hcju/isabelleps huggingface dataset:
intro: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "ring R addition multiplication zero unit"   and "comm_ring_axioms R multiplication" shows "comm_ring R addition multiplication zero unit"
eq_onp_live_step: fixes x :: "bool"   and y :: "bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "x = y" shows "(eq_onp P a a \<and> x) = (P a \<and> y)"
farkas_coefficients_ns_def: fixes ns :: "'a ns_constraint set"   and C :: "(rat \<times> 'a ns_constraint) list" shows "farkas_coefficients_ns ns C = (\<exists>c. (\<forall>(r, n)\<in>set C. n \<in> ns \<and> is_leq_ns (r *R n) \<and> r \<noteq> 0) \<and> (\<Sum>(r, n)\<leftarrow>C. lec_of_nsc (r *R n)) = Le_Constraint Leq_Rel 0 c \<and> c < (0::'a))"
axioms_2: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans \<eta>" shows "NatTransExt \<eta>"
lemma_not_dvd_hypnat_one: shows "\<not> (\<forall>n\<in>- {0}. is_unit (hypnat_of_nat n))"
neg_le_divideC_eq: fixes c :: "complex"   and a :: "'b"   and b :: "'b" assumes "c < 0" shows "(a \<le> b /\<^sub>C c) = (b \<le> c *\<^sub>C a)"
simps_5: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree"   and vd :: "'c"   and ve :: "'c Tree"   and vf :: "'c Tree" shows "siftDown (T v (T va vb vc) (T vd ve vf)) = (if val (T vd ve vf) \<le> val (T va vb vc) then if val (T va vb vc) \<le> v then T v (T va vb vc) (T vd ve vf) else T (val (T va vb vc)) (siftDown (set_val (T va vb vc) v)) (T vd ve vf) else if val (T vd ve vf) \<le> v then T v (T va vb vc) (T vd ve vf) else T (val (T vd ve vf)) (T va vb vc) (siftDown (set_val (T vd ve vf) v)))"
ntrunc_def: fixes k :: "nat"   and N :: "('c, 'd) node set" shows "ntrunc k N \<equiv> {n \<in> N. ndepth n < k}"
bt_augment_v_in_1: fixes v :: "'c"   and q :: "('c, 'd) pq"   and t :: "('d, 'c) bintree" assumes "v \<in> set |q|" shows "v \<in> set |bt_augment t q|"
eucl_of_list_image_lv_ivl: fixes xrs :: "real list"   and yrs :: "real list" assumes "length xrs = DIM('c)"   and "length yrs = DIM('c)" shows "eucl_of_list ` lv_ivl xrs yrs = {eucl_of_list xrs..eucl_of_list yrs}"
strip_while_eq_obtain_trailing: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "strip_while P xs = ys" obtains zs :: "'b list" where "xs = ys @ zs" and "\<And>z. z \<in> set zs \<Longrightarrow> P z" and "no_trailing P ys"
distinct_disc_20: fixes "term" :: "'a local.term" assumes "is_Tensor term" shows "term \<noteq> \<^bold>\<I>"
sup_comute1: fixes a :: "'a"   and b :: "'a" shows "pseudo_hoop_algebra.sup1 (\<sqinter>) (r\<rightarrow>) (l\<rightarrow>) a b = pseudo_hoop_algebra.sup1 (\<sqinter>) (r\<rightarrow>) (l\<rightarrow>) b a"
extended_AxE: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "F, \<Gamma> \<Rightarrow> F, \<Delta>"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflexive A less_eq"   and "antisymmetric A less_eq"   and "transitive A less_eq" shows "partially_ordered_set A less_eq"
valid_cube_valid_swap: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and C :: "real \<times> real \<Rightarrow> real \<times> real" assumes "R2 i j"   and "valid_two_cube C" shows "valid_two_cube (prod.swap \<circ> C \<circ> prod.swap)"
powers_5: fixes a :: "'a"   and b :: "'a" shows "a * b = b * a"
trivial_mute_fun_def: fixes x :: "'b"   and y :: "'b" shows "trivial_mute_fun x y = (x, y, 1::'b)"
acts_MapOps_def: shows "acts_MapOps \<equiv> \<lparr>MapOps.empty = empty_trie, lookup = acts_MapOps_lookup, update = acts_MapOps_update\<rparr>"
f_def: fixes n :: "nat" shows "f n = cdf (\<mu> n)"
