hcju/isabelleps huggingface dataset:
R_def: shows "R = (\<lambda>x. sum_upto (\<lambda>i. ind prime i * ln (real i) * frac (x / real i)) x)"
one_add_floor: fixes x :: "'a" shows "\<lfloor>x\<rfloor> + 1 = \<lfloor>x + (1::'a)\<rfloor>"
matrix_inv_mat: fixes c :: "'d" assumes "c \<noteq> (0::'d)" shows "mat c\<^sup>-\<^sup>1 = mat (inverse c)"
assms_2: shows "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1"
J_rel_coind: fixes R2 :: "'d JF1 \<Rightarrow> 'e JF1 \<Rightarrow> bool"   and R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R3 :: "'d JF2 \<Rightarrow> 'e JF2 \<Rightarrow> bool" assumes "\<forall>x2 y2. R2 x2 y2 \<longrightarrow> F1rel R1 R2 R3 (dtor1 x2) (dtor1 y2)"   and "\<forall>x3 y3. R3 x3 y3 \<longrightarrow> F2rel R1 R2 R3 (dtor2 x3) (dtor2 y3)" shows "R2 \<le> JF1rel R1 \<and> R3 \<le> JF2rel R1"
length_sorted_list_of_set: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set" assumes "class.linorder less_eq less" shows "length (linorder.sorted_list_of_set less_eq A) = card A"
word_eq_iff_unsigned: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and v :: "'c word"   and w :: "'c word" assumes "class.semiring_char_0 one times plus zero" shows "(v = w) = (semiring_1.unsigned one plus zero v = semiring_1.unsigned one plus zero w)"
lex_fun_lin: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "lex_fun s t \<or> lex_fun t s"
basis_card_eq_dim: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set"   and V :: "'b set" assumes "vector_space scale"   and "B \<subseteq> V"   and "V \<subseteq> module.span scale B"   and "\<not> module.dependent scale B" shows "card B = vector_space.dim scale V"
induct: fixes P :: "'a\<cdot>lifted \<Rightarrow> bool"   and x :: "'a\<cdot>lifted" assumes "P \<bottom>"   and "\<And>a. P (Lifted\<cdot>a)" shows "P x"
assign_Norm_Norm: fixes f :: "'a \<Rightarrow> state \<Rightarrow> state"   and v :: "'a"   and s :: "st"   and s' :: "st" assumes "f v (Norm s) = Norm s'" shows "assign f v (Norm s) = Norm s'"
delta_nonneg: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and deltaG :: "'b itself \<Rightarrow> real" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG" shows "0 \<le> deltaG TYPE('b)"
ptr_kinds_ptr_kinds_M: fixes ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" shows "(ptr \<in> set |h \<turnstile> shadow_root_ptr_kinds_M|\<^sub>r) = (ptr |\<in>| shadow_root_ptr_kinds h)"
map_add_Some: fixes f :: "'c \<Rightarrow> 'c option" shows "Some ++ f = map_id_on (- dom f) ++ f"
greatest_is_MaxR_opt: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A \<inter> Field r"   and "\<forall>y\<in>A \<inter> Field r. (y, x) \<in> r" shows "MaxR_opt A = Some x"
idc_cmaps: fixes \<Delta> :: "(coname \<times> ty) list"   and y :: "name"   and b :: "coname"   and x :: "name"   and M :: "trm" assumes "idc \<Delta> y cmaps b to Some (x, M)" shows "M = Ax x b"
minimal: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.pcpo below" shows "below (pcpo.bottom below) x"
perm_append_single: fixes a :: "'c"   and xs :: "'c list" shows "a # xs <~~> xs @ [a]"
word_less_two_pow_divI: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" assumes "x < 2 ^ (n - m)"   and "m \<le> n"   and "n < LENGTH('b)" shows "x < 2 ^ n div 2 ^ m"
induct: fixes x1 :: "('p, 'v, 's) configuration"   and x2 :: "('p, 'v, 's) configuration"   and P :: "('p, 'v, 's) configuration \<Rightarrow> ('p, 'v, 's) configuration \<Rightarrow> bool" assumes "reachable x1 x2"   and "\<And>cfg1. P cfg1 cfg1"   and "\<And>cfg1 cfg2 msg cfg3. \<lbrakk>reachable cfg1 cfg2; P cfg1 cfg2; cfg2 \<turnstile> msg \<mapsto> cfg3\<rbrakk> \<Longrightarrow> P cfg1 cfg3" shows "P x1 x2"
