hcju/isabelleps huggingface dataset:
is_sup_least: fixes x :: "'c"   and y :: "'c"   and sup :: "'c"   and z :: "'c" assumes "is_sup x y sup"   and "x \<sqsubseteq> z"   and "y \<sqsubseteq> z" shows "sup \<sqsubseteq> z"
rep_eq: fixes x :: "int" shows "Rep_mod_ring (of_int_mod_ring x) = x mod int CARD('a)"
distinct_19: fixes x3 :: "'a" shows "FalseF \<noteq> fm.Atom x3"
closed_subset: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T"   and "closed S" shows "closedin (top_of_set T) S"
uE_def: shows "uE \<equiv> E \<union> E\<inverse>"
hyps_2: assumes "s1 \<approx> s2" shows "\<exists>s2' tls2. s2 -2-tls2\<rightarrow>* s2' \<and> s1' \<approx> s2' \<and> tls1 [\<sim>] tls2"
function_ring_defs_6: shows "(\<odot>\<^bsub>F\<^esub>) = function_scalar_mult S R"
nth_take_lemma: fixes k :: "nat"   and xs :: "'g list"   and ys :: "'g list" assumes "k \<le> length xs"   and "k \<le> length ys"   and "\<And>i. i < k \<longrightarrow> xs ! i = ys ! i" shows "take k xs = take k ys"
trivial_code_def: fixes v :: "'val"   and vs :: "'val list" shows "trivial_code v vs = (the_trivial v vs \<noteq> None)"
n_undefined: fixes s :: "'s" assumes "s \<notin> S" shows "n s = undefined"
joins_sym: fixes a :: "'a"   and b :: "'a"   and e :: "'edge" shows "joins a b e = joins b a e"
simps_1: fixes x :: "'a \<Rightarrow> 'b \<times> 'a"   and ya :: "'a \<Rightarrow> 'b \<times> 'a" shows "(State x = State ya) = (x = ya)"
continuous_map_minus: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_map X euclidean f" shows "continuous_map X euclidean (\<lambda>x. - f x)"
taints_wellformedness: fixes ts :: "SINVAR_TaintingTrusted.taints" shows "untaints ts \<subseteq> taints ts"
bij_betw_same_card: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "card A = card B"
set_bd_1: fixes x :: "('l, 'k, 'm) F2" shows "|F2set1 x| \<le>o bd_F2"
npowf_expTr2: fixes x :: "'a"   and i :: "int"   and j :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>"   and "0 \<le> i + j" shows "x\<^bsub>K\<^esub>\<^bsup>(i + j)\<^esup> = x\<^bsub>K\<^esub>\<^bsup>i\<^esup> \<cdot>\<^sub>r x\<^bsub>K\<^esub>\<^bsup>j\<^esup>"
simps_1: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and M :: "'b ltln set" shows "\<phi> U\<^sub>n \<psi>[M]\<^sub>\<Sigma>\<^sub>2 = \<phi>[M]\<^sub>\<Sigma>\<^sub>2 U\<^sub>n \<psi>[M]\<^sub>\<Sigma>\<^sub>2"
norm_prod_def: fixes x :: "'c \<times> 'd" shows "norm x = sqrt ((norm (fst x))\<^sup>2 + (norm (snd x))\<^sup>2)"
prems: fixes x :: "'a" assumes "x \<in> insert x__ F" shows "0 < f x"
waryI: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and wt_sym :: "'a \<Rightarrow> nat"   and \<epsilon> :: "nat"   and extf :: "'a \<Rightarrow> (('a, 'b) tm \<Rightarrow> ('a, 'b) tm \<Rightarrow> bool) \<Rightarrow> ('a, 'b) tm list \<Rightarrow> ('a, 'b) tm list \<Rightarrow> bool"   and ground_heads_var :: "'b \<Rightarrow> 'a set"   and s :: "('a, 'b) tm" assumes "kbo_basic gt_sym wt_sym \<epsilon> extf ground_heads_var" shows "arity.wary (\<lambda>_. \<infinity>) (\<lambda>_. \<infinity>) s"
list_separated_toString_helper: shows "list_separated_toString = list_separated_toString_helper"
foldf: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ThinChamberComplexFolding X f"
finite_converse: fixes r :: "('c \<times> 'b) set" shows "finite (r\<inverse>) = finite r"
permute_fun_def: fixes p :: "perm"   and f :: "'a \<Rightarrow> 'b" shows "p \<bullet> f = (\<lambda>x. p \<bullet> f (- p \<bullet> x))"
map_of_eq: fixes xs :: "('a \<times> 'b) list" shows "AList_Upd_Del.map_of xs = Map.map_of xs"
