hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "'g"   and x2 :: "'g"   and r :: "('g \<times> 'g) set"   and P :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "(x1, x2) \<in> r\<^sup>*"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>(a, b) \<in> r\<^sup>*; P a b; (b, c) \<in> r\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
upper_triangular_upt_k_def: fixes A :: "(('d, 'e) vec, 'e) vec"   and k :: "nat" shows "upper_triangular_upt_k A k = (\<forall>i j. j < i \<and> mod_type_class.to_nat j < k \<longrightarrow> A $ i $ j = (0::'d))"
add_nonpos_eq_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq x zero"   and "less_eq y zero" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
table_eq_rel: fixes t1 :: "'a MFOTL.trm"   and t2 :: "'a MFOTL.trm"   and n :: "nat" assumes "trm.is_Const t1 \<or> trm.is_Const t2" shows "table n (fv_trm t1 \<union> fv_trm t2) (eq_rel n t1 t2)"
shift_prefix_cases: fixes xl :: "'c list"   and xs :: "'c stream"   and yl :: "'c list"   and ys :: "'c stream" assumes "xl @- xs = yl @- ys" shows "prefix xl yl \<or> prefix yl xl"
edge_succ_a_neq: shows "edge_succ M a \<noteq> a'"
viable_only_if_d: assumes "viable_market Mkt" shows "d < 1 + r"
category_axioms: shows "category (\<cdot>\<^sup>o\<^sup>p)"
dim_eq_card_independent: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B :: "'e set" assumes "vector_space scale"   and "\<not> module.dependent scale B" shows "vector_space.dim scale B = card B"
currentD_finite_OUT: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "current \<Gamma>' f" shows "d_OUT f x \<noteq> \<top>"
hdb: shows "hd vs # b = fst (splitAt ram1 vs)"
Arccos_Arcsin_csqrt_pos: fixes z :: "complex" assumes "0 < Re z \<or> Re z = 0 \<and> 0 \<le> Im z" shows "Arccos z = Arcsin (csqrt (1 - z\<^sup>2))"
S1_def: shows "S1 \<equiv> \<lambda>input\<^sub>1 (). let (b\<^sub>0, b\<^sub>1) = input\<^sub>1 in I \<bind> (\<lambda>(\<alpha>, \<tau>). local.etp.S \<alpha> \<bind> (\<lambda>y\<^sub>0. local.etp.S \<alpha> \<bind> (\<lambda>y\<^sub>1. return_spmf ((b\<^sub>0, b\<^sub>1), y\<^sub>0, y\<^sub>1))))"
is_cnf_encode_positive_transition_frame_axiom: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and v :: "'b" shows "is_cnf (encode_positive_transition_frame_axiom \<Pi> t v)"
simps_1: fixes A :: "char list"   and X :: "ctrm set" shows "NEGn (PR A) X = AXIOMSn (PR A) \<union> BINDINGn (PR A) X"
dom_inv: fixes f :: "'b" assumes "B.iso f" shows "B.dom (B.inv f) = B.cod f"
ide_char': fixes a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" shows "Cop_S.ide a = (Cop_S.arr a \<and> (Cop_S.dom a = a \<or> Cop_S.cod a = a))"
openin_prod_topology_alt: fixes X :: "'d topology"   and Y :: "'e topology"   and S :: "('d \<times> 'e) set" shows "openin (prod_topology X Y) S = (\<forall>x y. (x, y) \<in> S \<longrightarrow> (\<exists>U V. openin X U \<and> openin Y V \<and> x \<in> U \<and> y \<in> V \<and> U \<times> V \<subseteq> S))"
A: fixes x :: "rat" shows "Pri_AgrK x \<notin> U \<longrightarrow> Pri_AgrK x \<notin> parts (A \<union> spies evsFR4)"
keep_drop_sels_4: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "snd (R\<^sup>d) = invalid_assn R"
dg_dghm_id_is_dghm: shows "dghm_id \<CC> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
integral_fst: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "integrable (lborel \<Otimes>\<^sub>M lborel) (\<lambda>(x, y). f x y)" shows "LBINT x. integral\<^sup>L lborel (f x) = LINT (x, y)|(lborel \<Otimes>\<^sub>M lborel). f x y"
