hcju/isabelleps huggingface dataset:
eval_const: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and s :: "'e"   and r :: "'c" assumes "UP_pre_univ_prop R S h"   and "s \<in> carrier S"   and "r \<in> carrier R" shows "eval R S h s (monom (UP R) r 0) = h r"
invertible_row_add_iterate: fixes n :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b" assumes "n < nrows A" shows "\<exists>P. invertible P \<and> row_add_iterate A n i j = P ** A"
lemma_2_10_31: fixes b :: "'a"   and a :: "'a" shows "((b l\<rightarrow> a) r\<rightarrow> a) l\<rightarrow> a = b l\<rightarrow> a"
take_append_singl: fixes n :: "nat"   and al :: "'a list"   and a :: "'a" assumes "n < length al" shows "take n (al @ [a]) = take n al"
AbstrLevels_A9_A91: fixes i :: "AbstrLevelsID" assumes "sA9 \<in> AbstrLevel i" shows "sA91 \<notin> AbstrLevel i"
int_pow_pow: fixes x :: "'a"   and n :: "int"   and m :: "int" assumes "x \<in> carrier G" shows "(x [^] n) [^] m = x [^] (n * m)"
sd_ge_zero: shows "\<forall>c. 0 < sd c"
dfa_startnode_is_node: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat" assumes "wf_dfa A n" shows "dfa_is_node A 0"
prems_11: shows "x = Item (N, \<alpha> @ \<beta>) (length \<alpha>) (charslength (take r q)) k'"
nsets_singleton_iff: fixes a :: "'b"   and r :: "nat" shows "[{a}]\<^bsup>r\<^esup> = (if r = 0 then {{}} else if r = 1 then {{a}} else {})"
case_2: fixes f1 :: "'b"   and f2 :: "'c tree \<Rightarrow> 'b"   and f3 :: "'c tree \<Rightarrow> 'b"   and x2 :: "'c tree" shows "(case Bal x2 of Same \<Rightarrow> f1 | Bal x \<Rightarrow> f2 x | Unbal x \<Rightarrow> f3 x) = f2 x2"
simps_1: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
neq0: shows "v \<noteq> 0\<^sub>v (dim_vec v)"
joinD: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> A\<^sup>\<down>" shows "\<exists>c. (a, c) \<in> A\<^sup>* \<and> (b, c) \<in> A\<^sup>*"
Carmichael_divides: fixes n :: "nat"   and k :: "nat"   and a :: "nat" assumes "Carmichael n dvd k"   and "coprime n a" shows "[a ^ k = 1] (mod n)"
vectorspace: fixes U :: "'b set"   and V :: "'b set" assumes "Subspace.subspace U V"   and "vectorspace V" shows "vectorspace U"
cring_sub_deg_bound: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P" shows "deg R (Cring_Poly.compose R f g) \<le> deg R f * deg R g"
hom_swaprows: fixes hom :: "'c \<Rightarrow> 'd"   and i :: "nat"   and A :: "'c mat"   and j :: "nat" assumes "field_hom hom"   and "i < dim_row A"   and "j < dim_row A" shows "swaprows i j (map_mat hom A) = map_mat hom (swaprows i j A)"
rep_eq: fixes x :: "(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 xa :: "char list \<Rightarrow> ty option"   and xb :: "expr"   and xc :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list, val) mapping) set) option) \<times> (char list \<Rightarrow> val option)"   and xd :: "expr"   and xe :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list, val) mapping) set) option) \<times> (char list \<Rightarrow> val option)" shows "x,xa \<turnstile> \<langle>xb,xc\<rangle> \<Rightarrow>' \<langle>xd,xe\<rangle> = x,xa \<turnstile> \<langle>xb,map_prod (map_fun id (map_option (map_prod id ((`) (map_prod id mapping.rep))))) id xc\<rangle> \<Rightarrow> \<langle>xd,map_prod (map_fun id (map_option (map_prod id ((`) (map_prod id mapping.rep))))) id xe\<rangle>"