restr_in: fixes x :: "'a"   and A :: "'a set"   and al :: "('a \<times> 'b) list" assumes "x \<in> A" shows "map_of (restrictA A al) x = map_of al x"
simps_5: fixes r1 :: "'a rexp"   and r2 :: "'a rexp" shows "awidth (Times r1 r2) = awidth r1 + awidth r2"
run_detI: fixes c :: "'d Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'d"   and \<sigma> :: "heap option"   and s :: "'d" assumes "run c (Some h) (Some h') r"   and "run c (Some h) \<sigma> s" shows "\<sigma> = Some h' \<and> r = s"
finite_measure_Union: fixes A :: "real set"   and B :: "real set" assumes "A \<in> events"   and "B \<in> events"   and "A \<inter> B = {}" shows "prob (A \<union> B) = prob A + prob B"
increasing1: fixes ls :: "'a list"   and as :: "'a list"   and x :: "'a"   and cs :: "'a list"   and y :: "'a"   and bs :: "'a list" assumes "increasing ls"   and "ls = as @ x # cs @ y # bs" shows "x \<le> y"
some_snapshot_state_def: fixes t :: "('a, 'b, 'c) event list" shows "some_snapshot_state t \<equiv> SOME (t', i). trace init t final \<and> trace init t' final \<and> t' <~~> t \<and> state_equal_to_snapshot (S t' i) final"
canonical_retraction_retraction: fixes A :: "'a set set"   and C :: "'a set"   and v :: "'a" assumes "A \<in> \<A>"   and "chamber C"   and "C \<in> A"   and "v \<in> \<Union> A" shows "canonical_retraction A C v = v"
aeX2: shows "AE x in T \<Otimes>\<^sub>M P. 0 \<le> Pz (snd x)"
bit_iff_and_push_bit_not_eq_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit a n = (and a (push_bit n one) \<noteq> zero)"
gen_subalgebra_eq_space_sets: fixes M :: "'b measure"   and N :: "'b measure"   and P :: "'b set set"   and Q :: "'b set set" assumes "space M = space N"   and "P = Q"   and "P \<subseteq> Pow (space M)" shows "sets (gen_subalgebra M P) = sets (gen_subalgebra N Q)"
local_lipschitzE: fixes T :: "'a set"   and X :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and t :: "'a"   and x :: "'b" assumes "local_lipschitz T X f"   and "t \<in> T"   and "x \<in> X" obtains u :: "real" and L :: "real" where "0 < u" and "\<And>s. s \<in> cball t u \<inter> T \<Longrightarrow> L-lipschitz_on (cball x u \<inter> X) (f s)"
inject: fixes x21 :: "color"   and x22 :: "('a, 'b) rbt"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt"   and y21 :: "color"   and y22 :: "('a, 'b) rbt"   and y23 :: "'a"   and y24 :: "'b"   and y25 :: "('a, 'b) rbt" shows "(Branch x21 x22 x23 x24 x25 = Branch y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
assms_5: shows "F \<le> forest_components (F \<sqinter> v)"
filtermap_nhds_open_map: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "isCont f a"   and "\<And>S. open S \<Longrightarrow> open (f ` S)" shows "filtermap f (nhds a) = nhds (f a)"
cod_simp: fixes f :: "'a" assumes "Left.arr f" shows "Left.cod f = cod f"
subgraph_isomorphic_post_iso_closed: fixes H :: "nat set \<times> nat set set"   and G :: "nat set \<times> nat set set"   and G' :: "nat set \<times> nat set set" assumes "H \<sqsubseteq> G"   and "G \<simeq> G'" shows "H \<sqsubseteq> G'"
impP: shows "\<forall>x\<in>T1. x \<noteq> f a \<longrightarrow> x \<in> f ` S \<longrightarrow> P x"
cupcake_pmatch_eq: fixes pat :: "pat"   and envC :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and s :: "v store_v list"   and v0 :: "v"   and env :: "(char list \<times> v) list" assumes "is_cupcake_pat pat" shows "pmatch_single envC s pat v0 env = cupcake_pmatch envC pat v0 env"
of_nat_mod: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "semiring_1.of_nat one plus zero (m mod n) = modulo (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
list_augment_same_twice: fixes xs :: "'a list"   and k :: "nat"   and u :: "'a"   and v :: "'a" shows "list_augment (list_augment xs k u) k v = list_augment xs k v"
