hcju/isabelleps huggingface dataset:
norm_nn_equiv: fixes n :: "'a itrm"   and n' :: "'a itrm" assumes "n \<in> CF"   and "n' \<in> CF" shows "norm_nn n n' \<simeq> n \<diamondop> n'"
det_while_algo_intro: fixes DWA :: "'b det_while_algo" assumes "while_algo (det_wa_wa DWA)" shows "det_while_algo DWA"
emeasure_prod_count_space: fixes A :: "('c \<times> 'd) set"   and M :: "'d measure" assumes "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M M)" shows "emeasure (count_space UNIV \<Otimes>\<^sub>M M) A = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator A (x, y) \<partial>M \<partial>count_space UNIV"
feqm: fixes a :: "real"   and b :: "real"   and m :: "real"   and x :: "real" assumes "leftcut a b m \<le> x"   and "x \<le> rightcut a b m"   and "a \<le> m"   and "m \<le> b"   and "{a..b} \<subseteq> {0..1}" shows "f x = f m"
Rep_trace_inject: fixes x :: "'b trace"   and y :: "'b trace" shows "(Rep_trace x = Rep_trace y) = (x = y)"
elims_2: fixes x :: "interval_tree" assumes "tree_max_inv x" obtains "x = Tip"   | l :: "interval_tree" and it :: "nat idx_interval" and m :: "nat" and r :: "interval_tree" where "x = Node l it m r" and "tree_max_inv l \<and> tree_max_inv r \<and> m = max3 it (tmax l) (tmax r)"
zero_closed: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "RModule R smult M" shows "(0::'b) \<in> M"
i_join_singleton: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat" shows "f \<Join>\<^sub> {a} = (\<lambda>n. f a)"
inline1_unfold: fixes callee :: "'b \<Rightarrow> 'f \<Rightarrow> ('e \<times> 'b, 'c, 'd) gpv"   and gpv :: "('a, 'f, 'e) gpv"   and s :: "'b" shows "inline1 callee gpv s = bind_spmf (the_gpv gpv) (case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. bind_spmf (the_gpv (callee s out)) (case_generat (\<lambda>(x, y). inline1 callee (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))))"
fv_subset_Ands: fixes \<phi> :: "formula"   and \<phi>s :: "formula list" assumes "\<phi> \<in> set \<phi>s" shows "fv \<phi> \<subseteq> fv (Ands \<phi>s)"
lfilter_cong: fixes xs :: "'b llist"   and ys :: "'b llist"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> lset ys \<Longrightarrow> P x = Q x" shows "lfilter P xs = lfilter Q ys"
closure_of_subtopology: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set" shows "subtopology X U closure_of S = U \<inter> X closure_of (U \<inter> S)"
outstanding_non_volatile_writes_unshared_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "outstanding_non_volatile_writes_unshared \<S> ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> non_volatile_writes_unshared \<S> sb"
assms_3: shows "prio_emb_step prio t s \<noteq> s"
admits_2x2: shows "\<forall>A\<in>carrier_mat 2 2. admits_diagonal_reduction A"
simps_1: fixes uu :: "typ"   and t :: "Core.term"   and u :: "Core.term" shows "Abs uu t \<bullet> u = subst_bv u t"
SeqThrow: fixes P :: "(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 E :: "char list \<Rightarrow> ty option"   and e\<^sub>0 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "expr" assumes "P,E \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>"
sorted_iff_nth_mono_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> less_eq (xs ! i) (xs ! j))"
toplevel_summands_flatten: fixes X :: "'c rexp set" assumes "X \<noteq> {}"   and "finite X"   and "\<forall>x\<in>X. \<nexists>r s. x = Plus r s" shows "toplevel_summands (PLUS (sorted_list_of_set X)) = X"
homeomorphic_locally: fixes S :: "'b set"   and T :: "'c set"   and P :: "'b set \<Rightarrow> bool"   and Q :: "'c set \<Rightarrow> bool" assumes "S homeomorphic T"   and "\<And>X Y. X homeomorphic Y \<Longrightarrow> P X = Q Y" shows "locally P S = locally Q T"