RedT_def: fixes final :: "'c \<Rightarrow> bool"   and r :: "('a, 'b, 'c, 'd, 'e, 'f) semantics"   and convert_RA :: "'a \<Rightarrow>f nat \<Rightarrow> 'f list" shows "multithreaded_base.RedT final r convert_RA \<equiv> rtrancl3p (multithreaded_base.redT final r convert_RA)"
R'': fixes A :: "nat set" shows "R' `` A \<subseteq> {..n}"
minus_splus_sminus: fixes s :: "'a"   and t :: "'a"   and u :: "'a"   and v :: "'t" assumes "s adds t"   and "u adds\<^sub>p v" shows "(t - s) \<oplus> (v \<ominus> u) = t \<oplus> v \<ominus> (s + u)"
induct: fixes P :: "nat \<Rightarrow> Core.term \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "Core.term" assumes "\<And>uu_. P 0 uu_"   and "\<And>n all S T t. (all = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT \<Longrightarrow> P n t) \<Longrightarrow> P (Suc n) (Ct all S $ Abs T t)"   and "\<And>v va vb. P (Suc v) (Ct va vb)"   and "\<And>v va vb. P (Suc v) (Fv va vb)"   and "\<And>v va. P (Suc v) (Bv va)"   and "\<And>v va vb. P (Suc v) (Abs va vb)"   and "\<And>v vc vd vb. P (Suc v) (Fv vc vd $ vb)"   and "\<And>v vc vb. P (Suc v) (Bv vc $ vb)"   and "\<And>v vc vd vb. P (Suc v) (Abs vc vd $ vb)"   and "\<And>v vc vd vb. P (Suc v) (vc $ vd $ vb)"   and "\<And>v va vc vd. P (Suc v) (va $ Ct vc vd)"   and "\<And>v va vc vd. P (Suc v) (va $ Fv vc vd)"   and "\<And>v va vc. P (Suc v) (va $ Bv vc)"   and "\<And>v va vc vd. P (Suc v) (va $ (vc $ vd))" shows "P a0 a1"
independent_explicit_module: fixes s :: "'d set" shows "cindependent s = (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. u v *\<^sub>C v) = (0::'d) \<longrightarrow> v \<in> t \<longrightarrow> u v = 0)"
eq: fixes T' :: "'a set" shows "(\<exists>S'. T' = S' \<inter> T \<and> (\<exists>T. openin X T \<and> S' = T \<inter> S)) = (\<exists>Sa. T' = Sa \<inter> (S \<inter> T) \<and> openin X Sa)"
continuous_at_eps_delta: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "isCont f x = (\<forall>e>0. \<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
perp_up_def: fixes a :: "proj2"   and l :: "proj2_line" shows "perp_up a l \<equiv> if proj2_incident a l then SOME p. p \<in> S \<and> proj2_incident p (drop_perp a l) else endpoint_in_S (perp_foot a l) a"
int_pow_subgroup_generated: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and S :: "'a set"   and n :: "int" assumes "Group.group G"   and "x \<in> carrier (subgroup_generated G S)" shows "x [^]\<^bsub>subgroup_generated G S\<^esub> n = x [^]\<^bsub>G\<^esub> n"
onorm_def: fixes f :: "'b \<Rightarrow> 'c" shows "onorm f = (SUP x. (\<parallel>f x\<parallel>) / (\<parallel>x\<parallel>))"
pelims: fixes x :: "'b stream \<Rightarrow> 'a"   and xa :: "'b stream"   and y :: "'a" assumes "nxt x xa = y"   and "nxt_dom (x, xa)" obtains \<phi> :: "'b stream \<Rightarrow> 'a" and xs :: "'b stream" where "x = \<phi>" and "xa = xs" and "y = \<phi> (stl xs)" and "nxt_dom (\<phi>, xs)"
mem_affine_3_minus2: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and v :: "real" assumes "affine S"   and "x \<in> S"   and "y \<in> S"   and "z \<in> S" shows "x - v *\<^sub>R (y - z) \<in> S"
fmpredI: fixes m :: "('b, 'c) fmap"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x y" shows "fmpred P m"
Infs_def: fixes X :: "'a set" shows "Infs X = \<Squnion> {y. \<forall>x\<in>X. y \<le> x}"
X_plus_sub_deg: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier P" shows "deg R (Cring_Poly.compose R f (X_poly_plus R a)) = deg R f"
