hcju/isabelleps huggingface dataset:
bit_xor_iff: 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 bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (xor a b) n = (bit a n \<noteq> bit b n)"
finite_member: fixes dxs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "finite (Collect (DList_Set.member dxs))"
countable_subset: fixes A :: "'k set"   and B :: "'k set" assumes "A \<subseteq> B"   and "countable B" shows "countable A"
Sup_Un_distrib: fixes A :: "V set"   and B :: "V set" assumes "small A"   and "small B" shows "\<Squnion> (A \<union> B) = \<Squnion> A \<squnion> \<Squnion> B"
countable_image: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "countable A" shows "countable (f ` A)"
Tau: fixes P :: "pi" shows "\<tau>.P \<longmapsto> \<tau> \<prec> P"
toSet_fromList_set: fixes xs :: "'b list" shows "toSet (fromList xs) = set xs"
eq_impD: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and s :: "'f"   and s' :: "'f" assumes "eq_imp f g"   and "\<forall>x. f x s = f x s'" shows "g s = g s'"
typeBinOp1: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and e1 :: "expr"   and e2 :: "expr"   and bop :: "bop" assumes "\<Gamma> \<turnstile> e1 : Low"   and "\<Gamma> \<turnstile> e2 : Low" shows "\<Gamma> \<turnstile> e1 \<guillemotleft>bop\<guillemotright> e2 : Low"
symetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq> Q" shows "Q \<simeq> P"
ide_def: fixes a :: "'a \<times> 'a" shows "B.VV.ide a \<equiv> B.VV.comp a a \<noteq> B.VV.null \<and> (\<forall>f. (B.VV.comp f a \<noteq> B.VV.null \<longrightarrow> B.VV.comp f a = f) \<and> (B.VV.comp a f \<noteq> B.VV.null \<longrightarrow> B.VV.comp a f = f))"
axioms_1: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "eq_linorder_on D cmp" shows "linorder_on D cmp"
ord_strict_pE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "ord_strict_p p q" obtains v :: "'t" where "lookup p v = (0::'c)" and "lookup q v \<noteq> (0::'c)" and "\<And>u. lt_of_nat_term_order cmp_term v u \<Longrightarrow> lookup p u = lookup q u"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size"   and "incomplete_design_axioms point_set u_block_size" shows "incomplete_design point_set block_collection u_block_size"
assms: shows "prime (p\<^sup>2 + N * q\<^sup>2) \<and> 1 < N \<and> (p\<^sup>2 + N * q\<^sup>2)\<^sup>2 = r\<^sup>2 + N * s\<^sup>2 \<and> coprime r s"
Inv_in_Hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Inv C t \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = monoidal_language.Cod C t \<and> monoidal_language.Cod C t = monoidal_language.Dom C t}"
simps_13: shows "pratt1_max \<infinity>\<^sub>1 \<top>\<^sub>1 = \<top>\<^sub>1"
Pow: fixes X :: "V set" assumes "small X" shows "small (set ` Pow X)"
m_inv_monoid_consistent: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> Units (add_monoid R\<lparr>carrier := H\<rparr>)"   and "submonoid H (add_monoid R)" shows "inv\<^bsub>add_monoid R\<lparr>carrier := H\<rparr>\<^esub> x = inv\<^bsub>add_monoid R\<^esub> x"
prv_all_eql: fixes x :: "'var"   and y :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "y \<in> var"   and "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "prv (all x (imp (eql t1 t2) (imp (subst \<phi> t2 y) (subst \<phi> t1 y))))"
equivs_1: shows "mylens.base\<^sub>L \<approx>\<^sub>L x +\<^sub>L y"
triangle_not_betw_acb: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "\<triangle> a b c" shows "\<not> [[a c b]]"
le_diff_iff: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k \<le> m"   and "k \<le> n" shows "(m - k \<le> n - k) = (m \<le> n)"
