hcju/isabelleps huggingface dataset:
simps_2: fixes G :: "('b, 'a) labeled_graph"   and a :: "'b allegorical_term"   and b :: "'b allegorical_term" shows ":G:\<lbrakk>A_Cmp a b\<rbrakk> = (:G:\<lbrakk>a\<rbrakk>) O (:G:\<lbrakk>b\<rbrakk>)"
rball: shows "\<forall>x\<in>proots r. Im x = 0"
FVar: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and C :: "qtname"   and fn :: "vname"   and statDeclC :: "qtname"   and f :: "Decl.field" assumes "E,dt\<Turnstile>e\<Colon>-Class C"   and "accfield (prg E) (cls E) C fn = Some (statDeclC, f)" shows "E,dt\<Turnstile>{cls E,statDeclC,is_static f}e..fn\<Colon>=type f"
strong_cases: fixes a1 :: "lam"   and a2 :: "lam"   and x :: "name"   and xa :: "name" assumes " a1 \<longrightarrow>\<^sub>d a2" obtains x :: "name" where "a1 = Var x" and "a2 = Var x"   | t :: "lam" and s :: "lam" where "\<lbrakk>x__ \<sharp> a1; x__ \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = Lam [x__].t \<and> a2 = Lam [x__].s \<and>  t \<longrightarrow>\<^sub>d s"   | t1 :: "lam" and t2 :: "lam" and s1 :: "lam" and s2 :: "lam" where "a1 = App t1a__ s1__" and "a2 = App t2a__ s2__" and "\<nexists>y t'. t1a__ = Lam [y].t'" and " t1a__ \<longrightarrow>\<^sub>d t2a__" and " s1__ \<longrightarrow>\<^sub>d s2__"   | s1 :: "lam" and s2 :: "lam" and t1 :: "lam" and t2 :: "lam" where "\<lbrakk>xa \<sharp> a1; xa \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = App (Lam [xa].t1) s1 \<and> a2 = t2[xa::=s2] \<and> xa \<sharp> (s1, s2) \<and>  t1 \<longrightarrow>\<^sub>d t2 \<and>  s1 \<longrightarrow>\<^sub>d s2"
awalk_decomp: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and w :: "'a" assumes "G.awalk u p v"   and "w \<in> set (pawalk_verts u p)" shows "\<exists>q r. p = q @ r \<and> G.awalk u q w \<and> G.awalk w r v"
simps_4: fixes c :: "int"   and e :: "Ferrack.num" shows "plusinf (NEq (CN 0 c e)) = T"
simps_1: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and chan :: "chan set"   and bad :: "agent set"   and more :: "'a" shows "chan \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, chan = chan, bad = bad, \<dots> = more\<rparr> = chan"
induct: fixes P :: "recf \<Rightarrow> bool"   and a0 :: "recf" assumes "P Z"   and "P S"   and "\<And>m n. P (recf.Id m n)"   and "\<And>n f gs. P (Cn n f gs)"   and "\<And>n f g. P (Pr n f g)"   and "\<And>n f. P (Mn n f)" shows "P a0"
simps_7: fixes \<phi> :: "'b ltln"   and N :: "'b ltln set" shows "X\<^sub>n \<phi>[N]\<^sub>\<Pi>\<^sub>2 = X\<^sub>n \<phi>[N]\<^sub>\<Pi>\<^sub>2"
aeq: fixes f :: "nat \<Rightarrow> 'o"   and n :: "nat" shows "(\<Sum>i<n. T (f i) (state f i)) = sum (A f) {..<n} - \<Phi> (state f n)"
pow_ge_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and n :: "nat" assumes "class.ordered_semiring_1 one times plus zero less_eq"   and "less_eq zero a" shows "less_eq zero (power.power one times a n)"
return_def: fixes v :: "val" shows "return v \<equiv> {v'. v' \<sqsubseteq> v}"
annotateI: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and anno :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set"   and c :: "('f, 'e, 'g) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P anno Q,A"   and "c = anno" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
rcis_inj_on: fixes r :: "real" assumes "r \<noteq> 0" shows "inj_on (rcis r) {0..<2 * pi}"