sum_rep: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "sum (rep_nat \<circ> f) A = rep_nat (comm_monoid_add.sum plus zero f A)"
Supq_distr: fixes X :: "'a set"   and y :: "'a" shows "\<Sqinter> X \<cdot> y = (\<Sqinter>x\<in>X. x \<cdot> y)"
in_set_conv_decomp: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
discr_finished: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and c :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "proper c"   and "finished c" shows "PL_Indis.discr aval tval cval indis c"
simps_1: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b"   and v :: "'b" shows "horner_eval' f x v 0 = v"
per_init_Domain: fixes D :: "'c set" shows "Domain (per_init D) = D"
T_Sub: fixes \<Gamma> :: "binding list"   and t :: "trm"   and S :: "POPLmarkRecord.type"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> t : S"   and "\<Gamma> \<turnstile> S <: T" shows "\<Gamma> \<turnstile> t : T"
pw_leof_iff: fixes m :: "'a nres"   and m' :: "'a nres" shows "(m \<le>\<^sub>n m') = (nofail m \<longrightarrow> (\<forall>x. inres m x \<longrightarrow> inres m' x))"
Collect_case_prodD: fixes x :: "'b \<times> 'c"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "x \<in> {(x, y). A x y}" shows "A (fst x) (snd x)"
uminus_p_def: fixes p :: "int"   and x :: "int" shows "uminus_p p x = (if x = 0 then 0 else p - x)"
Mp_irreducible\<^sub>d_m: fixes f :: "int poly" shows "irreducible\<^sub>d_m (Mp f) = irreducible\<^sub>d_m f"
measurable_Sup2: fixes M :: "'b measure set"   and f :: "'c \<Rightarrow> 'b"   and N :: "'c measure" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M m"   and "\<And>m n. \<lbrakk>m \<in> M; n \<in> M\<rbrakk> \<Longrightarrow> space m = space n" shows "f \<in> N \<rightarrow>\<^sub>M Sup M"
alist_of_remove_min: fixes q :: "('c, 'd) pq" assumes "\<not> PQ.is_empty q" shows "pq.alist_of (remove_min q) = tl (pq.alist_of q)"
lt_setE: fixes t :: "'t"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "t \<in> lt_set F" obtains f :: "'t \<Rightarrow>\<^sub>0 'd" where "f \<in> F" and "f \<noteq> 0" and "lt f = t"
rbt_cases: fixes t :: "('a, 'b) rbt" obtains "t = Empty"   | l :: "('a, 'b) rbt" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "t = Branch R l__ k__ v__ r__"   | l :: "('a, 'b) rbt" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "t = Branch B l k v r"
rfn_tool16: fixes r :: "nat"   and s :: "nat"   and i :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "0 < s"   and "i \<le> s * r - Suc 0"   and "G \<guillemotright> f (i div s) "   and "\<natural>f (i div s) \<triangleright> f (Suc (i div s))"   and "\<natural>f (i div s) \<guillemotright> f (i div s) \<inter> g (s - Suc 0) " shows "\<natural>f (Suc (i div s)) \<diamondop> f (i div s) \<inter> g (s - Suc 0) \<triangleright> f (Suc (i div s))"
type_low_vars_low: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'AExp"   and mds :: "Mode \<Rightarrow> 'Var set"   and x :: "'Var" assumes "\<Gamma> \<turnstile>\<^sub>a e \<in> Low"   and "mds_consistent mds \<Gamma>"   and "x \<in> aexp_vars e" shows "to_total \<Gamma> x = Low"
o_assoc: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'd" shows "(f \<circ>\<circ>\<circ> (\<circ>)) g h = f \<circ> g \<circ> h"
simps_1: fixes x2 :: "'a"   and y2 :: "'a" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
