hcju/isabelleps huggingface dataset:
full_abstraction_impl_indRelRSTPO_to_SRel_and_TRel_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q" shows "\<forall>SP TQ. SP \<in>S P \<and> TQ \<in>T Q \<longrightarrow> (\<lbrakk>SP\<rbrakk>, TQ) \<in> TRel\<^sup>*"
hyps: assumes "M \<in> multisets A"   and "M \<in> multisets A"   and "M \<noteq> {#}" shows "mulex_on P A M (M + M)"
\<iota>\<^sub>1_is_iso: shows "D.iso \<iota>\<^sub>1"
Nml_HcompNml_1: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Nml (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
loc1_a_s_3: shows "a \<in>\<^sub>A frontier (c_pts c loc1)"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "('test, 'atom) com \<times> 'state" assumes "a1 \<rightarrow>c a2" obtains c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1a__ ;; c2__, sa__)" and "a2 = (c1'__ ;; c2__, s'a__)" and "(c1a__, sa__) \<rightarrow>c (c1'__, s'a__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1b__ ;; c2a__, sb__)" and "a2 = (c2a__, s'b__)" and "(c1b__, sb__) \<rightarrow>t s'b__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tsta__ then c1c__ else c2b__, sc__)" and "a2 = (c1c__, sc__)" and "tval tsta__ sc__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tstb__ then c1d__ else c2c__, sd__)" and "a2 = (c2c__, sd__)" and "\<not> tval tstb__ sd__"   | tst :: "'test" and s :: "'state" and c :: "('test, 'atom) com" where "a1 = (while tst do c, se__)" and "a2 = (c ;; while tst do c, se__)" and "tval tst se__"   | c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1e__ | c2d__, sf__)" and "a2 = (c1' | c2d__, s'c__)" and "(c1e__, sf__) \<rightarrow>c (c1', s'c__)"   | c2 :: "('test, 'atom) com" and s :: "'state" and c2' :: "('test, 'atom) com" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1f__ | c2e__, sg__)" and "a2 = (c1f__ | c2', s'd__)" and "(c2e__, sg__) \<rightarrow>c (c2', s'd__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1g__ | c2f__, sh__)" and "a2 = (c2f__, s'e__)" and "(c1g__, sh__) \<rightarrow>t s'e__"   | c2 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1 | c2, s)" and "a2 = (c1, s')" and "(c2, s) \<rightarrow>t s'"
convex_hull_insert: fixes S :: "'b set"   and a :: "'b" assumes "S \<noteq> {}" shows "convex hull insert a S = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. u + v = 1 \<and> b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b}"
card_0_eq: fixes A :: "'c set" assumes "finite A" shows "(card A = 0) = (A = {})"
takeWhileNot_path_entry: fixes E :: "(nat \<times> nat) set"   and p :: "nat list"   and x :: "nat"   and v :: "nat"   and c :: "nat list" assumes "path_entry E p x"   and "v \<in> set p"   and "takeWhile ((\<noteq>) v) (rev p) = c" shows "path_entry E (rev c) v"
Rxa_zero: shows "R \<diamondsuit>\<^sub>p \<zero> = {\<zero>}"
cardeq_rcoset: fixes M :: "'a set"   and g :: "'a" assumes "finite (carrier G)"   and "M \<subseteq> carrier G"   and "g \<in> carrier G" shows "card (M #> g) = card M"
simps: fixes a1 :: "(status \<times> 'x) \<times> 'm"   and a2 :: "('l, 't, status \<times> 'x, 'm, 'w, 'o action) thread_action"   and a3 :: "(status \<times> 'x) \<times> 'm" shows "init_fin_\<tau>move a1 a2 a3 = (\<exists>x m ta x' m'. a1 = ((Running, x), m) \<and> a2 = convert_TA_initial (convert_obs_initial ta) \<and> a3 = ((Running, x'), m') \<and> \<tau>move (x, m) ta (x', m'))"
