hcju/isabelleps huggingface dataset:
infinite_hyp_wrong_not_BC: fixes f :: "nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and s :: "nat \<Rightarrow> nat option" assumes "f \<in> U"   and "\<forall>n. \<exists>m>n. \<psi> (the (s (f \<triangleright> m))) \<noteq> f" shows "\<not> learn_bc \<psi> U s"
abc_app: assumes "2\<^sub>\<nat> \<in>\<^sub>\<circ> 3\<^sub>\<nat>" shows "abc\<lparr>2\<^sub>\<nat>\<rparr> \<in>\<^sub>\<circ> (if 2\<^sub>\<nat> = []\<^sub>\<circ> \<Rightarrow> \<AA> | 2\<^sub>\<nat> = 1\<^sub>\<nat> \<Rightarrow> \<BB> | otherwise \<Rightarrow> \<CC>)\<lparr>Obj\<rparr>"
preserves_dom: fixes f :: "'a \<times> 'a" assumes "C.VV.arr f" shows "C.dom (fst f \<star>\<^sub>C snd f) = fst (C.VV.dom f) \<star>\<^sub>C snd (C.VV.dom f)"
g_def: shows "g = Cn 4 r_ifless [recf.Id 4 1, len, recf.Id 4 1, Cn 4 r_ifeq [ph, nth, recf.Id 4 1, recf.Id 4 0]]"
continuous_on_open_invariant: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
simps_2: fixes uv :: "pnat" shows "pnat_plus pnat.PInfty uv = pnat.PInfty"
not_zero_jB: shows "\<not> is_zero_row_upt_k j k B"
diag_mat_mult_diag_mat: fixes B :: "'c mat"   and n :: "nat"   and D :: "'c mat" assumes "B \<in> carrier_mat n n"   and "D \<in> carrier_mat n n"   and "diagonal_mat B"   and "diagonal_mat D" shows "B * D = mat n n (\<lambda>(i, j). if i = j then B $$ (i, i) * D $$ (i, i) else (0::'c))"
get_match_def: fixes rule :: "'b rule" shows "get_match rule = (case rule of Rule x1 x2 \<Rightarrow> x1)"
urel64_power: fixes x :: "uint64"   and x' :: "int"   and y :: "uint64"   and y' :: "nat" assumes "urel64 x x'"   and "urel64 y (int y')" shows "urel64 (power_p64 pp x y) (power_p p x' y')"
periodic_mult: fixes a :: "nat"   and m :: "nat" shows "\<chi> (a + m * n) = \<chi> a"
4: shows "(w \<sqinter> - d)\<^sup>T * - (w\<^sup>T\<^sup>\<star> * d\<^sup>T * top) \<le> - (w\<^sup>T\<^sup>\<star> * d\<^sup>T * top)"
P3S'Q3: shows "(P3, S', Q3) = Smith_mxn H2_DR"
pos_unitl: fixes x :: "'a"   and y :: "'a"   and e :: "'a" assumes "D x y"   and "e \<in> E"   and "x \<oplus> y = e" shows "x = e"
smap_sconst: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "smap f (pure x) = pure (f x)"
gtr2gtrp_s: fixes s :: "'s"   and c :: "'s multiset"   and e :: "'l"   and c' :: "'s multiset" assumes "(add_mset s c, e, c') \<in> gtr gtrs" obtains s' :: "'s" and ce' :: "'s multiset" and ee :: "'l el_step" where "c' = add_mset s' ce'" and "e = le_rem_s ee" and "((s, c), ee, s', ce') \<in> gtrp gtrs"
q: shows "q = p - monom_mult (lookup (rep_list p) (t + punit.lt (rep_list f)) / punit.lc (rep_list f)) t f"
step_no_action: fixes s :: "'a list"   and q :: "'a" shows "step s q (0, []) = s"
set_cases_2: fixes e :: "'f"   and a :: "('g, 'f, 'h) new_thread_action" assumes "e \<in> set2_new_thread_action a" obtains z1 :: "'g" and z3 :: "'h" where "a = NewThread z1 e z3"
zero_in_succ: fixes i :: "hf" assumes "Ord i" shows "0 \<^bold>\<in> succ i"
bindK_def: shows "bindK \<equiv> \<Lambda> x g s f. x\<cdot>(\<Lambda> xv f'. g\<cdot>xv\<cdot>s\<cdot>f')\<cdot>f"
Ln_times_ii: fixes z :: "complex" assumes "z \<noteq> 0" shows "Ln (\<i> * z) = (if 0 \<le> Re z \<or> Im z < 0 then Ln z + \<i> * complex_of_real pi / 2 else Ln z - \<i> * complex_of_real (3 * pi / 2))"
sim_rel2_def: shows "sim_rel2 \<equiv> \<lambda>(c, n, ts) (ci, ni, tsi). (if is_REL_state ts then ci = replicate N False else ci = (replicate N False)[c := True]) \<and> ni = n \<and> tsi = ts"
valid_ENC_def: fixes n :: "nat"   and p :: "nat" shows "valid_ENC n p = (if n = 0 then Full else TIMES [Star (Atom (Arbitrary_Except p False)), Atom (Arbitrary_Except p True), Star (Atom (Arbitrary_Except p False))])"
