hcju/isabelleps huggingface dataset:
Dom_gf: shows "(\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Dom\<rparr> \<lparr>g, f\<rparr>\<^sub>\<bullet> = ab"
chain_subset_extension_on_Union: fixes R :: "('b \<times> 'b) set set"   and p :: "('b \<times> 'b) set" assumes "chain\<^sub>\<subseteq> R"   and "\<And>r. r \<in> R \<Longrightarrow> extension_on (Field r) r p" shows "extension_on (Field (\<Union> R)) (\<Union> R) p"
g_def: shows "g = (\<lambda>x. SOME a. a \<in> A \<and> f a = x)"
tranclp_converse: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<^sup>+\<^sup>+ = r\<^sup>+\<^sup>+\<inverse>\<inverse>"
well_base\<^sub>h_imp_well_base: fixes M :: "hmultiset" assumes "local.well_base\<^sub>h M" shows "local.well_base (hmsetmset M)"
Infinitesimal_interval2: fixes e :: "real star"   and e' :: "real star"   and x :: "real star" assumes "e \<in> Infinitesimal"   and "e' \<in> Infinitesimal"   and "e' \<le> x"   and "x \<le> e" shows "x \<in> Infinitesimal"
const_subterm_obtain': fixes t :: "('b, 'a) Term.term" assumes "fv t = {}" shows "\<exists>c. Fun c [] \<sqsubseteq> t"
multpw_listI: fixes xs :: "'a list"   and ys :: "'a list"   and X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "length xs = length ys"   and "X = mset xs"   and "Y = mset ys"   and "\<forall>i<length ys. (xs ! i, ys ! i) \<in> ns" shows "(X, Y) \<in> multpw ns"
y1: shows "(snd t', map (Pair True) ts) \<in> conv ars"
alpha_Tree_permute_rep_commute: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "p \<bullet> rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha> =\<^sub>\<alpha> rep_Tree\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
divides_irreducible_condition: fixes G :: "('c, 'd) monoid_scheme"   and r :: "'c"   and a :: "'c" assumes "irreducible G r"   and "a \<in> carrier G"   and "a divides\<^bsub>G\<^esub> r" shows "a \<in> Units G \<or> a \<sim>\<^bsub>G\<^esub> r"
prj1_ext: fixes f :: "'a"   and g :: "'a" assumes "\<not> C.cospan f g" shows "\<p>\<^sub>1[f, g] = C.null"
transpose_row_code: fixes A :: "(('b, 'c) vec, 'a) vec"   and i :: "'c" shows "($) (transpose_row A i) = (\<lambda>j. A $ j $ i)"
closed: fixes A :: "'a set" assumes "closed_csubspace A" shows "closed A"
cong2_conga_cong: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Cong A B A' B'"   and "Cong B C B' C'" shows "Cong A C A' C'"
zero_hom_axioms: shows "zero_hom (inv (inv (inv hom)))"
mnt: fixes R :: "('a, 'b) labeled_graph \<times> ('a, 'b) labeled_graph" assumes "R \<in> Rs" shows "maintained R (chain_sup S)"
galois_connection_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "galois_connection G \<equiv> connection G \<and> galois_connection_axioms G"
A: shows "ys' = filter (\<lambda>y. L y = Low) ys"
A: shows "comp_par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t public arity Ana \<Gamma> Pair A M C"
cardeq_lcoset: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and g :: "'c" assumes "Group.group G"   and "finite (carrier G)"   and "M \<subseteq> carrier G"   and "g \<in> carrier G" shows "card (g <#\<^bsub>G\<^esub> M) = card M"
nat_term_compare_inject: fixes x :: "'a nat_term_order"   and y :: "'a nat_term_order" shows "(nat_term_compare x = nat_term_compare y) = (x = y)"
restrict_spmf_return_pmf_None: fixes A :: "'a set" shows "return_pmf None \<upharpoonleft> A = return_pmf None"
fpr_objI: fixes P :: "cld list"   and ctx :: "ctx"   and ctxclds :: "(ctx \<times> cld) list" shows "find_path_rec P ctx cl_object ctxclds (Some ctxclds)"