hyps_2: shows "n_leaves (reflect t2) = n_leaves t2"
semilattice_neutr_axioms: shows "semilattice_neutr inf top"
obs_target_is_io_targets: fixes M :: "('d, 'e, 'f) FSM"   and io :: "('d \<times> 'e) list"   and tr :: "'f list"   and q :: "'f" assumes "observable M"   and "path M (io || tr) q"   and "length io = length tr" shows "io_targets M q io = {target (io || tr) q}"
strict_not_refl_weak: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier \<and> y \<in> carrier" shows "((y, x) \<notin> relation) = (x \<succeq> y \<and> (y, x) \<notin> relation)"
less_lsum_nchotomy: fixes al :: "'b list"   and f :: "'b \<Rightarrow> nat"   and i :: "nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "i < lsum f al" shows "\<exists>n j. n < length al \<and> j < f (al ! n) \<and> i = lsum f (take n al) + j"
separating_of_bipartite_aux: fixes x :: "'v"   and p :: "'v list"   and y :: "'v" assumes "path \<Gamma> x p y"   and "y \<in> B \<Gamma>"   and "x \<in> A \<Gamma> \<or> Inr x \<in> S" shows "(\<exists>z\<in>set p. z \<in> separating_of_bipartite S) \<or> x \<in> separating_of_bipartite S"
factorgroup_is_group: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme" assumes "H \<lhd> G" shows "Group.group (G Mod H)"
rref_upt_condition3_explicit: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and i :: "'d" assumes "reduced_row_echelon_form_upt_k A k"   and "i < i + (1::'d)"   and "\<not> is_zero_row_upt_k i k A"   and "\<not> is_zero_row_upt_k (i + (1::'d)) k A" shows "(LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ (i + (1::'d)) $ n \<noteq> (0::'b))"
simps_2: fixes n :: "nat"   and A :: "'b multiset" shows "repeat_mset (Suc n) A = A + repeat_mset n A"
conv_iso: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x\<^sup>\<smile> \<le> y\<^sup>\<smile>)"
"1_3": shows "(return_time p1 has_derivative dR1) (at x0 within S)"
ID_ccompare': fixes ccompare :: "('b \<Rightarrow> 'b \<Rightarrow> order) option"   and c :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "class.ccompare ccompare"   and "ID ccompare = Some c" shows "comparator c"
eval_fps_shift: fixes n :: "nat"   and f :: "'a fps"   and z :: "'a" assumes "n \<le> subdegree f"   and "ereal (norm z) < fps_conv_radius f" shows "eval_fps (fps_shift n f) z = (if z = (0::'a) then f $ n else eval_fps f z / z ^ n)"
bsmap_fs_dep_2: fixes e :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem"   and a :: "'b"   and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" assumes "(e, a) \<in># SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)" shows "a = BsSkewBinomialHeapStruc.eprio e"
fold_mset_union: fixes s :: "nat"   and M :: "('c \<times> 'd) multiset"   and N :: "('c \<times> 'd) multiset" shows "fold_mset ((+) \<circ> pair_size) s (M + N) = fold_mset ((+) \<circ> pair_size) (fold_mset ((+) \<circ> pair_size) s M) N"
poly_rel_minus: shows "(poly_rel ===> poly_rel ===> poly_rel) (minus_poly_i ops) (-)"
is_densein_def: fixes T :: "'b set set"   and A :: "'b set"   and B :: "'b set" shows "(A densein\<^bsub>T\<^esub> B) = (B \<subseteq> closure\<^bsub>T\<^esub> A)"
divide_poly_list: fixes f :: "'a poly"   and g :: "'a poly" shows "f div g = divide_poly_list f g"
ord_p_minimum_dgrad_p_set: fixes d :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> nat"   and x :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and Q :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and m :: "nat" assumes "dickson_grading d"   and "x \<in> Q"   and "Q \<subseteq> punit.dgrad_p_set d m" obtains q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "q \<in> Q" and "\<And>y. punit.ord_strict_p y q \<Longrightarrow> y \<notin> Q"