Suc_length_conv: fixes n :: "nat"   and xs :: "'f list" shows "(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
det_0_iff_vec_prod_zero_field: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(det A = (0::'b)) = (\<exists>v. v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> A *\<^sub>v v = 0\<^sub>v n)"
mprojTr2: fixes M1 :: "('e, 'a, 'f) Module_scheme"   and M2 :: "('e, 'a, 'f) Module_scheme"   and M :: "('e, 'a, 'f) Module_scheme"   and x :: "'e"   and x1 :: "'e"   and y1 :: "'e" assumes "R module M1"   and "R module M2"   and "R module M"   and "M1 \<Oplus>\<^bsub>R,M\<^esub> M2"   and "x \<in> carrier M"   and "x1 \<in> carrier M1"   and "x \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> x1 \<in> carrier M2"   and "y1 \<in> carrier M1"   and "x \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> y1 \<in> carrier M2" shows "x1 = y1"
subfield_Units: fixes K :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subfield K R" shows "Units (R\<lparr>carrier := K\<rparr>) = K - {\<zero>\<^bsub>R\<^esub>}"
steps_f_add': fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat"   and l :: "nat"   and u :: "nat"   and v :: "nat"   and m :: "nat" shows "steps A f l u v = steps A (\<lambda>x. f x + m) (l + m) u v"
ereal_of_enat_zero: shows "ereal_of_enat 0 = 0"
all_same_arities_add: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd1 :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and fd2 :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "Fubx_get F f = Some fd1"   and "arity fd1 = arity fd2" shows "all_same_arities (Fubx_get F) (Fubx_get (Fubx_add F f fd2))"
cfun_not_below: fixes f :: "'b \<rightarrow> 'a"   and x :: "'b"   and g :: "'b \<rightarrow> 'a" assumes "f\<cdot>x \<notsqsubseteq> g\<cdot>x" shows "f \<notsqsubseteq> g"
simps_1: fixes v :: "'b" shows "vars_of (Var v) = {v}"
delay_image: shows "\<tau> ` {\<delta>..} = {0..}"
ltake_LCons_Suc: fixes a :: "'b"   and l :: "'b llist"   and i :: "nat" shows "(a ## l) \<down> Suc i = a ## l \<down> i"
sums_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> nat \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'b" assumes "\<And>i. i \<in> I \<Longrightarrow> f i sums x i" shows "(\<lambda>n. \<Sum>i\<in>I. f i n) sums sum x I"
pr_ideal_def: fixes R :: "'b set"   and I :: "'b set"   and addition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and multiplication :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and unit :: "'b" shows "pr_ideal R I addition multiplication zero unit \<equiv> comm_ring R addition multiplication zero unit \<and> ideal I R addition multiplication zero unit \<and> pr_ideal_axioms R I multiplication"
in_subnet_def: fixes a :: "'b"   and S :: "'b set set" shows "a \<sqsubset> S = (\<exists>s\<in>S. a \<in> s)"
sum: fixes A :: "'b set"   and fa :: "'b \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>i. i \<in> A \<Longrightarrow> integrable M (fa i)"   and "\<And>i. i \<in> A \<Longrightarrow> P (fa i)" shows "P (\<lambda>x. \<Sum>i\<in>A. fa i x)"
divisors_int_pos_2: fixes x :: "int" shows "distinct (divisors_int_pos x)"
sorted_poly_list_rel_Cons_iff: fixes ys :: "char list list"   and n :: "int"   and p :: "(char list list \<times> int) list"   and a :: "(char list multiset \<times> int) multiset"   and S :: "char list list \<Rightarrow> char list list \<Rightarrow> bool" shows "(((ys, n) # p, a) \<in> sorted_poly_list_rel S) = ((p, remove1_mset (mset ys, n) a) \<in> sorted_poly_list_rel S \<and> (mset ys, n) \<in># a \<and> (\<forall>x\<in>set p. S ys (fst x)) \<and> sorted_wrt var_order ys \<and> distinct ys \<and> ys \<notin> set (map fst p) \<and> n \<noteq> 0 \<and> nonzero_coeffs a)"
