<le> freq t c))"
wrap_iso_arcs_in_funcsetI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "A \<subseteq> arcs G"   and "f \<in> A \<rightarrow> A" shows "wrap_iso_arcs hom f \<in> iso_arcs hom ` A \<rightarrow> iso_arcs hom ` A"
Sup_inner_def: fixes X :: "'c set"   and y :: "'c" shows "Sup_inner X y = SPEC (\<lambda>r. \<forall>x\<in>X. x \<bullet> y \<le> r)"
the_NF: fixes a :: "'a" shows "(a, the_NF A a) \<in> A\<^sup>!"
Texp_Lam: fixes y :: "Vars.var"   and e :: "exp"   and n :: "Arity" shows "without y (Texp e\<cdot>(pred\<cdot>n)) \<sqsubseteq> Texp (Lam [y]. e)\<cdot>n"
n_restrictl: fixes x :: "'a"   and y :: "'a" shows "\<lparr>!x\<rparr>y\<lparr>y\<rparr>"
map_mat_def: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c mat" shows "map_mat f A \<equiv> mat (dim_row A) (dim_col A) (\<lambda>ij. f (A $$ ij))"
MOST_iff_cofinite: fixes P :: "'b \<Rightarrow> bool" shows "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
inf_assoc: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) z = inf x (inf y z)"
ordLeq_csum2: fixes p2 :: "('a \<times> 'a) set"   and p1 :: "('b \<times> 'b) set" assumes "Card_order p2" shows "p2 \<le>o p1 +c p2"
almost_everywhere_avoid_finite: fixes I :: "'b set" assumes "finite I" shows "AE f in Pi\<^sub>M I (\<lambda>i. uniform_measure lborel {0..1}). inj_on f I"
ratfps_gcd: fixes f :: "'b ratfps"   and g :: "'b ratfps" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "gcd f g = ratfps_of_poly (monom (1::'b) (min (ratfps_subdegree f) (ratfps_subdegree g)))"
cases: fixes r :: "s_context" obtains s_inst :: "t_context list" and s_funcs :: "tf list" and s_tab :: "nat list" and s_mem :: "nat list" and s_globs :: "tg list" where "r = \<lparr>s_inst = s_inst, s_funcs = s_funcs, s_tab = s_tab, s_mem = s_mem, s_globs = s_globs\<rparr>"
direct_subtype_16: shows "ArrT BoolAT \<prec>1 CClassT Object"
two_composable_identity_arrows_of_spans_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" shows "two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<equiv> two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<and> identity_arrow_of_spans C \<mu> \<and> identity_arrow_of_spans C \<nu>"
Union_mono: fixes A :: "'b set set"   and B :: "'b set set" assumes "A \<subseteq> B" shows "\<Union> A \<subseteq> \<Union> B"
simps_25: fixes g1 :: "'a gexp"   and g2 :: "'a gexp"   and v :: "'a"   and va :: "value list" shows "local.less_gexp_aux (Nor g1 g2) (In v va) = False"
vcg_lift_comp: fixes f :: "('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> 'c"   and c :: "('answer, 'location, 'question, 'state) com"   and Q :: "'c \<Rightarrow> ('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool"   and P :: "'c \<Rightarrow> ('answer, 'location, 'proc, 'question, 'state) system_state \<Rightarrow> bool" assumes "\<And>P. coms, p, aft \<turnstile> \<lbrace>\<lambda>s. P (f s)\<rbrace> c"   and "\<And>x. \<lbrace>Q x\<rbrace> c \<lbrace>P x\<rbrace>" shows "\<lbrace>\<lambda>s. Q (f s) s\<rbrace> c \<lbrace>\<lambda>s. P (f s) s\<rbrace>"
sets_Collect_disj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "{x \<in> \<Omega>. P x} \<in> M"   and "{x \<in> \<Omega>. Q x} \<in> M" shows "{x \<in> \<Omega>. Q x \<or> P x} \<in> M"
valid_modif: shows "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<Turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)"
MGT_implies_complete: fixes c :: "com"   and P :: "state \<Rightarrow> state \<Rightarrow> bool"   and Q :: "state \<Rightarrow> state \<Rightarrow> bool" assumes "{} |\<turnstile>\<^sub>t {MGT\<^sub>t c}"   and "{} \<Turnstile>\<^sub>t {P} c {Q}" shows "{} \<turnstile>\<^sub>t {P} c {Q}"
representation_diff: fixes basis :: "'b set"   and v :: "'b"   and u :: "'b" assumes "independent basis"   and "v \<in> span basis"   and "u \<in> span basis" shows "representation basis (u - v) = (\<lambda>b. representation basis u b - representation basis v b)"