IH: assumes "\<And>x. x \<in> set xs \<Longrightarrow> g (f (fst x)) = fst x"   and "f (g x) = x" shows "map_of (map (\<lambda>a. case a of (k, a) \<Rightarrow> (f k, a)) xs) x = map_of xs (g x)"
subseq_seqseq: fixes n :: "nat" shows "strict_mono (seqseq n)"
exhaust_sel: fixes list_R1 :: "'c list_R1.list_R1" obtains "list_R1 = list_R1.list_R1 (list_R1.unR list_R1)"
oconf_blank: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and wt :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "is_class P C"   and "wf_prog wt P" shows "P,h \<turnstile> blank P C \<surd>"
interval_cart_2: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "cbox a b = {x. \<forall>i. a $ i \<le> x $ i \<and> x $ i \<le> b $ i}"
erf_of_real: fixes x :: "real" shows "erf (of_real x) = of_real (erf x)"
arrE: fixes f :: "'a \<times> 'a" assumes "B.VxV.arr f" obtains "B.arr (fst f) \<and> B.arr (snd f)"
S': fixes x :: "real" assumes "x \<in> {0..1} - ({0, 1} \<union> S)" shows "g differentiable at x"
freeD: shows "var \<notin> vars (2 * isolate_variable_sparse p var 2)"
rfb_irfb_galois: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and R :: "('d \<times> 'c) set" assumes "Inf_pres \<phi>" shows "(fb\<^sup>-\<^sub>\<R> \<phi> = R) = (fb\<^sub>\<R> R = \<phi>)"
Acc_to_rabin_accept: assumes "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi> (G \<psi>)) w"   and "the (\<pi> (G \<psi>)) < \<MM>.max_rank" shows "accept\<^sub>R (\<MM>.\<delta>\<^sub>\<R>, \<MM>.q\<^sub>\<R>, {(\<MM>.fail\<^sub>R \<union> \<MM>.merge\<^sub>R j, \<MM>.succeed\<^sub>R j) |j. j < \<MM>.max_rank}) w"
1: shows "\<forall>x\<in>S. g differentiable at x"
Cod_comp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "seq g f" shows "(if C1.arr (fst (g \<cdot> f)) \<and> C2.arr (snd (g \<cdot> f)) then (C1.cod (fst (g \<cdot> f)), C2.cod (snd (g \<cdot> f))) else (C1.null, C2.null)) = (if C1.arr (fst g) \<and> C2.arr (snd g) then (C1.cod (fst g), C2.cod (snd g)) else (C1.null, C2.null))"
fresh_list_elem: fixes a :: "atom"   and \<Gamma> :: "'b list"   and e :: "'b" assumes "a \<sharp> \<Gamma>"   and "e \<in> set \<Gamma>" shows "a \<sharp> e"
ex231d: fixes x :: "'a" shows "x \<le> x * top * x"
simps_5: fixes p :: "real" shows "- \<infinity> + ereal p = - \<infinity>"
ant_def: fixes m :: "int" shows "ant m = Abs_Ainteg (m, 0)"
simps_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "insort_key_rel f x [] = [x]"
normalize_canceled: fixes l :: "(bool \<times> 'c) list"   and l' :: "(bool \<times> 'c) list" assumes "cancels_to l l'" shows "Cancelation.normalize l = Cancelation.normalize l'"
DomainI: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
open_imp_subset: fixes U :: "'a set" assumes "is_open U" shows "U \<subseteq> S"
restrict_PiE_dflt: fixes A :: "'b set"   and dflt :: "'c"   and B :: "'b \<Rightarrow> 'c set" shows "(\<lambda>h. restrict h A) ` PiE_dflt A dflt B = Pi\<^sub>E A B"
ContrI: fixes A :: "formula"   and Gamma :: "formula list"   and R :: "(formula list \<times> formula list set) set" assumes "A # A # Gamma \<in> deductions R"   and "Contrs \<subseteq> R" shows "A # Gamma \<in> deductions R"
derivative_def: fixes x :: "nat"   and p :: "real mpoly" shows "derivative x p = (\<Sum>i = 0..MPoly_Type.degree p x - 1. isolate_variable_sparse p x (i + 1) * Var x ^ i * Const (real (i + 1)))"
