hcju/isabelleps huggingface dataset:
encrypt_oracle_def: fixes key :: "bool list"   and \<sigma> :: "unit"   and plain :: "bool list" shows "local.ind_cpa.encrypt_oracle key \<sigma> plain = encrypt key plain \<bind> (\<lambda>cipher. return_spmf (cipher, ()))"
independent_Basis: shows "\<not> module_on.dependent (*\<^sub>R) Basis"
le_INF_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and u :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq u (Inf (f ` A)) = (\<forall>i\<in>A. less_eq u (f i))"
bmod_comm: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" shows "\<otimes> \<rho> f g = \<otimes> \<rho> g f"
sel: fixes x2 :: "'k" shows "the (Some x2) = x2"
natural_isomorphism_axioms: shows "natural_isomorphism (\<bullet>) (\<bullet>) R local.map \<rho>.map"
min_less_iff_conj: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z < min x y) = (z < x \<and> z < y)"
intros_1: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> H" shows "H \<turnstile> A"
inf_bool_def: fixes a :: "bool"   and b :: "bool" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
lnth_LCons': fixes x :: "'b"   and xs :: "'b llist"   and n :: "nat" shows "lnth (LCons x xs) n = (if n = 0 then x else lnth xs (n - 1))"
simps_3: fixes xs :: "char list list"   and n :: "int"   and ys :: "char list list"   and m :: "int"   and p :: "(char list list \<times> int) list" shows "merge_coeffs0 ((xs, n) # (ys, m) # p) = (if xs = ys then if n + m \<noteq> 0 then merge_coeffs0 ((xs, n + m) # p) else merge_coeffs0 p else if n = 0 then merge_coeffs0 ((ys, m) # p) else (xs, n) # merge_coeffs0 ((ys, m) # p))"
assms_3: fixes s :: "char list"   and t :: "char list" shows "r s @ t = r (s @ t)"
r_inv: fixes x :: "'c" assumes "x \<in> carrier H" shows "x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> x = \<one>\<^bsub>H\<^esub>"
comp_arr_ide: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "H.VxV.ide a"   and "H.VxV.seq f a" shows "H.VxV.comp f a = f"
program_less_le: fixes F :: "'b program"   and H :: "'b program" shows "(F < H) = (F \<le> H \<and> F \<noteq> H)"
subset_fst_snd: fixes A :: "('d \<times> 'e) set" shows "A \<subseteq> fst ` A \<times> snd ` A"
heap_is_wellformed_children_distinct: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "distinct children"
prems_1: shows "list_comb f (x # xs) = list_comb g ys"
upper_density_eventually_measure: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and b :: "ennreal"   and a :: "real" assumes "\<And>n. {x \<in> space M. P x n} \<in> sets M"   and "b < emeasure M {x \<in> space M. upper_asymptotic_density {n. P x n} < a}" shows "\<exists>N. b < emeasure M {x \<in> space M. \<forall>n\<ge>N. real (card ({n. P x n} \<inter> {..<n})) < a * real n}"
divide_ereal_def: fixes x :: "ereal"   and y :: "ereal" shows "x / y = x * inverse y"
I: fixes y :: "'a" assumes "y \<in> S" shows "inv_into S f (f y) = y"
combined_size_bound_rat_log: fixes x :: "rat" assumes "x \<in> {gs.\<mu>' i j |i j. j \<le> i \<and> i < length fs} \<union> {gs.\<sigma> l i j |i j l. i < length fs \<and> j \<le> i \<and> l \<le> j}"   and "length fs \<noteq> 0"   and "x \<noteq> 0" shows "log 2 \<bar>real_of_rat x\<bar> \<le> log 2 (real (length fs)) + real (3 + 3 * length fs) * log 2 (real_of_rat gs.N)"
sl_def: fixes n :: "nat" shows "Listn.sl n = (\<lambda>(A, r, f). (list n A, Listn.le r, map2 f))"