mod_div_mult_eq: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (modulo a b) (times (divide a b) b) = a"
awalk_ends_eqD: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "G.awalk u p u"   and "G.awalk v p w" shows "v = w"
divide_fract: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "Fraction_Field.Fract a b / Fraction_Field.Fract c d = Fraction_Field.Fract (a * d) (b * c)"
subst_apply_stateful_strand_def: fixes S :: "('d, 'e) stateful_strand_step list"   and \<theta> :: "'e \<Rightarrow> ('d, 'e) Term.term" shows "S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
modifies_spec: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)"
guaranteesI: fixes F :: "'d program"   and X :: "'d program set"   and Y :: "'d program set" assumes "\<And>G. \<lbrakk>F ok G; F \<squnion> G \<in> X\<rbrakk> \<Longrightarrow> F \<squnion> G \<in> Y" shows "F \<in> X guarantees Y"
nodeToDigit_inv_2: fixes nd :: "('a, 'b) Node" assumes "FingerTreeStruc.is_measured_node nd" shows "FingerTreeStruc.is_measured_digit (FingerTreeStruc.nodeToDigit nd)"
qp_close_after_exexH_k: fixes P :: "complex mat"   and k :: "nat" assumes "is_quantum_predicate P"   and "k < n" shows "is_quantum_predicate (adjoint (exexH_k k) * P * exexH_k k)"
attract: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x"   and "y \<sqsubseteq> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubseteq> z"
simulation2: fixes s1 :: "'a"   and s2 :: "'c"   and tl2 :: "'d"   and s2' :: "'c" assumes "bisim s1 s2"   and "trsys2 s2 tl2 s2'"   and "\<not> False" shows "\<exists>s1' s1'' tl1. s1 -\<tau>1\<rightarrow>* s1' \<and> trsys1 s1' tl1 s1'' \<and> \<not> False \<and> bisim s1'' s2' \<and> tlsim tl1 tl2"
assms: shows "s' = user_reg_mod d w r s \<and> get_S (cpu_reg_val PSR s) = 0"
t_App: fixes \<Gamma> :: "(name \<times> ty) list"   and t\<^sub>1 :: "lam"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty"   and t\<^sub>2 :: "lam" assumes "\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 \<rightarrow> T\<^sub>2"   and "\<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1" shows "\<Gamma> \<turnstile> App t\<^sub>1 t\<^sub>2 : T\<^sub>2"
is_ground_Max: shows "is_ground_atm (Max (atms_of D \<union> set As))"
pathfinish_compose: fixes f :: "'c \<Rightarrow> 'b"   and p :: "real \<Rightarrow> 'c" shows "pathfinish (f \<circ> p) = f (pathfinish p)"
complement_eq: fixes x :: "i" shows "\<midarrow> x = (THE z. \<forall>w. P w z = (\<not> O w x))"
plane_of_def: fixes sctn :: "'a sctn" shows "plane_of sctn = plane (normal sctn) (pstn sctn)"
assms_7: shows "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"
dim_eq: fixes X :: "('a \<Rightarrow> real) set" assumes "X \<subseteq> dual_space S" shows "dual.dim X = dim X"
terminating_path_root_end_iff: fixes x :: "'a" shows "(\<exists>r e. terminating_path_root_end r x e) = terminating_path x"
ccFix_cong: fixes heap2 :: "(Vars.var \<times> exp) list"   and cccexp1 :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and cccexp2 :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and heap1 :: "(Vars.var \<times> exp) list" assumes "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> cccexp1 e = cccexp2 e"   and "heap1 = heap2" shows "CoCallArityAnalysis.CCfix cccexp1 heap1 = CoCallArityAnalysis.CCfix cccexp2 heap2"
successive_decomp: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list"   and zs :: "'b list"   and z :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "xs = x # ys @ zs @ [z]"   and "successive P xs"   and "\<not> P x \<or> \<not> P z" shows "successive P (zs @ [z] @ x # ys)"