a': shows "\<forall>y\<in>S. norm (v - a) \<le> norm (v - y) \<and> v - a \<in> orthogonal_complement S"
S_def: fixes base :: "int \<times> int poly list"   and elements :: "int poly list"   and n :: "nat" shows "S base elements n \<equiv> {foldr (\<lambda>x. map_prod (mul_const x) ((#) x)) ys base |ys. subseq_of_length n elements ys}"
ordermap_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "ordermap A r \<equiv> wfrec r (\<lambda>f x. set (f ` {y \<in> A. (y, x) \<in> r}))"
cases: fixes a1 :: "'d"   and a2 :: "'d list" assumes "ListMem a1 a2" obtains x :: "'d" and xs :: "'d list" where "a1 = x__" and "a2 = x__ # xs__"   | x :: "'d" and xs :: "'d list" and y :: "'d" where "a1 = x" and "a2 = y # xs" and "ListMem x xs"
same_level_path_aux_Append: fixes cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list" assumes "same_level_path_aux cs as"   and "same_level_path_aux (upd_cs cs as) as'" shows "same_level_path_aux cs (as @ as')"
oexp_0: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "0 \<up> \<alpha> = (if \<alpha> = 0 then 1 else 0)"
Suc: shows "snd (foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<k]) = foldl Gauss_Jordan_column_k (0, A) [0..<k]"
left_minus: fixes a :: "'a" shows "- a + a = (0::'a)"
jumpF_pathstart_part_circlepath: fixes st :: "real"   and tt :: "real"   and r :: "real"   and z :: "complex"   and z0 :: "complex" assumes "st < tt"   and "0 < r"   and "cmod (z - z0) \<noteq> r" shows "jumpF_pathstart (part_circlepath z r st tt) z0 = (if r * cos st + Re z - Re z0 = 0 then let \<Delta> = r * sin st + Im z - Im z0 in if (0 < sin st \<or> cos st = 1) \<and> \<Delta> < 0 \<or> (sin st < 0 \<or> cos st = - 1) \<and> 0 < \<Delta> then 1 / 2 else - 1 / 2 else 0)"
Poincare_recurrence_thm_1: fixes A :: "'a set" assumes "A \<in> sets MA" shows "A - A.recurrent_subset A \<in> null_sets MA"
sel_2: fixes x21 :: "'f"   and x22 :: "'h \<Rightarrow> ('g, 'f, 'h) Resumption.resumption" shows "Resumption.resumption.output (Resumption.resumption.Pause x21 x22) = x21"
hyps_1: shows "liftAll (\<lambda>a. a #[[x \<and> y]]_zs #[[x \<and> y]]_zs = a) inp"
foundedUn: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "'b \<Rightarrow> bool"   and A :: "(nat \<times> 'b) set"   and B :: "(nat \<times> 'b) set" shows "founded subs P (A \<union> B) = (founded subs P A \<and> founded subs P B)"
cases: fixes x :: "tm \<times> tm" obtains z :: "name" and t :: "tm" and r :: "tm" where "atom z \<sharp> (t, r)" and "x = (t, r)"
assms: shows "length v\<^sub>1 = length v\<^sub>2"
param_op_mop_iff: fixes Q :: "'e \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool"   and R :: "('e \<times> 'f) set"   and f :: "'e \<Rightarrow> 'g nres"   and g :: "'f \<Rightarrow> 'h nres"   and S :: "('g \<times> 'h) set" assumes "(Q, P) \<in> R \<rightarrow> bool_rel" shows "((f, g) \<in> [P]\<^sub>f R \<rightarrow> \<langle>S\<rangle>nres_rel) = ((mop Q f, mop P g) \<in> R \<rightarrow>\<^sub>f \<langle>S\<rangle>nres_rel)"
fold_A_fixespointwise1: shows "fixespointwise fold_A C"
even_succ_div_two: 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 a :: "'b" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "divide (plus a one) (numeral.numeral one plus (num.Bit0 num.One)) = divide a (numeral.numeral one plus (num.Bit0 num.One))"