two_comparisons_into_case_order_8: fixes x :: "'a"   and y :: "'a"   and Q :: "'c"   and P :: "'c"   and R :: "'c" shows "(if lt x y then Q else if y = x then P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
assms_3: shows "A1 \<noteq> 0 \<or> B1 \<noteq> 0"
b: shows "B \<in> \<gamma>\<^sub>T\<^sub>R K (\<phi> .\<and>. \<psi>)"
iT_Div_empty: fixes k :: "nat" shows "{} \<oslash> k = {}"
higher_differentiable_on_open_Union: fixes S :: "'b set set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "\<And>s. s \<in> S \<Longrightarrow> higher_differentiable_on s f n"   and "\<And>s. s \<in> S \<Longrightarrow> open s" shows "higher_differentiable_on (\<Union> S) f n"
convex_comb_dets: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real"   and s :: "real \<times> real" assumes "0 < det3 p q r" shows "s = (det3 s q r / det3 p q r) *\<^sub>R p + (det3 p s r / det3 p q r) *\<^sub>R q + (det3 p q s / det3 p q r) *\<^sub>R r"
set_tag_name_get_tag_name_different_pointers: fixes ptr :: "(_) element_ptr"   and ptr' :: "(_) element_ptr"   and w :: "((_) heap, exception, unit) prog"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr \<noteq> ptr'"   and "w \<in> set_tag_name_locs ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'"   and "r \<in> get_tag_name_locs ptr'" shows "r h h'"
div_power: fixes b :: "'a"   and a :: "'a"   and n :: "nat" assumes "b dvd a" shows "(a div b) ^ n = a ^ n div b ^ n"
conv_tail1_4: fixes s :: "'a"   and d :: "bool"   and \<alpha> :: "'b"   and t :: "'a"   and xs :: "(bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, (d, \<alpha>, t) # xs) \<in> conv ars" shows "lst_conv (s, (d, \<alpha>, t) # xs) = lst_conv (t, xs)"
zcount_uminus: fixes A :: "'a zmultiset"   and x :: "'a" shows "zcount (- A) x = - zcount A x"
set_nn_integral_cong: fixes M :: "'a measure"   and M' :: "'a measure"   and A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "M = M'"   and "A = B"   and "\<And>x. x \<in> space M \<inter> A \<Longrightarrow> f x = g x" shows "set_nn_integral M A f = set_nn_integral M' B g"
exp_conj_def: fixes a :: "'b \<Rightarrow> real"   and b :: "'b \<Rightarrow> real" shows "a && b \<equiv> \<lambda>s. a s .& b s"
prems_1: shows "steps0 (s, l, r) B (Suc stp) = (s', l', r')"
rel_Grp: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "list_all2 (BNF_Def.Grp A f) = BNF_Def.Grp {x. set x \<subseteq> A} (map f)"
simps_12: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and I :: "\<I>"   and \<phi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Prev I \<phi>) j = (if j = 0 then 0 else min (Suc (Monitor.progress \<sigma> P \<phi> j)) j)"
diff_add_assoc2: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b" shows "b + c - a = b - a + c"
vector_comp: fixes x :: "'a"   and y :: "'a" assumes "is_vector x"   and "is_vector y" shows "is_vector (x ; y)"
left_idem: fixes a :: "'d"   and b :: "'d" shows "sup a (sup a b) = sup a b"
indeps_8: shows "mrg_right \<bowtie> mrg_left"
1: shows "|\<Union> (range terms)| <o |UNIV|"
intros: fixes s :: "'b + 'c"   and x :: "'c" assumes "s = Inr x" shows "setrp s x"
homotopic_paths_linear: fixes g :: "real \<Rightarrow> 'a"   and h :: "real \<Rightarrow> 'a"   and S :: "'a set" assumes "path g"   and "path h"   and "pathstart h = pathstart g"   and "pathfinish h = pathfinish g"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> closed_segment (g t) (h t) \<subseteq> S" shows "homotopic_paths S g h"
9: shows "set s\<^sub>2 \<subseteq> invisible \<or> set s\<^sub>3 \<subseteq> invisible"