N'_A: shows "set_mset N' \<subseteq> A"
image_constant: fixes x :: "'c"   and A :: "'c set"   and c :: "'d" assumes "x \<in> A" shows "{c |. x \<in> A} = {c}"
assms: shows "set_pmf p \<inter> f -` s \<noteq> {}"
tendsto_def: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" assumes "class.topological_space open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> S))"
nth_list_update_neq: fixes i :: "nat"   and j :: "nat"   and xs :: "'d list"   and x :: "'d" assumes "i \<noteq> j" shows "xs[i := x] ! j = xs ! j"
assms: shows "isOK (check_allm_gen_index g f n xs)"
matrix_to_iarray_echelon_form_of_column_k: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and i :: "nat"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" assumes "k < ncols A"   and "i \<le> nrows A" shows "matrix_to_iarray (fst (echelon_form_of_column_k bezout (A, i) k)) = fst (echelon_form_of_column_k_iarrays (matrix_to_iarray A, i, bezout) k)"
tfr_stp_all_append: fixes S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list" assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p S"   and "list_all tfr\<^sub>s\<^sub>t\<^sub>p S'" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (S @ S')"
dfs_base: fixes g :: "(node \<times> node) list"   and ys :: "node list" shows "dfs g [] ys = ys"
normal_vectors_def: fixes ws :: "'a vec list" shows "normal_vectors ws = (let us = basis_extension ws in map (\<lambda>i. normal_vector (set (ws @ us) - {us ! i})) [0..<length us])"
power_0: fixes a :: "'a" shows "a ^ 0 = (1::'a)"
nothing_on_v1: shows "\<forall>c. \<parallel>len v1 ts c\<parallel> = 0 \<or> restrict v1 (clm ts) c = bot \<and> restrict v1 (res ts) c = bot"
compact_imp_fip: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and F :: "'a set set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "\<And>T. T \<in> F \<Longrightarrow> topological_space.closed open T"   and "\<And>F'. \<lbrakk>finite F'; F' \<subseteq> F\<rbrakk> \<Longrightarrow> S \<inter> \<Inter> F' \<noteq> {}" shows "S \<inter> \<Inter> F \<noteq> {}"
last_index_def: fixes xs :: "'b list"   and x :: "'b" shows "last_index xs x = (let i = index (rev xs) x; n = length xs in if i = n then i else n - (i + 1))"
"3_2": shows "\<forall>m\<in>set (v1 # v2 # va). \<forall>a\<in>set (normalize_match m). normalized_n_primitive (disc, sel) f a"
Sources_singleDSource: fixes i :: "AbstrLevelsID"   and S :: "CSet"   and C :: "CSet" assumes "DSources i S = {C}" shows "Sources i S = {C} \<union> Sources i C"
radiant_of_def: fixes p :: "(real, 2) vec" shows "radiant_of p = (THE \<omega>. 0 \<le> \<omega> \<and> \<omega> < 2 * pi \<and> of_radiant \<omega> = normalize p)"
disjoint_iff_in_not_in1: fixes A :: "'a set"   and B :: "'a set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. x \<notin> B)"
NextRecvCapD_2: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and t :: "'loc \<times> 't" assumes "NextRecvCap' c0 c1 p t" shows "prop_config c1 = prop_config c0"
gr_implies_not0: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "n \<noteq> 0"
show_law_list: fixes xs :: "'a list"   and s :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> show_law s x" shows "show_law (showsp_list s) xs"
app_vrestrictionD_3: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<circ> A" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
distinct_1: fixes x21 :: "'b"   and x22 :: "('a, 'b) Expression" shows "Vx \<noteq> x21 E@ x22"