that: fixes f :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and g :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "(f, g) \<in> set ps \<or> (g, f) \<in> set ps"   and "h = monom_mult ((1::'b) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f)" shows "thesis"
hyp_rref: shows "reduced_row_echelon_form_upt_k (Gauss_Jordan_upt_k A k) (Suc k)"
chain_mono: fixes Y :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "chain Y"   and "i \<le> j" shows "Y i \<sqsubseteq> Y j"
eq_f: shows "Twelvefold_Way_Entry10.functions_of A B N = domain_and_range_permutation A B `` {f}"
run_asset_i_ex: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "pred_asset_i n s (run_asset_i n s)"
prv_dsjL_fls: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (eqv (dsj \<phi> fls) \<phi>)"
small_le: fixes u :: "real"   and n :: "int" assumes "0 \<le> u"   and "u < 1" shows "(- u \<le> real_of_int n) = (0 \<le> n)"
realfun_mon_conv_times: fixes x :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and y :: "'b \<Rightarrow> real"   and z :: "real" assumes "x\<up>y"   and "0 \<le> z" shows "(\<lambda>m w. z * x m w)\<up>(\<lambda>w. z * y w)"
recvmsgE: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma> :: "'e"   and a :: "'d seq_action"   and \<sigma>' :: "'e" assumes "recvmsg (R \<sigma>) a"   and "\<And>m. R \<sigma> m \<Longrightarrow> R \<sigma>' m" shows "recvmsg (R \<sigma>') a"
preserves_cod: fixes f :: "'c" assumes "C.arr f" shows "D.cod (G f) = G (C.cod f)"
Nil_is_rev_conv: fixes xs :: "'a list" shows "([] = rev xs) = (xs = [])"
contraposition: fixes x :: "'a"   and y :: "'a" shows "inv_resr y (- x) = inv_resl (-' y) x"
pcp_def: fixes C :: "'b formula set set" shows "pcp C \<equiv> \<forall>S\<in>C. \<bottom> \<notin> S \<and> (\<forall>k. Atom k \<in> S \<longrightarrow> \<^bold>\<not> (Atom k) \<in> S \<longrightarrow> False) \<and> (\<forall>F G. F \<^bold>\<and> G \<in> S \<longrightarrow> F \<triangleright> G \<triangleright> S \<in> C) \<and> (\<forall>F G. F \<^bold>\<or> G \<in> S \<longrightarrow> F \<triangleright> S \<in> C \<or> G \<triangleright> S \<in> C) \<and> (\<forall>F G. F \<^bold>\<rightarrow> G \<in> S \<longrightarrow> \<^bold>\<not> F \<triangleright> S \<in> C \<or> G \<triangleright> S \<in> C) \<and> (\<forall>F. \<^bold>\<not> (\<^bold>\<not> F) \<in> S \<longrightarrow> F \<triangleright> S \<in> C) \<and> (\<forall>F G. \<^bold>\<not> (F \<^bold>\<and> G) \<in> S \<longrightarrow> \<^bold>\<not> F \<triangleright> S \<in> C \<or> \<^bold>\<not> G \<triangleright> S \<in> C) \<and> (\<forall>F G. \<^bold>\<not> (F \<^bold>\<or> G) \<in> S \<longrightarrow> \<^bold>\<not> F \<triangleright> \<^bold>\<not> G \<triangleright> S \<in> C) \<and> (\<forall>F G. \<^bold>\<not> (F \<^bold>\<rightarrow> G) \<in> S \<longrightarrow> F \<triangleright> \<^bold>\<not> G \<triangleright> S \<in> C)"
joiner_def: shows "joiner = (\<lambda>s. \<Union>k. s k \<inter> UNIV \<times> {v. k \<in> strategy v})"
sum_incoming_pointwise: fixes g :: "nat \<times> nat \<Rightarrow> 'c"   and u :: "nat" shows "sum g (incoming u) = (\<Sum>v\<in>E\<inverse> `` {u}. g (v, u))"
abs_eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(abs_fset x = abs_fset y) = (x \<approx> y)"
rep_eq: fixes x :: "\<kappa>" shows "proper x = (None \<noteq> eval\<kappa> x)"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>B b\<guillemotright>" shows "B.arr f"
relation_of_Stop: shows "action.relation_of Stop = R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> wait A')"