pos_mod_bound: fixes b :: "'d"   and a :: "'d" assumes "(0::'d) < b" shows "a mod b < b"
weight_simp: fixes f :: "'f"   and ts :: "('f, 'b) Term.term list" shows "weight (Fun f ts) = w (f, length ts) + sum_list (map weight (scf_list (scf (f, length ts)) ts))"
set_disconnected_nodes_get_child_nodes: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>Shadow_DOM.get_child_nodes_locs ptr'. r h h')"
annotateI: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "('b set \<times> 'a \<times> 'b set \<times> 'b set) set"   and F :: "'c set"   and P :: "'b set"   and anno :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set"   and c :: "('b, 'a, 'c) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P anno Q,A"   and "c = anno" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
hyps_6: fixes b :: "(name \<times> ty) list" assumes " (x, T\<^sub>1) # \<Gamma> \<subseteq> b "   and "valid b" shows "b \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2"
IH_1: assumes "send\<langle>t\<rangle> \<in> set S" shows "fv t \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
gauge_integral_Fubini_universe_x_1: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "integrable lborel f"   and "(\<lambda>x. integral UNIV (\<lambda>y. f (x, y))) \<in> borel_measurable lborel" shows "integral UNIV f = integral UNIV (\<lambda>x. integral UNIV (\<lambda>y. f (x, y)))"
dghm_tdghm_comp_NTMap_app: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V"   and x :: "V"   and \<HH> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "x \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN>)\<lparr>NTMap\<rparr>\<lparr>x\<rparr> = \<HH>\<lparr>ArrMap\<rparr>\<lparr>\<NN>\<lparr>NTMap\<rparr>\<lparr>x\<rparr>\<rparr>"
uint32_of_nat_def: shows "uint32_of_nat = uint32_of_int \<circ> int"
assms: fixes e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
well_def_subsets2: shows "all_list_constr_R subsets2 (length qs2)"
simps: fixes a1 :: "(nat \<times> bval) list"   and a2 :: "(nat \<times> exp) list" shows "bs_env a1 a2 = (a1 = [] \<and> a2 = [] \<or> (\<exists>w v \<rho> \<rho>' x. a1 = (x, w) # \<rho> \<and> a2 = (x, v) # \<rho>' \<and> bs_val w v \<and> bs_env \<rho> \<rho>'))"
is_orthogonal_sym: fixes \<psi> :: "'b"   and \<phi> :: "'b" shows "is_orthogonal \<psi> \<phi> = is_orthogonal \<phi> \<psi>"
oalist_inv_raw_update_by_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and kv :: "'a \<times> 'b" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.oalist_inv_raw comp (comparator.update_by_pair comp kv xs)"
SN_rel_defs_1: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" shows "SN_rel_on R S \<equiv> SN_on (S\<^sup>* O R O S\<^sup>*)"
prod_algebra_cong: fixes I :: "'c set"   and J :: "'c set"   and M :: "'c \<Rightarrow> 'd measure"   and N :: "'c \<Rightarrow> 'd measure" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i)" shows "prod_algebra I M = prod_algebra J N"
measurable_Sup2: fixes M :: "'c measure set"   and f :: "'d \<Rightarrow> 'c"   and N :: "'d measure" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M m"   and "\<And>m n. \<lbrakk>m \<in> M; n \<in> M\<rbrakk> \<Longrightarrow> space m = space n" shows "f \<in> N \<rightarrow>\<^sub>M Sup M"
P_def: shows "P = (\<lambda>x. jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0)"
next_fresh: fixes r :: "'a ref"   and h' :: "heap"   and x :: "'a"   and h :: "heap" assumes "(r, h') = Ref.alloc x h" shows "\<not> Ref.present h r"