c2: shows " App (Lam [a].t1) s1 \<longrightarrow>\<^sub>\<beta>\<^sup>* App (Lam [a].t2) s2"
matrix_vector_mult_0_right: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "A *v 0 = 0"
rel_simps_51: fixes n :: "num" shows "(0::'a) < numeral n"
Sup_eq_Inf: fixes A :: "('c \<times> 'd) set" shows "Sup_prod A = Inf_prod {b. \<forall>a\<in>A. less_eq_prod a b}"
pair_map_spmf2: fixes p :: "'a spmf"   and f :: "'c \<Rightarrow> 'b"   and q :: "'c spmf" shows "pair_spmf p (map_spmf f q) = map_spmf (apsnd f) (pair_spmf p q)"
cod_mkArr: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)" shows "partial_magma.cod S (set_category_given_img.mkArr S img \<AA> A B F) = set_category_given_img.mkIde S img \<AA> B"
fls_X_inv_nth: fixes n :: "int" shows "fls_nth fls_X_inv n = (if n = - 1 then 1::'a else (0::'a))"
assms_4: shows "f2 x1 y1 z1 = (0::'a)"
lookup_tail_when: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (tail p) v = (lookup p v when v \<prec>\<^sub>t lt p)"
Eq_Var1: fixes c :: "event_data"   and x :: "nat" shows "P (formula.Eq (Const c) (Var x))"
prob: shows "has_subprob_density (distr M borel (\<lambda>x. - inverse x)) lborel (\<lambda>x. f (- inverse x) * ennreal (inverse (x * x)))"
generate_valid_topology_mono_models: fixes m :: "'b SecurityInvariant_configured"   and M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set" shows "edges (generate_valid_topology (m # M) \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> edges (generate_valid_topology M \<lparr>nodes = V, edges = E\<rparr>)"
IH: fixes x :: "'a" assumes "0 < n" shows "exp (x / of_nat n) ^ n = exp x"
V01_not_in_R_cons: fixes s :: "nat \<Rightarrow> nat option" assumes "s \<in> \<R>" shows "\<not> learn_cons \<phi> V\<^sub>0\<^sub>1 s"
b: shows "effect_to_assignments op = as\<^sub>1 @ as\<^sub>2"
pause_writer_def: fixes pause :: "'g \<Rightarrow> ('h \<Rightarrow> 'f) \<Rightarrow> 'f"   and out :: "'g"   and c :: "'h \<Rightarrow> ('d, 'e, 'f) writerT" shows "pause_writer pause out c = WriterT (pause out (\<lambda>input. run_writer (c input)))"
v: shows "v = (\<lambda>i. if i < dim_vec (x @\<^sub>v y) then (x @\<^sub>v y) $ i else 0)"
exec_Normal_elim_cases_12: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> \<Rightarrow> t" obtains "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> \<Rightarrow> t"
diff_plus_axiom_def: shows "diff_plus_axiom \<equiv> Equals (Differential (Plus (state_fun fid1) (state_fun fid2))) (Plus (Differential (state_fun fid1)) (Differential (state_fun fid2)))"
UN_sets_chop: fixes n :: "nat"   and xs :: "'a list" assumes "0 < n" shows "\<Union> (set ` set (chop n xs)) = set xs"
inverse_arrowsE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "category.inverse_arrows C f g" obtains "partial_magma.ide C (C g f)" and "partial_magma.ide C (C f g)"
load_sub3_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and asi :: "word8"   and address :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (load_sub3 instr curr_win rd asi address s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
\<Gamma>_of_mds_tyenv_sec: fixes m :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val" shows "\<forall>x\<in>dom (\<Gamma>_of_mds m). x \<notin> m AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma>_of_mds m x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"
comp_ide_self: fixes a :: "'c" assumes "C.ide a" shows "a \<cdot>\<^sub>C a = a"
all_roots_real_const_iff: fixes p :: "real poly" assumes "all_roots_real p" shows "(degree p \<noteq> 0) = (\<exists>x. poly p x = 0)"