mk_minsky_seq: fixes P :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and R :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" assumes "mk_minsky P"   and "mk_minsky Q"   and "\<And>vs vs' vs''. \<lbrakk>P vs vs'; Q vs' vs''\<rbrakk> \<Longrightarrow> R vs vs''" shows "mk_minsky R"
distinct_5: fixes x41 :: "'b rexp"   and x42 :: "'b rexp" shows "Zero \<noteq> Plus x41 x42"
indep_varsD: fixes M' :: "'c \<Rightarrow> 'd measure"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> 'd"   and I :: "'c set"   and J :: "'c set"   and A :: "'c \<Rightarrow> 'd set" assumes "indep_vars M' X I"   and "J \<noteq> {}"   and "finite J"   and "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)" shows "prob (\<Inter>i\<in>J. X i -` A i \<inter> space M) = (\<Prod>i\<in>J. prob (X i -` A i \<inter> space M))"
rtrancl_refl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" shows "r\<^sup>*\<^sup>* a a"
matrix_inv_Gauss_Jordan_PA: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "invertible A" shows "matrix_inv A = fst (Gauss_Jordan_PA A)"
exhaust: fixes y :: "('a, 'b) Language" obtains x1 :: "'a" where "y = \<turnstile> x1 Type"   | x21 :: "'a" and x22 :: "('a, 'b) Expression" and x23 :: "'a" where "y = Vx : x21 \<turnstile> x22 : x23"   | x31 :: "'a" and x32 :: "('a, 'b) Expression" and x33 :: "('a, 'b) Expression" and x34 :: "'a" where "y = Vx : x31 \<turnstile> x32\<equiv>x33 : x34"
diff_numeral_special_9: shows "(1::'a) - (1::'a) = (0::'a)"
gcd_add1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and m :: "'a"   and n :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (plus m n) n = gcd m n"
sigma_sets_induct_disjoint: fixes G :: "'c set set"   and \<Omega> :: "'c set"   and A :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "Int_stable G"   and "G \<subseteq> Pow \<Omega>"   and "A \<in> sigma_sets \<Omega> G"   and "\<And>A. A \<in> G \<Longrightarrow> P A"   and "P {}"   and "\<And>A. \<lbrakk>A \<in> sigma_sets \<Omega> G; P A\<rbrakk> \<Longrightarrow> P (\<Omega> - A)"   and "\<And>A. \<lbrakk>disjoint_family A; range A \<subseteq> sigma_sets \<Omega> G; \<And>i. P (A i)\<rbrakk> \<Longrightarrow> P (\<Union> (range A))" shows "P A"
exhaust: fixes y :: "('b, 'a) hd" obtains x1 :: "'a" where "y = Var x1"   | x2 :: "'b" where "y = Sym x2"
MGFn_InitD: fixes n :: "nat"   and G :: "prog"   and A :: "state triple set"   and P :: "state \<Rightarrow> bool"   and C :: "qtname" assumes "\<forall>m. Suc m \<le> n \<longrightarrow> (\<forall>t. G,A\<turnstile>{=:m} t\<succ> {G\<rightarrow>})" shows "G,A\<turnstile>{Normal ((\<lambda>Y' s' s. s' = s \<and> P s) \<and>. G\<turnstile>init\<le>n)} .Init C. {(\<lambda>Y' s' s. G\<turnstile>s \<midarrow>In1r (Init C)\<succ>\<rightarrow> (Y', s') \<and> P s) \<and>. G\<turnstile>init\<le>n}"
IH: assumes "x \<in> set (independent_pdevs (filter (Not \<circ> coll 0 z) zs))"   and "y \<in> set (independent_pdevs (filter (Not \<circ> coll 0 z) zs))" shows "det3 0 x y \<noteq> 0"
bdd_plus: shows "bdd_above ((\<lambda>xa. L2_set (cmod \<circ> x) xa + L2_set (cmod \<circ> y) xa) ` Collect finite)"
substEnv_psubstEnv_idEnv: fixes ys :: "'varSort"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "rho &[Y / y]_ys = rho &[idEnv [y \<leftarrow> Y]_ys]"
sorted_list_of_fset_simps_2: fixes S :: "'b fset" shows "fset_of_list (sorted_list_of_fset S) = S"
sub_diff_mod_eq: fixes r :: "nat"   and t :: "nat"   and m :: "nat" assumes "r \<le> t" shows "(t - (t - r) mod m) mod m = r mod m"
