hcju/isabelleps huggingface dataset:
vpairwise_def: fixes R :: "V \<Rightarrow> V \<Rightarrow> bool"   and S :: "V" shows "vpairwise R S = (\<forall>x\<in>elts S. \<forall>y\<in>elts S. x \<noteq> y \<longrightarrow> R x y)"
null_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "null \<equiv> Abs_Bag\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<bottom>\<rfloor>"
assms_1: shows "valid ((Cn n f gs, xs, []) # rest)"
th: shows "\<forall>x. Ifm vs (x # bs) (plusinf p)"
rel_mono_strong: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option"   and y :: "'b option"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_option R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_option x; yb \<in> set_option y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_option Ra x y"
snd_dominant_term_ms_aux_MSLCons: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist" shows "snd (dominant_term_ms_aux (MSLCons (C, e) xs)) = e # snd (dominant_term C)"
finite_strict_prefix_set: fixes x :: "'a list" shows "finite {xa. strict_prefix xa x}"
mono_write_FD: fixes P :: "'b process"   and Q :: "'b process"   and c :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "P \<le> Q" shows "(c`!`x \<rightarrow> P) \<le> (c`!`x \<rightarrow> Q)"
distinct_23: fixes x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'d"   and x41 :: "nat"   and x42 :: "nat" shows "x31\<not>x32:arrive(x33) \<noteq> connect(x41, x42)"
abs_le_D1: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq (abs a) b" shows "less_eq a b"
inv_def: fixes f :: "'b \<Rightarrow> 'a" shows "inv f = (\<lambda>y. SOME x. f x = y)"
transfer: shows "rel_fun pcr_real (=) (\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n) positive"
\<Delta>_naturality: shows "((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>"
cas_takeI: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and n :: "nat"   and v' :: "'c" assumes "wf_digraph G"   and "pre_digraph.cas G u p v"   and "last (pre_digraph.awalk_verts G u (take n p)) = v'" shows "pre_digraph.cas G u (take n p) v'"
E_validD_2: fixes G :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "valid_graph G"   and "(v, e, v') \<in> edges G" shows "v' \<in> nodes G"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and f3 :: "nat \<Rightarrow> 'b"   and op :: "'c op" shows "P (case op of Empty \<Rightarrow> f1 | Push x \<Rightarrow> f2 x | Pop x \<Rightarrow> f3 x) = ((op = Empty \<longrightarrow> P f1) \<and> (\<forall>x2. op = Push x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. op = Pop x3 \<longrightarrow> P (f3 x3)))"
hyps: shows "r\<inverse> ^^ n = (r ^^ n)\<inverse>"
inject_1: fixes x11 :: "nat"   and x12 :: "'d"   and x13 :: "nat set"   and y11 :: "nat"   and y12 :: "'d"   and y13 :: "nat set" shows "(NodeS x11 x12 x13 = NodeS y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
swapidseq_id_iff: fixes p :: "'b \<Rightarrow> 'b" shows "swapidseq 0 p = (p = id)"
fr_out_2: shows "frontier (outside (path_image c2 \<union> path_image c)) = path_image c2 \<union> path_image c"
positively_invariant_trapC: shows "c.positively_invariant trapC"
h_f_g_equiv: fixes z :: "'c" shows "h (f z, g z) \<equiv> z"
member_of_inheritance: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and D :: "qtname"   and C :: "qtname" assumes "G \<turnstile> m member_of D"   and "G\<turnstile>D\<preceq>\<^sub>C C"   and "G\<turnstile>C\<preceq>\<^sub>C declclass m"   and "ws_prog G" shows "G \<turnstile> m member_of C"
ij: shows "(xi, xj) \<in> set (half_segments_of_aform X)"
uniformly_Cauchy_on_def: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" shows "uniformly_Cauchy_on X f = (\<forall>e>0. \<exists>M. \<forall>x\<in>X. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e)"
s1: fixes e :: "real" assumes "0 < e" shows "\<exists>y\<in>{norm (blinfun_apply f x) |x. x \<in> sphere (0::'a) 1}. norm (onorm (blinfun_apply f) - y) < e"
assms_3: fixes A :: "'a set" assumes "A \<in> sets M" shows "P (indicator A)"
