hcju/isabelleps huggingface dataset:
mat_max_carrier: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat" shows "ord.mat_max less_eq A B \<in> carrier_mat (dim_row A) (dim_col A)"
id2: shows "{n1..<n1 + n2} = (+) n1 ` {0..<n2}"
refine: fixes G :: "('b \<Rightarrow> bool, 'b \<Rightarrow> 'b list, 'b list, 'b \<Rightarrow> bool, 'c) gen_bg_impl_scheme" shows "RETURN (code_blue_dfs_ahs G) \<le> impl_blue_dfs_ahs G"
inverse_unique: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "Groups.group f z inverse"   and "f a b = z" shows "inverse a = b"
ns: shows "Gs = lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts"
basic_trans_rules_7: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "a \<le> b"   and "f b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
nodes_empty_iff_empty_1: fixes G :: "'b ugraph" shows "(nodes G = {}) = (G = graph {} {})"
tl_\<rho>s: shows "tl (renamings_apart [mset D, mset C]) = [\<rho>]"
gcd_code_int: fixes k :: "int"   and l :: "int" shows "gcd k l = \<bar>if l = 0 then k else gcd l (\<bar>k\<bar> mod \<bar>l\<bar>)\<bar>"
alphaBoundResidual: fixes x' :: "name"   and P :: "pi"   and a :: "subject"   and x :: "name" assumes "x' \<sharp> P" shows "a\<guillemotleft>x\<guillemotright> \<prec> P = a\<guillemotleft>x'\<guillemotright> \<prec> [(x, x')] \<bullet> P"
simps: fixes a1 :: "com \<times> (char list \<Rightarrow> val option)"   and a2 :: "com \<times> (char list \<Rightarrow> val option)" shows "red a1 a2 = ((\<exists>V e s. a1 = (V:=e, s) \<and> a2 = (Skip, s(V := \<lbrakk>e\<rbrakk>s))) \<or> (\<exists>c\<^sub>1 s c\<^sub>1' s' c\<^sub>2. a1 = (c\<^sub>1;; c\<^sub>2, s) \<and> a2 = (c\<^sub>1';; c\<^sub>2, s') \<and> \<langle>c\<^sub>1,s\<rangle> \<rightarrow> \<langle>c\<^sub>1',s'\<rangle>) \<or> (\<exists>c\<^sub>2 s. a1 = (Skip;; c\<^sub>2, s) \<and> a2 = (c\<^sub>2, s)) \<or> (\<exists>b s c\<^sub>1 c\<^sub>2. a1 = (if (b) c\<^sub>1 else c\<^sub>2, s) \<and> a2 = (c\<^sub>1, s) \<and> \<lbrakk>b\<rbrakk>s = Some true) \<or> (\<exists>b s c\<^sub>1 c\<^sub>2. a1 = (if (b) c\<^sub>1 else c\<^sub>2, s) \<and> a2 = (c\<^sub>2, s) \<and> \<lbrakk>b\<rbrakk>s = Some false) \<or> (\<exists>b s c. a1 = (while (b) c, s) \<and> a2 = (c;; while (b) c, s) \<and> \<lbrakk>b\<rbrakk>s = Some true) \<or> (\<exists>b s c. a1 = (while (b) c, s) \<and> a2 = (Skip, s) \<and> \<lbrakk>b\<rbrakk>s = Some false))"
Rep_inverse': fixes a :: "'a"   and r :: "'b" assumes "Rep a = r" shows "Abs r = a"
map: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "map_dual f (\<partial> x) = \<partial> (f x)"
that: fixes l :: "'a"   and u :: "'a" assumes "l \<in> S"   and "u \<in> S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> {f l..f u}" shows "thesis"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) (\<lambda>f. f \<otimes> \<I>) local.map \<rho>.map"
linear_order_on_acyclic: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "linear_order_on A r" shows "acyclic (r - Id)"
ground_range_vars: fixes s :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "ground (subst_range s)" shows "range_vars s = {}"
slp_vp: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^bsub>sl\<^esub>* n'" shows "n -as\<rightarrow>\<^sub>\<surd>* n'"
red_hext_incr: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>" shows "h \<unlhd> h'"
in_rel: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b H"   and b :: "'d H" shows "rel_H R a b = (\<exists>z. z \<in> {x. set_H x \<subseteq> {(x, y). R x y}} \<and> map_H fst z = a \<and> map_H snd z = b)"
wt_elim_cases_29: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and c1 :: "stmt"   and c2 :: "stmt"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (c1 Finally c2)\<Colon>x" obtains "x = Inl (PrimT Void)" and "E,dt\<Turnstile>c1\<Colon>\<surd>" and "E,dt\<Turnstile>c2\<Colon>\<surd>"
has_field_derivative_Ln: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "(Ln has_field_derivative inverse z) (at z)"
flip_def: fixes tup :: "'b \<times> 'a" shows "flip tup = (snd tup, fst tup)"