subset_imageE: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "B \<subseteq> f ` A" obtains C :: "'b set" where "C \<subseteq> A" and "B = f ` C"
reduced_word_for_length: fixes A :: "'a set"   and a :: "'a"   and as :: "'a list" assumes "reduced_word_for A a as" shows "length as = word_length A a"
integrate_insert_commute_axioms: shows "integrate_insert_commute M a s"
elims_3: fixes x :: "'e \<times> ('f, 'g, 'h, 'i) SLin_state_scheme" assumes "\<not> P2 x" obtains s1 :: "'e" and s2 :: "('f, 'g, 'h, 'i) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (\<forall>p. status s2 p \<noteq> Sleep \<longrightarrow> fst (pending s2 p) = p)"
Subring_pOp_ring_pOp: fixes S :: "('a, 'd) Ring_scheme"   and a :: "'a"   and b :: "'a" assumes "Subring R S"   and "a \<in> carrier S"   and "b \<in> carrier S" shows "a \<plusminus>\<^bsub>S\<^esub> b = a \<plusminus> b"
subset_imageE: fixes B :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "B \<subseteq> f ` A" obtains C :: "'e set" where "C \<subseteq> A" and "B = f ` C"
DisjointPaths_axioms: shows "DisjointPaths H_x v0 v1 Q"
assms_3: shows "ipresIGCons h' hA' MOD' MOD''"
Let_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "Let \<Gamma> e = LetA (heapToAssn \<Gamma>) e"
cnfi_rel_def: shows "cnfi_rel \<equiv> br cnf_\<alpha> is_2sati"
list_all2_rightE: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and y :: "'b" assumes "list_all2 P xs ys"   and "y \<in> set ys" obtains x :: "'a" where "x \<in> set xs" and "P x y"
sq_norm_vec_def: fixes v :: "'a vec" shows "\<parallel>v\<parallel>\<^sup>2 \<equiv> sum_list (map sq_norm (list_of_vec v))"
multi_nonempty_split: fixes M :: "'b multiset" assumes "M \<noteq> {#}" shows "\<exists>A a. M = a, A"
homeomorphic_ENR_iff_ENR: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "ENR S = ENR T"
numSt_transC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "noWhile c"   and "(c, s) \<rightarrow>c (c', s')" shows "numSt c' < numSt c"
similar_matrix_trans: fixes A :: "(('b, 'c) vec, 'c) vec"   and B :: "(('b, 'c) vec, 'c) vec"   and C :: "(('b, 'c) vec, 'c) vec" assumes "similar_matrix A B"   and "similar_matrix B C" shows "similar_matrix A C"
hyperb_ineq': fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and deltaG :: "'a itself \<Rightarrow> real"   and e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG" shows "min (Gromov_product_at e x y) (Gromov_product_at e y z) \<le> Gromov_product_at e x z + deltaG TYPE('a)"
rep_in_arr: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.rep f \<in> f"
interleave_butlast: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<in> ys \<otimes> zs" shows "butlast xs \<in> butlast ys \<otimes> zs \<or> butlast xs \<in> ys \<otimes> butlast zs"
conjunctionD1: fixes A :: "prop"   and B :: "prop" assumes "PROP A &&& PROP B" shows "PROP A"
axioms_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "abel_semigroup f" shows "semigroup f"
f_nxt_T_delete: fixes f :: "'b \<Rightarrow> 'b ttree"   and x :: "'b"   and T :: "'b set"   and x' :: "'b" assumes "f x = TTree.empty" shows "f_nxt f (T - {x}) x' = f_nxt f T x'"
1: fixes k :: "nat" shows "indicat_real (\<Union> (S ` {..k})) integrable_on UNIV"
powr_mult_base: fixes x :: "real"   and y :: "real" assumes "0 \<le> x" shows "x * x powr y = x powr (1 + y)"
that: fixes m :: "nat" assumes "m \<le> n"   and "normalize a = normalize (p ^ m)" shows "thesis"
part_circlepath_rotate_left: fixes c :: "complex"   and r :: "real"   and x :: "real"   and a :: "real"   and b :: "real" shows "part_circlepath c r (x + a) (x + b) = (\<lambda>z. c + cis x * (z - c)) \<circ> part_circlepath c r a b"