eq_Max_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and m :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(m = linorder.Max less_eq A) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq a m))"
fUnionM_finset: fixes x :: "'m"   and M :: "'m fset" shows "fUnionM (finsert x M) = funionM x (fUnionM M)"
zero_satisfies: fixes c :: "'a ns_constraint"   and v :: "nat \<Rightarrow> 'a" assumes "poly c = 0"   and "zero_satisfies c" shows "v \<Turnstile>\<^sub>n\<^sub>s c"
last_weak_ranking: fixes x :: "'a"   and y :: "'a" assumes "x \<in> last (weak_ranking le)"   and "y \<in> carrier" shows "le x y"
add_0_left: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and a :: "'f" assumes "class.monoid_add plus zero" shows "plus zero a = a"
cons_ap_list: fixes f :: "'d \<Rightarrow> 'c"   and fs :: "('d \<Rightarrow> 'c) list"   and xs :: "'d list" shows "(f # fs) \<diamondop> xs = map f xs @ fs \<diamondop> xs"
span_clauses_4: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and c :: "'a" assumes "module scale"   and "x \<in> module.span scale S" shows "scale c x \<in> module.span scale S"
derivative_norm_bounded_subset: fixes X' :: "'a set" assumes "X' \<noteq> {}"   and "X' \<subseteq> cball x0 r"   and "convex X'" shows "derivative_norm_bounded T X' f f' B B'"
assms_2: shows "- pi < \<alpha>' \<and> \<alpha>' \<le> pi"
drop_append: fixes n :: "nat"   and xs :: "'c list"   and ys :: "'c list" shows "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
wt_subst: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A"   and "ground_heads.wary_subst arity_sym arity_var ground_heads_var \<rho>"   and "arity.wary arity_sym arity_var s" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym (subst \<rho> s))) = eval_tpoly (tkbo_coefs.subst_zpassign ground_heads_var \<delta> arity_sym wt_sym coef_sym \<rho> A) (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s))"
insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<notin> A" shows "mset_set (insert x A) = add_mset x (mset_set A)"
delta: fixes S :: "'c set"   and a :: "'c"   and b :: "'c \<Rightarrow> 'd" assumes "finite S" shows "(\<Sum>k\<in>S. if k = a then b k else (0::'d)) = (if a \<in> S then b a else (0::'d))"
spec_ops_id_inc: fixes xs :: "('b \<times> 'c) list"   and oid :: "'b"   and oper :: "'c"   and deps :: "'c \<Rightarrow> 'b set"   and x :: "'b" assumes "spec_ops (xs @ [(oid, oper)]) deps"   and "x \<in> set (map fst xs)" shows "x < oid"
conj: fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "is_filter F"   and "F P"   and "F Q" shows "F (\<lambda>x. P x \<and> Q x)"
no_head_generates: fixes \<C> :: "'b list set"   and b :: "'b list" assumes "code \<C>"   and "b \<in> \<C>" shows "\<langle>no_head_gen \<C> b\<rangle> = no_head \<C> b"
lset_ltakeWhileD: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "x \<in> lset (ltakeWhile P xs)" shows "x \<in> lset xs \<and> P x"
assms_6: shows "h \<circ> u = u \<circ> g"
intro: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "comm_monoid f z" shows "comm_monoid_set f z"
subfield_iff_2: fixes K :: "'a set" assumes "subfield K R" shows "field (R\<lparr>carrier := K\<rparr>)"
chart_basis_inverse: fixes b :: "'b"   and x :: "'b proj_space" assumes "b \<in> Basis"   and "chart_basis_domainP b x" shows "chart_basis_inv b (chart_basis b x) = x"
FWdelay_bisimulation_diverge_flip: shows "FWdelay_bisimulation_diverge final2 r2 final1 r1 (\<lambda>t. flip (bisim t)) (flip bisim_wait) \<tau>move2 \<tau>move1"
rel_map_1: fixes Sb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'d \<Rightarrow> 'a"   and x :: "'d stream"   and y :: "'b stream" shows "stream_all2 Sb (smap i x) y = stream_all2 (\<lambda>x. Sb (i x)) x y"
awalkI_apath: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "apath u p v" shows "awalk u p v"
conseq_no_aux: 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 P' :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q' :: "'e set"   and A' :: "'e set"   and P :: "'e set"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' c Q',A'"   and "\<forall>s. s \<in> P \<longrightarrow> s \<in> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
