hcju/isabelleps huggingface dataset:
assign_def: fixes f :: "'b control_state_scheme \<Rightarrow> 'b control_state_scheme" shows "assign f = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some ((), \<sigma>) else Some ((), f \<sigma>))"
Union: fixes a :: "nat \<Rightarrow> 'a set"   and sp :: "'a set"   and A :: "'a set set" assumes "\<And>i. a i \<in> sigma_sets sp A" shows "\<Union> (range a) \<in> sigma_sets sp A"
completion_upper: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets (completion M)" obtains A' :: "'b set" where "A \<subseteq> A'" and "A' \<in> sets M" and "A' - A \<in> null_sets (completion M)" and "emeasure (completion M) A = emeasure M A'"
l2_inv8_def: shows "l2_inv8 \<equiv> {s. secret s \<subseteq> {NonceF (R $ sk) |R. R = test \<or> R \<in> partners}}"
executable: fixes p :: "'state"   and a :: "'transition" assumes "nodesp p"   and "enabled a p" shows "executablep a"
fmapU_stateT_def: shows "fmapU \<equiv> stateT_map\<cdot>ID"
row: shows "row B (n - 1) = 0\<^sub>v n"
F'_def: shows "F' \<equiv> (\<lambda>y. (2 / h\<^sup>2) *\<^sub>R (y - discrete_evolution (euler_increment f) (t + h) t (x t))) ` R"
assms_4: shows "i \<noteq> bounded_hashcode_nat (length l) k"
ltl_equivalence_def: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool" shows "ltl_equivalence eq \<equiv> equivp eq \<and> (\<sim>\<^sub>C) \<le> eq \<and> eq \<le> (\<sim>\<^sub>L)"
rot_y_ivl: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> 1"   and "0 \<le> y"   and "y \<le> 1" shows "0 \<le> x_coord x / (2 * circle_y (x_coord (x_coord_inv ((2 * y - 1) * circle_y (x_coord x))))) + 1 / 2 \<and> x_coord x / (2 * circle_y (x_coord (x_coord_inv ((2 * y - 1) * circle_y (x_coord x))))) + 1 / 2 \<le> 1"
\<psi>_def: fixes \<pi>l :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list" shows "\<psi> \<pi>l = eqOnGL (length \<pi>l) (Suc (length \<pi>l))"
group_closure_scalar_mult_left: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and s :: "'a"   and S :: "'a set"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "s \<in> ab_group_add.group_closure zero minus S" shows "times (semiring_1.of_nat one plus zero n) s \<in> ab_group_add.group_closure zero minus S"
sqrt_has_derivative: fixes f :: "'c \<Rightarrow> real"   and f' :: "'c \<Rightarrow> real"   and x :: "'c"   and S :: "'c set" assumes "(f has_derivative f') (at x within S)"   and "f x \<noteq> 0" shows "((\<lambda>x. sqrt (f x)) has_derivative (\<lambda>xa. (if 0 < f x then 1 else - 1) / (2 * sqrt (f x)) * f' xa)) (at x within S)"
MN: fixes J :: "'a set" assumes "J \<in> I"   and "finite J" shows "A \<in> Pi\<^sub>F {J} M \<rightarrow>\<^sub>M N"
assms_8: shows "r\<^sup>\<infinity> = r\<^sup>\<star>"
lemma_partial_derivatives: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and S :: "'a set"   and x :: "'a" assumes "linear f"   and "((\<lambda>x. f (x - a) /\<^sub>R norm (x - a)) \<longlongrightarrow> (0::'b)) (at a within S)"   and "\<And>v. v \<noteq> (0::'a) \<Longrightarrow> \<exists>k>0. \<forall>e>0. \<exists>x\<in>S - {a}. norm (a - x) < e \<and> k * norm (a - x) \<le> \<bar>v \<bullet> (x - a)\<bar>" shows "f x = (0::'b)"
snd_image_Sigma: fixes A :: "'e set"   and B :: "'e \<Rightarrow> 'd set" shows "snd ` Sigma A B = \<Union> (B ` A)"