prems_1: shows "\<Gamma>\<turnstile> \<langle>sequence Seq (x # xs),Normal s\<rangle> =n\<Rightarrow> s'"
LIM_NSLIM: fixes f :: "'c \<Rightarrow> 'd"   and L :: "'d"   and a :: "'c" assumes "f \<midarrow>a\<rightarrow> L" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
pick_injective_goodAbs: fixes A :: "('i, 'j, 'varSort, 'var, 'k) qAbs set"   and B :: "('i, 'j, 'varSort, 'var, 'k) qAbs set" assumes "goodAbs A"   and "goodAbs B" shows "(pick A = pick B) = (A = B)"
simps_3: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (CLEAR x[], s) = Some (SKIP, s(x := \<lambda>_. 0))"
INF_less_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and a :: "'b" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less (Inf (f ` A)) a = (\<exists>x\<in>A. less (f x) a)"
mem_simps_6: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
set_awalk_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "GS.awalk u p v" shows "set (GS.awalk_verts u p) = {u} \<union> set (map (tail (G \<restriction> S)) p) \<union> set (map (head (G \<restriction> S)) p)"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "arr (tensor f unity)"
bw_diff_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "bw_diff f x = f x - f (x - (1::'a))"
poly_newton_poly_xj: fixes j :: "nat"   and n :: "nat" assumes "j \<le> n" shows "poly (local.newton_poly n) (local.x j) = (\<Sum>i\<leftarrow>[0..<Suc j]. local.c i * poly (local.N i) (local.x j))"
assms: shows "\<guillemotleft>g : y' \<rightarrow>\<^sub>D y\<guillemotright>"
sigma_algebra_iff: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "sigma_algebra \<Omega> M = (algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> \<Union> (range A) \<in> M))"
bounded: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_clinear f" shows "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
elims_3: fixes x :: "'b heap" assumes "\<not> is_root x" obtains xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "\<not> no_Emptys hs"
simps_2: fixes ss :: "'t list"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and j :: "nat" shows "new_syz_sigs ss bs (Inr j) = (if is_pot_ord then filter_min_append (adds\<^sub>t) ss (filter_min (adds\<^sub>t) (map (\<lambda>b. term_of_pair (punit.lt (rep_list b), j)) bs)) else ss)"
intros: fixes I :: "'b set"   and P :: "'b set \<Rightarrow> ('b \<Rightarrow> 'c) measure"   and M :: "'b \<Rightarrow> 'c measure"   and J :: "'b set"   and X :: "('b \<Rightarrow> 'c) set" assumes "projective_family I P M"   and "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)" shows "prod_emb I M J X \<in> projective_family.generator I M"
Vd_ss_V: shows "Vd d \<subseteq> V"
Above_union: fixes x :: "'d"   and r :: "('d \<times> 'd) set"   and X :: "'d set"   and Y :: "'d set" shows "(x \<in> Above r (X \<union> Y)) = (x \<in> Above r X \<and> x \<in> Above r Y)"
concat_sing_list_pow': fixes ws :: "'b list list"   and u :: "'b list" assumes "ws \<in> lists {u}" shows "concat ws = u \<^sup>@ \<^bold>|ws\<^bold>|"
exec_Normal_elim_cases_1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and f :: "'f"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> \<Rightarrow> t" obtains "t = Fault f"
cpx_mat_cnj_def: fixes M :: "complex Matrix.mat" shows "M\<^sup>\<star> \<equiv> Matrix.mat (dim_row M) (dim_col M) (\<lambda>(i, j). cnj (M $$ (i, j)))"