intro: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G :: "'b inference set"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and Prec_F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Inf_FL :: "('a \<times> 'c) inference set" assumes "tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F"   and "labeled_tiebreaker_lifting_axioms Inf_F Inf_FL" shows "labeled_tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F Inf_FL"
finite_lderiv0: fixes a :: "'a" assumes "lformula0 a" shows "finite {fold lderiv xs (FBase a) |xs. True}"
segment_furthest_le: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and y :: "'a" assumes "x \<in> closed_segment a b" shows "norm (y - x) \<le> norm (y - a) \<or> norm (y - x) \<le> norm (y - b)"
S: shows "image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt q)) - {#Element ey ay qy#} + {#Element ey ay qy#} = image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt q))"
sublens_trans: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b"   and Z :: "'d \<Longrightarrow> 'b" assumes "X \<subseteq>\<^sub>L Y"   and "Y \<subseteq>\<^sub>L Z" shows "X \<subseteq>\<^sub>L Z"
eval_fps_shift: fixes n :: "nat"   and f :: "'a fps"   and z :: "'a" assumes "n \<le> subdegree f"   and "ereal (norm z) < fps_conv_radius f" shows "eval_fps (fps_shift n f) z = (if z = (0::'a) then fps_nth f n else eval_fps f z / z ^ n)"
poly_deg_monomial_zero: fixes c :: "'d" shows "poly_deg (monomial c 0) = (0::'b)"
m_ac_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes> y \<otimes> z = x \<otimes> (y \<otimes> z)"
ereal_divide_Infty_2: fixes x :: "ereal" shows "x / - \<infinity> = 0"
akra_bazzi_params_axioms: shows "akra_bazzi_params k as bs"
lepoll_empty_iff_empty: fixes A :: "'a set" shows "(A \<lesssim> {}) = (A = {})"
phis_disj_2: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list"   and n' :: "'node"   and vs' :: "'val list" assumes "Mapping.lookup (phis g) (n, v) = Some vs"   and "Mapping.lookup (phis g) (n', v) = Some vs'" shows "vs = vs'"
Abs_multiset_cases: fixes x :: "'k multiset" obtains y :: "'k \<Rightarrow> nat" where "x = Abs_multiset y" and "y \<in> multiset"
hd_append: fixes xs :: "'c list"   and ys :: "'c list" shows "lead_coeff (xs @ ys) = (if xs = [] then lead_coeff ys else lead_coeff xs)"
Pd_Xd: fixes x :: "'a"   and y :: "'a"   and d :: "'b" assumes "(x, y) \<in> Pd d" shows "Xd x = d \<and> Xd y = d"
keq: fixes u :: "real"   and v :: "'a" assumes "u = 1 / 2" shows "k1 (2 * u, v) = k2 (2 * u - 1, v)"
elims: fixes x :: "'d decision decision"   and y :: "'d decision" assumes "flat_orA x = y" obtains ya :: "'d" where "x = allow (allow ya__)" and "y = allow ya__"   | ya :: "'d" where "x = allow (deny yaa__)" and "y = allow yaa__"   | ya :: "'d" where "x = deny (allow yab__)" and "y = allow yab__"   | ya :: "'d" where "x = deny (deny ya)" and "y = deny ya"
unrest_uexpr_def: shows "unrest \<equiv> map_fun id (map_fun Rep_uexpr id) (\<lambda>x e. \<forall>b v. e (put\<^bsub>x\<^esub> b v) = e b)"
sig_inv_setI': fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "\<And>v. v \<in> keys r \<Longrightarrow> component_of_term v < j" shows "r \<in> sig_inv_set' j"
restrict_map_cancel: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set"   and T :: "'c set" shows "(m |` S = m |` T) = (dom m \<inter> S = dom m \<inter> T)"
\<sigma>W1_1: shows "strategy p** \<sigma>W1"
latestActEq: fixes nid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n' :: "nat"   and n :: "nat" assumes "\<parallel>nid\<parallel>\<^bsub>t n'\<^esub>"   and "\<not> (\<exists>n''>n'. n'' < n \<and> \<parallel>nid\<parallel>\<^bsub>t n'\<^esub>)"   and "n' < n" shows "\<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub> = n'"
equiv_class_eq: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "equiv A r"   and "(a, b) \<in> r" shows "r `` {a} = r `` {b}"
b_def: fixes n :: "nat" shows "b n = a + inverse (real (Suc n))"
RedTryCatch: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and 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 V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "hp s a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile> \<langle>try Throw a catch(C V) e\<^sub>2,s\<rangle> \<rightarrow> \<langle>{V:Class C; V:=addr a;; e\<^sub>2},s\<rangle>"