vpair_iff: fixes a :: "V"   and b :: "V"   and a' :: "V"   and b' :: "V" shows "(\<langle>a, b\<rangle> = \<langle>a', b'\<rangle>) = (a = a' \<and> b = b')"
map2_val_compat_Nil: fixes f :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'f) list" assumes "map2_val_compat f" shows "f [] = []"
simps: fixes a :: "'a llist" shows "maximal_path a = (a = LNil \<or> (\<exists>v. a = LCons v LNil \<and> deadend v) \<or> (\<exists>Ps v. a = LCons v Ps \<and> \<not> lnull Ps \<and> maximal_path Ps))"
point_measure_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> ennreal" shows "point_measure A f = density (count_space A) f"
WT_converter_comp: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter"   and \<I>'' :: "('m, 'n) \<I>"   and conv' :: "('k, 'l, 'm, 'n) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "\<I>', \<I>'' \<turnstile>\<^sub>C conv' \<surd>" shows "\<I>, \<I>'' \<turnstile>\<^sub>C conv \<odot> conv' \<surd>"
L_set: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and L_left :: "nat list"   and L_right :: "nat list" assumes "rtf_invar c s t f l n L_left L_right" shows "set L_left \<union> set L_right = Graph.V c - {s, t}"
empty_change_no_change: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "vars_change as vs s = []" shows "fmrestrict_set vs (exec_plan s as) = fmrestrict_set vs s"
wellorder_Least_lemma_1: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and k :: "'d" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
formulaTrueAndFormulaFalseImpliesInconsistent: fixes formula :: "Literal list list"   and valuation :: "Literal list" assumes "formulaTrue formula valuation"   and "formulaFalse formula valuation" shows "inconsistent valuation"
arr_char: fixes f :: "'a" shows "hom'.arr f = \<guillemotleft>f : trg (src f) \<rightarrow> trg e\<guillemotright>"
invertible_left_inverse: fixes u :: "'a" assumes "units.sub.invertible u"   and "u \<in> Units" shows "units.sub.inverse u \<cdot> u = \<one>"
c_def: shows "c \<equiv> min (\<Sqinter> (f ` results_gpv \<I> gpv)) fail"
simps: fixes dmus :: "'a iarray iarray"   and dmusi :: "'a iarray"   and dmusj :: "'a iarray"   and dll :: "'a"   and l :: "nat" shows "sigma_array dmus dmusi dmusj dll l = (if l = 0 then dmusi !! l * dmusj !! l else let l1 = l - 1; dll1 = dmus !! l1 !! l1 in (dll * sigma_array dmus dmusi dmusj dll1 l1 + dmusi !! l * dmusj !! l) div dll1)"
comm_monoid_add_hom_axioms: shows "comm_monoid_add_hom (\<lambda>a. [:a:])"
RRR_def: shows "RRR = {0<..<1} \<union> {1<..<2} \<union> {3} \<union> {5<..<7} \<inter> \<rat>"
Suffix_def: fixes L :: "'c list set" shows "Suffix L = {w. \<exists>u. u @ w \<in> L}"
alt: shows "qs = [x, y, y] \<or> qs = [y, y]"
subcocycle_cmult: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and c :: "real" assumes "subcocycle u"   and "0 \<le> c" shows "subcocycle (\<lambda>n x. c * u n x)"
subset_singletonD: fixes A :: "'k set"   and x :: "'k" assumes "A \<subseteq> {x}" shows "A = {} \<or> A = {x}"
get_norm_value: fixes pp :: "'a"   and a :: "'a"   and a0 :: "'a" assumes "(0::'a) < pp" obtains k :: "int" and a1 :: "'a" where "a = of_int k * pp + a1" and "a0 \<le> a1" and "a1 < a0 + pp"
shC: shows "sh C = \<lfloor>(sfs, Processing)\<rfloor>"
ofailE: fixes s :: "'e"   and v :: "'d"   and P :: "bool" assumes "ofail s = Some v" shows "P"
isometry_on_inverse_2: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "isometry_on X f"   and "x \<in> X" shows "inv_into X f (f x) = x"
suntil_def: shows "(suntil) \<equiv> \<lambda>\<phi> \<psi>. lfp (\<lambda>p x. (\<exists>\<omega>. x = \<omega> \<and> \<psi> \<omega>) \<or> (\<exists>\<omega>. x = \<omega> \<and> \<phi> \<omega> \<and> p (stl \<omega>)))"