select_convs_1: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "ip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = ip"
eqD_2: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and x :: "'b"   and y :: "'b" assumes "class.Eq_eq eq"   and "eq\<cdot>x\<cdot>y = FF" shows "x \<noteq> y"
iwlsFSbSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "iwlsFSbSw MOD \<equiv> iwlsFSb MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igSwapCls MOD"
real_root_decreasing: fixes n :: "nat"   and N :: "nat"   and x :: "real" assumes "0 < n"   and "n \<le> N"   and "1 \<le> x" shows "root N x \<le> root n x"
Enc: fixes a :: "cmsg"   and b :: "cmsg"   and c :: "cmsg"   and d :: "cmsg" assumes "eq a b"   and "eq c d" shows "eq (cEnc a c) (cEnc b d)"
row_transpose_scalar_prod_as_sum: fixes j :: "nat"   and A :: "'b mat"   and v :: "'b vec" assumes "j < dim_col A"   and "dim_vec v = dim_row A" shows "row A\<^sup>T j \<bullet> v = (\<Sum>i = 0..<dim_vec v. A $$ (i, j) * v $ i)"
fresh_bij: fixes pi :: "(atom \<times> atom) list"   and a :: "atom"   and x :: "'a" shows "pi \<bullet> a \<sharp> pi \<bullet> x = a \<sharp> x"
lexordp_eq_conv_lexord: fixes xs :: "'a word list"   and ys :: "'a word list" shows "signed.lexordp_eq xs ys = (xs = ys \<or> signed.lexordp xs ys)"
in_orbit_invI: fixes b :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "b \<in> orbit f a"   and "inj f" shows "a \<in> orbit (inv f) b"
angle_actrans_conj: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "|~ \<langle>F \<and> G\<rangle>_v = (\<langle>F\<rangle>_v \<and> \<langle>G\<rangle>_v)"
sum_choose_lower: fixes r :: "nat"   and n :: "nat" shows "(\<Sum>k\<le>n. r + k choose k) = Suc (r + n) choose n"
linf_norm_vec_greater_0: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "((0::'b) < \<parallel>v\<parallel>\<^sub>\<infinity>) = (v \<noteq> 0\<^sub>v n)"
restrictI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "restrict f A \<in> Pi A B"
MkIde_Cod: fixes a :: "('o, 'a) arr" assumes "arr a" shows "MkIde (Cod a) = cod a"
member_psubst: fixes p :: "('a, 'b) form"   and z :: "('a, 'b) form list"   and f :: "'a \<Rightarrow> 'c" assumes "p \<in> set z" shows "psubst f p \<in> set (map (psubst f) z)"
index_minus_vec_1: fixes i :: "nat"   and v\<^sub>2 :: "'b vec"   and v\<^sub>1 :: "'b vec" assumes "i < dim_vec v\<^sub>2" shows "(v\<^sub>1 - v\<^sub>2) $ i = v\<^sub>1 $ i - v\<^sub>2 $ i"
fls_subdegree_deriv': fixes f :: "'a fls" assumes "of_int (fls_subdegree f) \<noteq> (0::'a)" shows "fls_subdegree (fls_deriv f) = fls_subdegree f - 1"
sgn_zero_iff: fixes x :: "'a" shows "(sgn x = (0::'a)) = (x = (0::'a))"
AB: shows "\<forall>a\<in>A. \<forall>b\<in>B. a \<sqsubseteq> b"
regular_conv_closed: fixes x :: "'a" assumes "regular x" shows "regular (id x)"
remdups_adj_Cons': fixes x :: "'c"   and xs :: "'c list" shows "remdups_adj (x # xs) = x # remdups_adj (dropWhile (\<lambda>y. y = x) xs)"
any_sdominate_unreachable: fixes n :: "nat"   and d :: "nat" assumes "\<not> reachable n" shows "strict_dominate d n"
times_monomial_monomial: fixes t :: "'c"   and c :: "'d"   and v :: "'c"   and d :: "'d" shows "monomial c t * monomial d v = monomial (c * d) (t + v)"
Cons_le_Cons: fixes x :: "'a"   and xs :: "'a list"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and ys :: "'a list" shows "(x # xs [\<sqsubseteq>\<^bsub>r\<^esub>] y # ys) = (x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys)"