iagree_implies_niagree: shows "a0i_inv1_iagree \<subseteq> a0n_inv1_niagree"
STL5: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<box>F \<and> \<box>G) = (\<box>(F \<and> G))"
0: shows "x \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y = x mod p ^ k"
converse_converse: fixes r :: "('a \<times> 'b) set" shows "(r\<inverse>)\<inverse> = r"
sum_SDG_cdep_edge: fixes n :: "'node SDG_node"   and m :: "'node"   and n' :: "'node SDG_node"   and m' :: "'node" assumes "n = CFG_node m"   and "n' = CFG_node m'"   and "m controls m'" shows "n s\<longrightarrow>\<^bsub>cd\<^esub> n'"
to_rat: fixes x :: "'b" shows "to_rat x = (if x \<in> \<rat> then THE y. x = of_rat y else 0)"
cases: fixes a :: "('fenv, 'a, ('fun, 'b) frame) state" assumes "final a" obtains F :: "'fenv" and f :: "'fun" and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'num, 'bool) instr fundef" and pc :: "nat" and H :: "'a" and \<Sigma> :: "'b list" where "a = State F H [Frame f pc \<Sigma>]" and "F_get F f = Some fd" and "pc = length (body fd)"
graph_nodes: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and k :: "nat" shows "graph A w k \<subseteq> gunodes A w"
elem_fgs: fixes a :: "'a set"   and x :: "'a" assumes "a \<subseteq> carrier M"   and "x \<in> a" shows "x \<in> fgs R M a"
cgenideal_ideal: fixes a :: "'a" assumes "a \<in> carrier R" shows "ideal (PIdl a) R"
utility_weak_ranking_index: shows "is_vnm_utility (\<lambda>x. real (length (weak_ranking le) - weak_ranking_index x))"
len_greater_imp_nonempty: fixes x :: "nat"   and l :: "'a list" assumes "x < length l" shows "l \<noteq> []"
supp_Act: fixes \<alpha> :: "'g"   and x :: "('e, 'f, 'g) formula" shows "supp (Act \<alpha> x) = supp \<alpha> \<union> supp x - bn \<alpha>"
prod_2: fixes a :: "'a N"   and b :: "'a N"   and c :: "'a N" shows "(a r\<rightarrow> b) r\<rightarrow> c \<le> ((b r\<rightarrow> a) r\<rightarrow> c) r\<rightarrow> c"
order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<sqsubseteq> b) = (a = ord.min (\<sqsubseteq>) a b)"
1: shows "length (S @ t # T) = length (S @ s # T)"
dependent_def: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b set" assumes "vector_space scale" shows "module.dependent scale P = (\<exists>a\<in>P. a \<in> module.span scale (P - {a}))"
permutation_inverse: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "permutation (inv p)"
A_7: shows "v \<notin> \<lbrace>LT x c\<rbrace>"
idem: fixes a :: "'d upred" shows "(a \<and> a) = a"
measure_of_eq: fixes A :: "'a set set"   and \<Omega> :: "'a set"   and \<mu> :: "'a set \<Rightarrow> ennreal"   and \<mu>' :: "'a set \<Rightarrow> ennreal" assumes "A \<subseteq> Set.Pow \<Omega>"   and "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a" shows "measure_of \<Omega> A \<mu> = measure_of \<Omega> A \<mu>'"
naturality: fixes f :: "'a" assumes "B.arr f" shows "B.\<rr>'.map (B.cod f) \<cdot>\<^sub>B B.map f = B.R f \<cdot>\<^sub>B B.\<rr>'.map (B.dom f)"
is_OK_if_return_1: fixes b :: "bool"   and x :: "'d"   and m :: "'c + 'd" shows "isOK (if b then Inr x else m) = (b \<or> isOK m)"
dbm_entry_val_add_4: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and a :: "'d DBMEntry"   and d :: "'c"   and b :: "'d DBMEntry" assumes "dbm_entry_val u (Some c) None a"   and "dbm_entry_val u None (Some d) b" shows "dbm_entry_val u (Some c) (Some d) (dbm_add a b)"