assms_1: shows "M \<rightarrow>\<beta>\<^sup>* M'"
intros_1: fixes a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and volatile :: "bool"   and \<D> :: "bool"   and \<O>s :: "nat set list"   and i :: "nat"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat" assumes "a \<in> \<O> \<or> a \<in> read_only \<S> \<or> volatile \<and> a \<in> dom \<S>"   and "volatile \<longrightarrow> \<not> \<D>" shows "\<O>s,i\<turnstile> (Read volatile a t # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
ket_vec_def: fixes v :: "complex vec" shows "|v\<rangle> \<equiv> mat (dim_vec v) 1 (\<lambda>(i, j). v $ i)"
evaluate_determ_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and r3a :: "'b state \<times> (v, v) result"   and r3b :: "'b state \<times> (v, v) result" assumes "evaluate ck env s e r3a"   and "evaluate ck env s e r3b" shows "r3a = r3b"
rep_eq: fixes b :: "'f"   and x :: "('g, 'f) tllist"   and xa :: "('g, 'f) tllist" shows "tllist_ord b x xa = (case (llist_of_tllist x, terminal x) of (xs1, b1) \<Rightarrow> \<lambda>(xs2, b2). if lfinite xs1 then b1 = b \<and> lprefix xs1 xs2 \<or> xs1 = xs2 \<and> flat_ord b b1 b2 else xs1 = xs2) (llist_of_tllist xa, terminal xa)"
id: fixes i :: "nat" shows "([],  \<LM> At i  \<RM> \<Rightarrow>* At i) \<in> Ax"
degree_poly_mult_rat_main: fixes n :: "'a"   and d :: "'a"   and f :: "'a poly" assumes "n \<noteq> (0::'a)" shows "degree (poly_mult_rat_main n d f) = (if d = (0::'a) then 0 else degree f)"
nodes_def: shows "DGBA.nodes \<equiv> automaton.nodes' dgba.alphabet dgba.initial dgba.transition"
EnabledTrans_def: shows "EnabledTrans \<equiv> {(ST, SA, T). SA \<in> SAs (HA ST) \<and> T \<in> Delta SA \<and> source T \<in> Conf ST \<and> (Conf ST, Events ST, Value ST) |= label T}"
arrI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "J.arr f"
lt_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Lt C D" shows "A B Lt D C"
g_ind: fixes j :: "nat"   and apj :: "abc_inst list"   and arj :: "nat"   and ftj :: "nat"   and anything :: "nat list" assumes "j < i"   and "rec_ci (gs ! j) = (apj, arj, ftj)" shows "{\<lambda>nl. nl = args @ 0 \<up> (ftj - arj) @ anything} apj {\<lambda>nl. nl = args @ rec_exec (gs ! j) args # 0 \<up> (ftj - Suc arj) @ anything}"
constant_on_def: fixes f :: "'a \<Rightarrow> 'd"   and A :: "'a set" shows "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"
deg_zero: shows "deg R \<zero>\<^bsub>P\<^esub> = 0"
laplace_expansion_column: fixes A :: "'b mat"   and n :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "j < n" shows "det A = (\<Sum>i<n. A $$ (i, j) * cofactor A i j)"
hcomp_assoc_isomorphic: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "(f \<star> g) \<star> h \<cong> f \<star> g \<star> h"
l_zero: fixes A :: "('d, 'e) aGroup_scheme"   and a :: "'d" assumes "aGroup A"   and "a \<in> carrier A" shows "\<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> a = a"
mapping_impl_unit_def: shows "mapping_impl \<equiv> Phantom(unit) mapping_Assoc_List"
assms: shows "z' = k *\<^sub>s\<^sub>v z"
hcomp_simps\<^sub>W\<^sub>C_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "local.dom (\<nu> \<star> \<mu>) = local.dom \<nu> \<star> local.dom \<mu>"
gSwapAbsPresGWlsAbs_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSwapAbsPresGWlsAbs MOD \<equiv> \<forall>zs z1 z2 us s A' A. isInBar (us, s) \<and> wlsAbs (us, s) A' \<and> gWlsAbs MOD (us, s) A \<longrightarrow> gWlsAbs MOD (us, s) (gSwapAbs MOD zs z1 z2 A' A)"
demonic_def: fixes Q :: "'a \<Rightarrow> 'b"   and p :: "'b"   and s :: "'a" shows "[: Q :] p s = (Q s \<le> p)"
induct: fixes P :: "'e list \<Rightarrow> 'f list \<Rightarrow> bool"   and a0 :: "'e list"   and a1 :: "'f list" assumes "P [] []"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>x xs. P (x # xs) []"   and "\<And>y ys. P [] (y # ys)" shows "P a0 a1"
lemZEG: fixes z :: "'a"   and e :: "'a"   and g :: "'a" shows "z - e = g - e + (z - g)"
