hcju/isabelleps huggingface dataset:
exhaust: fixes y :: "poly" obtains x1 :: "int \<times> int" where "y = C x1"   | x2 :: "nat" where "y = Bound x2"   | x31 :: "poly" and x32 :: "poly" where "y = Add x31 x32"   | x41 :: "poly" and x42 :: "poly" where "y = Sub x41 x42"   | x51 :: "poly" and x52 :: "poly" where "y = Mul x51 x52"   | x6 :: "poly" where "y = Neg x6"   | x71 :: "poly" and x72 :: "nat" where "y = Pw x71 x72"   | x81 :: "poly" and x82 :: "nat" and x83 :: "poly" where "y = CN x81 x82 x83"
of_real_eq_minus_of_real_iff: fixes x :: "real"   and y :: "real" shows "(of_real x = - of_real y) = (x = - y)"
numeral_less_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "num"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (numeral.numeral one plus m) (numeral.numeral one plus n) = (m < n)"
ack_add_bound: fixes i1 :: "nat"   and j :: "nat"   and i2 :: "nat" shows "ack i1 j + ack i2 j < ack (4 + (i1 + i2)) j"
unless_def: fixes A :: "'d set"   and B :: "'d set" shows "A unless B \<equiv> A - B co A \<union> B"
vsubstBinp_def: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" shows "vsubstBinp ys y1 y2 \<equiv> substBinp ys (Var ys y1) y2"
dij_def: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and j :: "nat" shows "Linear_Diophantine_Equations.dij a b i j = lcm (a ! i) (b ! j) div a ! i"
C1_differentiable_on_minus: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set" assumes "f C1_differentiable_on S" shows "(\<lambda>x. - f x) C1_differentiable_on S"
eventually_uniformity_metric: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and P :: "'a \<times> 'a \<Rightarrow> bool" assumes "class.uniformity_dist dist uniformity" shows "eventually P uniformity = (\<exists>e>0. \<forall>x y. dist x y < e \<longrightarrow> P (x, y))"
Vfrom_succ_Ord: fixes i :: "V"   and a :: "V" assumes "Ord i" shows "Vfrom a (ZFC_in_HOL.succ i) = a \<union>\<^sub>\<circ> VPow (Vfrom a i)"
clw_rel_br: fixes a :: "'c \<Rightarrow> 'd set"   and I :: "'c \<Rightarrow> bool" shows "clw_rel (br a I) = br (\<lambda>xs. \<Union> (a ` set xs)) (\<lambda>xs. Ball (set xs) I)"
assms_5: fixes c :: "'a"   and b :: "bool" shows "P c (Bool b)"
span_closed: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and U :: "'d set"   and v :: "'d" assumes "vectorspace K V"   and "U \<subseteq> carrier V"   and "v \<in> LinearCombinations.module.span K V U" shows "v \<in> carrier V"
knows_emptyexpression: fixes C :: "specID" shows "knows C [] = True"
sel_1: fixes x1 :: "'b list"   and x2 :: "'b" shows "prems_of (Infer x1 x2) = x1"
Abs_fresh_star_iff_1: fixes as :: "atom set"   and bs :: "atom set"   and x :: "'e" shows "as \<sharp>* ([bs]set. x) = (as - bs) \<sharp>* x"
pcr_cr_eq: shows "pmf_as_function.pcr_pmf (=) = pmf_as_function.cr_pmf"
distinct_2: fixes x2 :: "nat"   and x1 :: "nat" shows "Regions.intv.Intv x2 \<noteq> Regions.intv.Const x1"
isDiagonal: fixes A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and b :: "'b" assumes "isDiagonal A"   and "to_nat a \<noteq> to_nat b" shows "A $ a $ b = (0::'a)"
pos_recurrent_def: fixes x :: "'s" shows "pos_recurrent x = (recurrent x \<and> U' x x \<noteq> \<infinity>)"
empty_concat: fixes ms :: "'d list" shows "concat (map (\<lambda>x. []) ms) = []"
