hcju/isabelleps huggingface dataset:
smult_assoc: fixes r :: "'f"   and s :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "s \<in> UNIV"   and "m \<in> V" shows "r \<cdot> s \<cdot> m = (r * s) \<cdot> m"
irreducible_smult_field: fixes c :: "'a"   and p :: "'a poly" shows "irreducible (smult c p) = (c \<noteq> (0::'a) \<and> irreducible p)"
wls_substAbs_simp: fixes ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "wls (asSort ys) Y"   and "wls s X"   and "xs \<noteq> ys \<or> x \<noteq> y"   and "fresh xs x Y" shows "Abs xs x X $[Y / y]_ys = Abs xs x (X #[Y / y]_ys)"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Atom x1 = Atom y1) = (x1 = y1)"
intros_2: fixes k :: "'b"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "Atom k \<in># \<Gamma>"   and "Atom k \<in># \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
trivial_code: fixes g :: "'g"   and v :: "'val"   and vs :: "'val list" assumes "phi g v = Some vs" shows "trivial g v = trivial_code v vs"
quat_inner_1_right: fixes x :: "quat" shows "x \<bullet> 1 = quat.Re x"
repeat_mset_replicate_mset: fixes n :: "nat"   and a :: "'k" shows "repeat_mset n {#a#} = replicate_mset n a"
permutes_def: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" shows "(p permutes S) = ((\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y))"
continuous_map_fst: fixes X :: "'b topology"   and Y :: "'c topology" shows "continuous_map (prod_topology X Y) X fst"
zero: shows "R (arith_ops_record.zero ff_ops) 0"
fract_content_eq_0_iff: fixes p :: "'a fract poly" shows "(fract_content p = 0) = (p = 0)"
readv: shows "v \<in> local.reads (path \<sigma> k)"
join_option_eq_None: fixes x :: "'c option option" shows "(join_option x = None) = (x = None \<or> x = Some None)"
prodinf_def: fixes f :: "nat \<Rightarrow> 'e" shows "prodinf f = (THE p. f has_prod p)"
mergeSibling_induct_consistent: fixes t :: "'a tree"   and P :: "'a tree \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "consistent t"   and "\<And>w b a. P (Leaf w b) a"   and "\<And>w w\<^sub>b b w\<^sub>c c a. b \<noteq> c \<Longrightarrow> P (Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; 0 < height t\<^sub>1 \<or> 0 < height t\<^sub>2; a \<in> alphabet t\<^sub>1; sibling t\<^sub>1 a \<in> alphabet t\<^sub>1; a \<notin> alphabet t\<^sub>2; sibling t\<^sub>1 a \<notin> alphabet t\<^sub>2; P t\<^sub>1 a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; 0 < height t\<^sub>1 \<or> 0 < height t\<^sub>2; a \<notin> alphabet t\<^sub>1; sibling t\<^sub>2 a \<notin> alphabet t\<^sub>1; a \<in> alphabet t\<^sub>2; sibling t\<^sub>2 a \<in> alphabet t\<^sub>2; P t\<^sub>2 a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a"   and "\<And>w t\<^sub>1 t\<^sub>2 a. \<lbrakk>consistent t\<^sub>1; consistent t\<^sub>2; alphabet t\<^sub>1 \<inter> alphabet t\<^sub>2 = {}; 0 < height t\<^sub>1 \<or> 0 < height t\<^sub>2; a \<notin> alphabet t\<^sub>1; a \<notin> alphabet t\<^sub>2\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 t\<^sub>2) a" shows "P t a"
distinct_getVars: fixes \<sigma>l :: "'tp list" shows "distinct (TE.getVars \<sigma>l)"
modes_respect_priv: shows "modes_respect_priv (map snd (fst gc\<^sub>C))"
irr: fixes f :: "int poly" assumes "f \<in> set res" shows "irreducible\<^sub>d f"
greaters_def: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and xs :: "'b list" shows "greaters R x xs = filter (\<lambda>y. (x, y) \<in> R \<and> (y, x) \<notin> R) xs"