Im2_sum: fixes f :: "'a \<Rightarrow> quat"   and S :: "'a set" shows "Im2 (sum f S) = (\<Sum>x\<in>S. Im2 (f x))"
that: fixes l :: "'a" assumes "x = Inl l" shows "thesis"
must_acquire_lock_append: fixes xs :: "lock_action list"   and ys :: "lock_action list" shows "must_acquire_lock (xs @ ys) = (if Lock \<in> set xs \<or> Unlock \<in> set xs then must_acquire_lock xs else must_acquire_lock ys)"
pqr_simps_2: shows "sturm p ! Suc i = sturm_squarefree' p ! (i + 1) * d"
instant_i: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^sub>i \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list"   and a2 :: "nat" assumes "\<And>cs fs n. (\<And>x. \<lbrakk>\<not> (cs = [] \<or> last cs = (0::'b) \<or> length fs < length cs - 1); \<not> n < length fs; x \<in> {..<length cs - 1}\<rbrakk> \<Longrightarrow> P cs fs (n + 1 - length cs + x)) \<Longrightarrow> P cs fs n" shows "P a0 a1 a2"
fps_inverse_code: fixes x :: "'b ratfps" shows "inverse (fps_of_ratfps x) = fps_of_ratfps (inverse x)"
satisfies_eqvt: fixes P :: "'state"   and \<phi> :: "'pred"   and p :: "perm" assumes "P \<turnstile> \<phi>" shows "p \<bullet> P \<turnstile> p \<bullet> \<phi>"
conditional_mutual_information_generic_eq: fixes S :: "'e measure"   and T :: "'f measure"   and P :: "'g measure"   and X :: "'a \<Rightarrow> 'e"   and Px :: "'e \<Rightarrow> real"   and Z :: "'a \<Rightarrow> 'g"   and Pz :: "'g \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'f"   and Pyz :: "'f \<times> 'g \<Rightarrow> real"   and Pxz :: "'e \<times> 'g \<Rightarrow> real"   and Pxyz :: "'e \<times> 'f \<times> 'g \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "sigma_finite_measure P"   and "distributed M S X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space S \<Longrightarrow> 0 \<le> Px x"   and "distributed M P Z (\<lambda>x. ennreal (Pz x))"   and "\<And>z. z \<in> space P \<Longrightarrow> 0 \<le> Pz z"   and "distributed M (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x)) (\<lambda>x. ennreal (Pyz x))"   and "\<And>y z. \<lbrakk>y \<in> space T; z \<in> space P\<rbrakk> \<Longrightarrow> 0 \<le> Pyz (y, z)"   and "distributed M (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) (\<lambda>x. ennreal (Pxz x))"   and "\<And>x z. \<lbrakk>x \<in> space S; z \<in> space P\<rbrakk> \<Longrightarrow> 0 \<le> Pxz (x, z)"   and "distributed M (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Y x, Z x)) (\<lambda>x. ennreal (Pxyz x))"   and "\<And>x y z. \<lbrakk>x \<in> space S; y \<in> space T; z \<in> space P\<rbrakk> \<Longrightarrow> 0 \<le> Pxyz (x, y, z)"   and "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Px x * Pyz (y, z))))"   and "integrable (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>(x, y, z). Pxyz (x, y, z) * log b (Pxz (x, z) / (Px x * Pz z)))" shows "conditional_mutual_information b S T P X Y Z = LINT (x, y, z)|(S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P). Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y, z) / Pz z)))"
adjoint_transpose_right_1: fixes u :: "'a"   and v :: "'a" assumes "ide u"   and "ide v"   and "src v = trg f\<^sup>*"   and "src u = trg f" shows "trnr\<^sub>\<eta> v \<in> hom (v \<star> f\<^sup>*) u \<rightarrow> hom v (u \<star> f)"
x_def: shows "x = [a, x\<lparr>1\<^sub>\<nat>\<rparr>]\<^sub>\<circ>"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "qSkel Y = qSkel X" shows "\<forall>X'. Y #= X' \<longrightarrow> qSkel Y = qSkel X'"
PDG: shows "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit (StrongPostdomination.weak_control_dependence sourcenode targetnode valid_edge Exit)"
le_rep_def: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "le_rep p1 p2 = (let (l1, h1) = p1; (l2, h2) = p2 in if is_empty_rep (l1, h1) then True else if is_empty_rep (l2, h2) then False else l2 \<le> l1 \<and> h1 \<le> h2)"
rcis_zero_mod: fixes a :: "real" shows "rcis 0 a = 0"
