hcju/isabelleps huggingface dataset:
rwof_step': fixes step :: "'c \<Rightarrow> 'c nres"   and s :: "'c"   and m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool" assumes "nofail (step s)"   and "rwof m0 cond step s"   and "cond s" shows "step s \<le> SPEC (rwof m0 cond step)"
term_nchotomy: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "good X" shows "(\<exists>xs x. X = Var xs x) \<or> (\<exists>delta inp binp. goodInp inp \<and> goodBinp binp \<and> X = Op delta inp binp)"
presWlsAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'c, 'd, 'i) Recursion.model_scheme" shows "presWlsAll h hA MOD \<equiv> presWls h MOD \<and> presWlsAbs hA MOD"
foldl_Gauss_Jordan_column_k_det_P_1: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" shows "fst (Gauss_Jordan_upt_k_det_P A k) = det (fst (Gauss_Jordan_upt_k_PA A k))"
line_extension_mem_iff: fixes u :: "'a list"   and K :: "'a list set"   and a :: "'a list"   and E :: "'a list set" shows "(u \<in> UP.line_extension K a E) = (\<exists>k\<in>K. \<exists>v\<in>E. u = k \<otimes>\<^bsub>K [X]\<^esub> a \<oplus>\<^bsub>K [X]\<^esub> v)"
assign_const_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and z :: "nat"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = (x \<leftarrow> aexp.Const z) ;; c\<^sub>A_tail"   and "c\<^sub>C = (Eg2_var\<^sub>C_of_Eg1 x \<leftarrow> aexp\<^sub>C.Const z) ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoWrite"   and "x \<notin> \<C>"   and "\<forall>mds\<^sub>A' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C'' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C'' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
delete_min_def: fixes xs :: "('c, 'd) bintree option list" shows "delete_min xs = (case find_min xs of None \<Rightarrow> [] | Some (Node a v ts) \<Rightarrow> Binomial_Queue.normalize (meld (map Some (rev ts)) (map (match a) xs)))"
simps_3: fixes a :: "'h"   and f :: "'i \<Rightarrow> 'g +\<^sub>\<bottom> 'h" shows "(try sum_bot.Right a catch f) = sum_bot.Right a"
sum_distr: fixes X :: "'c set"   and y :: "'c" assumes "finite X" shows "\<Sum> X \<cdot> y = \<Sum> {x \<cdot> y |x. x \<in> X}"
in_set_product_lists_length: fixes xs :: "'c list"   and xss :: "'c list list" assumes "xs \<in> set (product_lists xss)" shows "length xs = length xss"
2: shows "j < hd (dims (tensor_from_lookup [n] (\<lambda>x. if x = [i] then 1::'b else (0::'b))))"
empty_imp_negligible: fixes S :: "'b set" assumes "S = {}" shows "negligible S"
all_real_roots_mobius: fixes p :: "real poly"   and a :: "real"   and b :: "real" assumes "all_roots_real p"   and "a < b" shows "all_roots_real (fcompose p [:a, b:] [:1, 1:])"
ring_iso_imp_img_ring: fixes h :: "'a \<Rightarrow> 'd"   and S :: "('d, 'e) ring_scheme" assumes "h \<in> ring_iso R S" shows "ring (S\<lparr>zero := h \<zero>\<rparr>)"
eexp_ineq_1: fixes x :: "ereal" assumes "x \<le> 0" shows "eexp x \<le> 1"
intro: fixes 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\<^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"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) 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" assumes "l_new_document_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M 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\<^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"   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_new_document_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 get_child_nodes get_child_nodes_locs 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\<^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"
lookup_map_default': fixes x :: "'d"   and d :: "'c"   and f :: "'c \<Rightarrow> 'c"   and m :: "('d, 'c) mapping"   and y :: "'d" shows "Mapping.lookup (Mapping.map_default x d f m) y = (if x = y then Some (f (Mapping.lookup_default d m x)) else Mapping.lookup m y)"
