hcju/isabelleps huggingface dataset:
useful_tautologies_6: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi>) in v]"
fmap_strict: fixes f :: "'f \<rightarrow> 'e" assumes "f\<cdot>\<bottom> = \<bottom>" shows "fmap\<cdot>f\<cdot>\<bottom> = \<bottom>"
dvd_0_left: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times zero a" shows "a = zero"
bliftM_def: fixes f :: "'a \<rightarrow> 'b" shows "bliftM f \<equiv> \<Lambda> x. unbox\<cdot>x >>= (\<Lambda> x'. box\<cdot>(f\<cdot>x'))"
reciprocal_inf: shows "reciprocal \<infinity>\<^sub>h = 0\<^sub>h"
disjs: fixes s :: "'a set" assumes "s \<in> S" shows "A \<inter> B \<inter> s = {}"
append_vec_vNil: fixes v :: "'a vec" shows "vNil @\<^sub>v v = v"
ev_iff_ev_at: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "ev P \<omega> = (\<exists>n. ev_at P n \<omega>)"
p_def: shows "p = [inv lead_coeff q] \<otimes>\<^bsub>K [X]\<^esub> q"
continuous_at_imp_continuous_on: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>s. isCont f x" shows "continuous_on s f"
UnI2: fixes c :: "'c"   and B :: "'c set"   and A :: "'c set" assumes "c \<in> B" shows "c \<in> A \<union> B"
in_AboveS_underS: fixes a :: "'a"   and r :: "('a \<times> 'a) set" assumes "a \<in> Field r" shows "a \<in> AboveS r (underS r a)"
poly_lift_hom_cf: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and p :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "p \<in> carrier P" shows "poly_lift_hom R S \<phi> p k = \<phi> (p k)"
simple_function_measurable2: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "simple_function M f"   and "simple_function M g" shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
comp_R_1: shows "Maps.seq \<lbrakk>\<lbrakk>u\<^sub>1\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>"
mult_right_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq c one"   and "less_eq zero a" shows "less_eq (times c a) a"
sorted_list_of_set_insert: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A" shows "linorder.sorted_list_of_set less_eq (insert x A) = linorder.insort_key less_eq (\<lambda>x. x) x (linorder.sorted_list_of_set less_eq (A - {x}))"
u: shows "x \<notin> \<Union> (Fvars ` F)"
stateful_strand_step_subst_inI_5: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and A :: "('a, 'b) stateful_strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "delete\<langle>t,s\<rangle> \<in> set A" shows "delete\<langle>t \<cdot> \<theta>,s \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
infsetsum_nat': fixes f :: "nat \<Rightarrow> 'a" assumes "f abs_summable_on UNIV" shows "infsetsum f UNIV = (\<Sum>n. f n)"
X_mult_cf: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "p \<in> carrier (UP Zp)" shows "(p \<otimes>\<^bsub>UP Zp\<^esub> X_poly Zp) (k + 1) = p k"
is_pure_assn_starI: fixes a :: "assn"   and b :: "assn" assumes "is_pure_assn a"   and "is_pure_assn b" shows "is_pure_assn (a * b)"
assoc'_simps_4: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.dom \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = D.dom f \<star>\<^sub>D D.dom g \<star>\<^sub>D D.dom h"
hroot': shows "poly (map_poly complex_of_real P) x' = 0"
exhaust: fixes y :: "'b stream" obtains x1 :: "'b" and x2 :: "'b stream" where "y = x1 ## x2"
int_pow_integer_mod_group: fixes n :: "nat"   and x :: "int"   and m :: "int" shows "x [^]\<^bsub>Z n\<^esub> m = m * x mod int n"
simps: fixes a :: "ty" shows "is_refT a = (a = NT \<or> (\<exists>C. a = Class C) \<or> (\<exists>A. a = A\<lfloor>\<rceil>))"
sel_1: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs \<or> \<not> lnull ys" shows "lhd (lappend xs ys) = (case xs of LNil \<Rightarrow> lhd ys | LCons x xs' \<Rightarrow> x)"
iii: shows "\<forall>x. x \<in># E1 \<longrightarrow> (x, a) \<in> r"