mult_col_div_index_row: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and a :: "'b"   and k :: "nat" assumes "i < dim_row A"   and "i < dim_col A"   and "j < dim_row A"   and "j < dim_col A"   and "a \<noteq> (0::'b)" shows "mult_col_div_row a k A $$ (i, j) = (if i = k \<and> j \<noteq> i then inverse a * A $$ (i, j) else if j = k \<and> j \<noteq> i then a * A $$ (i, j) else A $$ (i, j))"
inject_8: fixes x9 :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and y9 :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" shows "(Tuple x9 = Tuple y9) = (x9 = y9)"
last_vFrom: fixes f :: "face"   and x :: "nat" assumes "distinct (vertices f)"   and "x \<in> \<V> f" shows "last (verticesFrom f x) = f\<^bsup>-1\<^esup> \<bullet> x"
exp_eq_one_iff: fixes x :: "real" shows "(exp x = 1) = (x = 0)"
monic_rhs: shows "monic (\<Prod>x\<in>UNIV. [:0, 1:] - [:x:])"
S_intersections_Col_1: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "proj2_Col (proj2_abs p) (proj2_abs q) (S_intersection1 p q)"
deliver_locally: fixes m :: "'msg"   and i :: "nat" assumes "Broadcast m \<in> set (history i)" shows "Broadcast m \<sqsubset>\<^sup>i Deliver m"
prod_encode_fst_inj: fixes j :: "nat" shows "inj (\<lambda>i. prod_encode (i, j))"
invertible_matrix_is_change_of_basis: fixes P :: "(('a, 'b) vec, 'b) vec"   and X :: "(('a, 'b) vec, 'b) vec" assumes "invertible P"   and "is_basis (set_of_vector X)" shows "\<exists>!Y. matrix_change_of_basis Y X = P \<and> is_basis (set_of_vector Y)"
forward_simulation_axioms: shows "forward_simulation Sstd.step Sinca.step Sstd.final Sinca.final (\<lambda>_ _. False) (\<lambda>_. (\<sim>))"
g: shows "(\<Sum>x\<in>UNIV. g x *s Y $ x) = f (X $ i)"
E: shows "filter (\<lambda>k. P (l ! k)) [0..<length l] = js @ [j]"
monoidal_functor_axioms: shows "monoidal_functor (\<cdot>\<^sub>C) T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D F \<phi>"
inv_translationD_9: fixes r :: "nat set" assumes "inv_translation r" shows "Suc (Suc (card r - 2)) = card r"
ddep_slice2: fixes n'' :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-V\<rightarrow>\<^sub>d\<^sub>d n'"   and "n' \<in> sum_SDG_slice2 n" shows "n'' \<in> sum_SDG_slice2 n"
isometry_on_inverse_1: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "isometry_on (f ` X) (inv_into X f)"
lucas_lehmer_hom_add: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "lucas_lehmer_hom m (lucas_lehmer_add' x y) = lucas_lehmer_add m (lucas_lehmer_hom m x) (lucas_lehmer_hom m y)"
eq: shows "g * (q1 - q2) = r2 - r1"
CptnModSeq1: fixes P0 :: "'b com"   and s :: "'b"   and xs :: "('b com option \<times> 'b) list"   and zs :: "('b com option \<times> 'b) list"   and P1 :: "'b com" assumes "(Some P0, s) # xs \<in> cptn_mod"   and "zs = map (lift P1) xs" shows "(Some (Seq P0 P1), s) # zs \<in> cptn_mod"
prems_3: fixes i :: "'a ifex"   and a :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" assumes "\<And>val. P (restrict_top i a val) (restrict_top t a val) (restrict_top e a val)"   and "lowest_tops [i, t, e] = Some a" shows "P i t e"
Suc: fixes x :: "real" shows "sin (x + real n * (2 * pi)) = sin x"
wellformed_2: fixes G :: "('b, 'c) pre_digraph"   and e :: "'c" assumes "wf_digraph G"   and "e \<in> arcs G" shows "head G e \<in> verts G"
eigenvector_hom: fixes A :: "real mat"   and n :: "nat"   and v :: "real Matrix.vec"   and ev :: "real" assumes "A \<in> carrier_mat n n"   and "eigenvector A v ev" shows "eigenvector (of_real_hom.mat_hom A) (of_real_hom.vec_hom v) (of_real ev)"
shd_def: fixes stream :: "'d stream" shows "shd stream = (case stream of x1 ## x2 \<Rightarrow> x1)"
integral_mult_left: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> integrable M f" shows "LINT x|M. f x * c = integral\<^sup>L M f * c"
bigo_elt_subset: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(g)" shows "O(f) \<subseteq> O(g)"
hyps: fixes ws :: "val list" assumes "map Val vs' = map Val ws @ throw ex # es" shows "False"
quot_of_fract_quot_to_fract: fixes x :: "'a \<times> 'a" shows "quot_of_fract (quot_to_fract x) = normalize_quot x"