det_def: fixes P :: "'a process"   and Q :: "'a process" shows "P [+] Q \<equiv> Abs_process ({(s, X). s = [] \<and> (s, X) \<in> F P \<inter> F Q} \<union> {(s, X). s \<noteq> [] \<and> (s, X) \<in> F P \<union> F Q} \<union> {(s, X). s = [] \<and> s \<in> D P \<union> D Q} \<union> {(s, X). s = [] \<and> tick \<notin> X \<and> [tick] \<in> T P \<union> T Q}, D P \<union> D Q)"
unit_mult: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "sound P"   and "bounded_by 1 P"   and "sound Q"   and "bounded_by 1 Q" shows "bounded_by 1 (\<lambda>s. P s * Q s)"
assms_8: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"
bitset_sum: fixes B :: "nat set" assumes "finite B" shows "bitset (sum ((^) 2) B) = B"
assms_3: fixes a :: "real" shows "continuous (at_right a) F"
disc_tree_orders: shows "h \<turnstile> map_M (to_tree_order \<circ> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r) disc_nodes_h3 \<rightarrow>\<^sub>r disc_tree_orders"
times_real_alg: fixes x :: "real_alg"   and y :: "real_alg" shows "real_of x * real_of y = real_of (x * y)"
rel_map_1: fixes Sb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'f \<Rightarrow> 'c"   and x :: "'f set['e]"   and y :: "'d set['e]" shows "rel_bset Sb (map_bset i x) y = rel_bset (\<lambda>x. Sb (i x)) x y"
csubspace_is_convex: fixes M :: "'b set" assumes "csubspace M" shows "convex M"
comp_Abs_inverse: fixes f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" assumes "f \<circ> Abs = g" shows "g \<circ> Rep = f"
Quotient_rel_abs2: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'e" assumes "Quotient R Abs Rep T"   and "R (Rep x) y" shows "x = Abs y"
COMP_def: fixes g :: "('a E.term, 'a) arr"   and f :: "('a E.term, 'a) arr" shows "g \<cdot> f \<equiv> if (f \<noteq> Null \<and> Dom f \<in> IDE \<and> Cod f \<in> IDE \<and> Map f \<in> HOM (Dom f) (Cod f)) \<and> (g \<noteq> Null \<and> Dom g \<in> IDE \<and> Cod g \<in> IDE \<and> Map g \<in> HOM (Dom g) (Cod g)) \<and> Dom g = Cod f then MkArr (Dom f) (Cod g) (Map g \<cdot>\<^sub>B Map f) else Null"
DF_Univ_freeness: fixes A :: "'d set"   and P :: "'d process" assumes "A \<noteq> {}"   and "DF A \<sqsubseteq>\<^sub>F\<^sub>D P" shows "deadlock_free P"
implicit_subspace_with: shows "subspace_with (+) (0::'a) (*\<^sub>R) = subspace.subspace"
Theorem_15_match: fixes ds :: "'b set" shows "TruePref.match (TruePref.gfp_F ds) = TruePref.cop ds"
norm_triangle_half_r: fixes y :: "'a"   and x1 :: "'a"   and e :: "real"   and x2 :: "'a" assumes "norm (y - x1) < e / 2"   and "norm (y - x2) < e / 2" shows "norm (x1 - x2) < e"
D_def: shows "D = (\<lambda>N. \<Union>n\<in>{1..N}. (\<lambda>m. (m, n)) ` totatives n)"
Hyp: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> H" shows "H \<turnstile> A"
if_eqvt: fixes pi :: "('b \<times> 'b) list"   and b :: "bool"   and c1 :: "'a"   and c2 :: "'a" shows "pi \<bullet> (if b then c1 else c2) = (if pi \<bullet> b then pi \<bullet> c1 else pi \<bullet> c2)"
length_digit_encode: fixes ds :: "nat list"   and a :: "nat" shows "length (digit_encode ds a) = length ds"
delete_shadow_root_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'aj" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
comm_group: shows "comm_group (add_monoid (S\<lparr>carrier := h ` carrier R, zero := h \<zero>\<rparr>))"
proj2_set_Col_endpoints_in_S: fixes a :: "proj2"   and b :: "proj2" assumes "a \<noteq> b"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S" shows "proj2_set_Col {endpoint_in_S a b, endpoint_in_S b a, a, b}"
length_replicate: fixes n :: "nat"   and x :: "'a" shows "List.length (replicate n x) = n"
prj_simps_3: shows "C.arr \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0"