map_index: fixes f :: "nat \<Rightarrow> 'd \<Rightarrow> 'c"   and xs :: "'d list" shows "map_index f xs = map2 f [0..<length xs] xs"
k: shows "fst (dioph_dfs n ks l) ! int_encode m = Some k"
map2_val_compat'_imp_map2_val_compat: fixes g :: "('a \<times> 'e) list \<times> 'o \<Rightarrow> ('a \<times> 'f) list \<times> 'o"   and ko :: "'o" assumes "map2_val_compat' g" shows "map2_val_compat ko (\<lambda>zs. fst (g (zs, ko)))"
solvable_imp_trivial_derived_seq: fixes H :: "'c set" assumes "solvable_seq H H" shows "\<exists>n. (derived H ^^ n) H = {\<one>\<^bsub>H\<^esub>}"
aval_ir_take: fixes A :: "nat"   and i :: "value list"   and a :: "vname aexp"   and r :: "nat \<Rightarrow>f value option"   and ra :: "nat \<Rightarrow>f value option" assumes "A \<le> length i"   and "enumerate_regs a = {}"   and "enumerate_aexp_inputs a \<noteq> {}"   and "Max (enumerate_aexp_inputs a) < A" shows "aval a (join_ir (take A i) r) = aval a (join_ir i ra)"
Cn_not: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(not_PL p \<in> Cn A) = (Cn (A \<union> {p}) = UNIV)"
fst_init_lvars: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "'a member_scheme"   and e :: "expr"   and a' :: "val"   and pvs :: "val list"   and x :: "abopt"   and s :: "st" shows "fst (init_lvars G C sig (invmode m e) a' pvs (x, s)) = (if is_static m then x else np a' x)"
Collect_neg_eq: fixes P :: "'a \<Rightarrow> bool" shows "{x. \<not> P x} = - {x. P x}"
dom_char: fixes f :: "'a" shows "hom.dom f = (if hom.arr f then local.dom f else null)"
space_prob_algebra: fixes M :: "'a measure" shows "space (prob_algebra M) = {N. sets N = sets M \<and> prob_space N}"
gb_schema_dummy_connectible2: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and data :: "nat \<times> nat \<times> 'd"   and f :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and g :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "struct_spec sel ap ab compl"   and "compl_conn compl"   and "dickson_grading d"   and "fst ` set gs \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis (fst ` set gs)"   and "fst ` set bs \<subseteq> dgrad_p_set d m"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "D \<subseteq> (set gs \<union> set bs) \<times> (set gs \<union> set bs)"   and "set ps \<inter>\<^sub>p D = {}"   and "unique_idx (gs @ bs) (snd data)"   and "\<And>B a b. \<lbrakk>set gs \<union> set bs \<subseteq> B; fst ` B \<subseteq> dgrad_p_set d m; (a, b) \<in>\<^sub>p D; fst a \<noteq> 0; fst b \<noteq> 0; \<And>x y. \<lbrakk>x \<in> set gs \<union> set bs; y \<in> set gs \<union> set bs; (x, y) \<notin>\<^sub>p D; fst x \<noteq> 0; fst y \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst x) (fst y)\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)"   and "\<And>x y. \<lbrakk>x \<in> set (fst (gb_schema_dummy data D bs ps)); y \<in> set (fst (gb_schema_dummy data D bs ps)); (x, y) \<notin>\<^sub>p snd (gb_schema_dummy data D bs ps); fst x \<noteq> 0; fst y \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` set (fst (gb_schema_dummy data D bs ps))) (fst x) (fst y)"   and "\<nexists>xs. fst (gb_schema_dummy data D bs ps) = full_gb xs"   and "(f, g) \<in>\<^sub>p snd (gb_schema_dummy data D bs ps)"   and "fst f \<noteq> 0"   and "fst g \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` set (fst (gb_schema_dummy data D bs ps))) (fst f) (fst g)"
simps_1: fixes uu :: "'a"   and B :: "'a"   and uv :: "'b" shows "Cop_S.Cod (Cop_S.MkArr uu B uv) = B"