netFlow_def: shows "netFlow \<equiv> sum f (outgoing' (V - {t})) - sum f (incoming' (V - {t}))"
boundOutputFresh: fixes x :: "name"   and xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput" shows "x \<sharp> \<lparr>\<nu>*xvec\<rparr>B = (x mem xvec \<or> x \<sharp> B)"
load_tmps_read_tmps_distinct: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "load_tmps is \<inter> read_tmps sb = {}"
leq_trans: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.quasi_order leq"   and "leq x y"   and "leq y z" shows "leq x z"
prime_elem_power_iff: fixes p :: "'a"   and n :: "nat" shows "prime_elem (p ^ n) = (prime_elem p \<and> n = 1)"
IH: fixes ys :: "('fun \<times> ('b, 'c, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef) list" assumes "Result.those (map (compile_env_entry \<A> \<O>) xs) = Ok ys" shows "rel_fundefs (Finca_get (Sinca.Fenv.from_list xs)) (Fubx_get (Subx.Fenv.from_list ys))"
var\<^sub>C_of_inj: shows "inj var\<^sub>C_of"
is_ubD: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and u :: "'b" assumes "is_ub r x y u" shows "(x, u) \<in> r \<and> (y, u) \<in> r"
pnPlus_PlusL: fixes t :: "'a rexp"   and r :: "'a rexp"   and s :: "'a rexp" assumes "t \<noteq> Zero" shows "pnPlus (Plus r s) t = pnPlus r (pnPlus s t)"
certify_def: fixes cert :: "'a pair_pre_digraph" shows "certify cert \<equiv> let x = gen_contr_graph (with_proj cert) (verts3 (with_proj cert)) in subgraph (with_proj cert) (with_proj G) \<and> (K\<^bsub>3,3\<^esub> (with_proj x) \<or> K\<^bsub>5\<^esub> (with_proj x))"
ring_iso_memE_4: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_iso R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
rel_restrict_def: fixes R :: "('a \<times> 'a) set"   and A :: "'a set" shows "rel_restrict R A \<equiv> {(v, w). (v, w) \<in> R \<and> v \<notin> A \<and> w \<notin> A}"
tsp_exists: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "\<not> Coplanar A B C P" shows "\<exists>Q. A B CTSP P Q"
real_inner_1_right: fixes x :: "real" shows "inner x 1 = x"
fup_Aexp_lookup_fresh: fixes v :: "Vars.var"   and e :: "exp"   and a :: "Arity\<^sub>\<bottom>" assumes "atom v \<sharp> e" shows "\<A>\<^sup>\<bottom>\<^bsub>a\<^esub> e v = \<bottom>"
open_exceptI_1: fixes x :: "'c"   and y :: "'c" assumes "x \<noteq> y" shows "open (open_except x y)"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg h h.tab (tab\<^sub>0 h) (tab\<^sub>1 h) h h.tab (tab\<^sub>0 h) (tab\<^sub>1 h) h"
saturated_def: fixes N :: "'f set" shows "q_calc.saturated N = (Inf_from N \<subseteq> Red_I_\<G>_q q N)"
containFalsum: fixes \<Gamma> :: "('a, 'b) form multiset"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and \<Delta> :: "('a, 'b) form multiset"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "ff \<in># \<Gamma>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* \<Delta>, 0) \<in> derivable (ext R R' M N)"
assms_2: shows "\<Theta>,\<Gamma> \<turnstile> B"
q_def: shows "q = (\<lambda>b. local.punit.decr_grading_p d m (q0 b))"
all_fulfill_def: fixes atom_ordering :: "('a \<times> 'a) set"   and P :: "'a Literal set \<Rightarrow> bool"   and S :: "'a Literal set set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.all_fulfill P S = (\<forall>C. C \<in> S \<longrightarrow> P C)"
correctness_def: fixes m1 :: "'a \<times> 'a"   and m2 :: "bool" shows "sim_def.correctness m1 m2 \<equiv> protocol m1 m2 = funct_OT_12 m1 m2"