span_redundant: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set" assumes "vector_space scale"   and "x \<in> module.span scale S" shows "module.span scale (insert x S) = module.span scale S"
flipidiflip: fixes y :: "nat"   and X :: "bool list"   and e :: "nat" assumes "y < length X"   and "e < length X" shows "flip e X ! y = (if e = y then \<not> X ! y else X ! y)"
getEnv_psubstEnv_None: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "rho xs x = None" shows "rho &[rho'] xs x = rho' xs x"
semiconnex: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "x \<sqsubset> y \<or> x = y \<or> y \<sqsubset> x"
power_inductl: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "z + x \<cdot> y \<le> y" shows "x ^ n \<cdot> z \<le> y"
simps_5: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b" shows "(case top.Top of top.Value x \<Rightarrow> f1 x | top.Top \<Rightarrow> f2) = f2"
norm_blinfun: fixes f :: "'c \<Rightarrow>\<^sub>L 'b"   and x :: "'c" shows "\<parallel>f *\<^sub>v x\<parallel> \<le> \<parallel>f\<parallel> * \<parallel>x\<parallel>"
skl1_inv6_derived: shows "skl1_inv4 \<inter> skl1_inv5 \<subseteq> skl1_inv6"
heap_get_def: shows "heap_get = Heap.Heap (\<lambda>heap. Some (heap, heap))"
list_of_pdevs_pdevs_of_list_eq: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> (0::'b)" shows "list_of_pdevs (pdevs_of_list xs) = zip (rev [0..<length xs]) (rev xs)"
F1rel_def: fixes R1 :: "'i \<Rightarrow> 'k \<Rightarrow> bool"   and R2 :: "'j \<Rightarrow> 'l \<Rightarrow> bool" shows "F1rel R1 R2 = (BNF_Def.Grp {x. F1set1 x \<subseteq> {(x, y). R1 x y} \<and> F2set x \<subseteq> {(x, y). R2 x y}} (F1map fst fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. F1set1 x \<subseteq> {(x, y). R1 x y} \<and> F2set x \<subseteq> {(x, y). R2 x y}} (F1map snd snd)"
Part_subset: fixes A :: "'b set"   and h :: "'c \<Rightarrow> 'b" shows "Part A h \<subseteq> A"
ctrm_is_poly: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "monom P (p 0) 0 \<in> carrier P"
Union: fixes a :: "nat \<Rightarrow> 'c set"   and sp :: "'c set"   and A :: "'c set set" assumes "\<And>i. a i \<in> sigma_sets sp A" shows "\<Union> (range a) \<in> sigma_sets sp A"
list_distinct_tl: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "equal_base.list_distinct equal xs" shows "equal_base.list_distinct equal (list.tl xs)"
greatest_LowerI: fixes A :: "'a set"   and L :: "('a, 'b) gorder_scheme"   and i :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> i \<sqsubseteq>\<^bsub>L\<^esub> x"   and "\<And>y. y \<in> Lower L A \<Longrightarrow> y \<sqsubseteq>\<^bsub>L\<^esub> i"   and "A \<subseteq> carrier L"   and "i \<in> carrier L" shows "is_glb L i A"
inf_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inf_class.inf (inf_class.inf x y) z = inf_class.inf x (inf_class.inf y z)"
term_rawInduct_fresh: fixes vars :: "'varSort \<Rightarrow> 'var set"   and terms :: "('a, 'b, 'varSort, 'var, 'c) qTerm set set"   and abs :: "('a, 'b, 'varSort, 'var, 'c) qAbs set set"   and envs :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option) set"   and phi :: "('a, 'b, 'varSort, 'var, 'c) qTerm set \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'varSort, 'var, 'c) qAbs set \<Rightarrow> bool"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "\<And>xs. (|vars xs| <o |UNIV| \<or> finite (vars xs)) \<and> (|terms| <o |UNIV| \<or> finite terms) \<and> Ball terms good \<and> (|abs| <o |UNIV| \<or> finite abs) \<and> Ball abs goodAbs \<and> (|envs| <o |UNIV| \<or> finite envs) \<and> Ball envs goodEnv"   and "\<And>xs x. phi (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>goodInp inp; goodBinp binp; liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (Op delta inp binp)"   and "\<And>xs x X. \<lbrakk>good X; x \<notin> vars xs; \<And>Y. Y \<in> terms \<Longrightarrow> fresh xs x Y; \<And>A. A \<in> abs \<Longrightarrow> freshAbs xs x A; \<And>rho. rho \<in> envs \<Longrightarrow> freshEnv xs x rho; phi X\<rbrakk> \<Longrightarrow> phiAbs (Abs xs x X)" shows "(good X \<longrightarrow> phi X) \<and> (goodAbs A \<longrightarrow> phiAbs A)"