effective_tensor_compose_distribution1: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "matrix_match A1 A2 B1 B2" shows "mat (row_length A1 * row_length B1) (length A2 * length B2) (A1 \<circ> A2 \<otimes> B1 \<circ> B2)"
induct: fixes P1 :: "POPLmarkRecord.type \<Rightarrow> bool"   and P3 :: "(char list \<times> POPLmarkRecord.type) list \<Rightarrow> bool"   and P2 :: "char list \<times> POPLmarkRecord.type \<Rightarrow> bool"   and char_list_type_prod :: "char list \<times> POPLmarkRecord.type" assumes "\<And>x. P1 (TVar x)"   and "P1 Top"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 \<rightarrow> x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (\<forall><:x1. x2)"   and "\<And>x. P3 x \<Longrightarrow> P1 (RcdT x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P2 char_list_type_prod"
upper_triangularD: fixes A :: "'c mat"   and j :: "nat"   and i :: "nat" assumes "upper_triangular A"   and "j < i"   and "i < dim_row A" shows "A $$ (i, j) = (0::'c)"
is_pr_iso_def: fixes \<phi> :: "'a \<Rightarrow> 'b"   and Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list" shows "is_pr_iso \<phi> Fs\<^sub>1 Fs\<^sub>2 \<equiv> is_pr_Iso \<phi> (set Fs\<^sub>1) (set Fs\<^sub>2)"
nGt_le_conv: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y \<noteq> Gt) = le_of_comp comp x y"
vifintersectionE3: fixes a :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)" obtains "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> a \<in>\<^sub>\<circ> f i"
D_pos: shows "\<integral>\<^sup>+ x. ennreal (D x) \<partial>M = 1"
dord_fun_degD: fixes ord :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" assumes "dord_fun ord s t" shows "deg_fun s \<le> deg_fun t"
step: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list"   and i :: "nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and cfg2 :: "('a, 'c, 'b) configuration" assumes "execution trans sends start exec trace"   and "i < length exec - 1"   and "cfg1 = exec ! i"   and "cfg2 = exec ! (i + 1)" shows "asynchronousSystem.steps trans sends cfg1 (trace ! i) cfg2"
f6: shows "a v \<noteq> \<zero>\<^bsub>cpx_rng\<^esub>"
valuation_some_in_prime_divisor1: fixes P :: "('a \<Rightarrow> ant) set" assumes "P \<in> Pds" shows "valuation K (SOME w. w \<in> P)"
take_bit_int_greater_eq_self_iff: fixes k :: "int"   and n :: "nat" shows "(k \<le> take_bit n k) = (k < 2 ^ n)"
elims_1: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b tree"   and y :: "bool" assumes "bst_wrt x xa = y" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" where "x = P__" and "xa = \<langle>\<rangle>" and "y"   | P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and l :: "'b tree" and a :: "'b" and r :: "'b tree" where "x = P" and "xa = \<langle>l, a, r\<rangle>" and "y = ((\<forall>x\<in>set_tree l. P x a) \<and> Ball (set_tree r) (P a) \<and> bst_wrt P l \<and> bst_wrt P r)"
knows_Spy_subset_knows_Spy_Notes: fixes evs :: "event list"   and A :: "agent"   and X :: "msg" shows "knows Spy evs \<subseteq> knows Spy (Notes A X # evs)"
prems_1: shows "valid_path_aux (upd_cs cs (a # as)) cs'"
transition_eqvt: fixes P :: "('state, 'effect) L_state"   and \<alpha>Q :: "(('act, 'effect) L_action, ('state, 'effect) L_state) residual"   and p :: "perm" assumes "P \<rightarrow>\<^sub>L \<alpha>Q" shows "p \<bullet> P \<rightarrow>\<^sub>L p \<bullet> \<alpha>Q"
product_integral_insert': fixes I :: "nat set"   and i :: "nat"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> real" assumes "finite I"   and "i \<notin> I"   and "integrable (Pi\<^sub>M (insert i I) (\<lambda>v. stock_measure (\<Gamma> v))) f" shows "integral\<^sup>L (Pi\<^sub>M (insert i I) (\<lambda>v. stock_measure (\<Gamma> v))) f = LINT y|stock_measure (\<Gamma> i). LINT x|Pi\<^sub>M I (\<lambda>v. stock_measure (\<Gamma> v)). f (x(i := y))"