get_const_distr: fixes a :: "'a DBMEntry"   and b :: "'a DBMEntry" assumes "a \<noteq> \<infinity>"   and "b \<noteq> \<infinity>" shows "get_const (a + b) = get_const a + get_const b"
intro: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" assumes "Contracts Xd Xh Pd Ch"   and "ContractsWithBilateralSubstitutes_axioms Xd Ch" shows "ContractsWithBilateralSubstitutes Xd Xh Pd Ch"
simps_4: fixes a :: "'b"   and b :: "'b" shows "sup (dRETURN a) (dRETURN b) = (if a = b then dRETURN b else dFAILi)"
arev_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(rev_G x = rev_G y) = (x = y)"
identify_blocks_rev: fixes A :: "'b mat"   and i :: "nat"   and k :: "nat" assumes "A $$ (i, Suc i) = (0::'b) \<and> Suc i < k \<or> Suc i = k"   and "k < n" shows "(identify_block A i, i) \<in> set (identify_blocks A k)"
s0g_defs_3: shows "s0g_trans \<equiv> \<Union>d A B G. (s0g_gen d A G \<union> s0g_learn d B \<union> s0g_leak d)\<^sup>="
delete_Some_all_distinct: fixes x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "delete x t = Some t'"   and "all_distinct t" shows "all_distinct t'"
fresh_atom_at_base: fixes a :: "atom"   and b :: "'b" shows "a \<sharp> atom b = a \<sharp> b"
maximal_wrt_def: fixes A :: "'a"   and DA :: "'a literal multiset" shows "maximal_wrt A DA = (DA = {#} \<or> A = Max (atms_of DA))"
ov: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and e :: "('a \<times> 'a) set"   and m :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and ov :: "('a \<times> 'a) set"   and d :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set"   and p :: "'a"   and q :: "'a" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> ov) = (\<exists>k l u v t. (meets k p \<and> meets p u \<and> meets u v) \<and> (meets k l \<and> meets l q \<and> meets q v) \<and> meets l t \<and> meets t u)"
diff_add_cancel: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "plus (minus a b) b = a"
scopeExtSumRight: fixes x :: "name"   and P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" assumes "x \<sharp> P"   and "\<And>y R. y \<sharp> R \<Longrightarrow> (R, \<lparr>\<nu>y\<rparr>R) \<in> Rel"   and "Id \<subseteq> Rel" shows "P \<oplus> \<lparr>\<nu>x\<rparr>Q \<leadsto>[Rel] \<lparr>\<nu>x\<rparr>(P \<oplus> Q)"
Rep_cpx_sqr_mat_cases: fixes y :: "complex mat" assumes "y \<in> {M |M. square_mat M}" obtains x :: "cpx_sqr_mat" where "y = Rep_cpx_sqr_mat x"
a: shows "a \<sharp> [c].([(c, a)] \<bullet> x)"
openin_topology\<^sub>N: fixes N :: "'a quasinorm"   and U :: "'a set" shows "openin (topology\<^sub>N N) U = (\<forall>x\<in>U. \<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U)"
filtrationI: fixes M :: "'d measure"   and F :: "'c \<Rightarrow> 'd measure" assumes "\<forall>t. subalgebra M (F t)"   and "\<forall>s t. s \<le> t \<longrightarrow> subalgebra (F t) (F s)" shows "Filtration.filtration M F"
order_tendsto_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a"   and F :: "'b filter" assumes "class.order_topology less_eq less open" shows "filterlim f (topological_space.nhds open x) F = ((\<forall>l. less l x \<longrightarrow> (\<forall>\<^sub>F x in F. less l (f x))) \<and> (\<forall>u. less x u \<longrightarrow> (\<forall>\<^sub>F x in F. less (f x) u)))"