LeftDerivationLadder_def: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and b :: "symbol list" shows "LeftDerivationLadder a D L b = (LeftDerivation a D b \<and> is_ladder D L \<and> LeftDerivationFix a (ladder_i L 0) (take (ladder_n L 0) D) (ladder_j L 0) (ladder_\<gamma> a D L 0) \<and> LeftDerivationIntros a D L)"
degree_uminus_pdevs: fixes x :: "'a pdevs" shows "degree (uminus_pdevs x) = degree x"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "inj_comm_monoid_add_hom hom" shows "map_poly_inj_comm_monoid_add_hom hom"
GreenThm_typeI_two_chain: shows "two_chain_integral two_chain (\<lambda>a. - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j a) = one_chain_line_integral F {i} (two_chain_boundary two_chain)"
nths_vars2': shows "nths b.dims0 (- b.vars1') = c.dims2"
major: shows "silent_movet t (x, shr s) (x', m')"
usubst_term: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<theta> :: "trm"   and I :: "interp" assumes "Uvariation \<nu> \<omega> U"   and "usubstappt \<sigma> U \<theta> \<noteq> undeft" shows "term_sem I (the (usubstappt \<sigma> U \<theta>)) \<nu> = term_sem (USubst.adjoint \<sigma> I \<omega>) \<theta> \<nu>"
observable_path_unique_ex: fixes M :: "('d, 'e, 'f) FSM"   and io :: "('d \<times> 'e) list"   and q :: "'f" assumes "observable M"   and "io \<in> LS M q" obtains tr :: "'f list" where "{t. path M (io || t) q \<and> length io = length t} = {tr}"
vsubset_if_subset: fixes A :: "V"   and B :: "V" assumes "elts A \<subseteq> elts B" shows "A \<subseteq>\<^sub>\<circ> B"
all_init_eq_imp_fun_eq: fixes f :: "nat \<Rightarrow> nat option"   and g :: "nat \<Rightarrow> nat option" assumes "f \<in> \<R>"   and "g \<in> \<R>"   and "\<And>n. f \<triangleright> n = g \<triangleright> n" shows "f = g"
typeof_box_operand: fixes x :: "('dyn, 'ubx1, 'ubx2) unboxed" shows "typeof (box_operand x) = None"
domain_unique: fixes a :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr"   and a' :: "(bool, 'arr list) concrete_category.arr" assumes "a \<in> domains f"   and "a' \<in> domains f" shows "a = a'"
Inf_lower: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq (Inf A) x"
LIMSEQ_n_over_Suc_n: shows "(\<lambda>n. of_nat n / of_nat (Suc n)) \<longlonglongrightarrow> (1::'a)"
index_mat_swaprows_mat_2: fixes n :: "nat"   and k :: "nat"   and l :: "nat" shows "dim_row (swaprows_mat n k l) = n"
simps_4: fixes uv :: "'b" shows "is_lit_plus (\<^bold>\<not> (Atom uv)) = True"
not_final_thread_cases2: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x2"   and ln :: "'a \<Rightarrow>f nat" assumes "r2.not_final_thread s t"   and "thr s t = \<lfloor>(x, ln)\<rfloor>" obtains "\<not> final2 x"   | "ln \<noteq> no_wait_locks"   | w :: "'d wait_set_status" where "wset s t = \<lfloor>w\<rfloor>"
lzip_lmap1: fixes f :: "'e \<Rightarrow> 'c"   and xs :: "'e llist"   and ys :: "'d llist" shows "lzip (lmap f xs) ys = lmap (\<lambda>(x, y). (f x, y)) (lzip xs ys)"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "satC \<xi> \<equiv> list_ex (satL \<xi>)"
trcl_empty_cons: fixes c :: "'a"   and c' :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" assumes "(c, [], c') \<in> trcl t" shows "c = c'"
subgroup_inverse_equality: fixes u :: "'a set" assumes "u \<in> induced.Ker" shows "contained.quotient.inverse u = induced.kernel.sub.inverse u"
subalg: fixes M :: "'b measure"   and F :: "'b measure" assumes "finite_measure_subalgebra M F" shows "subalgebra M F"