assms: shows "(B, C) \<in> s_mul_ext W S"
imp: shows "((\<lambda>s. True) \<longrightarrow>* P) h"
similar'_base_adm: shows "adm (\<lambda>x. similar'_base (fst x) (snd x))"
finite_has_maximal2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and a :: "'c" assumes "class.order less_eq less"   and "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. less_eq a m \<and> (\<forall>b\<in>A. less_eq m b \<longrightarrow> m = b)"
lsc_hull_of_proper_convex_proper: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f"   and "proper f" shows "proper (lsc_hull f)"
assms: fixes x :: "'a"   and y :: "'a" assumes "r x y" shows "r (f x) (f y)"
elims_1: fixes x :: "net_tree"   and y :: "bool" assumes "wf_net_tree x = y" obtains p1 :: "net_tree" and p2 :: "net_tree" where "x = p1 \<parallel> p2" and "y = (net_tree_ips p1 \<inter> net_tree_ips p2 = {} \<and> wf_net_tree p1 \<and> wf_net_tree p2)"   | i :: "nat" and R :: "nat set" where "x = \<langle>i; R\<rangle>" and "y"
simple_path_reversepath: fixes g :: "real \<Rightarrow> 'b" assumes "simple_path g" shows "simple_path (reversepath g)"
latestActless: fixes n\<^sub>s :: "nat"   and n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'\<ge>n\<^sub>s. n' < n \<and> active nid (t n')" shows "n\<^sub>s \<le> dynamic_component.latestAct active nid t n"
dom: shows "\<D>\<^sub>\<circ> vrat_inverse = \<rat>\<^sub>\<circ>"
assms_1: shows "proj2_line_abs v = proj2_line_abs w"
fag_aGroup: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set" assumes "commute_bpp f (aug_pm_set z i A)"   and "assoc_bpp (aug_pm_set z i A) f"   and "ipp_cond1 A i"   and "ipp_cond2 z A i f"   and "ipp_cond3 z i"   and "inv_ipp z i f A"   and "commute_bpp f (aug_pm_set z i A)"   and "zeroA z i f A z" shows "aGroup (fag_gen_by A f i z)"
prems: shows "\<not> is_final (steps0 c A (Suc n))"
select_convs_1: fixes carrier :: "'c set"   and eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and more :: "'d" shows "(.=\<^bsub>\<lparr>carrier = carrier, eq = eq, \<dots> = more\<rparr>\<^esub>) = eq"
assms_8: shows "F\<^sub>i\<^sub>n\<^sub>f = (\<forall>rel\<in>S. \<forall>(a, b, c)\<in>X rel. \<exists>x. \<forall>y<x. rel (a * y\<^sup>2 + b * y + c) 0)"
assms_4: shows "integral\<^sup>N N f \<le> 1"
assms_4: shows "(\<Sum>x\<in>X. \<beta> x * exp x) = 0"
factorize_polys_def: fixes ps :: "rat poly list" shows "factorize_polys ps = (let fact_ps = map factorize_rat_poly_monic ps; factors = remdups (map fst (concat (map snd fact_ps))) in Let (map (\<lambda>(c, fs). (c, map (\<lambda>(f, pow). (index_of factors f, pow)) fs)) fact_ps) (Pair factors))"
log_divide: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "0 < a"   and "a \<noteq> 1"   and "0 < x"   and "0 < y" shows "log a (x / y) = log a x - log a y"
in_diffD: fixes a :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "a \<in># M - N" shows "a \<in># M"
aux26: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set"   and p :: "int" assumes "twoNetsDistinct a b c d" shows "dom (C (AllowPortFromTo a b p)) \<inter> dom (C (DenyAllFromTo c d)) = {}"
not_IndP_VarP: fixes x :: "tm"   and A :: "fm" shows "{IndP x, VarP x} \<turnstile> A"
assms_4: shows "x = (SOME y. y \<in> U)"
IrrI: assumes "\<And>U. RModule FG (\<cdot>) U \<and> U \<subseteq> V \<Longrightarrow> U = 0 \<or> U = V" shows "IrrFinGroupRepresentation G (\<cdot>) V"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "\<^bold>\<lfloor>t \<^bold>\<cdot> u\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>"