intro: fixes B :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and g :: "'ag" assumes "partial_magma.arr B g" shows "constant_transformation_axioms B g"
sub_restrict_map_def: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "m `- S \<equiv> \<lambda>x. if x \<in> S then None else m x"
init: shows "\<forall>p. correct p 0 \<longrightarrow> 0 \<le> PC p 0 \<and> PC p 0 \<le> \<mu>"
nn_integral_embed_measure: fixes f :: "'k \<Rightarrow> 'l"   and g :: "'l \<Rightarrow> ennreal"   and M :: "'k measure" assumes "inj f"   and "g \<in> borel_measurable (embed_measure M f)" shows "integral\<^sup>N (embed_measure M f) g = \<integral>\<^sup>+ x. g (f x) \<partial>M"
q_def: shows "q \<equiv> \<lambda>n t. (1 - p t ^ n) ^ k ^ n"
Amortized_axioms: shows "Amortized arity exec inv cost \<Phi> U"
PlaneGraphs_def: shows "PlaneGraphs \<equiv> \<Union> (range PlaneGraphsP)"
sum_constant_scale: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and y :: "'d"   and A :: "'e set" assumes "module scale" shows "(\<Sum>x\<in>A. y) = scale (of_nat (card A)) y"
take_i_E_take_j: shows "LeftDerivation (take i \<alpha>) E (take j \<beta>)"
retract_of_UNIV: fixes S :: "'a set" shows "(S retract_of UNIV) = (AR S \<and> closed S)"
env_delete_fun_upd2: fixes x :: "'d"   and m :: "'d \<Rightarrow> 'e"   and v :: "'e" shows "(env_delete x m)(x := v) = m(x := v)"
set_intros_25: fixes aac :: "'c"   and x71 :: "'c list"   and x72 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and x73 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" assumes "aac \<in> set x71" shows "aac \<in> vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
term_ok_mk_eq_same_typ: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and s :: "Core.term"   and t :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> (mk_eq s t)" shows "typ_of s = typ_of t"
sum_gp0: fixes x :: "'b"   and n :: "nat" shows "sum ((^) x) {..n} = (if x = (1::'b) then of_nat (n + 1) else ((1::'b) - x ^ Suc n) / ((1::'b) - x))"
tagged_division_of_self: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> cbox a b" shows "{(x, cbox a b)} tagged_division_of cbox a b"
t''_zcount: shows "0 < zcount (c_imp c loc1) t''"
matrix_vector_mult_0_right: fixes A :: "(('d, 'f) vec, 'e) vec" shows "A *v 0 = 0"
rt_fresher_imp_nsqn_le: fixes ip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"   and "ip \<in> kD rt1"   and "ip \<in> kD rt2" shows "nsqn rt1 ip \<le> nsqn rt2 ip"
smult_add_right: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "Polynomial.smult a (p + q) = Polynomial.smult a p + Polynomial.smult a q"
column_set_Gram_Schmidt_upt_k: fixes k :: "nat"   and A :: "(('b, 'a) vec, 'c) vec" assumes "Suc k < CARD('a)" shows "{column i (Gram_Schmidt_upt_k A (Suc k)) |i. mod_type_class.to_nat i \<le> Suc k} = {column i (Gram_Schmidt_upt_k A k) |i. mod_type_class.to_nat i \<le> k} \<union> {column (mod_type_class.from_nat (Suc k)) (Gram_Schmidt_upt_k A k) - (\<Sum>x\<in>{column i (Gram_Schmidt_upt_k A k) |i. mod_type_class.to_nat i \<le> k}. (x \<bullet> column (mod_type_class.from_nat (Suc k)) (Gram_Schmidt_upt_k A k) / (x \<bullet> x)) *\<^sub>R x)}"
surjective_map_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" shows "surjective_map \<alpha> S T \<equiv> map \<alpha> S T \<and> surjective_map_axioms \<alpha> S T"
mat_mono_def: fixes P :: "'b \<Rightarrow> bool"   and sd :: "nat"   and A :: "'b mat" shows "mat_mono P sd A = (\<forall>j<sd. \<exists>i<sd. P (A $$ (i, j)))"
finite_card_zero_empty: fixes S :: "'a set" assumes "finite S"   and "card S \<le> 0" shows "S = {}"
pmdl_eqI_adds_lt_finite: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite G"   and "pmdl G \<subseteq> pmdl B"   and "\<And>f. \<lbrakk>f \<in> pmdl B; f \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g\<in>G. g \<noteq> 0 \<and> lt g adds\<^sub>t lt f" shows "pmdl G = pmdl B"