convergentI: fixes X :: "nat \<Rightarrow> 'd"   and L :: "'d" assumes "X \<longlonglongrightarrow> L" shows "convergent X"
sg_nsg1: fixes H :: "'a set"   and b :: "'a" assumes "G \<guillemotright> H "   and "\<forall>a\<in>carrier G. \<forall>h\<in>H. a \<cdot> h \<cdot> \<rho> a \<in> H"   and "b \<in> carrier G" shows "H \<bullet> b = b \<diamondsuit> H"
3: shows "inj_on (snd (fold (\<lambda>x (k, m). (Suc k, m(x \<mapsto> k))) tsl' (0, Map.empty))) (set tsl')"
ipath_to_trancl: fixes E :: "('b \<times> 'b) set"   and r :: "nat \<Rightarrow> 'b"   and i1 :: "nat"   and i2 :: "nat" assumes "ipath E r"   and "i1 < i2" shows "(r i1, r i2) \<in> E\<^sup>+"
len_space_left: fixes ts :: "traffic"   and v :: "view"   and c :: "cars" shows "left (space ts v c) \<le> right (ext v) \<longrightarrow> left (space ts v c) \<le> left (len v ts c)"
ring_zero: fixes R :: "('d, 'e) Ring_scheme" assumes "Ring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
md: shows "P \<turnstile> C\<^sub>0 sees M\<^sub>0, b :  Ts\<rightarrow>T = m in D"
is_node_ptr_kind_none: fixes ptr :: "(_) object_ptr" assumes "\<not> is_node_ptr_kind ptr" shows "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = None"
Cons_2: shows "(I, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set (hh # t)"
nonempty_imp_ex_extremal: fixes A :: "'b set"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b set" assumes "well_founded A less"   and "Q \<subseteq> A"   and "Q \<noteq> {}" shows "\<exists>z\<in>Q. \<forall>y\<in>Q. \<not> less y z"
finite_Di: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and y :: "nat list" shows "finite (hlde_ops.Di a b i y)"
nth_solution_def: fixes D :: "nat"   and n :: "nat" assumes "pell D" shows "pell.nth_solution D n = pell.pell_power D (pell.fund_sol D) n"
getNonWatchedUnfalsifiedLiteralSomeCharacterization_3: fixes clause :: "Literal list"   and w1 :: "Literal"   and w2 :: "Literal"   and M :: "(Literal \<times> bool) list"   and l :: "Literal" assumes "getNonWatchedUnfalsifiedLiteral clause w1 w2 M = Some l" shows "l \<noteq> w2"
transp_rtranclp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "transp R\<^sup>*\<^sup>*"
compile_s_correct: fixes ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and n :: "nat"   and r :: "nat"   and anything :: "nat list" assumes "rec_ci s = (ap, arity, fp)"   and "rec_exec s [n] = r" shows "{\<lambda>nl. nl = n # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = n # r # 0 \<up> (fp - Suc arity) @ anything}"
freshAbs_psubstAbs_E1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var"   and zs :: "'varSort"   and z :: "'var" assumes "goodAbs A"   and "goodEnv rho"   and "rho ys y = None"   and "freshAbs zs z A $[rho]" shows "freshAbs ys y A \<or> ys \<noteq> zs \<or> y \<noteq> z"
CC_def: shows "CC = (\<lambda>c. sum_list (map fst (filter (\<lambda>rc. snd rc = c) C)))"
ibody_def: fixes i :: "iface" shows "ibody i = \<lparr>access = access i, imethods = imethods i\<rparr>"
well_founded_def: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "well_founded A less \<equiv> \<forall>a P. a \<in> A \<longrightarrow> (\<forall>x. x \<in> A \<longrightarrow> (\<forall>y. y \<in> A \<longrightarrow> less y x \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> P a"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>X. \<lbrakk>X \<subseteq> A; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>e. extreme X (\<lambda>x y. less_eq y x) e" shows "well_related_set A less_eq"
Complex_in_Reals: fixes x :: "real" shows "Complex x 0 \<in> \<real>"
setmonotone_funpower: fixes f :: "'b set \<Rightarrow> 'b set"   and n :: "nat" assumes "setmonotone f" shows "setmonotone (funpower f n)"