simE2_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P \<leadsto>\<^sup>^<Rel> Q"   and "eqvt Rel"   and "(P, Q) \<in> Rel"   and "Q \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
RBTi2_axioms: shows "RBTi2 TYPE('b) (\<lambda>x xa. int xa \<le> \<lceil>c * log 2 (real x + 1)\<rceil>) (2 powr (1 / c) / (2 - 2 powr (1 / c)))"
hyps_2: fixes s :: "'a" assumes "\<Gamma>\<turnstile>mark_guards f c2 \<down> Normal s" shows "\<Gamma>\<turnstile>c2 \<down> Normal s"
simps: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" assumes "traces ind" shows "traces.eq_swap ind a1 a2 = (\<exists>a b u v. a1 = u @ [a] @ [b] @ v \<and> a2 = u @ [b] @ [a] @ v \<and> ind a b)"
bound: shows "\<forall>f\<in>\<F> g. |vertices f| \<le> 6"
not_shadows_vconsts_def: shows "not_shadows_vconsts \<equiv> value_pred.pred (\<lambda>_. list_all (\<lambda>(pat, t). fdisjnt all_consts (frees pat) \<and> \<not> shadows_consts t)) (\<lambda>_. True) (\<lambda>_. True)"
length_Casts_vs': fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Ts :: "ty list"   and vs :: "val list"   and vs' :: "val list" assumes "P \<turnstile> Ts Casts vs to vs' " shows "length Ts = length vs'"
testN_sum: shows "matrix_sum N (\<lambda>k. adjoint (testN k) * post * testN k) N = post"
is_sum_def: fixes z :: "V" shows "is_sum z = (\<exists>x. z = ZFC_Cardinals.Inl x \<or> z = ZFC_Cardinals.Inr x)"
lunit_def: fixes a :: "'c" shows "\<l>\<^sub>C[a] \<equiv> THE f. \<guillemotleft>f : \<I>\<^sub>C \<otimes>\<^sub>C a \<rightarrow>\<^sub>C a\<guillemotright> \<and> \<I>\<^sub>C \<otimes>\<^sub>C f = (\<iota>\<^sub>C \<otimes>\<^sub>C a) \<cdot>\<^sub>C C.assoc' \<I>\<^sub>C \<I>\<^sub>C a"
mset_factors_mult: fixes F :: "'a multiset"   and a :: "'a"   and G :: "'a multiset"   and b :: "'a" assumes "mset_factors F a"   and "mset_factors G b" shows "mset_factors (F + G) (a * b)"
ground_comp: fixes l :: "fterm literal" shows "ground\<^sub>l (l\<^sup>c) = ground\<^sub>l l"
rec_c: shows "dmu_array_row_main_cost fi i dmus' (Suc j) = (res, c_rec)"
assms: fixes Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3\<^sub>a {wp\<^sub>3\<^sub>' c Q} c { Q}"
conv_inf_bot_iff: fixes x :: "'a"   and y :: "'a" shows "(bot = x\<^sup>T \<sqinter> y) = (bot = x \<sqinter> y\<^sup>T)"
has_bochner_integral_scaleR_right: fixes c :: "real"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "c \<noteq> 0 \<Longrightarrow> has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. c *\<^sub>R f x) (c *\<^sub>R x)"
Ana_subst': fixes f :: "'fun"   and T :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and K :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "Ana (Fun f T) = (K, M)" shows "Ana (Fun f T \<cdot> \<delta>) = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>, M \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)"
f: fixes Z :: "'b set" assumes "openin (top_of_set (f ` S)) Z" shows "openin (top_of_set S) (S \<inter> f -` Z)"
select_convs_3: fixes states :: "hf set"   and init :: "hf set"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf set"   and eps :: "(hf \<times> hf) set"   and more :: "'c" shows "nfa.final \<lparr>nfa.states = states, init = init, final = final, nxt = nxt, eps = eps, \<dots> = more\<rparr> = final"
coprime_n_idealsTr2: fixes C :: "'a set"   and n :: "nat"   and J :: "nat \<Rightarrow> 'a set" assumes "ideal R C"   and "\<forall>k\<le>n. ideal R (J k)"   and "\<forall>i\<le>n. coprime_ideals R (J i) C" shows "coprime_ideals R (i\<Pi>\<^bsub>R,n\<^esub> J) C"