image_insert: fixes f :: "'i \<Rightarrow> 'h"   and a :: "'i"   and B :: "'i set" shows "f ` insert a B = insert (f a) (f ` B)"
interp_ins_distinct: fixes op_list :: "('b \<times> 'b option) list" assumes "insert_ops op_list" shows "distinct (interp_ins op_list)"
presheaf_of_rings_axioms: shows "presheaf_of_rings S is_open imS.im_sheaf imS.im_sheaf_morphisms b imS.add_im_sheaf imS.mult_im_sheaf imS.zero_im_sheaf imS.one_im_sheaf"
prime_ge_0_int: fixes p :: "int" assumes "prime p" shows "0 \<le> p"
lset_lnth: fixes x :: "'c"   and xs :: "'c llist" assumes "x \<in> lset xs" shows "\<exists>n. xs $ n = x"
rep_eq: fixes x :: "'m"   and xa :: "'m fset" shows "(x |\<in>| xa) = (x \<in> fset xa)"
candidate_values_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and z :: "'b trm"   and CC :: "'b eclause"   and C' :: "'b literal set"   and C :: "'b literal set"   and s :: "'b trm"   and L :: "'b literal"   and L' :: "'b literal"   and \<sigma> :: "('b \<times> 'b trm) list"   and t' :: "'b trm"   and s' :: "'b trm"   and t :: "'b trm"   and S :: "'b eclause set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.candidate_values trm_ord sel z CC C' C s L L' \<sigma> t' s' t S = (CC \<in> S \<and> t \<notin> subst_set (trms_ecl CC) \<sigma> \<and> basic_superposition.trms_irreducible trm_ord sel CC \<sigma> S t \<and> C' = cl_ecl CC \<and> (s, t) \<in> trm_ord \<and> ((s, t) \<in> trm_ord \<longrightarrow> z = basic_superposition.trm_rep trm_ord sel s S) \<and> basic_superposition.orient_lit_inst trm_ord L' t' s' pos \<sigma> \<and> sel C' = {} \<and> L' \<in> C' \<and> basic_superposition.maximal_literal trm_ord L C \<and> L = subst_lit L' \<sigma> \<and> C = subst_cl C' \<sigma> \<and> ground_clause C \<and> t = t' \<lhd> \<sigma> \<and> s = s' \<lhd> \<sigma> \<and> finite C' \<and> basic_superposition.smaller_lits_are_false trm_ord sel t C S \<and> basic_superposition.maximal_literal_is_unique trm_ord sel t s C' L' S \<sigma>)"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case True of True \<Rightarrow> f1 | False \<Rightarrow> f2) = f1"
sup_unit_def: fixes uu :: "unit"   and uv :: "unit" shows "uu \<squnion> uv = ()"
mvt_very_simple_closed_segmentE: fixes f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real" assumes " D f = f' on {a--b}" obtains y :: "real" where "y \<in> {a--b}" and "f b - f a = (b - a) * f' y"
PsiInv_beta2: shows "\<turnstile> beta2 \<and> $PsiInv \<longrightarrow> PsiInv$"
is_lfair_def: fixes ls :: "'c \<Rightarrow> 'b"   and ss :: "'c \<Rightarrow> 'a" shows "is_lfair ls ss \<equiv> \<forall>l i. \<exists>j\<ge>i. \<not> can_step l (ss j) \<or> ls j = l"
subdegree_leI: fixes f :: "'a fps"   and n :: "nat" assumes "fps_nth f n \<noteq> (0::'a)" shows "subdegree f \<le> n"
x: shows "x = lhd (ldropWhile (Not \<circ> P) ys)"
p_inf_sup_below: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (x \<squnion> y) \<le> y"
top_empty_eq: shows "top = (\<lambda>x. x \<in> UNIV)"
cop_F_range_inv: fixes ds :: "'b set" shows "invariant ds cop_F_range_inv"
in_carrier1: shows "(\<^bold>g [^] \<alpha>1) [^] r \<in> carrier \<G>"
assms_1: shows "i < row_length M1 * row_length M2"
all_neq_Inr_ex_eq_Inl: fixes x :: "'d + 'c" assumes "\<forall>r. x \<noteq> Inr r" shows "\<exists>l. x = Inl l"
weight_spmf_nonneg: fixes p :: "'a spmf" shows "0 \<le> weight_spmf p"
bound_at_worst_lemma: fixes x :: "real"   and a :: "('a, 'b) IEEE.float" assumes "\<bar>x\<bar> < threshold TYPE(('a, 'b) IEEE.float)"   and "is_finite a" shows "\<bar>valof (IEEE.round To_nearest x) - x\<bar> \<le> \<bar>valof a - x\<bar>"
countable_Union: fixes X :: "'c set set"   and M :: "'c measure" assumes "countable X"   and "X \<subseteq> sets M" shows "\<Union> X \<in> sets M"