forget_automatic: fixes x :: "name"   and L :: "lam"   and P :: "lam" assumes "x \<sharp> L" shows "L[x::=P] = L"
intros: fixes M :: "hmultiset" assumes "Multiset.Ball (hmsetmset M) local.well_base\<^sub>h"   and "local.well_base (hmsetmset M)" shows "local.well_base\<^sub>h M"
map_conv_bind_option: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c option" shows "map_option f x = Option.bind x (Some \<circ> f)"
subterms_singleton: fixes t :: "('b, 'a) Term.term" assumes "(\<exists>v. t = Var v) \<or> (\<exists>f. t = Fun f [])" shows "subterms t = {t}"
max_flow_min_cut: fixes \<Delta> :: "('c, 'd) network_scheme" assumes "countable_network \<Delta>" shows "\<exists>f S. flow \<Delta> f \<and> MFMC_Network.cut \<Delta> S \<and> orthogonal \<Delta> f S"
root_pow_root: fixes v :: "'b list"   and u :: "'b list"   and n :: "nat" assumes "v \<in> u*" shows "v \<^sup>@ n \<in> u*"
t_star_contr: fixes X :: "('a \<times> ('b \<times> 'a) list) set"   and Y :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod (t_star X) Y = (\<Union>n. t_prod (trace_dioid.power X n) Y)"
fM_fA_invL_def: shows "gc.fM_fA_invL = (\<lambda>s. (atS gc gc.fM_eq_locs s \<longrightarrow> gc_fM s\<down> = sys_fM s\<down>) \<and> (at gc gc.idle_store_fM s \<longrightarrow> gc_fM s\<down> \<noteq> sys_fM s\<down>) \<and> (at gc gc.idle_flip_noop_mfence s \<longrightarrow> sys_fM s\<down> \<noteq> gc_fM s\<down> \<longrightarrow> filter is_mw_fM (sys_mem_store_buffers gc s\<down>) \<noteq> []) \<and> (atS gc gc.fM_tso_empty_locs s \<longrightarrow> filter is_mw_fM (sys_mem_store_buffers gc s\<down>) = []) \<and> (atS gc gc.fA_eq_locs s \<longrightarrow> gc_fM s\<down> = sys_fA s\<down>) \<and> (atS gc gc.fA_neq_locs s \<longrightarrow> gc_fM s\<down> \<noteq> sys_fA s\<down>) \<and> (at gc gc.mark_noop_mfence s \<longrightarrow> gc_fM s\<down> \<noteq> sys_fA s\<down> \<longrightarrow> filter is_mw_fA (sys_mem_store_buffers gc s\<down>) \<noteq> []) \<and> (atS gc gc.fA_tso_empty_locs s \<longrightarrow> filter is_mw_fA (sys_mem_store_buffers gc s\<down>) = []))"
FV_def: fixes S :: "Core.term set" shows "FV S = \<Union> (fv ` S)"
simulation2_aux: fixes s1 :: "'s1"   and s2 :: "'s2"   and tl2 :: "'tl2"   and s2' :: "'s2" assumes "s1 \<approx> s2"   and "s2 -2-tl2\<rightarrow> s2'"   and "\<not> \<tau>move2 s2 tl2 s2'" shows "\<exists>s1' s1'' tl1. s1 -\<tau>1\<rightarrow>* s1' \<and> s1' -1-tl1\<rightarrow> s1'' \<and> \<not> \<tau>move1 s1' tl1 s1'' \<and> s1'' \<approx> s2' \<and> tl1 \<sim> tl2"
n2m_dtor_corec_coG_def: fixes s1 :: "'e \<Rightarrow> ('f, 'f coG coF + 'g) coG0"   and s2 :: "'g \<Rightarrow> ('f coG + 'e, 'f coG coF + 'g) coF0" shows "n2m_dtor_corec_coG s1 s2 = dtor_corec_coG (BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf \<circ> map_pre_coG\<^sub>M id id (case_sum (map_coF Inl) (dtor_corec_coF (BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf \<circ> map_pre_coGcoF\<^sub>M id id (map_sum (map_coF Inl) id) \<circ> s2))) \<circ> s1)"
nabla_le_s: fixes x :: "'a" shows "\<nabla> x \<le> 1\<^sub>\<sigma>"
Merge_insert: fixes x :: "transaction\<^sub>m"   and A :: "transaction\<^sub>m set" shows "transaction.Merge (insert x A) = (if A = {} then Some x else transaction.Merge A \<bind> merge_transaction x)"
rank_def: fixes a :: "V" shows "rank a \<equiv> transrec (\<lambda>f x. set (\<Union>y\<in>elts x. elts (ZFC_in_HOL.succ (f y)))) a"
assms_4: shows "the (rt2 dip) \<sqsubseteq> the (rt1 dip)"