rel_map_1: fixes Sb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e \<Rightarrow> 'c"   and x :: "'e llist"   and y :: "'d llist" shows "llist_all2 Sb (lmap i x) y = llist_all2 (\<lambda>x. Sb (i x)) x y"
islimpt_eq_acc_point: fixes l :: "'a"   and S :: "'a set" shows "l islimpt S = (\<forall>U. l \<in> U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
i: shows "isTrue (PPf n \<langle>\<phi>\<rangle>)"
Un_UNIV_right: fixes A :: "'d set" shows "A \<union> UNIV = UNIV"
listgcd_simps_1: shows "listgcd [] = (0::'a)"
word_not_simps_1: fixes x :: "'a word" shows "\<not> x < 0"
correctCompositionDiffLevelsA11: shows "correctCompositionDiffLevels sA11"
properfactor_mult_l: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G (c \<otimes> a) (c \<otimes> b) = properfactor G a b"
l2_inv8_aux: fixes s :: "l2_state"   and x :: "msg" assumes "s \<in> l2_inv8"   and "x \<in> secret s" shows "x \<notin> synth (analz generators)"
refine_trans_using_invariants: fixes R :: "('e \<times> 'f) set"   and I :: "'e set"   and J :: "'f set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme" assumes "{R \<inter> I \<times> J} TS.trans Ta, TS.trans Tc {> R}"   and "{I \<inter> Domain (R \<inter> UNIV \<times> J)} TS.trans Ta {> I}"   and "{J \<inter> Range (R \<inter> I \<times> UNIV)} TS.trans Tc {> J}" shows "{R \<inter> I \<times> J} TS.trans Ta, TS.trans Tc {> R \<inter> I \<times> J}"
ovsmfidiesi_d: shows "(ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>) O d \<subseteq> e \<union> s \<union> d \<union> ov \<union> ov\<inverse> \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d\<inverse>"
hmset_of_enat_minus_enat: fixes m :: "enat"   and n :: "nat" shows "hmset_of_enat (m - enat n) = hmset_of_enat m - of_nat n"
DFA_def: shows "Canon.DFA = \<lparr>dfa.states = index_f ` UNIV // eq_app_right language, init = index_f (eq_app_right language `` {[]}), final = {index_f (eq_app_right language `` {u}) |u. u \<in> language}, nxt = \<lambda>q x. index_f (\<Union>u\<in>Canon.hinv q. eq_app_right language `` {u @ [x]})\<rparr>"
degree_aform_inverse_aform': fixes X :: "real \<times> real pdevs"   and n :: "nat"   and p :: "nat" assumes "degree_aform X \<le> n" shows "degree_aform_err (inverse_aform' p X) \<le> n"
Qinter_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'a option"   and u :: "'v"   and v' :: "'v" shows "Qinter Q \<pi> u v' = (if upd_cond Q \<pi> u v' then enat (w {v', u}) else Q v')"
z_def: shows "z = (1 - t) * x + t * y"
BRANCH_mono: fixes rs :: "'f \<Rightarrow> ('g + 'h + 'i, 'f) Old_Datatype.node set"   and rs' :: "'f \<Rightarrow> ('g + 'h + 'i, 'f) Old_Datatype.node set"   and c :: "'g" assumes "\<And>d. rs d \<subseteq> rs' d" shows "BRANCH c rs \<subseteq> BRANCH c rs'"
oo_1: shows "Ord (\<omega> \<up> \<alpha>1)"
neutr_eq_iff: fixes a :: "'c"   and b :: "'c" shows "(bot = sup a b) = (a = bot \<and> b = bot)"
ord_pmf_increaseI: fixes p :: "'c spmf"   and q :: "'c spmf"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. spmf p x \<le> spmf q x"   and "\<And>x. x \<in> set_spmf p \<Longrightarrow> R x x" shows "ord_spmf R p q"
simps_2: fixes j :: "nat" shows "poly_of_pair (Inr j) = monomial (1::'b) (term_of_pair (0::'a, j))"
g_bounded: fixes x :: "real" assumes "x\<^sub>0 \<le> x"   and "x \<le> x\<^sub>1" shows "g x \<le> gb2"