well_base\<^sub>h_encode: fixes base :: "nat"   and e :: "nat"   and n :: "nat" assumes "2 \<le> base" shows "well_base\<^sub>h base (encode base e n)"
when_simps_2: fixes zero :: "'b"   and a :: "'b" shows "zero.when zero a False = zero"
greatest_not_zero_def: fixes A :: "'d iarray" shows "greatest_not_zero A = the (find (\<lambda>n. A !! n \<noteq> (0::'d)) (rev [0..<IArray.length A]))"
rep_eq: fixes x :: "('a, 'b) RBT.rbt" shows "lookup x = rbt_lookup (impl_of x)"
covector_comp_inf: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "covector z" shows "x * (y \<sqinter> z) = x * y \<sqinter> z"
runit_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg" shows "bicategory.runit V H \<a> \<i> src a \<equiv> prebicategory_with_homs_and_units.runit V H \<a> (bicategory.\<i>' V H \<i> src) src a"
get_proc_call: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "get_proc (targetnode a) = p"
qfmq: shows "qfree (simpfm (minusinf (fst (unit p))))"
mat_ring_def: fixes n :: "nat"   and b :: "'d" shows "mat_ring n b \<equiv> \<lparr>carrier = Collect (mat n n), mult = mat_mult n, one = mat1 n, zero = mat0 n n, add = mat_plus, \<dots> = b\<rparr>"
simps_11: fixes C' :: "char list"   and e :: "'a exp"   and M :: "char list"   and es :: "'a exp list" shows "not_init C' (e\<bullet>M(es)) = (not_init C' e \<and> not_inits C' es)"
positive_minus: fixes x :: "real" assumes "\<not> Real.positive x"   and "x \<noteq> 0" shows "Real.positive (- x)"
idem: fixes a :: "'d" shows "a \<sqinter> a = a"
drop_tl: fixes n :: "nat"   and xs :: "'c list" shows "drop n (tl xs) = tl (drop n xs)"
ground_subst: fixes t :: "('c, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('c, 'd) Term.term" shows "ground (t \<cdot> \<sigma>) = (\<forall>x\<in>vars_term t. ground (\<sigma> x))"
linears_strip_comb: fixes t :: "term" assumes "linear t" shows "linears (snd (strip_comb t))"
A: shows "Deliver (i, Create i e) \<in> set pre \<or> Deliver (i, Append i e) \<in> set pre \<or> (\<exists>mo. Deliver (i, Expunge e mo i) \<in> set pre) \<or> (\<exists>mo. Deliver (i, Store e mo i) \<in> set pre)"
inf_on_circline: fixes H :: "circline" shows "on_circline H \<infinity>\<^sub>h = is_line H"
sumAssocRight: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<oplus> (Q \<oplus> R) \<leadsto>[Rel] P \<oplus> Q \<oplus> R"
insertions_fst_subset: fixes ops :: "('c \<times> ('c, 'd) list_op) list" shows "set (map fst (insertions ops)) \<subseteq> set (map fst ops)"
meval_intro: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and n :: "nat"   and mem :: "'Var \<Rightarrow> 'Val"   and cm' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(cms ! n, mem) \<leadsto> (cm', mem')"   and "n < length cms" shows "mevalp cms mem n (cms[n := cm']) mem'"
prime_factorization_lcm: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "factorial_semiring.prime_factorization zero times one normalize (lcm a b) = factorial_semiring.prime_factorization zero times one normalize a \<union># factorial_semiring.prime_factorization zero times one normalize b"
gridI_def: fixes a :: "rat array"   and v :: "(nat \<times> int) list \<Rightarrow> real" shows "gridI a v = (\<exists>\<^sub>Axs. a \<mapsto>\<^sub>a xs * \<up> ((\<forall>p\<in>sparsegrid dm lm. v p = real_of_rat (xs ! pm p)) \<and> length xs = card (sparsegrid dm lm)))"
zero_le: fixes x :: "'d multiset" shows "{#} \<subseteq># x"