times_in_interval: fixes x :: "'a"   and X :: "'a interval"   and y :: "'a"   and Y :: "'a interval" assumes "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "x * y \<in>\<^sub>i X * Y"
\<phi>_def: fixes \<pi>l :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list" shows "\<phi> \<pi>l = eqButGH (length \<pi>l) (Suc (length \<pi>l))"
separating_hyperplane_closed_point: fixes S :: "'b set"   and z :: "'b" assumes "convex S"   and "closed S"   and "z \<notin> S" shows "\<exists>a b. a \<bullet> z < b \<and> (\<forall>x\<in>S. b < a \<bullet> x)"
bv_sliceI: fixes j :: "nat"   and i :: "nat"   and w :: "bit list"   and w1 :: "bit list"   and w2 :: "bit list"   and w3 :: "bit list" assumes "j \<le> i"   and "i < length w"   and "w = w1 @ w2 @ w3"   and "Suc i = length w2 + j"   and "j = length w3" shows "bv_slice w (i, j) = w2"
atoms_nderiv: fixes a :: "'a"   and r :: "'a rexp" shows "atoms (nderiv a r) \<subseteq> atoms r"
gx_def: fixes x :: "'a"   and y :: "'a" shows "gx x y = - x + (8::'a) / (10::'a)\<^sup>2 * y + x\<^sup>2 * y"
dom_char: fixes f :: "'b \<times> 'b" shows "B.VV.dom f = (if B.VV.arr f then B.VxV.dom f else B.VxV.null)"
finite_imp_compactin_eq: fixes S :: "'c set"   and X :: "'c topology" assumes "finite S" shows "compactin X S = (S \<subseteq> topspace X)"
dg_Set_def: fixes \<alpha> :: "V" shows "dg_Set \<alpha> = [Vset \<alpha>, ZFC_in_HOL.set {T. arr_Set \<alpha> T}, \<lambda>T\<in>\<^sub>\<circ>ZFC_in_HOL.set {T. arr_Set \<alpha> T}. T\<lparr>ArrDom\<rparr>, \<lambda>T\<in>\<^sub>\<circ>ZFC_in_HOL.set {T. arr_Set \<alpha> T}. T\<lparr>ArrCod\<rparr>]\<^sub>\<circ>"
elims: fixes x :: "(dir list \<times> dir) list"   and xa :: "('c, 'd) tm"   and y :: "('c, 'd) tm" assumes "emb_at x xa = y" obtains t :: "('c, 'd) tm" where "x = []" and "xa = t__" and "y = t__"   | p :: "dir list" and d :: "dir" and ps :: "(dir list \<times> dir) list" and t :: "('c, 'd) tm" where "x = (p, d) # ps" and "xa = t" and "y = emb_step_at p d (emb_at ps t)"
Iff_I: fixes A :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A H \<turnstile> B"   and "insert B H \<turnstile> A" shows "H \<turnstile> A IFF B"
hs: shows "h ` ({0..1} \<times> {0..1}) \<subseteq> s"
while_if2: fixes c' :: "com"   and s :: "loc \<Rightarrow> nat"   and t :: "loc \<Rightarrow> nat"   and e :: "exp"   and c :: "com" assumes "(c', s) -[eval]-> t"   and "c' = IF e THEN c;;WHILE e DO c ELSE SKIP" shows "(WHILE e DO c, s) -[eval]-> t"
lex_1: shows "Counterclockwise_2D_Arbitrary.lex q p"
Goal: fixes \<Psi>' :: "'b"   and \<Psi>\<^sub>Q' :: "'b"   and Pa :: "('a, 'b, 'c) psi"   and Ma :: "'a"   and Na :: "'a"   and P'a :: "('a, 'b, 'c) psi"   and A\<^sub>P' :: "name list"   and \<Psi>\<^sub>P' :: "'b"   and Qa :: "('a, 'b, 'c) psi"   and Ka :: "'a"   and xveca :: "name list"   and Q'a :: "('a, 'b, 'c) psi"   and A\<^sub>Q' :: "name list" assumes "\<Psi>' \<otimes> \<Psi>\<^sub>Q' \<rhd> Pa \<longmapsto> Ma\<lparr>Na\<rparr> \<prec> P'a"   and "extractFrame Pa = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>"   and "distinct A\<^sub>P'"   and "\<Psi>' \<otimes> \<Psi>\<^sub>P' \<rhd> Qa \<longmapsto> Ka\<lparr>\<nu>*xveca\<rparr>\<langle>Na\<rangle> \<prec> Q'a"   and "extractFrame Qa = \<langle>A\<^sub>Q', \<Psi>\<^sub>Q'\<rangle>"   and "distinct A\<^sub>Q'"   and "\<Psi>' \<otimes> \<Psi>\<^sub>P' \<otimes> \<Psi>\<^sub>Q' \<turnstile> Ma \<leftrightarrow> Ka"   and "A\<^sub>P' \<sharp>* \<Psi>'"   and "A\<^sub>P' \<sharp>* Pa"   and "A\<^sub>P' \<sharp>* Qa"   and "A\<^sub>P' \<sharp>* Ma"   and "A\<^sub>P' \<sharp>* A\<^sub>Q'"   and "A\<^sub>Q' \<sharp>* \<Psi>'"   and "A\<^sub>Q' \<sharp>* Pa"   and "A\<^sub>Q' \<sharp>* Qa"   and "A\<^sub>Q' \<sharp>* Ka"   and "xveca \<sharp>* Pa" shows "\<Psi>' \<rhd> Pa \<parallel> Qa \<longmapsto> \<tau> \<prec> \<lparr>\<nu>*xveca\<rparr>P'a \<parallel> Q'a"
less_eq_multiset_plus_right: fixes M :: "'b multiset"   and N :: "'b multiset" shows "M \<le> M + N"
comp_cod_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and b :: "'b" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.cod C f = b" shows "C b f = f"
ennreal_divide_self: fixes a :: "ennreal" assumes "a \<noteq> 0"   and "a < top" shows "a / a = 1"
root_2_def: fixes n :: "nat"   and x :: "real_alg_2" shows "root_2 n x = root_1 n (real_alg_1 x)"
map_add_comm_weak: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" assumes "\<forall>k\<in>dom m1 \<inter> dom m2. m1 k = m2 k" shows "m1 ++ m2 = m2 ++ m1"
simps_8: fixes f1 :: "'c \<times> sort_dir \<Rightarrow> 'c sort_key \<Rightarrow> 'b \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "rec_sort_key f1 f2 (Final x2) = f2 x2"
standard_liouville_summable: fixes q :: "int"   and p :: "nat \<Rightarrow> int" assumes "1 < q"   and "range p \<subseteq> {0..<q}" shows "summable (\<lambda>k. real_of_int (p k) / real_of_int q ^ fact (Suc k))"
real_of_float_in_real_interval_of: fixes x :: "float"   and X :: "float interval" assumes "x \<in>\<^sub>i X" shows "real_of_float x \<in>\<^sub>r X"
vec_simps_17: fixes al :: "'a"   and bl :: "'a"   and cj :: "'a" shows "al - bl - cj = al - (bl + cj)"
bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_finite: fixes x :: "('d, 'c) stateful_strand_step" shows "finite (set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p x))"
order_trans: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" assumes "x \<le>s y"   and "y \<le>s z" shows "x \<le>s z"
cong_iff_dvd_diff: 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 division_segment :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and m :: "'c" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo a b m = dvd.dvd times m (minus a b)"
prema: shows "a \<in> set (wordinterval_CIDR_split_prefixmatch rs)"
ub_exp_horner_nonneg: fixes x :: "float"   and prec :: "nat"   and n :: "nat" assumes "real_of_float x \<le> 0" shows "0 \<le> real_of_float (ub_exp_horner prec (get_odd n) (Suc 0) (Suc 0) x)"
projection_eqI': fixes M :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "convex M"   and "is_projection_on f M" shows "projection M = f"
poly_y_x_poly_x_minus_y: fixes p :: "'b poly" shows "poly_y_x (poly_x_minus_y p) = poly_y_minus_x p"
c_f_list_def: shows "c_f_list = (\<lambda>f. let g = \<lambda>x. c_cons (f 0 x) 0 in Let (\<lambda>a b c. c_cons (f (Suc a) c) b) (PrimRecOp g))"
count_image_mset_le_count_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b multiset"   and y :: "'c" assumes "inj_on f (set_mset M)" shows "count (image_mset f M) y \<le> count M (inv_into (set_mset M) f y)"
mat_distl: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<otimes> g \<oplus> h = (f \<otimes> g) \<oplus> (f \<otimes> h)"