preserves_trg: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "trg\<^sub>C (F \<mu>) = F.map\<^sub>0 (trg\<^sub>D \<mu>)"
sr_imp_eigen_vector_main: fixes u :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "u \<in> X"   and "r u = sr" shows "sr *s u = A *v u"
mn_loop: fixes B :: "abc_inst list"   and arity :: "nat"   and fap :: "abc_inst list"   and ft :: "nat"   and fft :: "nat"   and xs :: "nat list"   and far :: "nat"   and x :: "nat"   and f :: "recf"   and anything :: "nat list" assumes "B = [Dec (Suc arity) (length fap + 5), Dec (Suc arity) (length fap + 3), Goto (Suc (length fap)), Inc arity, Goto 0]"   and "ft = max (Suc arity) fft"   and "length xs = arity"   and "far = Suc arity"   and "\<forall>xc. {\<lambda>nl. nl = xs @ x # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (fft - Suc far) @ xc}"   and "0 < rec_exec f (xs @ [x])"   and "rec_ci f = (fap, far, fft)" shows "\<exists>stp>0. abc_steps_l (0, xs @ x # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp = (0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything)"
iso5: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> U \<cdot> 1\<^sub>\<pi> = x \<cdot> 1\<^sub>\<pi>"
NCol_perm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B C \<and> \<not> Tarski_neutral_dimensionless.Col Bet A C B \<and> \<not> Tarski_neutral_dimensionless.Col Bet B A C \<and> \<not> Tarski_neutral_dimensionless.Col Bet B C A \<and> \<not> Tarski_neutral_dimensionless.Col Bet C A B \<and> \<not> Tarski_neutral_dimensionless.Col Bet C B A"
transpose_infmatrix_inject: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and B :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "(transpose_infmatrix A = transpose_infmatrix B) = (A = B)"
crsp_step: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and ins :: "abc_inst" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "abc_fetch as ap = Some ins" shows "\<exists>stp>0. crsp ly (abc_step_l (as, lm) (Some ins)) (steps0 (s, l, r) tp stp) ires"
splus_left_commute: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and s :: "'b"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.splus pair_of_term term_of_pair s (term_powerprod.splus pair_of_term term_of_pair t v) = term_powerprod.splus pair_of_term term_of_pair t (term_powerprod.splus pair_of_term term_of_pair s v)"
multiplicity_2_diff_pow_4div: fixes x :: "int"   and y :: "int"   and n :: "nat" assumes "odd x"   and "odd y"   and "4 dvd x - y"   and "0 < n"   and "x \<noteq> y" shows "multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 n"
a'_def: shows "a' \<equiv> fst (normalize_quot (a, b))"
FIN: shows "\<forall>A\<in>set Flist. finite A"
mult_right_less_imp_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_semiring plus minus less_eq less zero times"   and "less (times a c) (times b c)"   and "less_eq zero c" shows "less a b"
mono_nest: fixes r :: "('a \<times> 'a) set" shows "mono (nest r)"
nneg_bounded_by_nneg: fixes b :: "real"   and P :: "'b \<Rightarrow> real" assumes "bounded_by b P"   and "nneg P" shows "0 \<le> b"
analz_valid: fixes H :: "msg set"   and x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> sym.valid \<union> range LtK \<union> Tags"   and "implem_sym (Chan x A B M) \<in> analz H" shows "implem_sym (Chan x A B M) \<in> H"
nn_integral_PInf_AE: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "f \<in> borel_measurable M"   and "integral\<^sup>N M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
nfv_simps_1: fixes \<phi> :: "'b formula" shows "nfv (Neg \<phi>) = nfv \<phi>"