elimation_rules_strictly_decreasing: fixes \<Gamma>\<^sub>1 :: "'c constr list"   and n\<^sub>1 :: "nat"   and \<Psi>\<^sub>1 :: "'c TESL_atomic list"   and \<Phi>\<^sub>1 :: "'c TESL_atomic list"   and \<Gamma>\<^sub>2 :: "'c constr list"   and n\<^sub>2 :: "nat"   and \<Psi>\<^sub>2 :: "'c TESL_atomic list"   and \<Phi>\<^sub>2 :: "'c TESL_atomic list" assumes "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>e \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2" shows "\<mu> \<Psi>\<^sub>2 < \<mu> \<Psi>\<^sub>1"
Sup_least: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and z :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
ctr_1: fixes xs :: "'c llist"   and ys :: "'d llist" assumes "lnull xs \<or> lnull ys" shows "lzip xs ys = LNil"
neg_numeral_le_iff: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus (numeral.numeral one plus m)) (uminus (numeral.numeral one plus n)) = (n \<le> m)"
less_Suc_eq: fixes m :: "nat"   and n :: "nat" shows "((m::nat) < Suc (n::nat)) = (m < n \<or> m = n)"
nodes_graph': fixes V :: "'a set"   and E :: "('a \<times> 'a) set" assumes "nodes_edges_consistent V E" shows "nodes (graph V E) = V"
runit'_simps_3: fixes f :: "'d" assumes "D.ide f" shows "trg\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[f] = trg\<^sub>D f"
wt_App_ge_fun: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym (App s t)) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s)"
simps_2: fixes M :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term list"   and TI :: "('sets set \<times> 'sets set) list"   and f :: "('fun, 'b, 'sets) prot_fun"   and T :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term list" shows "intruder_synth_mod_timpls M TI (Fun f T) = (list_ex (\<lambda>t. timpls_transformable_to TI t (Fun f T)) M \<or> public f \<and> length T = arity f \<and> list_all (intruder_synth_mod_timpls M TI) T)"
divide_out_primepow: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and n :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "n \<noteq> zero"   and "\<not> dvd.dvd times n one" obtains p :: "'a" and k :: "nat" and n' :: "'a" where "normalization_semidom.prime zero times one normalize p" and "dvd.dvd times p n" and "\<not> dvd.dvd times p n'" and "0 < k" and "n = times (power.power one times p k) n'"
A: shows "dfa_accepting M (dfa_steps M 0 w) = dfa_accepting M (snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! (fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! dfa_steps M 0 w))"
hk: fixes y :: "'a" assumes "y \<in> sphere z 1 \<inter> U" shows "h (k y) = y"
simps_1: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "higher_differentiable_on S f 0 = continuous_on S f"
rep_eq: fixes x :: "uint64" shows "int_of_uint64 x = uint (Rep_uint64 x)"
step_2: fixes x :: "nat" assumes "j + 2 < card X"   and "x \<in> {j + 3..card X - 1}"   and "0 < x"   and "x + 1 \<le> card X" shows "aa x + bb x \<le> Dube_aux (card X) (poly_deg f) x"
cases: fixes a1 :: "ccs"   and a2 :: "ccs"   and Rel :: "(ccs \<times> ccs) set" assumes "(a1, a2) \<in> bangRel Rel" obtains P :: "ccs" and Q :: "ccs" where "a1 = !P__" and "a2 = !Q__" and "(P__, Q__) \<in> Rel"   | R :: "ccs" and T :: "ccs" and P :: "ccs" and Q :: "ccs" where "a1 = R \<parallel> P" and "a2 = T \<parallel> Q" and "(R, T) \<in> Rel" and "(P, Q) \<in> bangRel Rel"
sound_mode_use_preservation: fixes gc\<^sub>A :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and gc\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" assumes "length (fst gc\<^sub>A) = length cms"   and "length (fst gc\<^sub>C) = length cms"   and "\<And>i. i < length cms \<Longrightarrow> ((fst gc\<^sub>A ! i, snd gc\<^sub>A), fst gc\<^sub>C ! i, snd gc\<^sub>C) \<in> \<R>_rel (cms ! i)"   and "abs.sound_mode_use gc\<^sub>A"   and "modes_respect_priv (map snd (fst gc\<^sub>C))" shows "conc.sound_mode_use gc\<^sub>C"