summ_zg: fixes A :: "nat set" shows "summable (zero_on g A)"
ext_field_simp: fixes f :: "qtname \<times> 'b decl_scheme" shows "(declclass f, fld f) = f"
Pd_above_irrelevant: fixes Pd' :: "'b \<Rightarrow> ('a \<times> 'a) set"   and d' :: "'b"   and X :: "'a set"   and x :: "'a"   and ds :: "'b set" assumes "Linear_order (Pd' d')"   and "Field (Pd' d') \<subseteq> {y. Xd y = d'}"   and "\<forall>d. d \<noteq> d' \<longrightarrow> Pd' d = Pd d"   and "Singleton_for_d.dX X d' \<subseteq> Field (Pd' d')"   and "Above (Pd' d') (Singleton_for_d.dX X d') \<subseteq> Above (Pd d') (Singleton_for_d.dX X d')"   and "x \<in> X"   and "stable_on ds X" shows "\<exists>y\<in>Singleton_for_d.CH (Contracts.fp_cop_F Pd' Ch ds). (x, y) \<in> Pd' (Xd x)"
Abrupt: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and t :: "Term.term"   and n :: "nat" shows "G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (undefined3 t, (Some xc, s))"
HInv3_def: fixes s :: "state" shows "HInv3 s = (\<forall>p q d. HInv3_inner s p q d)"
rADnMT: fixes p :: "('b, 'c) Combinators list" assumes "p \<noteq> []" shows "removeAllDuplicates p \<noteq> []"
split_sel_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> option = Some (the option) \<and> \<not> P (f2 (the option))))"
elims: fixes x :: "'a Point"   and xa :: "'a"   and y :: "'a Cone" assumes "mkCone x xa = y" obtains v :: "'a Point" and s :: "'a" where "x = v" and "xa = s" and "y = \<lparr>vertex = v, slope = s\<rparr>"
fplus_assoc: fixes a :: "'e\<cdot>'f"   and b :: "'e\<cdot>'f"   and c :: "'e\<cdot>'f" shows "fplus\<cdot>(fplus\<cdot>a\<cdot>b)\<cdot>c = fplus\<cdot>a\<cdot>(fplus\<cdot>b\<cdot>c)"
ivl_disj_un_14: fixes l :: "'c"   and u :: "'c" assumes "l \<le> u" shows "{l..<u} \<union> {u..} = {l..}"
s_max: fixes x :: "real" shows "delayed_safe_distance.other.s x \<le> delayed_safe_distance.other.s_stop"
validF: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
pole_polar: fixes p :: "proj2" shows "pole (polar p) = p"
Abs_fset_inverse: fixes y :: "'a set" assumes "y \<in> {A. finite A}" shows "fset (Abs_fset y) = y"
rec_all2_def: fixes f :: "recf" shows "rec_all2 f = CN rec_sign [rec_accum2 f]"
sg_l_unit: fixes H :: "'a set"   and h :: "'a" assumes "G \<guillemotright> H "   and "h \<in> H" shows "\<one> \<cdot> h = h"
assms_4: fixes M :: "nat \<Rightarrow> 'a" assumes "incseq M"   and "\<And>i. P (M i)" shows "\<alpha> (Sup (range M)) = (SUP i. \<alpha> (M i))"
swap_usubst_inj: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x"   and "vwb_lens y"   and "x \<bowtie> y" shows "inj [x \<mapsto>\<^sub>s &y, y \<mapsto>\<^sub>s &x]"
core_mono: fixes a :: "'a set"   and b :: "'a set" assumes "a \<subseteq> b" shows "core a \<subseteq> core b"
absabc: shows "\<bar>a\<bar> = (int m)\<^sup>2 - (int n)\<^sup>2 \<and> \<bar>b\<bar> = 2 * int m * int n \<and> \<bar>c\<bar> = (int m)\<^sup>2 + (int n)\<^sup>2"
abs_pdevs_val_less_tdev: fixes e :: "nat \<Rightarrow> real"   and x :: "'a pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1<..<1}"   and "0 < degree x" shows "\<bar>pdevs_val e x\<bar> < tdev x"
kop_idem: shows "op\<^sub>K \<circ> op\<^sub>K = id"
abs_le_D1: fixes a :: "'b"   and b :: "'b" assumes "\<bar>a\<bar> \<le> b" shows "a \<le> b"
simps_2: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "wellformed_policy1_strong (x # xs) = (x = DenyAll \<and> DenyAll \<notin> set xs)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (f, cod \<iota>)) local.map \<rho>.map"