path_component_of_equiv: fixes X :: "'c topology"   and x :: "'c"   and y :: "'c" shows "path_component_of X x y = (x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y)"
queue_invar_rev: fixes q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.queue_invar q" shows "BinomialHeapStruc.queue_invar (rev q)"
arc_regular: fixes x :: "'a" assumes "bijective (x * top) \<and> bijective (x[[top]])" shows "regular x"
def: shows "roots3_generic depressed_solver p = map (\<lambda>y. y - b / ((3::'a) * a)) (depressed_solver e f)"
N: fixes n :: "nat"   and a :: "'a" assumes "(n, a) \<in> set n_as"   and "norm a = 1" shows "n \<le> N"
sylow_axioms_def: fixes G :: "('a, 'b) 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)"
congdd_base_gen_congdd_base_aux: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and R' :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "rel (gen_cong R) x y"   and "R \<le> R'"   and "local.cong R'" shows "R' (eval x) (eval y)"
\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>_complete: fixes opubx :: "'opubx"   and xs :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and y :: "('dyn, 'ubx1, 'ubx2) unboxed" assumes "\<UU>\<bb>\<xx>\<OO>\<pp> opubx xs = Some y" shows "\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> opubx = (map typeof xs, typeof y)"
dtor1_o_unfold1: fixes s1 :: "'d \<Rightarrow> ('e, 'd, 'f) F1"   and s2 :: "'f \<Rightarrow> ('e, 'd, 'f) F2" shows "dtor1 \<circ> unfold1 s1 s2 = F1map id (unfold1 s1 s2) (unfold2 s1 s2) \<circ> s1"
interior_of_empty: fixes X :: "'c topology" shows "X interior_of {} = {}"
subseq_Cons2: fixes xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "subseq xs ys" shows "subseq (x # xs) (x # ys)"
perm2: fixes p :: "perm" assumes "supp p \<sharp>* c" shows "p \<bullet> f b y c = f (p \<bullet> b) (p \<bullet> y) c"
index_of_c_pair_n_def: shows "index_of_c_pair_n = (\<lambda>n. pair_by_index (index_of_const n) index_of_id)"
assms_1: shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec'"
indep_region_coregion: fixes S :: "<'d, 'e> \<Longleftrightarrow> 'f" assumes "psym_lens S" shows "\<V>\<^bsub>S\<^esub> \<bowtie> \<C>\<^bsub>S\<^esub>"
bij_betw_insert: fixes b :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and A' :: "'b set" assumes "b \<notin> A"   and "f b \<notin> A'"   and "bij_betw f A A'" shows "bij_betw f (insert b A) (insert (f b) A')"
index_size_conv: fixes xs :: "'e list"   and n :: "nat"   and x :: "'e" assumes "length xs = n" shows "(index xs x = n) = (x \<notin> set xs)"
neg_0_equal_iff_equal: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.group_add minus plus zero uminus" shows "(zero = uminus a) = (zero = a)"
neq_LNil_conv: fixes xs :: "'a llist" shows "(xs \<noteq> []\<^sub>l) = (\<exists>x xs'. xs = x #\<^sub>l xs')"
rep_eq: fixes x :: "'b"   and xa :: "'b ttree" shows "Rep_ttree (without x xa) = filter (\<lambda>x'. x' \<noteq> x) ` Rep_ttree xa"
finite_nodup: fixes I :: "'a set" assumes "finite I" shows "\<exists>l. set l = I \<and> nodup_all l"