simps_1: fixes s :: "'b sparc_state" shows "delayed_pool_rm_list [] s = s"
hyps: shows "Q \<parallel> S \<longmapsto> \<alpha> \<prec> QT'"
simps: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" shows "floor_1 (p, l, r) = (let (l', r', sr') = tighten_poly_bounds_epsilon p (1 / 2) l r (sgn (ipoly p r)); fr = \<lfloor>r'\<rfloor>; fl = \<lfloor>l'\<rfloor>; fr' = rat_of_int fr in if fr = fl then fr else let (l'', r'', sr'') = tighten_poly_bounds_for_x p fr' l' r' sr' in if fr' < l'' then fr else fl)"
sets_eP: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "sets (eP n x) = sets (Pi\<^sub>M {0..<Suc n} M)"
all_subfrmls_def: fixes n :: "'a node" shows "all_subfrmls n \<equiv> \<Union> (subfrmlsr ` (new n \<union> old n \<union> next n))"
c: shows "c \<in> {0..<n} \<rightarrow> carrier_vec n"
irreducible\<^sub>d_smult_int: fixes c :: "int"   and p :: "int poly" assumes "c \<noteq> 0" shows "irreducible\<^sub>d (smult c p) = irreducible\<^sub>d p"
assms_1: shows "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"
shift1_degree: fixes p :: "'b poly" shows "Polynomial_Expression.degree (shift1 p) = 1 + Polynomial_Expression.degree p"
may_lock_may_lock_t_eq: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and t' :: "'b" assumes "may_lock l t"   and "may_lock l t'" shows "l = None \<or> t = t'"
matrix_vector_column: fixes A :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec" shows "A *v x = (\<Sum>i\<in>UNIV. x $ i *s Finite_Cartesian_Product.transpose A $ i)"
IH: assumes "range_vars \<delta> \<inter> bvars\<^sub>s\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta> \<circ>\<^sub>s \<theta> = (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>) \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>"
inf_scene_def: fixes X :: "'a scene"   and Y :: "'a scene" shows "X \<sqinter>\<^sub>S Y = - (- X \<squnion>\<^sub>S - Y)"
elims: fixes x :: "('sequent, 'rule, 'link) ctree"   and y :: "'sequent" assumes "seqOf x = y" obtains l :: "'link" where "x = Link l" and "y = seqOf (pointsTo l)"   | s :: "'sequent" and r :: "'rule" and uu_ :: "('sequent, 'rule, 'link) ctree fset" where "x = cNode (s, r) uu_" and "y = s"
group_right_inverse: fixes x :: "'a" shows "x * inverse x = (1::'a)"
inter_Gp_nsg: fixes N :: "'a set"   and H :: "'a set" assumes "G \<triangleright> N"   and "G \<guillemotright> H " shows "\<natural>H \<triangleright> H \<inter> N"
key_channel_send_def: fixes s :: "bool list option \<times> bool list cstate"   and m :: "bool list" shows "local.key_channel_send s m \<equiv> (key.key_oracle (key \<eta>))\<dagger> s () \<bind> (\<lambda>(k, s). enc \<eta> k m \<bind> (\<lambda>c. \<dagger>channel.send_oracle s c \<bind> (\<lambda>(uu_, s). return_spmf ((), s))))"
cf_list_entries: fixes i :: "nat"   and R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "i \<le> deg R p" shows "cf_list R p ! i = p i"
homogeneous_hom_component: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "homogeneous (hom_component p n)"
orthogonalD: fixes vs :: "'b vec list"   and i :: "nat"   and j :: "nat" assumes "orthogonal vs"   and "i < length vs"   and "j < length vs" shows "(vs ! i \<bullet> vs ! j = (0::'b)) = (i \<noteq> j)"
sorted_def: fixes j :: "nat" shows "sorted j = (\<forall>x\<in>{1..j}. \<forall>y\<in>{1..x}. t x \<le> t y)"
vertex_subtree_connected: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and v :: "'e"   and s :: "'g"   and t :: "'g"   and xs :: "'g list" assumes "TreeDecomposition G T bag"   and "v \<in> V\<^bsub>G\<^esub>"   and "s \<in> TreeDecomposition.vertex_subtree T bag v"   and "t \<in> TreeDecomposition.vertex_subtree T bag v"   and "s \<leadsto>xs\<leadsto>\<^bsub>T\<^esub> t" shows "set xs \<subseteq> TreeDecomposition.vertex_subtree T bag v"