subcomplex_gallery: fixes Y :: "'a set set"   and Cs :: "'a set list" assumes "ChamberSubcomplex Y"   and "SimplicialComplex.maxsimpchain Y Cs" shows "gallery Cs"
Im_complex_div_ge_0: fixes a :: "complex"   and b :: "complex" shows "(0 \<le> Im (a / b)) = (0 \<le> Im (a * cnj b))"
pair_pair_pmf: fixes u :: "'d pmf"   and v :: "'e pmf"   and w :: "'f pmf" shows "pair_pmf (pair_pmf u v) w = map_pmf (\<lambda>(x, y, z). ((x, y), z)) (pair_pmf u (pair_pmf v w))"
simple_ownership_distinct: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'p"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'sb"   and \<D>\<^sub>i :: "'dirty"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "'rels"   and p\<^sub>j :: "'p"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'sb"   and \<D>\<^sub>j :: "'dirty"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "'rels" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "\<O>\<^sub>i \<inter> \<O>\<^sub>j = {}"
nn_integral_count_space_indicator: fixes X :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "NO_MATCH UNIV X" shows "integral\<^sup>N (count_space X) f = set_nn_integral (count_space UNIV) X f"
pivot_eq_rvars: fixes x :: "nat"   and e :: "nat \<times> linear_poly"   and v :: "nat" assumes "x \<in> rvars_eq (pivot_eq e v)"   and "x \<noteq> lhs e"   and "coeff (rhs e) v \<noteq> 0"   and "v \<noteq> lhs e" shows "x \<in> rvars_eq e"
Q: shows "snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)))) \<in> carrier_mat 2 2"
seqsim_refl: fixes s :: "nat \<Rightarrow> 'a" shows "s \<approx> s"
rel_resource_map_resource'1: shows "rel_resource A\<inverse>\<inverse> B\<inverse>\<inverse> (map_resource' f res2) res1 = rel_resource A\<inverse>\<inverse> (\<lambda>x. B\<inverse>\<inverse> (f x)) res2 res1"
ln_Gamma_real_pos: fixes x :: "real" assumes "0 < x" shows "ln_Gamma x = ln (Gamma x)"
atLeastAtMost_iff: fixes i :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and u :: "'b" shows "(i \<in> ord.atLeastAtMost less_eq l u) = (less_eq l i \<and> less_eq i u)"
res_e: shows "ord_resolve S CAs DA AAs As \<sigma> E"
llist_all2_lmap2: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and g :: "'g \<Rightarrow> 'f"   and ys :: "'g llist" shows "llist_all2 P xs (lmap g ys) = llist_all2 (\<lambda>x y. P x (g y)) xs ys"
strict_trans: fixes b :: "'b multiset"   and a :: "'b multiset"   and c :: "'b multiset" assumes "b \<subset># a"   and "c \<subset># b" shows "c \<subset># a"
Bonk_Schramm_extension_unfolded_wo_props_2: shows "\<forall>x\<in>range basepoint. \<forall>y\<in>- range basepoint. (x, y) \<in> Bonk_Schramm_extension_unfolded_wo"
s_delayed_has_field_derivative: fixes \<delta> :: "real"   and t :: "real" assumes "\<delta> < t"   and "0 \<le> v"   and "a < 0" shows "((\<lambda>x. s (x - \<delta>)) has_real_derivative s' (t - \<delta>)) (at t within {\<delta><..})"
typeI_regions_integral: shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} \<gamma>"
good: shows "qGoodInp qinp \<or> qGoodInp qinp'"
FF_def: shows "F.FF \<equiv> \<lambda>\<mu>\<nu>. if B.VV.arr \<mu>\<nu> then (F (fst \<mu>\<nu>), F (snd \<mu>\<nu>)) else C.VV.null"
map_disc_iff: fixes f1 :: "'j \<Rightarrow> 'g"   and f2 :: "'k \<Rightarrow> 'h"   and f3 :: "'l \<Rightarrow> 'i"   and a :: "('j, 'k, 'l) generat" shows "is_Pure (map_generat f1 f2 f3 a) = is_Pure a"