pwd_owner_iii_ex: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "pred_owner_iii n (pwd_spy_ii n) (pwd_owner_iii n)"
PSQ: shows "(P, S, Q) = Smith_2xn A\<^sup>T"
times_eq_iff: fixes A :: "'c set"   and B :: "'d set"   and C :: "'c set"   and D :: "'d set" shows "(A \<times> B = C \<times> D) = (A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
distributed_integral: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "distributed M N X (\<lambda>x. ennreal (f x))"   and "g \<in> borel_measurable N"   and "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x" shows "LINT x|N. f x * g x = LINT x|M. g (X x)"
scaleR_left: fixes r :: "real"   and a :: "'b \<Rightarrow>\<^sub>L 'a"   and b :: "'b" shows "blinfun_apply (r *\<^sub>R a) b = r *\<^sub>R blinfun_apply a b"
V0_ss: shows "G'.V0 \<subseteq> G'.V"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VV.seq g f = (VV.arr f \<and> VV.arr g \<and> VxV.seq g f)"
remdups_mset_def: fixes S :: "'a multiset" shows "remdups_mset S = mset_set (set_mset S)"
psubstEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho'' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "goodEnv rho"   and "goodEnv rho'"   and "goodEnv rho''" shows "rho &[rho'] &[rho''] = rho &[rho' &[rho'']]"
\<kappa>\<^sub>l_def: shows "M.sumlist (map (\<lambda>j. - \<mu> l j \<cdot>\<^sub>v gso j) [0..<l]) = M.sumlist (map (\<lambda>j. \<kappa>\<^sub>l j \<cdot>\<^sub>v fs ! j) [0..<l])"
apply_adaptation_def: fixes first :: "bool"   and mem :: "'a \<Rightarrow> 'b"   and A :: "'a \<Rightarrow> ('b \<times> 'b) option" shows "apply_adaptation first mem A = (\<lambda>x. case A x of None \<Rightarrow> mem x | Some (v\<^sub>1, v\<^sub>2) \<Rightarrow> if first then v\<^sub>1 else v\<^sub>2)"
strengthen_postR: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} c { Q}"   and "\<And>s. Q s \<Longrightarrow> Q' s" shows "\<turnstile>\<^sub>3\<^sub>a {P} c { Q'}"
a_distinct_lists_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_distinct_lists get_child_nodes get_disconnected_nodes h = distinct (concat (map (\<lambda>ptr. |h \<turnstile> get_child_nodes ptr|\<^sub>r) |h \<turnstile> object_ptr_kinds_M|\<^sub>r @ map (\<lambda>document_ptr. |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r) |h \<turnstile> document_ptr_kinds_M|\<^sub>r))"
ltl_prop_equiv_def: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" shows "\<phi> \<equiv>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> = \<A> \<Turnstile>\<^sub>P \<psi>"
set_of_abs_interval: fixes x :: "'a interval" shows "set_of (abs_interval x) = abs ` set_of x"
K_def: shows "K = {V. open V \<and> V \<subseteq> U}"
perron_frobenius_def: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" shows "perron_frobenius A \<equiv> pf_nonneg_mat A \<and> perron_frobenius_axioms A"
regular: assumes "\<And>v u. \<lbrakk>v \<in> V; u \<in> V; v \<noteq> u\<rbrakk> \<Longrightarrow> \<exists>!n. adjacent v n \<and> adjacent u n"   and "finite E"   and "finite V"   and "\<not> (\<exists>v\<in>V. degree v G = 2)" shows "\<exists>k. \<forall>v\<in>V. degree v G = k"
path_from_given_point_1: fixes x :: "'a"   and p :: "'a" assumes "path x"   and "point p" shows "path (x\<^sup>T\<^sup>\<star> ; p \<cdot> x)"
App_apps: fixes s :: "('a, 'b) tm"   and ts :: "('a, 'b) tm list"   and t :: "('a, 'b) tm" shows "App (apps s ts) t = apps s (ts @ [t])"