assms_3: shows "S1 \<sqsubseteq> S2"
path_trans_single_neg: fixes SES :: "('c, 'd) SES_rec"   and s1 :: "'c"   and t :: "'d list"   and s2 :: "'c"   and e :: "'d"   and sn :: "'c" assumes "s1 t\<Longrightarrow>\<^bsub>SES\<^esub> s2"   and "T\<^bsub>SES\<^esub> s2 e \<noteq> Some sn" shows "path SES s1 (t @ [e]) \<noteq> Some sn"
narrow1: fixes narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.wn narrow widen less_eq less"   and "less_eq y x" shows "less_eq y (narrow x y)"
splitFace_add_vertices_direct: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and n :: "nat" shows "vertices (snd (snd (splitFace g ram1 ram2 oldF [countVertices g..<countVertices g + n]))) = vertices g @ [countVertices g..<countVertices g + n]"
prime_power_cancel_less: fixes p :: "'a"   and m :: "'a"   and k :: "nat"   and m' :: "'a"   and k' :: "nat" assumes "prime p"   and "m * p ^ k = m' * p ^ k'"   and "k < k'"   and "\<not> p dvd m" shows "False"
quasi_sturm_seq_def: fixes ps :: "real poly list" shows "quasi_sturm_seq ps \<equiv> (\<forall>x y. sgn (poly (last ps) x) = sgn (poly (last ps) y)) \<and> ps \<noteq> [] \<and> (\<forall>i x. i < length ps - 2 \<longrightarrow> poly (ps ! (i + 1)) x = 0 \<longrightarrow> poly (ps ! (i + 2)) x * poly (ps ! i) x < 0)"
equiv_app: fixes s :: "dB"   and s' :: "dB"   and t :: "dB"   and t' :: "dB" assumes "s \<leftrightarrow> s'"   and "t \<leftrightarrow> t'" shows "s \<degree> t \<leftrightarrow> s' \<degree> t'"
dirichlet_prod'_mangoldt_floor_asymptotics: shows "(\<lambda>x. dirichlet_prod' mangoldt (\<lambda>x. real_of_int \<lfloor>x\<rfloor>) x - x * ln x + x) \<in> O(ln)"
insort_insert_insort_key: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "f x \<notin> f ` set xs" shows "insort_insert_key f x xs = insort_key f x xs"
basis_null_space_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "basis_null_space A = {row i (P_Gauss_Jordan (Finite_Cartesian_Product.transpose A)) |i. rank A \<le> mod_type_class.to_nat i}"
these_empty_eq: fixes B :: "'b option set" shows "(Option.these B = {}) = (B = {} \<or> B = {None})"
Quotient_af: shows "Quotient (=) pure ??.Abstract_AF.af.Rep_af cr_af"
assms: fixes x :: "'a" shows "convergent (\<lambda>n. f n *\<^sub>v x)"
f_shrink_nth_eq_f_last_message_hold_last: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" assumes "n < length xs div k" shows "xs \<div> k ! n = last (xs \<longmapsto> k \<up> (n * k) \<down> k)"
maximal_def: fixes S :: "'b set"   and C :: "'b set set" shows "maximal S C = (\<forall>S'\<in>C. S \<subseteq> S' \<longrightarrow> S = S')"
assignsV_good_approx: fixes Env :: "'a env_scheme"   and s0 :: "state"   and v :: "Term.var"   and vf :: "vvar"   and s1 :: "state" assumes "prg Env\<turnstile>s0 \<midarrow>v=\<succ>vf\<rightarrow> s1"   and "normal s1" shows "assignsV v \<subseteq> dom (locals (snd s1))"
binomial_plus_le: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "m choose k + (n choose k) \<le> m + n choose k"
unity_root_def: fixes k :: "nat"   and n :: "int" shows "unity_root k n = cis (2 * pi * real_of_int n / real k)"
image_mset_cong2: fixes M :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and N :: "'b multiset" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x"   and "M = N" shows "image_mset f M = image_mset g N"
cases: fixes t :: "'t"   and a1 :: "'x2 \<times> 'm2"   and a2 :: "'x2 \<times> 'm2" assumes "r2.silent_move t a1 a2" obtains s :: "'x2 \<times> 'm2" and s' :: "'x2 \<times> 'm2" and tl :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action" where "a1 = s" and "a2 = s'" and "t \<turnstile> s -2-tl\<rightarrow> s'" and "\<tau>move2 s tl s'"