sfirst_eq_enat_iff: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and n :: "nat" shows "(sfirst P \<omega> = enat n) = ev_at P n \<omega>"
atLeast1_atMost_eq: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {Suc 0..n} = (\<Sum>k<n. g (Suc k))"
lookup_map_default_neq: fixes x :: "'c"   and y :: "'c"   and d :: "'d"   and f :: "'d \<Rightarrow> 'd"   and m :: "('c, 'd) mapping" assumes "x \<noteq> y" shows "Mapping.lookup (Mapping.map_default x d f m) y = Mapping.lookup m y"
assms_4: fixes K :: "'a set" assumes "subfield K R"   and "H \<subseteq> K" shows "E \<subseteq> K"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(Regions.intv.Const x1 = Regions.intv.Const y1) = (x1 = y1)"
strict_iff_order: fixes b :: "'c list"   and a :: "'c list" shows "strict_prefix b a = (prefix b a \<and> a \<noteq> b)"
SSpec_def: fixes c :: "OBJ"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" shows "SSpec c s t = s , c \<Down> t "
simps: fixes t :: "('b, 'sz) trm"   and i :: "'sz" shows "singleton t i = (if i = vid1 then t else Const 0)"
E_def: shows "E = addrow_mat m (- rat_of_int c) i j"
induct: fixes P :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<Rightarrow> bool"   and a0 :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" assumes "\<And>Pa A Q. \<lbrakk>\<And>x xa xb xc xd. \<lbrakk>x = A $h 0 $h 0; xa = A $h 0 $h 1; xb = A $h 1 $h 0; xc = A $h 1 $h 1; x \<noteq> (0::'b) \<and> \<not> x dvd xa; xd = Finite_Cartesian_Product.transpose (bezout_matrix (Finite_Cartesian_Product.transpose A) 0 1 0 euclid_ext2)\<rbrakk> \<Longrightarrow> P (Pa, A ** xd, Q ** xd); \<And>x xa xb xc xd. \<lbrakk>x = A $h 0 $h 0; xa = A $h 0 $h 1; xb = A $h 1 $h 0; xc = A $h 1 $h 1; \<not> (x \<noteq> (0::'b) \<and> \<not> x dvd xa); x \<noteq> (0::'b) \<and> \<not> x dvd xb; xd = bezout_matrix A 0 1 0 euclid_ext2\<rbrakk> \<Longrightarrow> P (xd ** Pa, xd ** A, Q)\<rbrakk> \<Longrightarrow> P (Pa, A, Q)" shows "P a0"
assms_2: shows "bdd_below {u n / real n |n. N0 < n}"
axioms_1: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "equivalence A equiv" shows "symmetric A equiv"
case_nat_in_state_measure: fixes x :: "val"   and t1 :: "pdf_type"   and \<sigma> :: "nat \<Rightarrow> val"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" assumes "x \<in> type_universe t1"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "case_nat x \<sigma> \<in> space (state_measure (shift_var_set V) (case_nat t1 \<Gamma>))"
x1: fixes M :: "real" assumes "\<And>t. finite t \<Longrightarrow> (\<Sum>i\<in>t. (cmod (x i))\<^sup>2) \<le> M" shows "\<exists>B. \<forall>F. finite F \<longrightarrow> (\<Sum>s\<in>F. (cmod (x s))\<^sup>2) < B"
gen_idem: fixes P :: "'a set set"   and I :: "'a set" shows "gen P (gen P I) = gen P I"
t_n_closed: fixes x :: "'a" shows "n n n x = n x"
meld_mset: fixes q :: "('c, 'd) SkewBinomialTree list"   and q' :: "('c, 'd) SkewBinomialTree list" assumes "queue_invar q"   and "queue_invar q'" shows "queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'"
hcomp_simps_4: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.hseq \<nu> \<mu>" shows "S.cod (\<nu> \<star>\<^sub>S \<mu>) = S.cod \<nu> \<star>\<^sub>S S.cod \<mu>"
filterlim_at_top_to_right: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter" shows "filterlim f F at_top = (LIM x at_right 0. f (inverse x) :> F)"
Fun_repeatable: fixes e :: "exp" assumes "isVal e" shows "repeatable (Texp e\<cdot>0)"