CondSwap: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P1 :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P2 :: "'e set"   and c2 :: "('e, 'd, 'f) com"   and P :: "'e set"   and b :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P1 c1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P2 c2 Q,A"   and "P \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P1) \<and> (s \<notin> b \<longrightarrow> s \<in> P2)}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Cond b c1 c2 Q,A"
assms: shows "negs (mset As) = SDA' \<cdot> \<eta>"
Recv: shows "ev' = Recv i'' (occurs_on ev') s u'' u''' m'"
assms_4: shows "(ir, Rem is e) \<in> set (node_deliver_messages xs)"
mult_2: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<omega>[F](g1)"   and "f2 \<in> \<Omega>[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> \<omega>[F](\<lambda>x. g1 x * g2 x)"
path_from_to_ends: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and w :: "'c"   and ys :: "'c list" assumes "Digraph G"   and "v \<leadsto>(xs @ w # ys)\<leadsto>\<^bsub>G\<^esub> w" shows "ys = []"
hf_of_nat_def: shows "hf_of \<equiv> ord_of"
key_gen_def: fixes \<G> :: "'a cyclic_group" shows "elgamal_base.key_gen \<G> = sample_uniform (order \<G>) \<bind> (\<lambda>x. return_spmf (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> x, x))"
strand_sem_subst_c': fixes \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and M :: "('fun, 'var) Term.term set"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"   and "\<lbrakk>M; S \<cdot>\<^sub>s\<^sub>t \<delta>\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M; S\<rbrakk>\<^sub>c (\<delta> \<circ>\<^sub>s \<theta>)"
i: shows "\<And>s. (P s) \<Longrightarrow> (Q s)"
triple_schroeder_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" shows "(x \<sqinter> y \<sqinter> z \<le> - w) = (id x \<sqinter> w \<sqinter> id z \<le> - y)"
F: shows "arr F \<and> (\<forall>f. f \<in> Map F \<longrightarrow> B.equivalence_map f)"
VectorSpace_FGModuleHomSet: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModule G smult V"   and "FGModule G smult' W" shows "VectorSpace (\<lambda>a T v. aezfun_scalar_mult.fsmult smult' a (T v)) (FGModuleHomSet G smult V smult' W)"
simps: fixes atom_ordering :: "('a \<times> 'a) set"   and P1 :: "'a Literal set"   and P2 :: "'a Literal set"   and A :: "'a" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.resolvent_upon P1 P2 A = P1 - {Pos A} \<union> (P2 - {Neg A})"
interp_subseteq_Interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.interp S N C \<subseteq> ground_resolution_with_selection.Interp S N C"
minus_mult_right: fixes a :: "'b"   and b :: "'b" shows "- (a * b) = a * - b"
signed_take_bit_Suc_bit1: fixes n :: "nat"   and k :: "num" shows "signed_take_bit (Suc n) (numeral (num.Bit1 k)) = signed_take_bit n (numeral k) * 2 + 1"
dflt_size_integer_def: shows "dflt_size_integer \<equiv> integer_of_int (int dflt_size)"
RPCFail_enabled: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and p :: "PrIds"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState" assumes "basevars (rtrner send!p, caller rcv!p, rst!p)" shows "\<turnstile> \<not> Calling rcv p \<and> Calling send p \<longrightarrow> Enabled RPCFail send rcv rst p"
intros_1: fixes xs :: "'varSort"   and x :: "'var" shows "wls (asSort xs) (Var xs x)"