resPres: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "P \<sim>\<^sub>e Q" shows "<\<nu>x>P \<sim>\<^sub>e <\<nu>x>Q"
tendsto_norm_zero: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "(f \<longlongrightarrow> (0::'c)) F" shows "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
2: shows "(w \<sqinter> - d) * d\<^sup>T * top = bot"
of_nat_eq_0_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = zero) = (m = 0)"
subset_DFS_def: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat" shows "subset_DFS A n \<equiv> wf_nfa A n"
simps_1: fixes f :: "'a" shows "\<lbrace>\<^bold>\<langle>f\<^bold>\<rangle>\<^sub>0\<rbrace> = (if arr f then f else null)"
time_combine_tm: fixes p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list" shows "time (combine_tm (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps) \<le> 3 + 18 * length ps"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "comm_monoid_add.sum (\<squnion>) bot g A = comm_monoid_add.sum (\<squnion>) bot h B"
irr: fixes q :: "int poly"   and r :: "int poly" assumes "degree q < degree p"   and "degree r < degree p" shows "p \<noteq> q * r"
zero_ell2_def: shows "0 \<equiv> Abs_ell2 (\<lambda>_. 0)"
irreducible_prod_rI: fixes a :: "'a"   and b :: "'a" assumes "irreducible G a"   and "b \<in> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "irreducible G (a \<otimes> b)"
simps_8: shows "T_f32 \<noteq> T_i64"
cases_scheme: fixes r :: "('a, 'b, 'c, 'd) bar_scheme" obtains bar1 :: "'a" and bar2 :: "'b" and bar3 :: "'c" and bar21 :: "'b \<times> 'a" and bar32 :: "'c \<times> 'b" and bar31 :: "'c \<times> 'a" and more :: "'d" where "r = \<lparr>bar1 = bar1, bar2 = bar2, bar3 = bar3, bar21 = bar21, bar32 = bar32, bar31 = bar31, \<dots> = more\<rparr>"
dghm_op_dghm_op_dghm: shows "op_dghm (op_dghm \<GG>) = \<GG>"
normal_intersection_hom_kernel: shows "kernel (G\<lparr>carrier := S\<rparr>) (G\<lparr>carrier := H <#> S\<rparr> Mod H) ((#>) H) = H \<inter> S"
atLeast_Suc_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..Suc n} = sum (g \<circ> Suc) {m..n}"
weak_ranking_eqclass1: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "A \<in> set (weak_ranking le)"   and "x \<in> A"   and "y \<in> A" shows "le x y"
refl_r: fixes x :: "'a" assumes "x \<in> A" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x"
preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "trg\<^sub>D (H \<mu>) = H.map\<^sub>0 (trg\<^sub>C \<mu>)"
complement_def: fixes x :: "'b word" shows "complement x \<equiv> NOT x"
sp_append_one: fixes g :: "('a, 'b) rgraph_scheme"   and v1 :: "'a"   and es :: "'a edge list"   and e :: "'a edge"   and v2 :: "'a" shows "subpath g v1 (es @ [e]) v2 = (subpath g v1 es (src e) \<and> e \<in> edges g \<and> tgt e = v2)"
load_word_mem_select_10: fixes s1 :: "'b sparc_state"   and addra :: "word32"   and s2 :: "'c sparc_state" assumes "fst (case load_word_mem s1 addra 10 of None \<Rightarrow> (None, s1) | Some w \<Rightarrow> (Some w, add_data_cache s1 addra w 15)) = fst (case load_word_mem s2 addra 10 of None \<Rightarrow> (None, s2) | Some w \<Rightarrow> (Some w, add_data_cache s2 addra w 15))" shows "load_word_mem s1 addra 10 = load_word_mem s2 addra 10"
BIT_inv_def: fixes s :: "('a list \<times> bool list \<times> 'a list) pmf"   and x :: "'a"   and i :: "'a list" shows "BIT_inv s x i \<equiv> s = type0 i x (hd (filter (\<lambda>y. y \<noteq> x) i))"