ball_biholomorphism_exists: fixes a :: "complex" assumes "a \<in> ball 0 1" obtains f :: "complex \<Rightarrow> complex" and g :: "complex \<Rightarrow> complex" where "f a = 0" and "f holomorphic_on ball 0 1" and "f ` ball 0 1 \<subseteq> ball 0 1" and "g holomorphic_on ball 0 1" and "g ` ball 0 1 \<subseteq> ball 0 1" and "\<And>z. z \<in> ball 0 1 \<Longrightarrow> f (g z) = z" and "\<And>z. z \<in> ball 0 1 \<Longrightarrow> g (f z) = z"
assms_4: shows "\<not> P (a, hd xs) \<or> \<not> P (last xs, a)"
as_sem_env_def: fixes C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> sterm) list"   and env :: "v sem_env" shows "as_sem_env C_info rs env = make_sem_env (build_rec_env (cakeml.mk_letrec_body C_info (fset_of_list (map fst rs) |\<union>| constructors.C C_info) rs) env nsEmpty) nsEmpty"
OUTfromVCorrect1_data7: shows "OUTfromVCorrect1 data7"
Red_I_of_subset: fixes Bot :: "'b set"   and Inf :: "'b inference set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I :: "'b set \<Rightarrow> 'b inference set"   and Red_F :: "'b set \<Rightarrow> 'b set"   and N :: "'b set"   and N' :: "'b set" assumes "calculus Bot Inf entails Red_I Red_F"   and "N \<subseteq> N'" shows "Red_I N \<subseteq> Red_I N'"
ltake_all: fixes xs :: "'b llist"   and m :: "enat" assumes "llength xs \<le> m" shows "ltake m xs = xs"
unitE: fixes Y :: "'b ptrm" assumes "PUnit \<approx> Y" obtains "Y = PUnit"
trans_Restr: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "trans r" shows "trans (Restr r A)"
WTrt_binop_simps_8: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>Subtract\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
simps_1: fixes onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and i :: "nat"   and R\<^sub>i :: "nat set" shows "opnet onp \<langle>i; R\<^sub>i\<rangle> = \<langle>i : onp i : R\<^sub>i\<rangle>\<^sub>o"
terminates_while_lemma: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and w :: "('e, 'd, 'f) com"   and fk :: "('e, 'f) xstate"   and f :: "nat \<Rightarrow> 'e"   and k :: "nat"   and b :: "'e set"   and c :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile>w \<down> fk"   and "fk = Normal (f k)"   and "w = While b c"   and "\<forall>i. \<Gamma>\<turnstile> \<langle>c,Normal (f i)\<rangle> \<Rightarrow> Normal (f (Suc i))" shows "\<exists>i. f i \<notin> b"
ssaBS_nonempty: fixes g :: "'g"   and s :: "'val \<Rightarrow> 'node option" shows "\<not> g \<turnstile> []\<Down>\<^sub>s s"
prime_power_inj'': 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 p :: "'a"   and q :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q" shows "(power.power one times p m = power.power one times q n) = (m = 0 \<and> n = 0 \<or> p = q \<and> m = n)"
eq_name_eq_pres: fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and pres' :: "(nat \<times> nat) list"   and effs' :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer' :: "nat" assumes "(name, pres, effs, layer) \<in> set ast\<delta>"   and "(name, pres', effs', layer') \<in> set ast\<delta>" shows "pres = pres'"
simps: fixes N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "P_of_state (N, P, Q) = P"
kmp2_correct: fixes s :: "'a list"   and t :: "'a list" assumes "s \<noteq> []" shows "kmp2 s t \<le> kmp_SPEC s t"
monotonic: fixes P :: "ccs"   and A :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and B :: "(ccs \<times> ccs) set" assumes "P \<leadsto>[A] Q"   and "A \<subseteq> B" shows "P \<leadsto>[B] Q"
l_cancel: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Dgrp prod one" shows "(prod x y = prod x z) = (y = z)"