enum_max_le: fixes n :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d" assumes "0 < n" shows "Max (set (enum xs index key n mi ma)) \<le> length xs"
zip_eq_conv: fixes xs :: "'c list"   and ys :: "'d list"   and zs :: "('c \<times> 'd) list" assumes "length xs = length ys" shows "(xs || ys = zs) = (map fst zs = xs \<and> map snd zs = ys)"
trms\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d, 'e) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
parallel_fix_ind_inc: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and G :: "'a \<rightarrow> 'a"   and H :: "'b \<rightarrow> 'b" assumes "adm (\<lambda>x. P (fst x) (snd x))"   and "\<And>y. P \<bottom> y"   and "\<And>x. P x \<bottom>"   and "\<And>x y. \<lbrakk>P x y; P (G\<cdot>x) y; P x (H\<cdot>y)\<rbrakk> \<Longrightarrow> P (G\<cdot>x) (H\<cdot>y)" shows "P (fix\<cdot>G) (fix\<cdot>H)"
that: fixes x :: "'a"   and y :: "'a"   and p1 :: "('a \<times> 'a) list"   and p2 :: "('a \<times> 'a) list" assumes "(x, y) \<in> set p"   and "x \<in> S"   and "y \<notin> S"   and "path (restrict_edges g (- {(x, y), (y, x)})) u p1 x"   and "path (restrict_edges g (- {(x, y), (y, x)})) y p2 v" shows "thesis"
intros_1: fixes l :: "int"   and uu :: "int" shows "valid_intv' l uu (Smaller' (- l))"
hyps_2: fixes t2 :: "lam" assumes " t \<longrightarrow>\<^sub>1 t2" shows "\<exists>t3.  s1 \<longrightarrow>\<^sub>1 t3 \<and>  t2 \<longrightarrow>\<^sub>1\<^sup>* t3"
assms_4: shows "g \<in> space\<^sub>N (\<LL> (ennreal p) M)"
fset_strong_cases: fixes xs :: "'c fset" obtains "xs = {||}"   | ys :: "'c fset" and x :: "'c" where "x |\<notin>| ys" and "xs = finsert x ys"
trms_of_deg_leq_id': fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "f \<in> carrier P" shows "f = trms_of_deg_leq k f \<oplus>\<^bsub>P\<^esub> shift (Suc k) f \<otimes>\<^bsub>P\<^esub> monom P \<one> (Suc k)"
floor_diff_of_int: fixes x :: "'a"   and z :: "int" shows "\<lfloor>x - of_int z\<rfloor> = \<lfloor>x\<rfloor> - z"
affine_diffs_subspace: fixes S :: "'a set"   and a :: "'a" assumes "affine S"   and "a \<in> S" shows "subspace ((+) (- a) ` S)"
lunit_agreement: fixes a :: "'a" assumes "ide a" shows "lunit a = C.lunit a"
association_end_det: fixes classes :: "'a fset"   and associations :: "String.literal \<rightharpoonup>\<^sub>f String.literal \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and \<C> :: "'a"   and "from" :: "String.literal option"   and role :: "String.literal"   and \<D>\<^sub>1 :: "'a"   and end\<^sub>1 :: "'a \<times> nat \<times> enat \<times> bool \<times> bool"   and \<D>\<^sub>2 :: "'a"   and end\<^sub>2 :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" assumes "object_model classes associations"   and "unique_closest_association_end classes associations \<C> from role \<D>\<^sub>1 end\<^sub>1"   and "unique_closest_association_end classes associations \<C> from role \<D>\<^sub>2 end\<^sub>2" shows "\<D>\<^sub>1 = \<D>\<^sub>2 \<and> end\<^sub>1 = end\<^sub>2"
vquot_dbtm_fresh: fixes V :: "name set"   and t :: "dbtm" assumes "atom ` V \<sharp>* t" shows "vquot_dbtm V t = quot_dbtm t"
in_face_cycle_setD: fixes y :: "'b"   and x :: "'b" assumes "y \<in> H.face_cycle_set x"   and "x \<in> arcs H" shows "y \<in> arcs H"
outputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a{b}.P \<longmapsto> Rs" obtains aa :: "name" and ba :: "name" and Pa :: "pi" where "a = aa \<and> b = ba \<and> P = Pa" and "Rs = aa[ba] \<prec> Pa"