a: shows "a \<in> T ` A \<rightarrow> carrier K"
vsubstAll_swapAll: fixes P :: "('a, 'b, 'varSort, 'var, 'c) param"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> (\<forall>y1 y2. {y1, y2} \<subseteq> varsOf P \<and> fresh ys y1 X \<longrightarrow> X #[y1 // y2]_ys = X #[y1 \<and> y2]_ys)) \<and> (goodAbs A \<longrightarrow> (\<forall>y1 y2. {y1, y2} \<subseteq> varsOf P \<and> freshAbs ys y1 A \<longrightarrow> A $[y1 // y2]_ys = A $[y1 \<and> y2]_ys))"
intro: fixes \<phi> :: "'b \<Rightarrow> 'c"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "bij_betw \<phi> (Collect (category.terminal S)) (Collect (category.terminal S'))" shows "two_set_categories_bij_betw_Univ_axioms S S' \<phi>"
1: shows "(SOME a. a \<in> targets \<mu>) \<in> targets \<mu>"
hyps: fixes ys :: "'a list" shows "length (subtract xs ys) \<le> length ys"
abs_sqrt_wlog: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "\<And>x. (0::'b) \<le> x \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
cblinfun_compose_add_right: fixes a :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and b :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c"   and c :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "a o\<^sub>C\<^sub>L b + c = (a o\<^sub>C\<^sub>L b) + (a o\<^sub>C\<^sub>L c)"
id_apply: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "lam" shows "SN.id \<Gamma><t> = t"
IH: fixes D :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "u' \<in> [D]\<^bsub>v,n\<^esub>" shows "\<exists>D'. A \<turnstile> \<langle>l', D\<rangle> \<leadsto>*\<^bsub>v,n\<^esub> \<langle>l'', D'\<rangle> \<and> u'' \<in> [D']\<^bsub>v,n\<^esub>"
S_def: fixes \<alpha> :: "'index" shows "S \<alpha> = spmf_of_set (range \<alpha>)"
is_lfair_def: fixes ls :: "'b \<Rightarrow> 'l"   and ss :: "'b \<Rightarrow> 'a" shows "A.is_lfair ls ss \<equiv> \<forall>l i. \<exists>j\<ge>i. \<not> A.can_step l (ss j) \<or> ls j = l"
B_def: shows "B \<equiv> \<Prod>j. d (t + 1 + j)"
assms: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "y \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "x \<noteq> y" shows "r\<lparr>x\<rparr> \<noteq> r\<lparr>y\<rparr>"
heap_appended_el: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int"   and heap' :: "int \<Rightarrow> 'a" assumes "is_heap cmp heap l r"   and "heap = heap' on {l..<r}" shows "is_heap_except_up cmp heap' l (r + 1) r"
interpret_floatariths_Var: fixes xs :: "nat list"   and vs :: "real list" shows "interpret_floatariths (map Var xs) vs = map ((!) vs) xs"
exe_complete_tcsigs_def: fixes cs :: "('a \<times> 'a) list"   and arss :: "('b \<times> ('a \<times> 'c) list) list" shows "exe_complete_tcsigs cs arss \<equiv> \<forall>ars\<in>snd ` set arss. \<forall>(c\<^sub>1, c\<^sub>2)\<in>set cs. c\<^sub>1 \<in> fst ` set ars \<longrightarrow> c\<^sub>2 \<in> fst ` set ars"
assms_2: fixes x :: "'a" assumes "x \<in> carrier - X" shows "\<not> indep (insert x X)"
valid_typeII_div: shows "valid_typeII_division s two_chain_typeII"
alw_HLD_iff_streams: fixes X :: "'b set"   and \<omega> :: "'b stream" shows "alw (HLD X) \<omega> = (\<omega> \<in> streams X)"
fourier_expansion_periodic_arithmetic_1: fixes k :: "nat"   and f :: "nat \<Rightarrow> complex" assumes "0 < k"   and "periodic_arithmetic f k" shows "periodic_arithmetic (\<lambda>x. 1 / of_nat k * (\<Sum>m<k. f m * unity_root k (- int x * int m))) k"