in_set_impl_in_set_zip2: fixes xs :: "'a list"   and ys :: "'b list"   and y :: "'b" assumes "length xs = length ys"   and "y \<in> set ys" obtains x :: "'a" where "(x, y) \<in> set (zip xs ys)"
GrpE: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" assumes "BNF_Def.Grp A f x y" obtains "f x = y" and "x \<in> A"
IN_k: fixes x :: "'v" shows "d_IN k x = (if x \<in> RF (TER f) then d_IN h x else d_IN g x)"
convolution_symm: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "convolution f g = (\<lambda>x. \<Sum>y | y \<in> supp f \<and> - y + x \<in> supp g. f y * g (- y + x))"
wf_2: shows "R \<equiv> prefs_from_table yss"
mirror_elem_Max: fixes I :: "nat set" assumes "finite I"   and "I \<noteq> {}" shows "mirror_elem (Max I) I = iMin I"
if_elim: fixes b :: "bexp\<^sub>C"   and p :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and q :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "eval_abv\<^sub>C \<langle>Stmt.If b p q, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C" obtains "c' = p" and "mem' = mem" and "mds' = mds" and "ev\<^sub>B\<^sub>C mem b"   | "c' = q" and "mem' = mem" and "mds' = mds" and "\<not> ev\<^sub>B\<^sub>C mem b"
left_neutral: fixes a :: "'r" shows "(0::'r) + a = a"
adj_not_same: fixes a :: "'a" assumes "(a, a) \<in> parcs (list_digraph G)" shows "False"
Prefixes_altdef: fixes A :: "'a list set" shows "Prefixes A = (\<Union>w\<in>A. set (prefixes w))"
nn_integral_spmf: fixes A :: "'h set"   and p :: "'h spmf" shows "(\<Sum>\<^sup>+ x\<in>A. ennreal (spmf p x)) = emeasure (measure_spmf p) A"
pi_approx: fixes n :: "nat" assumes "even n"   and "0 < n" shows "\<bar>pi - pi_approx n\<bar> \<le> inverse (2 ^ (13 * n - 2))"
Method_def: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D \<equiv> \<exists>Mm. P \<turnstile> C sees_methods Mm \<and> Mm M = \<lfloor>((b, Ts, T, m), D)\<rfloor>"
coprime_normalize_quot: fixes x :: "'b \<times> 'b" shows "coprime (fst (normalize_quot x)) (snd (normalize_quot x))"
aux7a: fixes a :: "'b set set"   and b :: "'b set set"   and p :: "('b set set, 'c) Combinators list"   and c :: "'b set set"   and d :: "'b set set"   and po :: "'c" assumes "DenyAllFromTo a b \<in> set p"   and "allNetsDistinct (AllowPortFromTo c d po # p)"   and "a \<noteq> c \<or> b \<noteq> d" shows "twoNetsDistinct a b c d"
lnull_cons'_def: fixes g :: "('d, 'c) lgenerator'" shows "lnull_cons' g = lnull_cons (lgenerator g)"
geom_proc_eq_snth: fixes n :: "nat"   and x :: "bool stream"   and y :: "bool stream"   and m :: "nat" assumes "\<And>m. m \<le> Suc n \<Longrightarrow> geom_proc m x = geom_proc m y"   and "m \<le> n" shows "x !! m = y !! m"
wf_rexp_of_alt': fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> n \<phi>" shows "alphabet.wf (wf_atom \<Sigma>) n (formula.rexp_of_alt' \<Sigma> n \<phi>)"
hcomp_simps_3: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "B.hseq \<nu> \<mu>" shows "B.dom (\<nu> \<star>\<^sub>B \<mu>) = B.dom \<nu> \<star>\<^sub>B B.dom \<mu>"
piecewise_differentiable_on_imp_continuous_on: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f piecewise_differentiable_on S" shows "continuous_on S f"
order_iff_strict: fixes b :: "'a"   and a :: "'a" shows "le b a = (lt b a \<or> a = b)"
smooth_compat_le: fixes k :: "enat"   and c1 :: "('b, 'c) chart"   and c2 :: "('b, 'c) chart"   and l :: "enat" assumes "k-smooth_compat c1 c2"   and "l \<le> k" shows "l-smooth_compat c1 c2"
distinct_5: fixes x1 :: "agent"   and x4 :: "nat" shows "Agent x1 \<noteq> Key x4"