rel_sel: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('a, 'c) hd"   and b :: "('b, 'd) hd" shows "rel_hd R1 R2 a b = (is_Var a = is_Var b \<and> (is_Var a \<longrightarrow> is_Var b \<longrightarrow> R2 (var a) (var b)) \<and> (is_Sym a \<longrightarrow> is_Sym b \<longrightarrow> R1 (hd.sym a) (hd.sym b)))"
Sup_nres_def: fixes X :: "'z nres set" shows "Sup X \<equiv> if FAILi \<in> X then FAILi else RES (\<Union> {x. RES x \<in> X})"
exprel_imp_eqv_freeargs: fixes U :: "freeExp"   and V :: "freeExp" assumes "U \<sim> V" shows "(freeargs U, freeargs V) \<in> listrel exprel"
get_owner_document_wf_get_root_node_wf_is_l_get_owner_document_wf_get_root_node_wf: shows "l_get_owner_document_wf_get_root_node_wf heap_is_wellformed DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_root_node get_owner_document"
simps: fixes n :: "nat"   and uu :: "atom list"   and uv :: "atom fm list" shows "IdentityHeuristic n uu uv = n"
agent_abs_def: fixes a :: "'a"   and t :: "('a, 'es, 'as) BEState Trace" shows "agent_abs a t \<equiv> rel_ext (\<lambda>uu_. \<exists>t'. uu_ = (tFirst t', tLast t') \<and> t' \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jview a t)"
extremum_unique: fixes a :: "'d" shows "(a \<le> Orderings.bot_class.bot) = (a = Orderings.bot_class.bot)"
complete_base: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "R.dimension n K E"   and "R.independent K Us"   and "set Us \<subseteq> E" shows "\<exists>Vs. length (Vs @ Us) = n \<and> R.independent K (Vs @ Us) \<and> R.Span K (Vs @ Us) = E"
InterE: fixes A :: "'c"   and C :: "'c set set"   and X :: "'c set" assumes "A \<in> \<Inter> C" obtains "X \<notin> C"   | "A \<in> X"
methd_rec': fixes C :: "qtname"   and c :: "class" assumes "class tprg C = Some c" shows "methd tprg C = (if C = Object then Map.empty else filter_tab (\<lambda>sig m. tprg \<turnstile> C inherits method sig m) (methd tprg (super c))) ++ table_of (map (\<lambda>(s, m). (s, C, m)) (methods c))"
supp_def: fixes f :: "'b \<Rightarrow> 'c" shows "Formal_Puiseux_Series.supp f = f -` (- {0::'c})"
b_has_ticket_unique: fixes ci :: "nat"   and ni :: "nat"   and tsi :: "nat \<Rightarrow> thread"   and t :: "nat"   and tki :: "nat"   and t' :: "nat" assumes "B.reachable (ci, ni, tsi)"   and "t < N"   and "has_ticket (tsi t) tki"   and "t' < N"   and "has_ticket (tsi t') tki" shows "t' = t"
rel_optionT'_phantom: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_optionT' A = rel_optionT' \<top>"
monom_power: fixes a :: "'a"   and b :: "nat"   and n :: "nat" shows "Polynomial.monom a b ^ n = Polynomial.monom (a ^ n) (b * n)"
all_partitions_equivalence': fixes xs :: "'c list" assumes "distinct xs" shows "set (map set (all_partitions_list xs)) = all_partitions (set xs) \<and> (\<forall>ps\<in>set (all_partitions_list xs). distinct ps)"
left_commute: fixes b :: "'d multiset"   and a :: "'d multiset"   and c :: "'d multiset" shows "b \<union># (a \<union># c) = a \<union># (b \<union># c)"
restriction_card_somewhere_mon: fixes v :: "view"   and vl :: "view"   and v1 :: "view"   and v2 :: "view"   and vr :: "view"   and vu :: "view"   and v3 :: "view"   and v' :: "view"   and vd :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=vl\<parallel>v1 \<and> v1=v2\<parallel>vr \<and> v2=vu--v3 \<and> v3=v'--vd \<and> |restrict v' f c| = 1 \<longrightarrow> 1 \<le> |restrict v f c|"