two_singl_Rcons: fixes a :: "'a"   and b :: "'a" shows "[a, b] = [a] ## b"
reflpE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "reflp r" obtains "r x x"
simps_2: fixes uv :: "'b list" shows "longest_common_prefix [] uv = []"
7: fixes x :: "'a" shows "x \<squnion> x = x"
eff_mono: fixes p :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and pc :: "nat"   and s :: "(ty list \<times> ty err list) option"   and t :: "(ty list \<times> ty err list) option" assumes "wf_prog p P"   and "pc < length is"   and "s \<sqsubseteq>\<^bsub>sup_state_opt P\<^esub> t"   and "app pc t" shows "set (eff pc s) {\<sqsubseteq>\<^bsub>sup_state_opt P\<^esub>} set (eff pc t)"
u: shows "u \<in> elts (A \<Uplus> B)"
ring_of_setsI: fixes M :: "'b set set"   and \<Omega> :: "'b set" assumes "M \<subseteq> Pow \<Omega>"   and "{} \<in> M"   and "\<And>a b. \<lbrakk>a \<in> M; b \<in> M\<rbrakk> \<Longrightarrow> a \<union> b \<in> M"   and "\<And>a b. \<lbrakk>a \<in> M; b \<in> M\<rbrakk> \<Longrightarrow> a - b \<in> M" shows "ring_of_sets \<Omega> M"
rep_eq: fixes x :: "'a \<Rightarrow>f 'b"   and xa :: "'a"   and xb :: "'b" shows "($) x(xa $:= xb) = (($) x)(xa := xb)"
pad_disjoint: fixes A :: "'b vec set"   and n :: "nat"   and B :: "'b vec set"   and m :: "nat" assumes "A \<subseteq> carrier_vec n"   and "0\<^sub>v n \<notin> A"   and "B \<subseteq> carrier_vec m" shows "(\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` A \<inter> (@\<^sub>v) (0\<^sub>v n) ` B = {}"
atLeastLessThan_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "class.order less_eq less"   and "less_eq b a" shows "ord.atLeastLessThan less_eq less a b = {}"
concat_bit_of_zero_2: fixes n :: "nat"   and k :: "int" shows "concat_bit n k 0 = take_bit n k"
is_float_t_def: fixes t :: "t" shows "is_float_t t = (case t of T_f32 \<Rightarrow> True | T_f64 \<Rightarrow> True | _ \<Rightarrow> False)"
diff_numeral_simps_2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (numeral.numeral one plus m) (uminus (numeral.numeral one plus n)) = numeral.numeral one plus (m + n)"
integral_one_over_x_ln_x_squared_1: fixes a :: "real" assumes "1 < a" shows "set_integrable lborel {a<..} (\<lambda>t. 1 / (t * (ln t)\<^sup>2))"
d_2: fixes x :: "'a" assumes "x \<in> cbox (0::'a) One" shows "d \<le> norm (f x - x)"
implies_increasing: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<leadsto> x"
intro_elim_4_e: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>"   and \<Theta> :: "\<o>" assumes "[\<phi> \<^bold>\<or> \<psi> in v]"   and "[\<phi> \<^bold>\<equiv> \<chi> in v]"   and "[\<psi> \<^bold>\<equiv> \<Theta> in v]" shows "[\<chi> \<^bold>\<or> \<Theta> in v]"
ide_def: fixes a :: "nat" shows "J.ide a \<equiv> J.comp a a \<noteq> J.null \<and> (\<forall>f. (J.comp f a \<noteq> J.null \<longrightarrow> J.comp f a = f) \<and> (J.comp a f \<noteq> J.null \<longrightarrow> J.comp a f = f))"
UN_Un: fixes M :: "'e \<Rightarrow> 'd set"   and A :: "'e set"   and B :: "'e set" shows "\<Union> (M ` (A \<union> B)) = \<Union> (M ` A) \<union> \<Union> (M ` B)"
Deq_def: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "Deq ic q oc \<equiv> $q \<noteq> #[] \<and> oc$ = hd<$q> \<and> q$ = tl<$q> \<and> ic$ = $ic"