prems_2: fixes \<AA> :: "V" assumes "semicategory \<alpha> \<AA>" shows "\<exists>!\<FF>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
assms: shows "wf_effect (ty_term Q constT) \<phi>"
uniformity_sym: fixes E :: "'a \<times> 'a \<Rightarrow> bool" assumes "eventually E uniformity" shows "\<forall>\<^sub>F (x, y) in uniformity. E (y, x)"
connected_in_V_1: fixes v :: "'c"   and w :: "'c" assumes "T.connected v w" shows "v \<in> V\<^bsub>T\<^esub>"
semiring_normalization_rules_7: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "times a b = times b a"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs"   and "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M" shows "l_set_mode_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
subst_apply_fv_union: fixes \<delta> :: "'e \<Rightarrow> ('d, 'c) Term.term"   and V :: "'e set"   and t :: "('d, 'e) Term.term" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V) \<union> fv (t \<cdot> \<delta>) = fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` (V \<union> fv t))"
iso_lunit: 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 f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "category.iso V (bicategory.lunit V H \<a> \<i> src trg f)"
varible_swap: fixes a :: "'b"   and b :: "'b" shows "rdom \<lceil>\<lambda>s. s ''x'' = a \<and> s ''y'' = b\<rceil> \<subseteq> wp ((''z'' ::= (\<lambda>s. s ''x'')) ; (''x'' ::= (\<lambda>s. s ''y'')) ; (''y'' ::= (\<lambda>s. s ''z''))) \<lceil>\<lambda>s. s ''x'' = b \<and> s ''y'' = a\<rceil>"
add_num_frac: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and y :: "'b"   and z :: "'b"   and x :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero" shows "plus z (divide x y) = divide (plus x (times z y)) y"
prod_sum_PiE: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> finite (B x)" shows "(\<Prod>x\<in>A. sum (f x) (B x)) = (\<Sum>g\<in>Pi\<^sub>E A B. \<Prod>x\<in>A. f x (g x))"
abs_eq: fixes xc :: "complex"   and xb :: "complex"   and xa :: "complex"   and x :: "complex" shows "mk_moebius_mat xc xb xa x = Abs_moebius_mat (mk_moebius_cmat xc xb xa x)"
simps_9: fixes i :: "int"   and c :: "int"   and e :: "MIR.num" shows "\<zeta> (Dvd i (CN 0 c e)) = c"
a_star_refl: fixes M :: "trm" shows "M \<longrightarrow>\<^sub>a* M"
Cons_1: assumes "distinct (map fst xs)" shows "set (sort_oalist xs) = {kv \<in> set xs. snd kv \<noteq> (0::'b)}"
currk_Check_in: fixes s\<^sub>2 :: "event list"   and g :: "guest"   and r :: "room"   and k :: "key"   and k' :: "key"   and s\<^sub>1 :: "event list"   and r' :: "room" assumes "hotel (s\<^sub>2 @ Check_in g r (k, k') # s\<^sub>1)"   and "k' = currk (s\<^sub>2 @ Check_in g r (k, k') # s\<^sub>1) r'" shows "r' = r"
simps_33: shows "cospan_shape.comp cospan_shape.AA cospan_shape.TT = cospan_shape.Null"
nextl_state: fixes Q :: "hf set"   and xs :: "'a list" shows "ST.nextl Q xs \<subseteq> nfa.states \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr>"
substitutes_on_def2: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "substitutes_on A f = (\<forall>B\<subseteq>A. \<forall>a\<in>A. \<forall>b\<in>A. b \<notin> f (insert b B) \<longrightarrow> b \<notin> f (insert a (insert b B)))"
sams123231: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "A \<noteq> C"   and "B \<noteq> C" shows "Tarski_neutral_dimensionless.SAMS Bet Cong A B C B C A"
simps_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'c"   and a :: "('a, 'b) atm" assumes "U.Struct wtFsym wtPsym arOf parOf D intF intP" shows "U.Struct.satL intF intP \<xi> (Pos a) = U.Struct.satA intF intP \<xi> a"