mk_vec_def: fixes f :: "'c \<Rightarrow> 'b" shows "HMA_Connect.mk_vec f = (\<chi>i. f i)"
arr_mkIde: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set" shows "arr (mkIde A) = (A \<subseteq> Univ \<and> |A| <o natLeq)"
bij_swap_comp: fixes p :: "'a \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "bij p" shows "Fun.swap a b id \<circ> p = Fun.swap (inv' p a) (inv' p b) p"
prefix_match_to_CIDR_def: fixes pfx :: "'b prefix_match" shows "prefix_match_to_CIDR pfx \<equiv> (pfxm_prefix pfx, pfxm_length pfx)"
bounded_sesquilinear_cinner: shows "bounded_sesquilinear (\<bullet>\<^sub>C)"
lsu_fres_closed: fixes x :: "'b"   and y :: "'b" shows "\<nu>\<^sup>\<natural> (\<nu>\<^sup>\<natural> x \<leftarrow> \<nu>\<^sup>\<natural> y) = \<nu>\<^sup>\<natural> x \<leftarrow> \<nu>\<^sup>\<natural> y"
finite: fixes A :: "'c set" assumes "class.finite TYPE('c)" shows "finite A"
integral_neg: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "integral S (\<lambda>x. - f x) = - integral S f"
whisker_left: fixes B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and K :: "'c \<Rightarrow> 'd"   and A :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'e \<Rightarrow> 'c"   and G :: "'e \<Rightarrow> 'c"   and \<tau> :: "'e \<Rightarrow> 'c"   and H :: "'e \<Rightarrow> 'c"   and \<tau>' :: "'e \<Rightarrow> 'c" assumes "functor B C K"   and "natural_transformation A B F G \<tau>"   and "natural_transformation A B G H \<tau>'" shows "K \<circ> vertical_composite.map A B \<tau> \<tau>' = vertical_composite.map A C (K \<circ> \<tau>) (K \<circ> \<tau>')"
As0'_def: shows "As0' = As0 \<cdot>al \<rho>"
CollectD: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "s2p {x. P x} a" shows "P a"
a_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "ad x \<cdot> (ad y \<oplus> ad z) = ad x \<cdot> ad y \<oplus> ad x \<cdot> ad z"
ring_mat_simps_5: fixes ty :: "'b itself"   and n :: "nat"   and b :: "'c" shows "carrier (ring_mat ty n b) = carrier_mat n n"
DomainE: fixes a :: "'b"   and r :: "('b \<times> 'd) set" assumes "a \<in> Domain r" obtains b :: "'d" where "(a, b) \<in> r"
assms: fixes x :: "V"   and y :: "V" assumes "\<langle>x, y\<rangle> \<in>\<^sub>\<circ> r" shows "y \<in>\<^sub>\<circ> A"
assms_4: shows "\<forall>idn\<in>set fresh_idns. idn \<notin> fst ` (fv t \<union> \<Union> (fv ` snd ` set insts) \<union> fst ` set insts)"
Stable_reachable: fixes F :: "'d program" shows "F \<in> Stable (reachable F)"
index_uminus_mat_2: fixes A :: "'b mat" shows "dim_row (- A) = dim_row A"
Pair_eqvt: fixes p :: "perm"   and x :: "'a"   and y :: "'b" shows "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
cross_edges_finished_decr: fixes u :: "'v"   and v :: "'v" assumes "(u, v) \<in> cross_edges s"   and "u \<in> dom (finished s)" shows "\<phi> s v < \<phi> s u"
assms_2: shows "\<R>\<^sub>\<circ> r = set {b}"
equivp_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "equivp R = (\<forall>x y. R x y = (R x = R y))"
rHom_tOp: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and x :: "'c"   and y :: "'c"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "f \<in> rHom A R" shows "f (x \<cdot>\<^sub>r\<^bsub>A\<^esub> y) = f x \<cdot>\<^sub>r\<^bsub>R\<^esub> f y"
altst: shows "altst \<equiv> \<lambda>n. alternating_list n s t"
distinct_14: fixes x5 :: "char list" shows "Class x5 \<noteq> Boolean"
pseudo_mod_impl: fixes f :: "'b poly"   and g :: "'b poly" shows "pseudo_mod f g = poly_of_list (pseudo_mod_list (coeffs f) (coeffs g))"
upper_trivial: fixes x :: "'b word" assumes "x \<noteq> 2 ^ LENGTH('b) - 1" shows "x < 2 ^ LENGTH('b) - 1"