timpl_closure_wf_trms: fixes m :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and TI :: "('sets set \<times> 'sets set) set" assumes "wf\<^sub>t\<^sub>r\<^sub>m m" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (timpl_closure m TI)"
rstep_def: fixes l :: "nat"   and s :: "nat \<Rightarrow> nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and i :: "nat" shows "A.rstep l s i \<equiv> alstep l (s i) (s (Suc i))"
finite_llast_V: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0"   and "lfinite P" shows "llast P \<in> V\<^bsub>G\<^esub>"
surj: fixes x :: "'b set" assumes "x \<subseteq> \<pi> ` b1"   and "x \<subseteq> \<pi> ` b2" shows "x \<in> {\<pi> ` xa |xa. xa \<subseteq> b1 \<and> xa \<subseteq> b2 \<and> card xa = card x}"
runit_eqI: fixes f :: "'a"   and b :: "'a"   and \<mu> :: "'a" assumes "Right.in_hom f (src f) b"   and "Right.in_hom \<mu> (Right.R f) f"   and "Right.R \<mu> = Right.comp (f \<star> \<i>[src f]) \<a>[f, src f, src f]" shows "\<mu> = Right.runit f"
emb_step_under_args_emb_step: fixes p :: "dir list"   and t :: "('c, 'd) tm"   and d :: "dir" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p"   and "position_of t (p @ [d])" obtains i :: "nat" where "i < num_args t" and "args t ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b args (emb_step_at p d t) ! i" and "\<And>j. \<lbrakk>j < num_args t; i \<noteq> j\<rbrakk> \<Longrightarrow> args t ! j = args (emb_step_at p d t) ! j"
nn_integral_monotone_convergence_INF_decseq: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure"   and i :: "nat" assumes "decseq f"   and "\<And>i. f i \<in> borel_measurable M"   and "integral\<^sup>N M (f i) < \<infinity>" shows "\<integral>\<^sup>+ x. (INF i. f i x) \<partial>M = (INF i. integral\<^sup>N M (f i))"
h'_def: shows "h' = h 0 \<oplus> (\<lambda>(x, y). g (\<langle>y\<rangle>, \<langle>x\<rangle>))"
fully_abstract_wrt_trans_source_relation: fixes RelS :: "('procS \<times> 'procS) set" assumes "\<forall>S1 S2. \<lbrakk>S1\<rbrakk> = \<lbrakk>S2\<rbrakk> \<longrightarrow> (S1, S2) \<in> RelS"   and "trans RelS" shows "\<exists>RelT. \<forall>S1 S2. ((S1, S2) \<in> RelS) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> RelT)"
AE_pair_measure: fixes P :: "'b \<times> 'c \<Rightarrow> bool" assumes "{x \<in> space (S \<Otimes>\<^sub>M T). P x} \<in> sets (S \<Otimes>\<^sub>M T)"   and "AE x in S. AE y in T. P (x, y)" shows "almost_everywhere (S \<Otimes>\<^sub>M T) P"
subset_insertI2: fixes A :: "'b set"   and B :: "'b set"   and b :: "'b" assumes "(A::'b set) \<subseteq> (B::'b set)" shows "(A::'b set) \<subseteq> (b::'b) \<triangleright> (B::'b set)"
mC_C_finer: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set"   and Q :: "nat set set" assumes "mC_C theta c d s t P F"   and "trans theta"   and "finer P Q"   and "finite Q"   and "{} \<notin> Q"   and "part {..<brn c} Q"   and "partCompat Q indis (eff c s)"   and "partCompat Q theta (local.cont c s)" shows "mC_C theta c d s t Q (lift P F)"
likelihood_forward: fixes s :: "'s"   and os :: "'t list" assumes "s \<in> \<S>" shows "likelihood s os = (\<Sum>t\<in>\<S>. forward s t os)"
intros: fixes p :: "perm"   and x :: "'a" shows "(p \<bullet> x, x) \<in> hull_rel"
atc_io_reduction_on_subset: fixes M1 :: "('d, 'e, 'f) FSM"   and T :: "'d list set"   and \<Omega> :: "('d, 'e) ATC set"   and M2 :: "('d, 'e, 'g) FSM"   and T' :: "'d list set" assumes "M1 \<preceq>\<lbrakk>T.\<Omega>\<rbrakk> M2"   and "T' \<subseteq> T" shows "M1 \<preceq>\<lbrakk>T'.\<Omega>\<rbrakk> M2"