preserves_src: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<mu> :: "'c" assumes "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"   and "partial_magma.arr C \<mu>" shows "src\<^sub>D (F \<mu>) = weak_arrow_of_homs.map\<^sub>0 src\<^sub>D F (src\<^sub>C \<mu>)"
elims_3: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list" assumes "\<not> distinct_on x xa" obtains cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xb :: "'a" and xs :: "'a list" where "x = cmp" and "xa = xb # xs" and "\<not> (\<not> Bex (set xs) (cmp xb) \<and> distinct_on cmp xs)"
nodes_sim: fixes a :: "'b"   and b :: "'a" assumes "a \<in> b.V"   and "(a, b) \<in> R\<inverse>" shows "b \<in> a.V"
unx_none_simps_2: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unapp (free name) = None"
subspace_sums: fixes S :: "'d set"   and T :: "'d set" assumes "csubspace S"   and "csubspace T" shows "csubspace {x + y |x y. x \<in> S \<and> y \<in> T}"
transpose_apply_matrix: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c matrix" assumes "f (0::'c) = (0::'b)" shows "transpose_matrix (apply_matrix f A) = apply_matrix f (transpose_matrix A)"
rbt_keys_generator_def: shows "rbt_keys_generator \<equiv> Generator (rbt_has_next, rbt_keys_next)"
Nmul_commute: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "isnormNum x"   and "isnormNum y" shows "x *\<^sub>N y = y *\<^sub>N x"
signOnly_def: fixes K :: "nat"   and X :: "msg" shows "signOnly K X = Crypt K (Hash X)"
int: shows "set_integrable lborel {0..} (\<lambda>t. exp (- t\<^sup>2))"
processed_alt: fixes a :: "'b"   and b :: "'b"   and xs :: "'b list"   and ps :: "('b \<times> 'b) list" shows "processed (a, b) xs ps = (a \<in> set xs \<and> b \<in> set xs \<and> (a, b) \<notin>\<^sub>p set ps)"
assms_2: shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A"
hyps_4: shows "vinsert x F = \<D>\<^sub>\<circ> r"
lookup_update_neq: fixes k :: "'b"   and k' :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" assumes "k \<noteq> k'" shows "lookup (update k v m) k' = lookup m k'"
minus_frag_cmul: fixes k :: "int"   and x :: "'b \<Rightarrow>\<^sub>0 int" shows "- frag_cmul k x = frag_cmul (- k) x"
simps_1: fixes \<theta> :: "trm" shows "Differentialo (Aterm \<theta>) = Aterm (Differential \<theta>)"
code: fixes z :: "complex" shows "round_complex z = Gauss_Int (round (Re z)) (round (Im z))"
INF_cong_simp: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d \<Rightarrow> 'e"   and D :: "'d \<Rightarrow> 'e"   and Inf :: "'e set \<Rightarrow> 'e" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
takeWhile_append: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" shows "takeWhile P (xs @ ys) = (if \<forall>x\<in>set xs. P x then xs @ takeWhile P ys else takeWhile P xs)"
SF_always: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<box>SF(A)_v = SF(A)_v"
convex_hull_insert_segments: fixes a :: "'a"   and S :: "'a set" shows "convex hull insert a S = (if S = {} then {a} else \<Union> (closed_segment a ` (convex hull S)))"
mat_times_vec_leqD_1: fixes A :: "'a mat"   and x :: "'a vec"   and b :: "'a vec" assumes "[A*\<^sub>vx]\<le>b" shows "dim_row A = dim_vec b"
antisymp_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "antisymp r = (\<forall>x y. r x y \<longrightarrow> r y x \<longrightarrow> x = y)"
awalk_rev_path': fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v" shows "awalk v (rev_path p) u"
correctCompositionDiffLevelsA81: shows "correctCompositionDiffLevels sA81"
prod_fun_def: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and k :: "'b" shows "prod_fun f1 f2 k = (\<Sum>l. f1 l * (\<Sum>q. f2 q when k = l + q))"
Mem_SUCC_EH_2: fixes u :: "tm"   and t :: "tm"   and A1 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert (u IN t) (insert A1 H) \<turnstile> B"   and "insert (u EQ t) (insert A1 H) \<turnstile> B" shows "insert A1 (insert (u IN SUCC t) H) \<turnstile> B"