rep_eq: fixes x :: "'b" shows "blinfun_apply (blinfun_inner_left x) = (\<lambda>y. y \<bullet> x)"
strictified_bicategory_def: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b" shows "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<equiv> bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"
proj_poly_syz_sum_list: fixes n :: "nat"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "proj_poly_syz n (sum_list xs) = sum_list (map (proj_poly_syz n) xs)"
ap_spmf_homo: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "Applicative.ap (return_spmf f) (return_spmf x) = return_spmf (f x)"
tensor_mat_add1: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and m3 :: "'a mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d1 d1"   and "m3 \<in> carrier_mat d2 d2" shows "tensor_mat (m1 + m2) m3 = tensor_mat m1 m3 + tensor_mat m2 m3"
eq: fixes x :: "real" shows "ennreal \<bar>c\<bar> * (f x / ennreal \<bar>c\<bar>) = f x"
identity_functor_axioms: shows "identity_functor (local.Left a')"
C14: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "class.C_algebra plus times one zero less_eq less star" shows "star x = times (star (power.power one times x (n + 1))) (dioid_one_zero.powsum plus times one zero x 0 n)"
attract_mono_imp_least_qfp: assumes "attractive B (\<sqsubseteq>)"   and "well_complete B (\<sqsubseteq>)"   and "monotone_on B (\<sqsubseteq>) (\<sqsubseteq>) f" shows "\<exists>c. extreme {p \<in> B. f p \<sim> p \<or> f p = p} (\<lambda>x y. y \<sqsubseteq> x) c \<and> f c \<sim> c"
iT_defs_1: fixes n :: "nat" shows "[n\<dots>] \<equiv> {n..}"
causal_broadcast: fixes m2 :: "'a"   and j :: "nat"   and m1 :: "'a"   and i :: "nat" assumes "Deliver m2 \<in> set (history j)"   and "Deliver m1 \<sqsubset>\<^sup>i Broadcast m2" shows "Deliver m1 \<sqsubset>\<^sup>j Deliver m2"
less_set_def: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subset> B) = ((\<lambda>x. x \<in> A) < (\<lambda>x. x \<in> B))"
nqfree_map_fm: fixes f :: "'b \<Rightarrow> 'a"   and \<phi> :: "'b fm" shows "nqfree (map\<^sub>f\<^sub>m f \<phi>) = nqfree \<phi>"
that: fixes B :: "'b set" assumes "B \<subseteq> V"   and "independent B"   and "V \<subseteq> span B"   and "card B = dim V" shows "thesis"
insert_not_empty: fixes a :: "'c"   and A :: "'c set" shows "insert a A \<noteq> {}"
runit'_in_vhom: fixes f :: "'b"   and a :: "'b" assumes "D.ide f"   and "src\<^sub>D f = a" shows "\<guillemotleft>\<r>\<^sub>D\<^sup>-\<^sup>1[f] : f \<Rightarrow>\<^sub>D f \<star>\<^sub>D a\<guillemotright>"
non_volatile_owned_or_read_only_drop: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "non_volatile_owned_or_read_only False \<S> \<O> sb" shows "non_volatile_owned_or_read_only True (share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S>) (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
in_multiset_1: shows "(\<lambda>a. 0) \<in> multiset"
excess_s_non_pos: shows "excess f s \<le> (0::'a)"
polyn_degreeTr: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "(polyn_expr R X k c = \<zero>) = ({j. j \<le> k \<and> snd c j \<noteq> \<zero>\<^bsub>S\<^esub>} = {})"
arcs_ends_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "arcs_ends (app_iso hom G) = (\<lambda>(u, v). (iso_verts hom u, iso_verts hom v)) ` arcs_ends G"
sum_representation_eq: fixes basis :: "'a set"   and v :: "'a"   and B :: "'a set" assumes "independent basis"   and "v \<in> span basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. representation basis v b *\<^sub>R b) = v"
padic_int_simps_2: fixes p :: "int" shows "\<zero>\<^bsub>padic_int p\<^esub> = padic_zero p"
sums_ereal: fixes f :: "nat \<Rightarrow> real"   and x :: "real" shows "(\<lambda>x. ereal (f x)) sums ereal x = f sums x"
from_nat_to_nat_id: fixes x :: "'b" shows "from_nat (to_nat x) = x"