assms_3: fixes s0 :: "'a list"   and rs :: "'a list" shows "length (Aoff s0 rs) = length rs"
wtl_take: fixes "is" :: "'b list"   and c :: "'a list"   and pc :: "nat"   and s :: "'a"   and pc' :: "nat" assumes "wtl is c pc s \<noteq> \<top>" shows "wtl (take pc' is) c pc s \<noteq> \<top>"
dy_fake_msg_insert_dy_fake_msg: fixes M :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and CH :: "chan set" assumes "M \<in> dy_fake_msg bad IK CH" shows "dy_fake_msg bad (insert M IK) CH \<subseteq> dy_fake_msg bad IK CH"
rep_eq: fixes x :: "\<Pi>\<^sub>1" shows "d\<^sub>1 x = Some (eval\<Pi>\<^sub>1 x)"
oexp_\<omega>_Limit: fixes \<beta> :: "V" assumes "Limit \<beta>" shows "\<omega> \<up> \<beta> = \<Squnion> ((\<up>) \<omega> ` elts \<beta>)"
solutions_linorder: fixes a :: "nat"   and b :: "nat"   and x :: "nat"   and y :: "nat" assumes "solution (a, b)"   and "solution (x, y)" shows "a \<le> x \<and> b \<le> y \<or> x \<le> a \<and> y \<le> b"
precedesTotalOrder: fixes a :: "'b"   and l :: "'b list"   and b :: "'b" assumes "a \<in> set l"   and "b \<in> set l" shows "a = b \<or> precedes a b l \<or> precedes b a l"
lookup_mult_scalar: fixes p :: "'b \<Rightarrow>\<^sub>0 'a"   and q :: "'b \<Rightarrow>\<^sub>0 'a"   and v :: "'b" shows "lookup (punit.mult_scalar p q) v = lookup (p * punit.proj_poly (punit.component_of_term v) q) (punit.pp_of_term v)"
Un_open_segment: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<in> open_segment a c" shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c"
env_restr_eqD: fixes S :: "'a set"   and m\<^sub>1 :: "'a \<Rightarrow> 'b"   and m\<^sub>2 :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "m\<^sub>1 f|` S = m\<^sub>2 f|` S"   and "x \<in> S" shows "m\<^sub>1 x = m\<^sub>2 x"
supported_strong_def: fixes a :: "'b zmultiset"   and t :: "'b" shows "supported_strong a t = (\<exists>s<t. zcount a s < 0 \<and> nonpos_upto a s)"
sum_list_ge_length_times: fixes xs :: "'a list"   and a :: "'a" assumes "\<forall>i<length xs. a \<le> xs ! i" shows "of_nat (length xs) * a \<le> sum_list xs"
P1: shows "Col B A C0 \<and> B A Perp C C0"
foldri_cons_id: fixes l :: "'a list" shows "foldri l (\<lambda>_. True) (#) [] = l"
le: shows "int_of_uint32 x * int_of_uint32 x' < 4294967296"
simps_2: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and v :: "char list" shows "eval \<sigma> (Var v) = \<sigma> v"
Cd_def: fixes d :: "'b" shows "TruePref_tax.Cd d \<equiv> set_option \<circ> MaxR.MaxR_opt (Pd'_tax d)"
l2_dy_fake_msg_def: fixes m :: "msg" shows "l2_dy_fake_msg m \<equiv> {(s, s'). m \<in> dy_fake_msg (l2_state.bad s) (ik s) (chan s) \<and> s' = s\<lparr>ik := {m} \<union> ik s\<rparr>}"
fmdrop_fset_add_distrib: fixes A :: "'c fset"   and m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "fmdrop_fset A (m ++\<^sub>f n) = fmdrop_fset A m ++\<^sub>f fmdrop_fset A n"
exec_iff_execn: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and c :: "('f, 'e, 'g) com"   and s :: "('f, 'g) xstate"   and t :: "('f, 'g) xstate" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t = (\<exists>n. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t)"
perp_in_sym: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "X PerpAt C D A B "
less_eq_name_def: fixes xs :: "name"   and ys :: "name" shows "(xs \<le> ys) = (xs < ys \<or> xs = ys)"
wf_rel_while_option_Some: fixes R :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'c"   and s :: "'c" assumes "wf R"   and "\<And>s. P s \<and> b s \<Longrightarrow> (c s, s) \<in> R"   and "\<And>s. P s \<and> b s \<Longrightarrow> P (c s)"   and "P s" shows "\<exists>t. while_option b c s = Some t"