that: fixes a :: "'a" assumes "a \<in> A"   and "\<And>b. rel b a \<Longrightarrow> b \<notin> A" shows "thesis"
sccs_verts_subsets: fixes S :: "'a set" assumes "S \<in> S.sccs_verts" shows "S \<subseteq> pverts (mk_symmetric G)"
finfun_update_def: shows "finfun_update \<equiv> map_fun finfun_apply (map_fun id (map_fun id Abs_finfun)) fun_upd"
pmf_of_list_wf_def: fixes xs :: "('b \<times> 'c) list" shows "pmf_of_list_wf xs = ((\<forall>x\<in>set (map snd xs). (0::'c) \<le> x) \<and> sum_list (map snd xs) = (1::'c))"
diff_add: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.ordered_cancel_comm_monoid_diff plus zero less_eq less minus"   and "less_eq a b" shows "plus (minus b a) a = b"
orbit_cyclic_eq3: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and y :: "'c" assumes "cyclic_on f S"   and "y \<in> S" shows "orbit f y = S"
measurable_integrable: fixes S :: "'c set" shows "(S \<in> lmeasurable) = (indicat_real S integrable_on UNIV)"
lunit'_in_hom_2: fixes f :: "'d" assumes "D.ide f" shows "\<guillemotleft>\<l>\<^sub>D\<^sup>-\<^sup>1[f] : f \<Rightarrow>\<^sub>D trg\<^sub>D f \<star>\<^sub>D f\<guillemotright>"
range_coeff: fixes p :: "'b poly" shows "range (coeff p) = insert (0::'b) (set (coeffs p))"
inverse_less_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse a) (inverse b) = ((less zero (times a b) \<longrightarrow> less b a) \<and> (less_eq (times a b) zero \<longrightarrow> less a b))"
cidr_split_prefix_single: fixes start :: "'a word"   and "end" :: "'a word" shows "\<Union> (uncurry ipset_from_cidr ` set (cidr_split (iprange_interval (start, end)))) = {start..end}"
Le_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Le Bet Cong A B C D \<equiv> \<exists>E. Bet C E D \<and> Cong A B C E"
perron_frobenius_spectral_radius: fixes A :: "real mat"   and n :: "nat"   and d :: "nat" assumes "A \<in> carrier_mat n n"   and "nonneg_mat A"   and "\<forall>x. poly (char_poly A) x = 0 \<longrightarrow> x \<le> 1"   and "\<forall>x. cmod x = 1 \<longrightarrow> order x (map_poly complex_of_real (char_poly A)) \<le> d" shows "\<exists>c1 c2. \<forall>k a. a \<in> elements_mat (A ^\<^sub>m k) \<longrightarrow> \<bar>a\<bar> \<le> c1 + c2 * real k ^ (d - 1)"
CatIdDomCod_2: fixes X :: "'a" assumes "X \<in> obj" shows "cod (id X) = X"
length_concat_acc_lengths: fixes ls :: "'a list list"   and k :: "nat" assumes "ls \<noteq> []" shows "k + length (concat ls) \<in> list.set (acc_lengths k ls)"
UINF_cong_eq: fixes P\<^sub>1 :: "'d \<Rightarrow> 'e upred"   and P\<^sub>2 :: "'d \<Rightarrow> 'e upred"   and Q\<^sub>1 :: "'d \<Rightarrow> ('f, 'e) uexpr"   and Q\<^sub>2 :: "'d \<Rightarrow> ('f, 'e) uexpr" assumes "\<And>x. P\<^sub>1 x = P\<^sub>2 x"   and "\<And>x. `P\<^sub>1 x \<Rightarrow> Q\<^sub>1 x =\<^sub>u Q\<^sub>2 x`" shows "UINF P\<^sub>1 Q\<^sub>1 = UINF P\<^sub>2 Q\<^sub>2"
headp': shows "head ((shift1 ^^ (Reflected_Multivariate_Polynomial.degree s - n)) p) = head p"
f7: shows "t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = \<zero>"
i_append_nth_Cons: fixes x :: "'d"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "([x] \<frown> f) n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> f k)"