good_prefix_lemma: fixes ws :: "letter list list"   and f :: "nat \<Rightarrow> letter list" assumes "bar ws"   and "is_prefix ws f" shows "\<exists>vs. is_prefix vs f \<and> good vs"
simps_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" shows "power.power one times a 0 = one"
gap_invar_def: shows "gap_invar \<equiv> \<lambda>(l, r, buf). l \<le> r \<and> r \<le> length buf"
remove_child_child_in_heap: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> remove_child ptr' child \<rightarrow>\<^sub>h h'" shows "child |\<in>| node_ptr_kinds h"
axioms_4: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<eta> :: "'d \<Rightarrow> 'd"   and \<epsilon> :: "'c \<Rightarrow> 'c" assumes "unit_counit_adjunction C D F G \<eta> \<epsilon>" shows "functor C D G"
is_tiny_tdghmI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<Z> \<alpha>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"
d_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "d x \<le> d y"
less_add_same_cancel1: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subset># a + b) = ({#} \<subset># b)"
idempotency_L: fixes A :: "'a set" shows "Cn (Cn A) = Cn A"
assms_1: shows "A \<in>\<^sub>\<circ> VPow B"
proj_Cons_5: fixes l :: "'d strand_label"   and n :: "'d"   and a :: "'e"   and A :: "('d strand_label \<times> 'e) list" assumes "l = \<star>" shows "proj n ((l, a) # A) = (l, a) # proj n A"
take_add: fixes i :: "nat"   and j :: "nat"   and xs :: "'b list" shows "take (i + j) xs = take i xs \<cdot> take j (drop i xs)"
supapartmentD_1: fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "chamber D" shows "supapartment C D \<in> \<A>"
seg_betw: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "(x \<in> MinkowskiBetweenness.segment \<P> betw a b) = betw a x b"
remdups_adj_append_dropWhile: fixes xs :: "'g list"   and y :: "'g"   and ys :: "'g list" shows "remdups_adj (xs @ y # ys) = remdups_adj (xs @ [y]) @ remdups_adj (dropWhile (\<lambda>x. x = y) ys)"
simps_5: fixes qe :: "atom list \<Rightarrow> atom fm" shows "R.lift_dnf_qe qe TrueF = TrueF"
ex_in_conv: fixes A :: "'f set" shows "(\<exists>x. x \<in> A) = (A \<noteq> {})"
c: fixes n :: "nat" shows "of_int (c n) = poly.coeff p n"
igbg_impl_rel_eext_internal_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "igbg_impl_rel_eext Rm Rv \<equiv> \<langle>Rm, Rv \<rightarrow> \<langle>nat_rel\<rangle>bs_set_rel\<rangle>gen_igbg_impl_rel_eext"
congruentP_def: fixes \<phi> :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" shows "f respectsP \<phi> \<equiv> \<forall>x y. \<phi> x y \<longrightarrow> f x = f y"
CastNull: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"
cong_gcd_eq: fixes a :: "'c"   and b :: "'c"   and m :: "'c" assumes "[a = b] (mod m)" shows "gcd a m = gcd b m"
edges_are_in_V_2: fixes v :: "nat"   and w :: "nat" assumes "v \<rightarrow> w" shows "w \<in> V"
infsetsum_cmult_left: fixes c :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "c \<noteq> (0::'d) \<Longrightarrow> f abs_summable_on A" shows "(\<Sum>\<^sub>ax\<in>A. f x * c) = infsetsum f A * c"
less_supI2: fixes x :: "'a"   and b :: "'a"   and a :: "'a" assumes "x < b" shows "x < a \<squnion> b"
ir_If_True: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> bval b s) c\<^sub>1 c' Q" shows "ir_hoare P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) c' Q"