pcr_spmf_gpv_def: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "pcr_spmf_gpv A = cr_spmf_gpv OO rel_gpv A (=)"
of_int_numeral_less_iff: fixes n :: "num"   and z :: "int" shows "(numeral n < of_int z) = (numeral n < z)"
assms: shows "nop_bij A (2\<^sub>\<nat>) f"
varDFalse: fixes b :: "BExpr"   and s :: "Var \<Rightarrow> Val"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and t :: "Var \<Rightarrow> Val" assumes "\<not> evalB b s"   and "A s t" shows "(b, A, s, t) \<in> varD"
successor_point: fixes x :: "'a"   and r :: "'a" assumes "is_inj x"   and "point r"   and "x ; r \<noteq> 0" shows "point (x ; r)"
assms_1: shows "\<forall>i. num_syms (f i) = num_syms (f (Suc i))"
sumlist_carrier: fixes xs :: "rat vec list" assumes "set xs \<subseteq> Rn" shows "gs.M.sumlist xs \<in> Rn"
i1: shows "(a, b) \<in> Partitions (v @ z)"
simps_1: fixes a :: "'a"   and b :: "'a" shows "a l-0\<rightarrow> b = b"
prio_def: fixes BinomialTree :: "('d, 'c) BinomialTree" shows "prio BinomialTree = (case BinomialTree of Node x1 x2 x3 x4 \<Rightarrow> x2)"
otherwithI: fixes I :: "'a set"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "'a \<Rightarrow> 'b"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "\<And>j. j \<notin> I \<Longrightarrow> Q (\<sigma> j) (\<sigma>' j)"   and "P \<sigma> a" shows "otherwith Q I P \<sigma> \<sigma>' a"
cases: fixes a1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "a1 \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> a2" obtains lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" where "a1 = lc__" and "a2 = lc'__" and "(case lc__ of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) lc'__"   | lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" where "a1 = lc" and "a2 = lc'" and "lc \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> lc'"
r_take: fixes x :: "nat"   and es :: "nat list"   and n :: "nat" assumes "x = Partial_Recursive.list_encode es" shows "eval r_take [n, x] \<down>= Partial_Recursive.list_encode (take n es)"
star_left_unfold: fixes y :: "'b" shows "Rf y (y\<^sup>\<star>) \<le> y\<^sup>\<star>"
Suc_less: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and i :: "'d"   and j :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less i j"   and "plus i one \<noteq> j" shows "less (plus i one) j"
ldropn_all: fixes xs :: "'c llist"   and m :: "nat" assumes "llength xs \<le> enat m" shows "ldropn m xs = LNil"
compact_uniformly_equicontinuous: fixes S :: "'b set"   and \<F> :: "('b \<Rightarrow> 'c) set"   and e :: "real" assumes "compact S"   and "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>f\<in>\<F>. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"   and "0 < e" obtains d :: "real" where "0 < d" and "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
field_differentiable_derivI: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "f field_differentiable at x" shows "(f has_field_derivative deriv f x) (at x)"
strict_boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < inf b c" obtains "a < b" and "a < c"
exhaust_sel: fixes \<Gamma> :: "('k, 'l) web_scheme"   and vertex :: "'m bi.vertex" assumes "countable_bipartite_web \<Gamma>" obtains "vertex = bi.SOURCE"   | "vertex = bi.SINK"   | "vertex = bi.Inner (bi.inner vertex)"
vec_nth_inverse: fixes x :: "('c, 'd) vec" shows "vec_lambda (($) x) = x"
put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) document_ptr"   and setter :: "('w \<Rightarrow> 'w) \<Rightarrow> ('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> ('n, 'l, 'm, 'u option) RDocument_scheme"   and v :: "'w" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
expectation_gpv2_def: shows "expectation_gpv2 \<equiv> expectation_gpv 1 \<I>'"