fmap_add_commute: fixes A :: "'a \<rightharpoonup>\<^sub>f 'b"   and B :: "'a \<rightharpoonup>\<^sub>f 'b" assumes "fmdom A |\<inter>| fmdom B = {||}" shows "A ++\<^sub>f B = B ++\<^sub>f A"
unatSuc: fixes n :: "'b word" assumes "1 + n \<noteq> 0" shows "unat (1 + n) = Suc (unat n)"
runit_simp: fixes f :: "'a" assumes "ide f" shows "runit f = B.runit f"
S_transform_formula_Act: fixes \<alpha> :: "'b"   and x :: "('a, 'c, 'b) formula" shows "S_transform_formula (Formula.Act \<alpha> x) = Act\<^sub>\<alpha> (S_action.Act \<alpha>) (S_transform_formula x)"
xor3_def: fixes a :: "bool"   and b :: "bool"   and c :: "bool" shows "xor3 a b c = (a = (b = c))"
sbis_lsbis: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2" shows "sbis B1 B2 s1 s2 (lsbis1 B1 B2 s1 s2) (lsbis2 B1 B2 s1 s2)"
c: shows "properdiv_set 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24}"
mycasex_def: fixes x :: "'a"   and y :: "'a" shows "mycasex x y = Plus (Atom y) (Atom x)"
winding_number_homotopic_paths_eq: fixes p :: "real \<Rightarrow> complex"   and \<zeta> :: "complex"   and q :: "real \<Rightarrow> complex" assumes "path p"   and "\<zeta> \<notin> path_image p"   and "path q"   and "\<zeta> \<notin> path_image q"   and "pathstart q = pathstart p"   and "pathfinish q = pathfinish p" shows "(winding_number p \<zeta> = winding_number q \<zeta>) = homotopic_paths (- {\<zeta>}) p q"
inf_mono: fixes a :: "'c multiset"   and c :: "'c multiset"   and b :: "'c multiset"   and d :: "'c multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<inter># b \<subseteq># c \<inter># d"
simps_1: fixes x :: "'b" shows "del_list x [] = []"
pmf_not_neg: fixes p :: "'a pmf"   and x :: "'a" shows "\<not> pmf p x < 0"
comm_neutral: fixes f :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and z :: "'f"   and a :: "'f" assumes "comm_monoid f z" shows "f a z = a"
inject_7: fixes x9 :: "('a, 'b) form"   and y9 :: "('a, 'b) form" shows "(Exists x9 = Exists y9) = (x9 = y9)"
aless_linear: fixes x :: "ant"   and y :: "ant" shows "x < y \<or> x = y \<or> y < x"
stop_doesnt_access: fixes x :: "'Var" shows "doesnt_modify Stop x \<and> doesnt_read_or_modify Stop x"
surjective: fixes r :: "'a control_state_scheme" shows "r = \<lparr>break_status = break_status r, return_status = return_status r, \<dots> = more r\<rparr>"
simps_1: fixes pverts :: "'b set"   and parcs :: "('b \<times> 'b) set"   and more :: "'c" shows "pverts \<lparr>pverts = pverts, parcs = parcs, \<dots> = more\<rparr> = pverts"
induct_list012: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y zs. \<lbrakk>P zs; P (y # zs)\<rbrakk> \<Longrightarrow> P (x # y # zs)" shows "P xs"
assms: shows "nsqn\<^sub>r r < nsqn\<^sub>r r'"
inv_comp_right_2: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.iso (CC.comp g f)"   and "CC.iso f" shows "CC.iso g"
adj_not_same: fixes a :: "'a" assumes "a \<rightarrow> a" shows "False"
simps_7: fixes i :: "String.literal"   and p :: "fol fm"   and q :: "fol fm" shows "free_in i (Or p q) = (free_in i p \<or> free_in i q)"
multiple_unfold_1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and n :: "nat" assumes "incidence_system point_set block_collection" shows "repeat_mset (Suc n) block_collection = repeat_mset n block_collection + block_collection"
word_lsb_def: fixes a :: "'b word" shows "lsb a = odd (uint a)"
bij_betw_inv_into_subset: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and B :: "'b set"   and B' :: "'c set" assumes "bij_betw f A A'"   and "B \<subseteq> A"   and "f ` B = B'" shows "bij_betw (inv_into A f) B' B"
ccHeap_rec_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "ccHeap_rec \<Gamma> e = (\<Lambda> a. CCfix \<Gamma>\<cdot>(Afix \<Gamma>\<cdot>(Aexp_syn' a e \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), ccExp_syn a e))"