imp_foreach_def: fixes tsl :: "'g \<Rightarrow> 'h list Heap"   and s :: "'g"   and c :: "'f \<Rightarrow> bool Heap"   and f :: "'h \<Rightarrow> 'f \<Rightarrow> 'f Heap"   and \<sigma> :: "'f" shows "imp_foreach tsl s c f \<sigma> \<equiv> tsl s \<bind> (\<lambda>l. imp_nfoldli l c f \<sigma>)"
sing_word_concat: fixes us :: "'b list list" assumes "\<^bold>|us\<^bold>| = 1" shows "[concat us] = us"
preserves_dom: fixes f :: "'s" assumes "S.arr f" shows "S'.dom (\<Phi> f) = \<Phi> (S.dom f)"
that_1: assumes "bdd_above S"   and "\<And>y. y \<in> S \<Longrightarrow> y \<le> x" shows "thesis"
measure_measure_3: shows "measure_space \<Omega> A \<mu>"
local: fixes \<sigma> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and l :: "'e"   and a :: "'f"   and \<sigma>' :: "'a"   and l' :: "'e" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "l \<in> labels \<Gamma> p"   and "P (\<sigma>, l)"   and "Q ((\<sigma>, l), a, \<sigma>', l')"   and "S \<sigma> \<sigma>' a" shows "P (\<sigma>', l')"
vec_eq_iff: fixes x :: "'c vec"   and y :: "'c vec" shows "(x = y) = (dim_vec x = dim_vec y \<and> (\<forall>i<dim_vec y. x $ i = y $ i))"
times_divide_times_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide x y) (divide z w) = divide (times x z) (times y w)"
root_nat: fixes p :: "nat"   and x :: "nat" assumes "p \<noteq> 0 \<or> x \<noteq> 1" shows "set (root_nat p x) = {y. y ^ p = x}"
OUadmitFO_def: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'f) trm"   and \<theta> :: "('e + 'd, 'f) ODE"   and U :: "('f + 'f) set" shows "OUadmitFO \<sigma> \<theta> U = ((\<Union>i\<in>{i. Inl (Inr i) \<in> SIGO \<theta>}. FVT (\<sigma> i)) \<inter> U = {})"
nn_integral_liminf: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "\<And>i. u i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. liminf (\<lambda>n. u n x) \<partial>M \<le> liminf (\<lambda>n. integral\<^sup>N M (u n))"
left_adjoint_is_ide: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.is_left_adjoint f" shows "S.ide f"
powr_less_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(b powr y < x) = (y < log b x)"
CSP2_notok_a: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and a :: "('c, 'd) alpha_rp_scheme"   and b :: "('c, 'd) alpha_rp_scheme" assumes "CSP2 P (a, b\<lparr>ok := False\<rparr>)" shows "P (a, b\<lparr>ok := False\<rparr>)"
preserves_diagrams: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 'c" assumes "diagram J (\<cdot>) D" shows "diagram J (\<cdot>) (C.map \<circ> D)"
simps_3: fixes n :: "'a"   and t1 :: "'a ifex"   and t2 :: "'a ifex"   and s :: "'a \<Rightarrow> bool" shows "val_ifex (IF n t1 t2) s = (if s n then val_ifex t1 s else val_ifex t2 s)"
eq: shows "(\<lambda>x. H (f x) (g x)) = (\<lambda>x. H (fst (f x, g x)) (snd (f x, g x)))"
dom_char: fixes f :: "(bool, unit) J.arr" shows "J.dom f = (if J.arr f then J.MkIde (J.Dom f) else J.null)"
OrI2': fixes P :: "bool"   and Q :: "bool" assumes "\<not> P \<Longrightarrow> Q" shows "P \<or> Q"
r_limr_def: shows "r_limr \<equiv> SOME g. recfn 1 g \<and> Partial_Recursive.total g \<and> (\<forall>i. \<phi> (the (eval g [i])) \<in> \<R> \<and> (\<forall>U \<psi>. learn_lim \<psi> U (\<phi> i) \<longrightarrow> learn_lim \<psi> U (\<phi> (the (eval g [i])))))"
l: fixes x :: "real" assumes "x \<le> l" shows "y \<le> f x"