simps_1: fixes xs :: "'b list" shows "plus_coeffs xs [] = xs"
per_supset_rel_dom: fixes p1 :: "('a \<times> 'a) set"   and p2 :: "('a \<times> 'a) set" assumes "(p1, p2) \<in> SeprefUF.per_supset_rel" shows "Domain p2 \<subseteq> Domain p1"
neg_int_cases: fixes k :: "int" assumes "k < 0" obtains n :: "nat" where "k = - int n" and "0 < n"
sel_1: fixes t :: "'a tree" shows "root (mirror t) = root t"
p2r_disj_hom: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "p2r P \<union> p2r Q = p2r (P \<squnion> Q)"
iprefix_eq_iprefix_take: fixes xs :: "'d list"   and f :: "nat \<Rightarrow> 'd" shows "(xs \<sqsubseteq> f) = (f \<Down> length xs = xs)"
eq0: shows "comp (fst (f (k, v))) k'' = comp k k'"
uncaught_xcpt_correct: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "jvm_prog"   and h :: "heap"   and xcp :: "nat"   and obj :: "obj"   and f :: "frame"   and frs :: "frame list"   and sh :: "sheap" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "h xcp = \<lfloor>obj\<rfloor>"   and "P,\<Phi> |- (None, h, f # frs, sh) [ok]"   and "curr_method f \<noteq> clinit" shows "P,\<Phi> |- find_handler P xcp h frs sh [ok]"
qtables: shows "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) j \<phi>)) \<phi>s rels"
intersect_list'_language: fixes AA :: "('a, 'b) nba list" shows "NGBA.language (intersect_list' AA) = \<Inter> (NBA.language ` set AA)"
lunit_closed': fixes f :: "'a" assumes "Arr f" shows "Arr (B.\<ll> f)"
Ar: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "\<not> ewp y"   and "x = x \<cdot> y + z" shows "x = z \<cdot> y\<^sup>\<star>"
A2_def: shows "A2 = mat_of_rows (dim_col A) (map (Matrix.row A) [1..<dim_row A])"
assms_1: shows "homotopic_with p X1 Y1 f f'"
axioms_4: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" assumes "inverse_transformations A B F G \<tau> \<tau>'" shows "functor A B G"
cofactor_transpose: fixes A :: "('c, 'd) sq_matrix" shows "cofactor (Square_Matrix.transpose A) = Square_Matrix.transpose (cofactor A)"
rev_map_nonempty: fixes A :: "ant set" assumes "A \<noteq> {}" shows "rev_o ` A \<noteq> {}"
simps_2: fixes x2 :: "'d"   and y2 :: "'d" shows "(deny x2 = deny y2) = (x2 = y2)"
continuous_image_closed_interval: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a \<le> b"   and "continuous_on {a..b} f" shows "\<exists>c d. f ` {a..b} = {c..d} \<and> c \<le> d"
even_plus_one_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a one) = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
a_mult_add: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_ad x ; (y \<union> x) = rel_ad x ; y"
semiring_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.semiring plus times \<equiv> class.ab_semigroup_add plus \<and> class.semigroup_mult times \<and> class.semiring_axioms plus times"
QP1: shows "four_block_mat Q1 (0\<^sub>m n m) (0\<^sub>m m n) Q2 * four_block_mat P1 (0\<^sub>m n m) (0\<^sub>m m n) P2 = 1\<^sub>m (n + m)"
prems_3: shows "a \<in> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (x # sb))"
default_reduce_argument_spec_def: shows "default_reduce_argument_spec = SPEC (\<lambda>x. True)"
dom_simp: fixes f :: "'d \<times> 'c" assumes "DopxC.arr f" shows "DopxC.dom f = (Dop.dom (fst f), C.dom (snd f))"
downset_set_iso: shows "mono (ord.downset_set (\<lambda>x y. y \<le> x))"
DirProd_list_carrier_mem_1: fixes gs :: "'c list"   and Gs :: "('c, 'd) monoid_scheme list" assumes "gs \<in> carrier (DirProd_list Gs)" shows "length gs = length Gs"
wf_ci_def: fixes ci :: "'c instr \<Rightarrow> (nat \<times> nat \<times> 'c instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'd \<Rightarrow> 'c val list \<Rightarrow> 'c val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'c frame list \<Rightarrow> bool" shows "wf_ci ci = (ci_stk_offer ci \<and> ci \<le> check_instr' \<and> (\<forall>ins P h stk loc C M pc pc' frs. ci ins P h stk loc C M pc frs \<longrightarrow> ins_jump_ok ins pc' \<longrightarrow> ci ins P h stk loc C M pc' frs))"