gamma2_differentiable_3: shows "(\<forall>x\<in>{0..1} - s' \<inter> {0..1}. (\<gamma>2 has_vector_derivative D__ x) (at x)) \<and> continuous_on ({0..1} - s' \<inter> {0..1}) D__"
type_II_chain_vert_bound_valid: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeII_chain i j F two_chain s" shows "\<forall>(k, \<gamma>)\<in>two_chain_vertical_boundary two_chain. valid_path \<gamma>"
singular_relboundary_minus: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_relboundary p X S (- c) = singular_relboundary p X S c"
FDOM_state_succ: fixes a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and s :: "('a, 'b) fmap" assumes "fmdom' (snd a) \<subseteq> fmdom' s" shows "fmdom' (state_succ s a) = fmdom' s"
add_0_right: fixes a :: "'c" shows "a \<squnion> \<bottom> = a"
f_bind: fixes m :: "'m"   and x :: "'k"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'm" shows "bind m (\<lambda>a. bind (f x) (g a)) = bind (f x) (\<lambda>b. bind m (\<lambda>a. g a b))"
order: fixes a :: "'a"   and n :: "nat"   and p :: "'a list" shows "([- a, 1::'a] %^ n divides p \<and> \<not> [- a, 1::'a] %^ Suc n divides p) = (n = order a p \<and> poly p \<noteq> poly [])"
tickFree_append: fixes s :: "'a event list"   and t :: "'a event list" shows "tickFree (s @ t) = (tickFree s \<and> tickFree t)"
ordIso_transitive: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and r'' :: "('f \<times> 'f) set" assumes "r =o r'"   and "r' =o r''" shows "r =o r''"
mat_of_rows_map: fixes vs :: "'a vec list"   and n :: "nat"   and f :: "'a \<Rightarrow> 'b" assumes "set vs \<subseteq> carrier_vec n" shows "mat_of_rows n (map (map_vec f) vs) = map_mat f (mat_of_rows n vs)"
finsert_fsubset: fixes x :: "'b"   and A :: "'b fset"   and B :: "'b fset" shows "(finsert x A |\<subseteq>| B) = (x |\<in>| B \<and> A |\<subseteq>| B)"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>statename = statename r, datastate = datastate r, action = action r, output = output r, \<dots> = state.more r\<rparr>"
elims: fixes x :: "event_data option list set list \<Rightarrow> 'a \<Rightarrow> 'a"   and xa :: "'a"   and xb :: "event_data option list set list list"   and y :: "'a \<times> event_data option list set list list" assumes "mbufn_take x xa xb = y" obtains f :: "event_data option list set list \<Rightarrow> 'a \<Rightarrow> 'a" and z :: "'a" and buf :: "event_data option list set list list" where "x = f" and "xa = z" and "xb = buf" and "y = (if buf = [] \<or> [] \<in> set buf then (z, buf) else mbufn_take f (f (map hd buf) z) (map tl buf))"
le_rem_simps_2: fixes a :: "'c" shows "le_rem_s (ENV a) = a"
is_lub_least: fixes P :: "'a prog"   and U :: "ty"   and V :: "ty"   and T :: "ty"   and T' :: "ty" assumes "P \<turnstile> lub(U, V) = T"   and "P \<turnstile> U \<le> T'"   and "P \<turnstile> V \<le> T'" shows "P \<turnstile> T \<le> T'"
outside_reduces_domain: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "Domain (P outside X) = Domain P - X"
prems_4: shows "{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>} \<in> set (wall_crossings (C # (E # Es) @ [D]))"
simps_1: fixes c :: "'c"   and \<sigma> :: "'d \<Rightarrow> ('b, 'c) trm" shows "OsubstFO (OVar c) \<sigma> = OVar c"
order_prod_list: fixes ps :: "'b poly list"   and x :: "'b" assumes "\<And>p. p \<in> set ps \<Longrightarrow> p \<noteq> 0" shows "order x (prod_list ps) = sum_list (map (order x) ps)"
T1: shows "C A TS P R \<and> C A TS B R"