evContainsExval: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and body :: "formula"   and fs :: "formula list" assumes "EV (contains f (i, FAll Neg body))"   and "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))" shows "\<forall>v. EV (contains f (0, instanceF v body))"
subset_iprev_in_imp_in: fixes n :: "nat"   and A :: "nat set"   and B :: "nat set" assumes "iprev n A \<in> B"   and "A \<subseteq> B" shows "n \<in> B"
f9: shows "low_equal (snd (fst (get_curr_win () s1))) (snd (fst (get_curr_win () s2)))"
shortest_path_inf: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c"   and f :: "'d \<Rightarrow> real" assumes "wf_digraph G"   and "\<not> u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "wf_digraph.\<mu> G f u v = \<infinity>"
discI_2: fixes llist :: "'a llist"   and x21 :: "'a"   and x22 :: "'a llist" assumes "llist = LCons x21 x22" shows "\<not> lnull llist"
systemOUT_def: fixes x :: "chanID"   and i :: "AbstrLevelsID" shows "systemOUT x i \<equiv> (\<forall>C1\<in>AbstrLevel i. x \<notin> IN C1) \<and> (\<exists>C2\<in>AbstrLevel i. x \<in> OUT C2)"
spans_image: fixes V :: "'b set"   and B :: "'b set" assumes "V \<subseteq> vs1.span B" shows "construct B f ` V \<subseteq> vs2.span (construct B f ` B)"
drop_bin2bl_aux: fixes m :: "nat"   and n :: "nat"   and bin :: "int"   and bs :: "bool list" shows "drop m (bin_to_bl_aux n bin bs) = bin_to_bl_aux (n - m) bin (drop (m - n) bs)"
induct: fixes P :: "'a \<Rightarrow> 'a tree \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x l y r. \<lbrakk>\<And>xa. \<lbrakk>xa \<in> set_pmf (bernoulli_pmf (1 / real (size l + size r + 2))); \<not> xa; x < y\<rbrakk> \<Longrightarrow> P x l; \<And>xa. \<lbrakk>xa \<in> set_pmf (bernoulli_pmf (1 / real (size l + size r + 2))); \<not> xa; \<not> x < y; y < x\<rbrakk> \<Longrightarrow> P x r\<rbrakk> \<Longrightarrow> P x \<langle>l, y, r\<rangle>" shows "P a0 a1"
assms_1: shows "A \<TTurnstile>\<^sub>A onll \<Gamma> P"
binary_functor_Prod_1: shows "binary_functor (\<cdot>) (\<cdot>) (\<cdot>) local.Prod"
set_mult_closed: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "H <#> K \<subseteq> carrier G"
streamsE: fixes s :: "'b stream"   and A :: "'b set" assumes "s \<in> streams A" obtains "shd s \<in> A" and "stl s \<in> streams A"
column_add_iarray_def: fixes A :: "'b iarray iarray"   and n :: "nat"   and m :: "nat"   and q :: "'b" shows "column_add_iarray A n m q = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>i j. if j = n then A !! i !! n + A !! i !! m * q else A !! i !! j)"
comp_wb_lens: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'c \<Longrightarrow> 'd" assumes "wb_lens x"   and "wb_lens y" shows "wb_lens (x ;\<^sub>L y)"
wtA_hOfFax: fixes f :: "'fsym" assumes "wtFsym f"   and "protCl (resOf f)" shows "list_all GE.wtA (hOfFax f)"
assms: fixes i :: "nat" shows "P (\<sigma> |\<^sub>s i)"
distinct_list_split: fixes xs :: "'b list"   and xa :: "'b list"   and x :: "'b"   and ya :: "'b list"   and xb :: "'b list"   and yb :: "'b list" assumes "distinct xs"   and "xs = xa @ x # ya"   and "xs = xb @ x # yb" shows "xa = xb \<and> ya = yb"
asig_outputs_def: shows "outputs \<equiv> fst \<circ> snd"
ptrm_alpha_equiv_transitive: fixes X :: "'b ptrm"   and Y :: "'b ptrm"   and Z :: "'b ptrm" assumes "X \<approx> Y"   and "Y \<approx> Z" shows "X \<approx> Z"
monotonic: fixes P :: "pi"   and A :: "(pi \<times> pi) set"   and P' :: "pi"   and B :: "(pi \<times> pi) set" assumes "P \<leadsto><A> P'"   and "A \<subseteq> B" shows "P \<leadsto><B> P'"
collapse_1: fixes tm :: "('c, 'd) tm" assumes "is_Hd tm" shows "Hd (head tm) = tm"
cblinfun_assoc_left_2: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and S :: "'c ccsubspace" shows "A *\<^sub>S B *\<^sub>S S = (A o\<^sub>C\<^sub>L B) *\<^sub>S S"
drop_tl: fixes n :: "nat"   and xs :: "'b list" shows "drop n (tl xs) = tl (drop n xs)"
prems_3: shows "poly (sturm p ! (i + 1)) x = 0"