interval_inner_leI_1: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b"   and i :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "x \<in> ord.atLeastAtMost less_eq a b"   and "less_eq zero i" shows "inner a i \<le> inner x i"
deriv: fixes x :: "real" assumes "a \<le> x" shows "(g has_real_derivative g' x) (at x within {a..})"
lfilter_eq_LNil: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "(lfilter P xs = LNil) = (\<forall>x\<in>lset xs. \<not> P x)"
induct: fixes P :: "(('a \<rightarrow> ('b, 'a) Step) \<rightarrow> 'a \<rightarrow> 'b LList) \<Rightarrow> bool" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (\<Lambda> h v. Fixrec.run\<cdot>(match_bottom\<cdot>v\<cdot>(Fixrec.succeed\<cdot>\<bottom>) +++ Fixrec.succeed\<cdot>(case h\<cdot>v of Done \<Rightarrow> LNil | Skip\<cdot>s' \<Rightarrow> x\<cdot>h\<cdot>s' | Yield\<cdot>xa\<cdot>s' \<Rightarrow> LCons\<cdot>xa\<cdot>(x\<cdot>h\<cdot>s'))))" shows "P unfold"
uniform_groups: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and u_group_size :: "nat"   and G :: "'b set" assumes "uniform_group_design point_set block_collection groups u_group_size"   and "G \<in> groups" shows "card G = u_group_size"
abs_moebius_mu_le: fixes n :: "nat" shows "\<bar>moebius_mu n\<bar> \<le> (1::'a)"
lead_coeff_not_zero: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and a :: "'c"   and p :: "'c list" assumes "ring R"   and "polynomial\<^bsub>R\<^esub> K (a # p)" shows "a \<in> K - {\<zero>\<^bsub>R\<^esub>}"
eucl_truncate_up_le: fixes x :: "'a"   and y :: "'a"   and e :: "nat" assumes "x \<le> y" shows "x \<le> eucl_truncate_up e y"
rel_mono: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "R \<le> Ra" shows "rel_fun (=) R \<le> rel_fun (=) Ra"
return_def: fixes a :: "'e" shows "return a \<equiv> \<lambda>s. ((a, s), False)"
filter_pdevs_raw_def: fixes I :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'd" shows "filter_pdevs_raw I X = (\<lambda>i. if I i (X i) then X i else (0::'d))"
primroot_nemp: fixes x :: "'b list" assumes "x \<noteq> \<epsilon>" shows "\<rho> x \<noteq> \<epsilon>"
exhaustive_fusion_def: fixes proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and g :: "('b, 'c) generator"   and s :: "'c" shows "proper_intrvl.exhaustive_fusion proper_interval g s = proper_intrvl.exhaustive proper_interval (list.unfoldr g s)"
erH_rHom_nonzero: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and f :: "'a \<Rightarrow> 'e"   and p :: "'a" assumes "PolynRg A B Y"   and "f \<in> rHom S B"   and "p \<in> carrier R"   and "erH R S X A B Y f p \<noteq> \<zero>\<^bsub>A\<^esub>" shows "p \<noteq> \<zero>"
assms_5: fixes i :: "nat" assumes "i < n"   and "p A i k" shows "local.diff_ev A i k \<and> local.uppert A i k"
fUnionM_funionM: fixes f :: "'b \<Rightarrow> 'm"   and g :: "'b \<Rightarrow> 'm"   and A :: "'b fset" shows "fUnionM ((\<lambda>y. funionM (f y) (g y)) |`| A) = funionM (fUnionM (f |`| A)) (fUnionM (g |`| A))"
to_language_Int: fixes A :: "'b list set"   and B :: "'b list set" shows "to_language (A \<inter> B) = Coinductive_Language.Inter (to_language A) (to_language B)"
eval_and: fixes a :: "atom fm"   and b :: "atom fm"   and L :: "real list" shows "eval (and a b) L = (eval a L \<and> eval b L)"
rbd_def: shows "bd\<^sub>\<R> = bd\<^sub>\<F> \<circ> \<F>"