fset_of_list_append: fixes xs :: "'b list"   and ys :: "'b list" shows "fset_of_list (xs @ ys) = fset_of_list xs |\<union>| fset_of_list ys"
nv_OfWax_1: fixes x :: "Preliminaries.var"   and \<sigma>' :: "'d" shows "x \<notin> nvT (lOfWax \<sigma>')"
FactGroup_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "G Mod H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
analz_Un_PairSet: fixes G :: "msg set"   and G' :: "msg set"   and H :: "msg set" shows "analz (PairSet G G' \<union> H) \<subseteq> PairSet G G' \<union> analz (G \<union> G' \<union> H)"
cases: fixes a :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "resumptionp a" obtains aa :: "'h" where "a = TERMINAL aa"   | r :: "('f + 'g + 'h, 'i) Old_Datatype.node set" and b :: "'g" where "a = LINEAR b r" and "resumptionp r"   | rs :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" and c :: "'f" where "a = BRANCH c rs" and "\<And>d. resumptionp (rs d)"
IH: shows "mapping_generator V ks = {m. Mapping.keys m = set ks \<and> (\<forall>k\<in>set ks. the (Mapping.lookup m k) \<in> set (V k))}"
subgraph_trans: fixes G\<^sub>1 :: "('d, 'e) labeled_graph"   and G\<^sub>2 :: "('d, 'e) labeled_graph"   and G\<^sub>3 :: "('d, 'e) labeled_graph" assumes "subgraph G\<^sub>1 G\<^sub>2"   and "subgraph G\<^sub>2 G\<^sub>3" shows "subgraph G\<^sub>1 G\<^sub>3"
t_insert_height: fixes x :: "'b"   and t :: "'b tree" shows "t_insert x t \<le> height t + 1"
assms: shows "u \<rightarrow>\<^bsup>n\<^esup> v"
lookup_monom_mult: fixes c :: "'b"   and t :: "'c"   and p :: "'c \<Rightarrow>\<^sub>0 'b"   and v :: "'c" shows "lookup (MPoly_Type_Class.punit.monom_mult c t p) v = (if MPoly_Type_Class.punit.adds_pp t v then c * lookup p (MPoly_Type_Class.punit.sminus v t) else (0::'b))"
ih_6: shows "InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
pderiv_smult: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier P" shows "pderiv (a \<odot>\<^bsub>P\<^esub> f) = a \<odot>\<^bsub>P\<^esub> pderiv f"
sup_continuous_def: fixes F :: "'a \<Rightarrow> 'b" shows "sup_continuous F = (\<forall>M. incseq M \<longrightarrow> F (\<Squnion> range M) = (\<Squnion>i. F (M i)))"
disjoint_iff_not_equal: fixes A :: "'g set"   and B :: "'g set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
function_zero_eval: fixes R :: "('a, 'b) ring_scheme"   and a :: "'c"   and S :: "'c set" assumes "ring_functions R"   and "a \<in> S" shows "\<zero>\<^bsub>function_ring S R\<^esub> a = \<zero>\<^bsub>R\<^esub>"
FactRing_iso_set_aux: shows "(\<lambda>X. the_elem (h ` X)) \<in> ring_iso (R Quot a_kernel R S h) (S\<lparr>carrier := h ` carrier R\<rparr>)"
simps: fixes uu :: "'c"   and uv :: "'d"   and ts :: "('c, 'd) bintree list" shows "children (Node uu uv ts) = ts"
ahm_delete_correct_1: fixes hm :: "('a, 'b) ArrayHashMap_Impl.hashmap"   and k :: "'a" assumes "ArrayHashMap_Impl.ahm_invar hm" shows "ArrayHashMap_Impl.ahm_invar (ArrayHashMap_Impl.ahm_delete k hm)"
sep_globalise: fixes P :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool"   and h :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "(P \<and>* R) h"   and "\<And>h. P h \<Longrightarrow> Q h" shows "(Q \<and>* R) h"
weak_unit_self_composable_2: fixes a :: "'a" assumes "weak_unit a" shows "ide (a \<star> a)"
pos_recurrentI_communicating: fixes y :: "'s"   and x :: "'s" assumes "pos_recurrent y"   and "(y, x) \<in> communicating" shows "pos_recurrent x"
awhd_of_awalk: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v" shows "hd (pawalk_verts u p) = u"
con_rews_1: fixes cfun :: "Value \<rightarrow> Value" shows "Fn\<cdot>cfun \<noteq> \<bottom>"