plus_sat_def: fixes x :: "'a sat"   and y :: "'a sat" shows "x + y = Abs_sat' (nat_of x + nat_of y)"
cases: fixes x :: "nat \<times> nat" obtains k :: "nat" and m :: "nat" where "x = (k, m)"
simps_9: fixes t :: "MIR.num" shows "not (NEq t) = Eq t"
functionEquivalenceOnSets: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x\<in>X. f x = g x" shows "f ` X = g ` X"
not_nonzero_memE: fixes a :: "'a" assumes "a \<notin> nonzero R"   and "a \<in> carrier R" shows "a = \<zero>"
permute: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "p permutes S" shows "sum g S = sum (g \<circ> p) S"
gcsort_stab_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and t :: "nat \<times> nat list \<times> 'c list"   and p :: "nat"   and t' :: "nat \<times> nat list \<times> 'c list"   and f :: "'d \<Rightarrow> 'c list" assumes "index_less index key"   and "index_same index key"   and "add_inv n t"   and "n \<le> p"   and "t' \<in> gcsort_set index key p t"   and "stab_inv f key t" shows "stab_inv f key t'"
rtrancl_while_finite_Some: fixes f :: "'c \<Rightarrow> 'c list"   and x :: "'c"   and p :: "'c \<Rightarrow> bool" assumes "finite ({(x, y). y \<in> set (f x)}\<^sup>* `` {x})" shows "\<exists>y. rtrancl_while p f x = Some y"
cSUP_upper2: 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 f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'d"   and u :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq (f ` A)"   and "x \<in> A"   and "less_eq u (f x)" shows "less_eq u (Sup (f ` A))"
suntil_mono: fixes P :: "'b stream \<Rightarrow> bool"   and Q1 :: "'b stream \<Rightarrow> bool"   and Q2 :: "'b stream \<Rightarrow> bool"   and R1 :: "'b stream \<Rightarrow> bool"   and R2 :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" assumes "\<And>\<omega>. \<lbrakk>P \<omega>; Q1 \<omega>\<rbrakk> \<Longrightarrow> Q2 \<omega>"   and "\<And>\<omega>. \<lbrakk>P \<omega>; R1 \<omega>\<rbrakk> \<Longrightarrow> R2 \<omega>"   and "(Q1 suntil R1) \<omega>"   and "alw P \<omega>" shows "(Q2 suntil R2) \<omega>"
MakeFaceFinal_edges_disj: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "edges_disj (makeFaceFinal f g)"
inverse_map_identity: fixes S :: "'d set" shows "inverse_map (identity S) S S = identity S"
ereal_mult_divide: fixes b :: "ereal"   and a :: "ereal" assumes "0 < b"   and "b < \<infinity>" shows "b * (a / b) = a"
abs_eq_0_iff: fixes a :: "'a" shows "(\<bar>a\<bar> = (0::'a)) = (a = (0::'a))"
finite_lattice_complete_Inf_insert: fixes x :: "'a"   and A :: "'a set" shows "Inf (insert x A) = inf x (Inf A)"
example: fixes n :: "nat" shows "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. enat (2 + 3 * n) + \<up> (s ''x'' = int n)} ''y'' ::= N 0;; wsum {\<lambda>s. 0}"
finite: shows "finite {(p, i). prime p \<and> 1 \<le> i \<and> p ^ i \<le> n}"
essential_vertex: fixes G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "essential G B S x"   and "x \<notin> B" shows "vertex G x"
simps_9: fixes v :: "nat" shows "unpower v FalseF = FalseF"
nn_integral_T_lfp: fixes g :: "'s \<Rightarrow> ennreal \<Rightarrow> ennreal"   and s :: "'s" assumes "(\<lambda>(x, y). g x y) \<in> borel_measurable (count_space UNIV \<Otimes>\<^sub>M borel)"   and "\<And>s. sup_continuous (g s)"   and "\<And>f s. f \<in> borel_measurable (stream_space (count_space UNIV)) \<Longrightarrow> \<integral>\<^sup>+ \<omega>. g s (f \<omega>) \<partial>T s = g s (integral\<^sup>N (T s) f)" shows "integral\<^sup>N (T s) (lfp (\<lambda>f \<omega>. g (shd \<omega>) (f (stl \<omega>)))) = lfp (\<lambda>f s. \<integral>\<^sup>+ t. g t (f t) \<partial>measure_pmf (K s)) s"
intro: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'a" assumes "GroupHom G T"   and "GroupEnd_axioms G T" shows "GroupEnd G T"
internal_inv_Branching_Right: fixes b :: "bool list"   and l :: "tree"   and r :: "tree" shows "internal (Right # b) (Branching l r) = internal b r"