max2_equals2: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "wo_rel r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(max2 r a b = b) = ((a, b) \<in> r)"
binomial_0_Suc: fixes k :: "nat" shows "0 choose Suc k = 0"
Conf_ChiRel: fixes S :: "'d"   and T :: "'d"   and ST :: "('d, 'e, 'f) status" assumes "(S, T) \<in> ChiRel (HA ST)"   and "T \<in> Conf ST" shows "S \<in> Conf ST"
ch_equiv2: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" assumes "long_ch_by_ord2 f X"   and "finite X" shows "long_ch_by_ord f X"
some: fixes y :: "'b" assumes "fmlookup m x = Some y"   and "P x y" shows "thesis"
integrable_dominated_convergence: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and s :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and w :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>i. s i \<in> borel_measurable M"   and "integrable M w"   and "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"   and "\<And>i. AE x in M. norm (s i x) \<le> w x" shows "integrable M f"
near_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "near_ordered_set A less_eq \<equiv> antisymmetric A less_eq \<and> transitive A less_eq"
o_eq_elim: fixes a :: "'d \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'd"   and c :: "'e \<Rightarrow> 'c"   and d :: "'b \<Rightarrow> 'e" assumes "a \<circ> b = c \<circ> d" obtains "\<And>v. a (b v) = c (d v)"
arr_char: fixes f :: "('obj, 'obj) concrete_category.arr" shows "arr f = (Dom f \<in> Obj \<and> f = MkIde (Dom f))"
0: shows "y \<ominus> x = \<ominus> (x \<ominus> y)"
Chain_finite_wf: shows "wf {(f ((f ^^ k) a), (f ^^ k) a) |k. f ((f ^^ k) a) \<noteq> (f ^^ k) a}"
coprime_cong_prime_factors: fixes x :: "'a"   and y :: "'a"   and x' :: "'a"   and y' :: "'a" assumes "x \<noteq> (0::'a)"   and "y \<noteq> (0::'a)"   and "x' \<noteq> (0::'a)"   and "y' \<noteq> (0::'a)"   and "prime_factors x = prime_factors x'"   and "prime_factors y = prime_factors y'" shows "coprime x y = coprime x' y'"
assms_9: shows "rk {A', B', P, Q} = 4"
augment_indep_in: fixes \<E> :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "\<E> \<subseteq> carrier"   and "indep_in \<E> X"   and "indep_in \<E> Y"   and "card Y < card X" shows "\<exists>x\<in>X - Y. indep_in \<E> (insert x Y)"
assoc'_eq_inv_assoc: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<a>\<^sup>-\<^sup>1[f, g, h] = local.inv \<a>[f, g, h]"
is_2sati_def: fixes cnfi :: "(lit \<times> lit) list" shows "is_2sati cnfi \<equiv> \<forall>(l1, l2)\<in>set cnfi. l1 \<noteq> l2"
hyps: fixes s' :: "nat"   and r :: "nat \<Rightarrow>f value option" assumes "recognises_execution e s' r t" shows "\<exists>c1 s. obtains s c1 e s' r t"
synthp_synth_eq: fixes H :: "msg set" shows "synthp (\<lambda>x. x \<in> H) = (\<lambda>x. x \<in> synth H)"
ide_unity: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b" assumes "monoidal_category C T \<alpha> \<iota>" shows "partial_magma.ide C (monoidal_category.unity C \<iota>)"
measurable_restrict_space2_iff: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and \<Omega> :: "'e set" shows "(f \<in> M \<rightarrow>\<^sub>M restrict_space N \<Omega>) = (f \<in> M \<rightarrow>\<^sub>M N \<and> f \<in> space M \<rightarrow> \<Omega>)"
unit_l_inv_ex: fixes x :: "'a" assumes "unit x" shows "\<exists>y. y ** x = one"
Iface_single_not_1: fixes \<beta> :: "'a common_primitive \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and X :: "iface"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (MatchNot (Match (IIface X))) a p = (\<not> match_iface X (p_iiface p))"