assms: shows "sum_list (map (num_syms \<circ> (\<lambda>t. t \<cdot> \<sigma>)) ts) = sum_list (map num_syms ts)"
19: fixes x :: "'a" shows "- x \<sqinter> x = bot"
mod_minus_cong: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and a' :: "'c" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a b = modulo a' b" shows "modulo (uminus a) b = modulo (uminus a') b"
gcd_smult_left: fixes c :: "'a"   and f :: "'a poly"   and g :: "'a poly" assumes "c \<noteq> (0::'a)" shows "gcd (smult c f) g = gcd f g"
elims: fixes x :: "real poly"   and xa :: "real poly list \<times> rat mat \<times> nat list list \<times> rat list list"   and xb :: "real poly list \<times> rat mat \<times> nat list list \<times> rat list list"   and y :: "real poly list \<times> rat mat \<times> nat list list \<times> rat list list" assumes "combine_systems x xa xb = y" obtains p :: "real poly" and qs1 :: "real poly list" and m1 :: "rat mat" and sub1 :: "nat list list" and sgn1 :: "rat list list" and qs2 :: "real poly list" and m2 :: "rat mat" and sub2 :: "nat list list" and sgn2 :: "rat list list" where "x = p" and "xa = (qs1, m1, sub1, sgn1)" and "xb = (qs2, m2, sub2, sgn2)" and "y = smash_systems p qs1 qs2 sub1 sub2 sgn1 sgn2 m1 m2"
Max_eqI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq y x"   and "x \<in> A" shows "linorder.Max less_eq A = x"
spp_inv_def: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "spp_inv sp = Ex (spp_rel sp)"
approx_mult_star_of: fixes a :: "'a star"   and b :: "'a"   and c :: "'a star"   and d :: "'a" assumes "a \<approx> star_of b"   and "c \<approx> star_of d" shows "a * c \<approx> star_of b * star_of d"
simps_2: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree" shows "ltree \<langle>l, (a, n), r\<rangle> = (min_height r \<le> min_height l \<and> n = min_height r + 1 \<and> ltree l \<and> ltree r)"
start_heap_obs_def: fixes empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set" shows "heap_base.start_heap_obs empty_heap allocate = map2 (\<lambda>C a. NewHeapElem a (Class_type C)) initialization_list (fst (snd (heap_base.start_heap_data empty_heap allocate)))"
apx_empty_iff: fixes Z :: "('c \<Rightarrow> real) set"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "Z \<subseteq> V"   and "Z = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n" shows "(Z = {}) = (Approx\<^sub>\<beta> Z = {})"
dj_tyvrs_vrs: shows "disjoint TYPE(tyvrs) TYPE(vrs)"
trans: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word" assumes "a \<le>s b"   and "b \<le>s c" shows "a \<le>s c"
id_func_preserves_arrows: shows "id_func AA \<^bsub>\<a>\<^esub> \<in> Ar \<rightarrow> Ar"
continuous_logarithm_on_ball: fixes a :: "'a"   and r :: "real"   and f :: "'a \<Rightarrow> complex" assumes "continuous_on (ball a r) f"   and "\<And>z. z \<in> ball a r \<Longrightarrow> f z \<noteq> 0" obtains h :: "'a \<Rightarrow> complex" where "continuous_on (ball a r) h" and "\<And>z. z \<in> ball a r \<Longrightarrow> f z = exp (h z)"
sign_changes_mult_aux: fixes d :: "real"   and f :: "'a \<Rightarrow> real"   and xs :: "'a list" assumes "d \<noteq> 0" shows "length (remdups_adj (filter (\<lambda>x. x \<noteq> 0) (map ((*) d \<circ> f) xs))) = length (remdups_adj (filter (\<lambda>x. x \<noteq> 0) (map f xs)))"
sum_closed: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'r" assumes "finite A"   and "f ` A \<subseteq> R" shows "sum f A \<in> R"
order_iff: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (a = sup a b)"
b: shows "F @@ X \<in> obj\<^bsub>CatDom G\<^esub>"
Suc_k_less_card: shows "Suc k < ncols A"
list_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp R" shows "transp (list_all2 R)"