prop_equiv_finite: fixes P :: "'b ltln set" assumes "finite P" shows "finite {abs_ltln\<^sub>P \<psi> |\<psi>. prop_atoms \<psi> \<subseteq> P}"
prv_dsj_com_imp3: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<phi>3 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla" shows "prv (imp (dsj \<phi>1 (dsj \<phi>2 \<phi>3)) (dsj \<phi>2 (dsj \<phi>1 \<phi>3)))"
slpa_preds: fixes cs :: "'edge list"   and as :: "'edge list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cfsx :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cfs' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "same_level_path_aux cs as"   and "s = cfsx @ cf # cfs"   and "s' = cfsx @ cf # cfs'"   and "length cfs = length cfs'"   and "\<forall>a\<in>set as. valid_edge a"   and "length cs = length cfsx"   and "preds (kinds as) s" shows "preds (kinds as) s'"
PRECOND_tagI: fixes P :: "bool" assumes "P" shows "PRECOND_tag P"
csmset: shows "fmset G cs = fmset G as \<inter># fmset G bs"
rtranclp_False: shows "(\<lambda>a b. False)\<^sup>*\<^sup>* = (=)"
bind_spmf_def: fixes x :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "bind_spmf x f = bind_pmf x (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
left_totalI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. \<exists>y. R x y" shows "left_total R"
weakly_preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.isomorphic (F (C'.trg \<mu>)) (D'.trg (F \<mu>))"
assms_2: shows "h \<cdot> g \<noteq> null"
Cauchy_Binet: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n" shows "Determinant.det (A * B) = (\<Sum>I | I \<subseteq> {0..<m} \<and> card I = n. Determinant.det (submatrix A UNIV I) * Determinant.det (submatrix B I UNIV))"
le_max3I1: fixes n2 :: "nat"   and n1 :: "nat"   and n3 :: "nat" shows "n2 \<le> max n1 (max n2 n3)"
simulation: shows "strong_reduction_simulation Rel (Calculus CWB) \<and> rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a CWB)"
image_comp: fixes f :: "'c::type \<Rightarrow> 'b::type"   and g :: "'d::type \<Rightarrow> 'c::type"   and r :: "'d::type set" shows "(f::'c::type \<Rightarrow> 'b::type) ` (g::'d::type \<Rightarrow> 'c::type) ` (r::'d::type set) = (f \<circ> g) ` r"
wf_fut: fixes idx :: "'n"   and \<psi> :: "('a, 'k) aformula"   and b :: "bool" assumes "local.wf idx \<psi>" shows "local.wf idx (fut b idx \<psi>)"
sets_Least: fixes M :: "'c measure"   and P :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "nat set" assumes "\<And>i. {x \<in> space M. P i x} \<in> sets M" shows "(\<lambda>x. LEAST j. P j x) -` A \<inter> space M \<in> sets M"
pointermap_insert_p_validI: fixes p :: "nat"   and m :: "('a, 'b) pointermap_scheme"   and a :: "'a" assumes "pointermap_p_valid p m" shows "pointermap_p_valid p (pointermap_insert a m)"
simps: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and ax :: "Core.term list" shows "exe_wf_theory' (ExeTheory (ExeSignature cto tao sa) ax) = (exe_sig_conds (ExeSignature cto tao sa) \<and> (\<forall>p\<in>set ax. exeterm_ok' (ExeSignature cto tao sa) p \<and> typ_of p = Some propT) \<and> exeis_std_sig (ExeSignature cto tao sa) \<and> exe_wf_sig (ExeSignature cto tao sa) \<and> eq_axs \<subseteq> set ax)"
simps: fixes l :: "nat"   and r :: "linear_poly" shows "rhs (l, r) = r"
counit_in_vhom: shows "\<guillemotleft>D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) : F f \<star>\<^sub>D F g \<Rightarrow>\<^sub>D src\<^sub>D (F g)\<guillemotright>"
comp_null_1: fixes f :: "'a" shows "B.null \<cdot>\<^sub>B f = B.null"
ACI_norm_deriv: fixes as :: "'a"   and r :: "'a rexp" shows "\<guillemotleft>deriv as \<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>deriv as r\<guillemotright>"
cod_simp: fixes f :: "'a1 \<times> 'a2" assumes "arr f" shows "cod f = (C1.cod (fst f), C2.cod (snd f))"