holomorphic_hurwitz_zeta': fixes f :: "complex \<Rightarrow> complex"   and A :: "complex set"   and a :: "real" assumes "f holomorphic_on A"   and "0 < a"   and "\<And>z. z \<in> A \<Longrightarrow> f z \<noteq> 1" shows "(\<lambda>x. hurwitz_zeta a (f x)) holomorphic_on A"
combine_altdef: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and m1 :: "('c, 'd) mapping"   and m2 :: "('c, 'd) mapping" shows "Mapping.combine f m1 m2 = combine_with_key (\<lambda>_. f) m1 m2"
trans_strong: fixes zs :: "'a list"   and ys :: "'a list"   and xs :: "'a list"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "ext gt zs ys"   and "ext gt ys xs" shows "ext gt zs xs"
ZOretr_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set" assumes "PL_Indis cval indis" shows "PL_Indis.ZOretr aval tval cval indis theta \<equiv> {(c, d). PL_Indis.matchC_LC aval tval cval indis theta c d}"
d: fixes z :: "'b" assumes "norm (z - y) < d" shows "norm (g z - g y - g' (z - y)) \<le> 1 / 2 * norm (g z - g y)"
sample_uniform_units_def: fixes q :: "nat" shows "sample_uniform_units q = spmf_of_set ({..<q} - {0})"
LeastI_ex: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x" shows "P (ord.Least less_eq P)"
non_special_solution_non_minimal: fixes x :: "nat list"   and y :: "nat list"   and i :: "nat"   and j :: "nat" assumes "(x, y) \<in> local.Solutions - Special_Solutions"   and "i < m"   and "j < n"   and "local.dij i j \<le> x ! i"   and "local.eij i j \<le> y ! j" shows "(x, y) \<notin> local.Minimal_Solutions"
inj_comm_monoid_add_hom_axioms: shows "inj_comm_monoid_add_hom (\<lambda>a. [:a:])"
prefix_def: fixes l1 :: "'c list"   and l2 :: "'c list" shows "l1 \<preceq> l2 \<equiv> \<exists>l3. l1 @ l3 = l2"
zero_neq_numeral: fixes n :: "num" shows "(0::'a) \<noteq> numeral n"
Cons_prefix_Cons: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
CCCD: fixes f :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "NatTrans f" shows "CatCod (NTCod f) = CatCod (NTDom f)"
U_def: fixes s :: "'s \<times> 's option"   and t :: "'s \<times> 's option" shows "KN.U s t = KN.T.prob s {\<omega> \<in> space (KN.T s). ev (HLD {t}) \<omega>}"
lsemimodule4: fixes a :: "'a" shows "\<nn> \<circ> a = a"
antisympI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>r x y; r y x\<rbrakk> \<Longrightarrow> x = y" shows "antisymp r"
less_cblinfun_def: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(A < B) = (A \<le> B \<and> \<not> B \<le> A)"
flip_def: fixes xs :: "bool list"   and i :: "nat" shows "flip xs i = xs[i := \<not> xs ! i]"
strict_order_iff: fixes a :: "'c word"   and b :: "'c word" shows "(a <s b) = (a = signed.min a b \<and> a \<noteq> b)"
2: shows "\<guillemotleft>incl_of (img f) (cod f) : img f \<rightarrow> cod f\<guillemotright>"
butlast_idx: fixes y :: "'b"   and xs :: "'b list" assumes "y \<in> set (butlast xs)" obtains i :: "nat" where "xs ! i = y" and "i < length xs - 1"
R1E: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> \<R>" obtains r :: "recf" where "recfn 1 r" and "Partial_Recursive.total r" and "f = (\<lambda>x. eval r [x])"
V: fixes p :: "'a" shows "\<forall>(c, aft)\<in>vcg_fragments (PGMs sys p). PGMs sys, p, aft \<turnstile> \<lbrace>I\<rbrace> c"
diff_cancel: fixes a :: "'b" shows "a - a = (zero_class.zero::'b)"
keys_monom_mult_subset: fixes c :: "'d"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'd" shows "keys (monom_mult c t p) \<subseteq> (\<oplus>) t ` keys p"
smap_streams: fixes s :: "'a stream"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "s \<in> streams A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "smap f s \<in> streams B"