intro: fixes n :: "nat"   and program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and c\<^sub>0 :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and safe :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool" assumes "\<And>k c l. \<lbrakk>k \<le> n; program_trace program_step c 0 k; c 0 = c\<^sub>0; l \<le> k\<rbrakk> \<Longrightarrow> safe (c l)" shows "program_safe_reach_upto program_step n safe c\<^sub>0"
mask_eqs_4: fixes a :: "'b word"   and b :: "'b word"   and n :: "nat" shows "a - (b AND mask n) AND mask n = a - b AND mask n"
inj_monoid_mult_hom_axioms: shows "inj_monoid_mult_hom hom"
lcoset_relI: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" assumes "- x + y \<in> A" shows "(x, y) \<in> lcoset_rel A"
enumAppend_length1: fixes ls :: "nat list"   and n :: "nat"   and nmax :: "nat"   and lss :: "nat list list"   and k :: "nat" assumes "ls \<in> set ((enumAppend nmax ^^ n) lss)"   and "\<forall>l\<in>set lss. |l| = k" shows "|ls| = k + n"
linorder_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "class.linorder less_eq less \<equiv> class.order less_eq less \<and> class.linorder_axioms less_eq"
distinct_18: fixes x61 :: "nat"   and x62 :: "recf" shows "Mn x61 x62 \<noteq> S"
strong_induct: fixes P :: "'a \<Rightarrow> Late_Semantics.freeRes \<Rightarrow> bool"   and z :: "'a"   and freeRes :: "Late_Semantics.freeRes" assumes "\<And>name1 name2 z. P z (name1[name2])"   and "\<And>z. P z (\<tau>)" shows "P z freeRes"
compl_eq_compl_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "(uminus x = uminus y) = (x = y)"
great_Picard_alt: fixes M :: "complex set"   and z :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "open M"   and "z \<in> M"   and "f holomorphic_on M - {z}"   and "\<And>l. \<not> f \<midarrow>z\<rightarrow> l"   and "\<And>l. \<not> (inverse \<circ> f) \<midarrow>z\<rightarrow> l" obtains a :: "complex" where "- {a} \<subseteq> f ` (M - {z})"
convert_perm_def: shows "convert_perm = (\<lambda>\<sigma> x. if x \<in> Roots then Root (\<sigma> (unRoot x)) else x)"
simps_7: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and e1' :: "'a aexp"   and e2' :: "'a aexp" shows "local.less_gexp_aux (Eq e1 e2) (Eq e1' e2') = (e1 < e1' \<or> e1 = e1' \<and> e2 < e2')"
smc_SemiCAT_Obj_iff: fixes \<AA> :: "V"   and \<alpha> :: "V" shows "(\<AA> \<in>\<^sub>\<circ> smc_SemiCAT \<alpha>\<lparr>Obj\<rparr>) = semicategory \<alpha> \<AA>"
sdp_PC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "sub_distrib_pconj a"   and "sub_distrib_pconj b"   and "healthy (wp a)"   and "healthy (wp b)"   and "nearly_healthy (wlp b)"   and "unitary P" shows "sub_distrib_pconj (a \<^bsub>P\<^esub>\<oplus> b)"
"2_3": shows "length slist_2 = Suc (length as_2)"
coplanar_perm_23: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D" shows "Coplanar D C B A"
assm: shows "|restrict v f c| = |lan v| \<and> restrict v f c \<le> lan v"
fv_regex_alt: fixes fv :: "'e \<Rightarrow> 'd set"   and r :: "'e regex" shows "fv_regex fv r = \<Union> (fv ` atms r)"
simps_2: fixes A :: "'b set"   and B :: "'b set"   and v :: "'b"   and va :: "'b list" shows "merge A B (v # va) [] = v # va"
Nv: shows "\<forall>n\<ge>Nv. \<forall>m\<ge>Nv. dist (v n) (v m) < e / 2"
in_r: fixes \<P>1 :: "'a set set"   and \<P>2 :: "'a set set" shows "((\<P>1, \<P>2) \<in> r) = (\<P>1 \<subseteq> \<P>2 \<and> \<P>1 \<in> Field_r \<and> \<P>2 \<in> Field_r)"
not_numeral_le_zero: fixes n :: "num" shows "\<not> numeral n \<le> (0::'a)"
to_int': fixes y :: "'i" assumes "0 \<le> arith_ops_record.to_int ff_ops y"   and "arith_ops_record.to_int ff_ops y < p" shows "R y (of_int (arith_ops_record.to_int ff_ops y))"