shift_autonomous_solution: fixes x :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set"   and t0 :: "real" assumes "(x solves_ode f) T X"   and "\<And>s t. s \<in> T \<Longrightarrow> f s (x s) = f t (x s)" shows "((\<lambda>t. x (t + t0)) solves_ode f) ((\<lambda>t. t - t0) ` T) X"
delete_point_index: fixes G1 :: "'a set"   and x :: "'a"   and G2 :: "'a set"   and y :: "'a"   and z :: "'a" assumes "G1 \<in> {g - {x} |g. g \<in> \<G> \<and> g \<noteq> {x}}"   and "G2 \<in> {g - {x} |g. g \<in> \<G> \<and> g \<noteq> {x}}"   and "G1 \<noteq> G2"   and "y \<in> G1"   and "z \<in> G2" shows "int (del_point_blocks x index {y, z}) = \<Lambda>"
simps_1: fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list" shows "qe_eq_repeat_helper var [] L F = list_conj (map fm.Atom L @ F)"
cong_modulus_minus_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo b c (uminus a) = unique_euclidean_semiring.cong modulo b c a"
le_halfspace_def: fixes sctn :: "'c sctn"   and x :: "'c" shows "le_halfspace sctn x = (x \<bullet> normal sctn \<le> pstn sctn)"
intros: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "0 < f e" shows "support_flowp f e"
rcosets_finite: fixes R :: "nat set"   and H :: "nat set" assumes "R \<in> rcosets\<^bsub>G\<^esub> H"   and "H \<subseteq> carrier G"   and "finite H" shows "finite R"
ldc_tuple: fixes C :: "char list"   and Cs :: "char list list" shows "ldc (C, Cs) = last Cs"
assms: shows "comp_tfr\<^sub>s\<^sub>t arity Ana \<Gamma> (SMP0 Ana \<Gamma> (trms_list\<^sub>s\<^sub>t S)) S"
union_assoc: fixes A :: "'s"   and B :: "'s"   and C :: "'s" shows "A \<^bold>\<union> B \<^bold>\<union> C = A \<^bold>\<union> (B \<^bold>\<union> C)"
1: shows "D.dom (unit a) = map\<^sub>0 a"
renormalize: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' 0 (a - b) (c - a)" shows "ccw' b a c"
simps_3: fixes x :: "Vars.var"   and S :: "stack_elem list" shows "upds_list (Upd x # S) = x # upds_list S"
mkPoint_def: fixes a :: "'s"   and t :: "'s" shows "mkPoint a t \<equiv> mkArr {unity} (local.set a) (\<lambda>_. t)"
ChiRel_ChiPlus_NoCycles: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(S, T) \<in> ChiRel A" shows "(T, S) \<notin> ChiPlus A"
map_poly_pdivmod: fixes p :: "'a poly"   and q :: "'a poly" shows "map_prod (map_poly hom) (map_poly hom) (p div q, p mod q) = (map_poly hom p div map_poly hom q, map_poly hom p mod map_poly hom q)"
right_cancel: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier H"   and "y \<in> carrier H"   and "z \<in> carrier H" shows "(y \<otimes>\<^bsub>H\<^esub> x = z \<otimes>\<^bsub>H\<^esub> x) = (y = z)"
moebius_mu_aux_def: fixes n :: "nat"   and ps :: "unit \<Rightarrow> nat list" shows "moebius_mu_aux n ps = (if n \<noteq> 0 \<and> \<not> 4 dvd n \<and> \<not> 9 dvd n then let ps = ps () in if distinct ps then if even (length ps) then 1 else - 1 else 0 else 0)"
iso_bij_betw_individual_blocks_inv: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "bij_betw (inv_into \<V> \<pi>) (\<pi> ` bl) bl"
botE: fixes x :: "'a"   and P :: "bool" assumes "pred.eval \<bottom> x" shows "P"
cfs_times_X: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "g \<in> carrier P" shows "(X_poly R \<otimes>\<^bsub>P\<^esub> g) (Suc n) = g n"
setsetmapim_cong: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set set"   and B :: "'b set set" assumes "fun_eq_on g f (\<Union> A)"   and "order.greater_eq A B" shows "g \<turnstile> B = f \<turnstile> B"
unit_simps_3: fixes a :: "'b" assumes "B.obj a" shows "trg\<^sub>C (H.unit a) = H.map\<^sub>0 a"