assms: fixes s :: "'a" assumes "fds_abs_converges g s" shows "fds_abs_converges f s"
to_g: shows "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. cmod (f n x) \<le> h n"
tan_cot: fixes x :: "real" shows "tan (pi / 2 - x) = inverse (tan x)"
fact_gt_zero: fixes n :: "nat" shows "(0::'b) < fact n"
element_power_order_eq_1: fixes x :: "'a mod_ring poly" assumes "x \<in> carrier (mult_of R)" shows "x [^]\<^bsub>mult_of R\<^esub> order (mult_of R) = \<one>\<^bsub>mult_of R\<^esub>"
distinct_entries: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "distinct (list.map fst (RBT_Impl.entries t))"
assms: shows "z1 * cnj z1 = z2 * cnj z2"
permute_Abs_set: fixes p :: "perm"   and as :: "atom set"   and x :: "'e" shows "p \<bullet> ([as]set. x) = [p \<bullet> as]set. p \<bullet> x"
inj_p_fun: fixes x :: "'a" shows "is_inj x = is_p_fun (x\<^sup>T)"
M_carrier: fixes x :: "'a"   and l :: "nat" assumes "x \<in> M (V l)" shows "x \<in> carrier"
assms_2: shows "\<forall>P. Col C D P \<longrightarrow> A B OS C P"
test_d_fixpoint_iff: fixes x :: "'a" shows "test x = (d x = x)"
graph_def: fixes F :: "'c set"   and f :: "'c \<Rightarrow> real" shows "graph F f = {(x, f x) |x. x \<in> F}"
eq: fixes p :: "nat" assumes "prime p" shows "(\<Sum>\<^sub>an\<in>{p..}. f n p) = complex_of_real (ln (real p) / real p) * hurwitz_zeta (real p) s"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_loop_check m rs (l, r) = (\<exists>ml mr ln rn. l = Oc # Bk \<up> ln @ Bk # Oc # Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> r = Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc rs)"
dist_s_mtf: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list"   and n :: "nat" assumes "MTF_Off as rs s0" shows "distinct (MTF_Off.s_mtf rs s0 n)"
tendsto_compose: fixes g :: "'a \<Rightarrow> 'b"   and l :: "'a"   and f :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "g \<midarrow>l\<rightarrow> g l"   and "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
lim_proc_is_ub: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "range S <| lim_proc (range S)"
axioms_1: fixes M :: "real measure" assumes "finite_borel_measure M" shows "finite_measure M"
gcsrp_map_gbijec: fixes N :: "'a set" assumes "G \<triangleright> N" shows "gbij\<^bsub>(G / N),Gcsrp G N\<^esub> gcsrp_map G N"
le_supI2: fixes x :: "'b"   and b :: "'b"   and a :: "'b" assumes "x \<le> b" shows "x \<le> sup a b"
res: shows "ord_resolve (CAs \<cdot>\<cdot>cl \<rho>s) (DA \<cdot> \<rho>) (AAs \<cdot>\<cdot>aml \<rho>s) (As \<cdot>al \<rho>) \<sigma> E"
key: shows "(flow t\<^sub>0 s usolves_ode f from t\<^sub>0) (ex_ivl s) S"
pair_bidirected_digraphI_bidirected_digraph: fixes G :: "'a pair_pre_digraph" assumes "bidirected_digraph (with_proj G) (swap_in (parcs G))" shows "pair_bidirected_digraph G"
particular_affirmation_def: fixes A :: "'a set"   and B :: "'a set" shows "A I B \<equiv> \<exists>b\<in>B. b \<in> A"
tickFree_append: fixes s :: "'b event list"   and t :: "'b event list" shows "tickFree (s @ t) = (tickFree s \<and> tickFree t)"
rep_eq: shows "apply_commute Uprod.not_equal = (\<noteq>)"
norm_mult: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.real_normed_div_algebra divide inverse one times plus zero minus uminus scaleR dist norm sgn uniformity open" shows "norm (times x y) = norm x * norm y"
relative_to_inc: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and U :: "'c set" assumes "P S" shows "(P relative_to U) (U \<inter> S)"
assms: fixes x :: "'a"   and s :: "'b" shows "RETURN (f x s) \<le> f' x s"