ln_upper_13_neg: fixes x :: "real" assumes "0 < x"   and "x \<le> 1" shows "ln x \<le> ln_upper_13 x"
T_Mprefix: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "T (Mprefix A P) = {s. s = [] \<or> (\<exists>a. a \<in> A \<and> s \<noteq> [] \<and> hd s = ev a \<and> tl s \<in> T (P a))}"
set_s_A: fixes n :: "nat" shows "set (s_A n) = set init"
hyps_2: shows "M' = (\<lambda>i' j'. if i' = i \<and> j' = 0 then Le d else if i' = 0 \<and> j' = i then Le (- d) else M i' j')"
set_def: fixes a :: "'s" shows "local.set a = some_img ` hom unity a"
finite_cyclic_subgroup_int: fixes x :: "'a" assumes "x \<in> carrier G" shows "finite (carrier (subgroup_generated G {x})) = (\<exists>i. i \<noteq> 0 \<and> x [^] i = \<one>)"
ASSUME_eq: shows "ASSUME \<equiv> iASSUME RETURN"
lsided_fres_eq: fixes x :: "'a" shows "lsd x = (x \<leftarrow> x = \<top>)"
int_rw1: shows "int (x ^ a) - 1 = int (x ^ a - 1)"
split_aform_def: fixes X :: "'a \<times> 'a pdevs"   and i :: "nat" shows "split_aform X i = (let xi = pdevs_apply (snd X) i /\<^sub>R 2 in ((fst X - xi, pdev_upd (snd X) i xi), fst X + xi, pdev_upd (snd X) i xi))"
pol_coeff_cartesian: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "(fst c, snd c) = c"
group_comm_groupI: assumes "\<And>x y. \<lbrakk>x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n)); y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y = y \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x" shows "comm_group (relative_homology_group (int n) (nsphere n) (equator n))"
WCFG_targetlabel_less_num_nodes: fixes prog :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind"   and l :: "nat" assumes "prog \<turnstile> n -et\<rightarrow> (_ l _)" shows "l < #:prog"
is_UP_monomE_1: fixes f :: "nat \<Rightarrow> 'a" assumes "is_UP_monom f" shows "f \<in> carrier P"
mprod_expR_mem: fixes R :: "('c, 'd) Ring_scheme"   and e :: "nat \<Rightarrow> nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "Ring R"   and "e \<in> {j. j \<le> n} \<rightarrow> {j. 0 \<le> j}"   and "f \<in> {j. j \<le> n} \<rightarrow> carrier R" shows "mprod_expR R e f n \<in> carrier R"
derive_if_remove_subsumed: fixes D :: "'a literal multiset"   and St :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and C :: "'a literal multiset" assumes "D \<in> N_of_state St \<union> P_of_state St \<union> Q_of_state St"   and "subsumes D C" shows "sr_ext.derive (grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St) \<union> grounding_of_cls C) (grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St))"
simps_2: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and e :: "'a"   and x :: "'a"   and xs :: "'a list" shows "insert_sort_insert_by cmp e (x # xs) = (if cmp e x then e # x # xs else x # insert_sort_insert_by cmp e xs)"
lc_setE: fixes c :: "'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "c \<in> lc_set F" obtains f :: "'t \<Rightarrow>\<^sub>0 'c" where "f \<in> F" and "f \<noteq> 0" and "lc f = c"
IH: shows "edges (generate_valid_topology_SOME M G) = edges G - (\<Union>m\<in>set M. if c_sinvar m G then {} else SOME F. F \<in> c_offending_flows m G)"
r2f_def: fixes R :: "('c \<times> 'd) set" shows "\<F> R = (``) R \<circ> \<eta>"
tiling_Un: fixes t :: "'a set"   and A :: "'a set set"   and u :: "'a set" assumes "t \<in> tiling A"   and "u \<in> tiling A"   and "t \<inter> u = {}" shows "t \<union> u \<in> tiling A"
thetaZOSeqI_def: shows "thetaZOSeqI \<equiv> {(e ;; c, e ;; d) |e c d. siso e \<and> c \<approx>01 d}"
not_eq_order_implies_strict: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<noteq> b"   and "b \<subseteq># a" shows "b \<subset># a"