assms_2: shows "X \<in> sets (stock_measure (\<Gamma> x))"
segment_reverse: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C" shows "\<exists>B'. Bet A B' C \<and> Cong C B' A B"
abs_of_nonneg: fixes abs :: "'d \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq zero a" shows "abs a = a"
complex_quadratic_equation_monic_only_two_roots: fixes \<xi> :: "complex"   and b :: "complex"   and c :: "complex" assumes "\<xi>\<^sup>2 + b * \<xi> + c = 0" shows "\<xi> = (- b + ccsqrt (b\<^sup>2 - 4 * c)) / 2 \<or> \<xi> = (- b - ccsqrt (b\<^sup>2 - 4 * c)) / 2"
Replace_0: fixes R :: "hf \<Rightarrow> hf \<Rightarrow> bool" shows "Replace 0 R = 0"
simps: fixes base :: "nat"   and a :: "'a multiset" assumes "2 \<le> base" shows "well_base base a = (\<exists>M. a = M \<and> (\<forall>n. count M n < base))"
rPar1: fixes \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)" shows "Prop C \<Psi> (P \<parallel> !P) \<alpha> (P' \<parallel> !P)"
at_within_def: fixes a :: "'c"   and s :: "'c set" shows "at a within s = inf (nhds a) (principal (s - {a}))"
bres_galois: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "less_eq_prod (times_prod' x y) z = less_eq_prod y (prod.bres x z)"
discI_3: fixes thread :: "thread"   and x4 :: "nat" assumes "thread = REL x4" shows "is_REL thread"
fmpred_fmmap_keys: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and m :: "('a, 'c) fmap" shows "fmpred P (fmmap_keys f m) = fmpred (\<lambda>a b. P a (f a b)) m"
orientable_12_implies_11: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.stone_relation_algebra inf less_eq less sup bot top uminus one times conv"   and "inf x (conv x) = bot \<and> sup x (conv x) = uminus one" shows "inf (sup x one) (conv (sup x one)) = one \<and> sup (sup x one) (conv (sup x one)) = top"
gauss_int_norm_cnj: fixes z :: "gauss_int" shows "gauss_int_norm (gauss_cnj z) = gauss_int_norm z"
comp_red_basis_aux_nonzero: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_minimal_basis (set (xs @ ys))"   and "distinct (xs @ ys)"   and "p \<in> set (comp_red_basis_aux xs ys)" shows "p \<noteq> 0"
Int_Un_eq_4: fixes T :: "'d set"   and S :: "'d set" shows "T \<union> S \<inter> T = T"
scopeExtSum: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<oplus> Q) \<approx> P \<oplus> <\<nu>x>Q"
older_seniors_tower: fixes x :: "nat"   and n :: "nat"   and y :: "nat"   and m :: "nat" assumes "x \<le> n"   and "y \<in> older_seniors x n"   and "older_seniors x n = older_seniors x (n + m)" shows "older_seniors y n = older_seniors y (n + m)"
powr_one_eq_one: fixes a :: "'a" shows "(1::'a) powr a = (1::'a)"
assms_1: shows "P \<Longrightarrow>\<^sub>t Q"
dg_Rel_Arr_iff: fixes x :: "V"   and \<alpha> :: "V" shows "(x \<in>\<^sub>\<circ> dg_Rel \<alpha>\<lparr>Arr\<rparr>) = arr_Rel \<alpha> x"
F12_1: fixes F :: "nat \<Rightarrow> 'a poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "F 1 \<noteq> 0"
converse_rtrancl_into_rtrancl: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>*"
inf_step_table_imp_inf_step: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g"   and stls :: "('g \<times> 'h \<times> 'g) llist" assumes "trsys.inf_step_table trsys s stls" shows "trsys.inf_step trsys s (lmap (fst \<circ> snd) stls)"
A_def: shows "A = insert (\<a> ps) (poly_deg ` fst ` set ps)"