inj_onD: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and x :: "'f"   and y :: "'f" assumes "inj_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
assms: shows "P \<longmapsto> a[b] \<prec> P'"
valid_path_aux_intra_path: fixes as :: "'edge list"   and cs :: "'edge list" assumes "\<forall>a\<in>set as. intra_kind (kind a)" shows "valid_path_aux cs as"
rtrancl_converse: fixes r :: "('a \<times> 'a) set" shows "(r\<inverse>)\<^sup>* = (r\<^sup>*)\<inverse>"
assms_1: shows "\<forall>x y z. le x y \<and> le y z \<longrightarrow> le x z"
next_ge_all: fixes S :: "'a set"   and s :: "'a" assumes "finite S"   and "\<forall>y\<in>S. y \<le> s" shows "next s \<notin> S"
in_indetsI: fixes x :: "'c"   and t :: "'c \<Rightarrow>\<^sub>0 nat"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "x \<in> keys t"   and "t \<in> keys p" shows "x \<in> indets p"
assms_2: shows "t \<cdot> \<theta> = t' \<cdot> \<theta>"
leadsTo_cancel1: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and A' :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> B \<union> A'"   and "F \<in> B \<longmapsto> B'" shows "F \<in> A \<longmapsto> B' \<union> A'"
IH: fixes E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P,E,h \<turnstile> e : T"   and "P,h \<turnstile> l(V := None) (:\<le>) E" shows "P,h' \<turnstile> l' (:\<le>) E"
sel_2: fixes x1 :: "int"   and x2 :: "int" shows "ImZ (Gauss_Int x1 x2) = x2"
mset_concat_merge_pairs: fixes xs :: "'a list list" shows "mset (concat (local.merge_pairs xs)) = mset (concat xs)"
ln_inverse_approx_ge: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "x < y" shows "2 * (y - x) / (x + y) \<le> ln y - ln x"
aui_lemma: fixes pa :: "'a list"   and qa :: "'a list"   and i :: "'a" assumes "cycle pa"   and "cycle qa"   and "set pa \<inter> set qa = {}"   and "i \<in> set pa"   and "i \<notin> set qa" shows "(cycle_of_list pa \<circ> cycle_of_list qa) i = (cycle_of_list qa \<circ> cycle_of_list pa) i"
nn_integral_completion: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N (completion M) f = integral\<^sup>N M f"
step: fixes m :: "enat"   and n :: "enat" assumes "(m, n) \<in> X" shows "m = 0 \<or> (\<exists>m' n' k. m = eSuc m' \<and> n = n' + enat k \<and> k \<noteq> 0 \<and> ((m', n') \<in> X \<or> m' \<le> n'))"
rel_frontier_deformation_retract_of_punctured_convex: fixes S :: "'a set"   and T :: "'a set"   and a :: "'a" assumes "convex S"   and "convex T"   and "bounded S"   and "a \<in> rel_interior S"   and "rel_frontier S \<subseteq> T"   and "T \<subseteq> affine hull S" obtains r :: "'a \<Rightarrow> 'a" where "homotopic_with_canon (\<lambda>x. True) (T - {a}) (T - {a}) id r" and "retraction (T - {a}) (rel_frontier S) r"
prime_power_inj'_1: fixes p :: "'a"   and q :: "'a"   and m :: "nat"   and n :: "nat" assumes "prime p"   and "prime q"   and "p ^ m = q ^ n"   and "0 < m"   and "0 < n" shows "p = q"
p_q_2: shows "local.my_U q \<in> {0..1}"
infliveness_dense_iff: fixes P :: "'a llist set" assumes "A \<noteq> {}"   and "P \<subseteq> A\<^sup>\<omega>" shows "infliveness A P = (P dense)"
unfinished_nodes_decr: fixes u :: "'v" assumes "u \<notin> S"   and "D u < \<infinity>" shows "unfinished_dnodes (Set.insert u S) \<subset> unfinished_dnodes S"
mbisim: shows "s1 \<approx>m (ls2, (ts2, m2), ws2, is2)"
lhd_LCons_ltl: fixes llist :: "'b llist" assumes "\<not> lnull llist" shows "lhd llist % ltl llist = llist"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc (C\<^sub>1;; C\<^sub>2) Q = (vc C\<^sub>1 (pre C\<^sub>2 Q) \<and> vc C\<^sub>2 Q)"