obj_def: fixes a :: "'a" shows "C.obj a \<equiv> C.arr a \<and> src\<^sub>C a = a"
dghm_const_is_tm_dghm: fixes \<alpha> :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" assumes "tiny_digraph \<alpha> \<CC>"   and "digraph \<alpha> \<DD>"   and "f : a \<mapsto>\<^bsub>\<DD>\<^esub> a" shows "dghm_const \<CC> \<DD> a f : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<DD>"
map_o_corec: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and ga :: "'c \<Rightarrow> bool"   and gb :: "'c \<Rightarrow> 'e stream"   and gc :: "'c \<Rightarrow> 'c" shows "smap f \<circ> corec_stream g ga gb gc = corec_stream (f \<circ> g) ga (smap f \<circ> gb) gc"
simps_4: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com" shows "lvars (IF b THEN c1 ELSE c2) = lvars c1 \<union> lvars c2"
t_scast: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and D :: "nat"   and C :: "nat" assumes "CT;\<Gamma> \<turnstile> e0 : D"   and "CT \<turnstile> C \<not><: D"   and "CT \<turnstile> D \<not><: C" shows "CT;\<Gamma> \<turnstile> Cast C e0 : C"
rotate8: fixes A8 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A8 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H))))))) \<turnstile> B"
rb_aux_inv_D7: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and b :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "rb_aux_inv (bs, ss, ps)"   and "b \<in> set bs"   and "p \<in> set ps" shows "lt b \<preceq>\<^sub>t sig_of_pair p"
fold_congs_7: fixes r :: "'a t_context_scheme"   and r' :: "'a t_context_scheme"   and v' :: "t list list"   and f :: "t list list \<Rightarrow> t list list"   and f' :: "t list list \<Rightarrow> t list list" assumes "r = r'"   and "label r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "label_update f r = label_update f' r'"
prems_3: shows "LLL_Impl.basis_reduction_main \<alpha> n m upw i state = state'"
freeword_funlift_Abs_freeword: fixes xs :: "('b \<times> bool) list"   and f :: "'b \<Rightarrow> 'c" assumes "proper_signed_list xs" shows "freeword_funlift f (Abs_freeword xs) = sum_list (map (apply_sign f) xs)"
is_ground_atms_union: fixes AA :: "'a set"   and BB :: "'a set" shows "is_ground_atms (AA \<union> BB) = (is_ground_atms AA \<and> is_ground_atms BB)"
intros_1: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a" assumes "graph weights"   and "l1 = l2" shows "graph.path weights l1 l2 []"
simps: fixes e :: "'c"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "'a"   and a2 :: "'c" shows "Rec e r a1 a2 = (a1 = zero \<and> a2 = e \<or> (\<exists>m n. a1 = succ m \<and> a2 = r m n \<and> Rec e r m n))"
category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "category C \<equiv> partial_magma C \<and> category_axioms C"
list_rel_imp_same_length: fixes l :: "'g list"   and l' :: "'h list"   and R :: "('g \<times> 'h) set" assumes "(l, l') \<in> \<langle>R\<rangle>list_rel" shows "length l = length l'"
measure_on_def: fixes f :: "'b \<Rightarrow> nat"   and A :: "'b set" shows "measure_on f A = inv_image_betw (<) f A UNIV"
pconj_lzero: fixes b :: "real" assumes "b \<le> 1" shows "0 .& b = 0"
unProg_inject: fixes x :: "Prog"   and y :: "Prog" shows "(unProg x = unProg y) = (x = y)"
adopt_node_locs_def: fixes parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr" shows "adopt_node_locs parent owner_document document_ptr = (if parent = None then {} else remove_child_locs (the parent) owner_document) \<union> set_disconnected_nodes_locs document_ptr \<union> set_disconnected_nodes_locs owner_document"