SD_singleton_right: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and q :: "'b pmf" assumes "Order_Predicates.preorder_on carrier le"   and "x \<in> carrier"   and "q \<in> lotteries_on carrier" shows "SD le q (return_pmf x) = (\<forall>y\<in>set_pmf q. le y x)"
base_in_hom_1: shows "\<guillemotleft>r : src\<^sub>C r \<rightarrow>\<^sub>C trg\<^sub>C r\<guillemotright>"
assms_12: shows "2 ^ (5 * (degree F - 1) * (degree F - 1)) * \<parallel>F\<parallel>\<^sup>2 ^ (2 * (degree F - 1)) < pl\<^sup>2"
eq_matI: fixes B :: "'a Matrix.mat"   and A :: "'a Matrix.mat" assumes "\<And>i j. \<lbrakk>i < dim_row B; j < dim_col B\<rbrakk> \<Longrightarrow> A $$ (i, j) = B $$ (i, j)"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
map_simp_ide: fixes a :: "'c" assumes "C.ide a" shows "\<epsilon>.map a = \<epsilon>o a"
sorted_list_of_set_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq = folding.F (linorder.insort_key less_eq (\<lambda>x. x)) []"
gkernTr9: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and a :: "'e"   and b :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "a \<in> carrier F"   and "b \<in> carrier F" shows "(gker\<^bsub>F,G\<^esub> f \<bullet>\<^bsub>F\<^esub> a = gker\<^bsub>F,G\<^esub> f \<bullet>\<^bsub>F\<^esub> b) = (f a = f b)"
simps_3: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b dres" shows "dRETURN x \<bind> f = f x"
intF_def: shows "intF \<equiv> ProblemIkTpartG_GEModel.intF eintF"
simps_11: fixes f1 :: "agent \<Rightarrow> agent \<Rightarrow> msg \<Rightarrow> 'a"   and f2 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and f3 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and x21 :: "agent"   and x22 :: "msg" shows "(case Gets x21 x22 of Says x xa xb \<Rightarrow> f1 x xa xb | Gets x xa \<Rightarrow> f2 x xa | Notes x xa \<Rightarrow> f3 x xa) = f2 x21 x22"
fps_zeror: fixes f :: "('c, 'd) fps" shows "0 + f = f"
least_dual: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and A :: "'a set" shows "least (inv_gorder L) x A = greatest L x A"
Suc_leD: fixes m :: "nat"   and n :: "nat" assumes "order.greater_eq n (Suc m)" shows "order.greater_eq n m"
le: fixes r :: "rat"   and m :: "nat" assumes "0 < r" shows "\<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
map_le_implies_dom_le: fixes f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" assumes "f \<subseteq>\<^sub>m g" shows "dom f \<subseteq> dom g"
reverse_iterateoi_rule: fixes m :: "'s" assumes "invar m" shows "map_iterator_rev_linord (reverse_iterateoi m) (\<alpha> m)"
sum_outgoing_extend: fixes u :: "nat"   and g :: "nat \<times> nat \<Rightarrow> 'b" assumes "\<And>v. \<lbrakk>v \<in> cf.V; (u, v) \<notin> cf.E\<rbrakk> \<Longrightarrow> g (u, v) = (0::'b)" shows "sum g (cf.outgoing u) = (\<Sum>v\<in>cf.V. g (u, v))"
interchange_rows_mat_1: fixes a :: "'f"   and b :: "'f"   and A :: "(('d, 'e) vec, 'f) vec" shows "interchange_rows (mat (1::'d)) a b ** A = interchange_rows A a b"
Gromov_extension_continuous: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c Gromov_completion" assumes "lambda C -quasi_isometry f"   and "x \<in> Gromov_boundary" shows "isCont (Gromov_extension f) x"
exec1_def: shows "exec1 \<equiv> {((xa, x), xc, xb). exec1p xa x xc xb}"
4: shows "X \<in> obj\<^bsub>NTCatDom f\<^esub>"
akiyama_tanigawa_row_Suc: fixes n :: "nat"   and l :: "nat"   and u :: "nat" shows "akiyama_tanigawa_row (Suc n) l u = akiyama_tanigawa_step_aux (Suc l) (akiyama_tanigawa_row n l (Suc u))"
pm_pth_append: fixes p :: "nat"   and m :: "('a, 'b) pointermap_scheme"   and a :: "'a" assumes "pointermap_p_valid p m" shows "pm_pth (pointermap_insert a m) p = pm_pth m p"