\<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_labels: fixes \<xi> :: "state"   and p :: "(state, msg, pseqp, pseqp label) seqp"   and i :: "nat" assumes "(\<xi>, p) \<in> \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i" shows "labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p = {PAodv-:0}"
is_scc_connected: fixes E :: "('b \<times> 'b) set"   and U :: "'b set"   and x :: "'b"   and y :: "'b" assumes "is_scc E U"   and "x \<in> U"   and "y \<in> U" shows "(x, y) \<in> E\<^sup>*"
stirling_sum_complex_of_real: fixes j :: "nat"   and m :: "nat"   and x :: "real" shows "stirling_sum j m (complex_of_real x) = complex_of_real (stirling_sum j m x)"
permute_rep_abs_Tree\<^sub>p: fixes t :: "('e, 'f, 'g) Tree" obtains p :: "perm" where "rep_Tree\<^sub>p (abs_Tree\<^sub>p t) = p \<bullet> t"
Joints_independent_aforms_eq: fixes x :: "'b \<times> 'b pdevs"   and xs :: "('b \<times> 'b pdevs) list" shows "Joints (independent_aforms x xs) = set_Cons (Affine x) (Joints xs)"
order_trans: fixes less_eq :: "'n \<Rightarrow> 'n \<Rightarrow> bool"   and less :: "'n \<Rightarrow> 'n \<Rightarrow> bool"   and x :: "'n"   and y :: "'n"   and z :: "'n" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
is_sum2sq_int_def: fixes n :: "int" shows "is_sum2sq_int n = (\<exists>a b. n = sum2sq_int (a, b))"
update_convs_2: fixes arcs' :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and verts :: "'a set"   and arcs :: "('a \<times> 'a) set"   and more :: "'b" shows "arcs_update arcs' \<lparr>verts = verts, arcs = arcs, \<dots> = more\<rparr> = \<lparr>verts = verts, arcs = arcs' arcs, \<dots> = more\<rparr>"
sorted_sorted_wrt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq = sorted_wrt less_eq"
fix_eqI: fixes F :: "'c \<rightarrow> 'c"   and x :: "'c" assumes "F\<cdot>x = x"   and "\<And>z. F\<cdot>z = z \<Longrightarrow> x \<sqsubseteq> z" shows "fix\<cdot>F = x"
iso_tuple_update_accessor_cong_from_eq: fixes upd :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and ac :: "'b \<Rightarrow> 'a"   and v :: "'b"   and f :: "'a \<Rightarrow> 'a"   and v' :: "'b"   and x :: "'a" assumes "iso_tuple_update_accessor_eq_assist upd ac v f v' x" shows "iso_tuple_update_accessor_cong_assist upd ac"
obs_strong_postdominate: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and n :: "'b"   and n' :: "'b"   and S :: "'b set" assumes "WeakControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"   and "n \<in> CFG.obs sourcenode targetnode valid_edge n' (WeakControlDependencePDG.PDG_BS_w sourcenode targetnode valid_edge Def Use Exit S)"   and "n \<noteq> Exit" shows "StrongPostdomination.strong_postdominate sourcenode targetnode valid_edge Exit n n'"
card_of_Pow_cong: fixes A :: "'c set"   and B :: "'d set" assumes "|A| =o |B|" shows "|Pow A| =o |Pow B|"
mg_eq: shows "map_graph_fn G' (g \<circ> f) = map_graph_fn G' h"
comp_inv_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "category.inverse_arrows C f g" shows "C g f = partial_magma.dom C f"
moebius_inv_translation_translation: fixes v :: "complex"   and z :: "complex_homo" shows "moebius_pt (moebius_translation (- v)) (moebius_pt (moebius_translation v) z) = z"
gr_one_powr: fixes x :: "real"   and y :: "real" assumes "1 < x"   and "0 < y" shows "1 < x powr y"
measurable_density_eq1: fixes g :: "'b \<Rightarrow> 'c"   and Mg :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and Mg' :: "'c measure" shows "(g \<in> density Mg f \<rightarrow>\<^sub>M Mg') = (g \<in> Mg \<rightarrow>\<^sub>M Mg')"
ord_less_eq_trans: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "less a b"   and "b = c" shows "less a c"
card_dom_random_oracle: fixes \<A> :: "('b, 'a, '\<gamma>) gpv"   and q :: "nat"   and y :: "'b"   and \<sigma>' :: "'a \<Rightarrow> '\<gamma> option"   and \<sigma> :: "'a \<Rightarrow> '\<gamma> option" assumes "interaction_bounded_by' (\<lambda>_. True) \<A> q"   and "(y, \<sigma>') \<in> set_spmf (exec_gpv prf.random_oracle \<A> \<sigma>)"   and "finite (dom \<sigma>)" shows "card (dom \<sigma>') \<le> q + card (dom \<sigma>)"
min_def: fixes a :: "'c"   and b :: "'c" shows "min a b = (if a \<le> b then a else b)"
scaleR_mono: fixes a :: "real"   and b :: "real"   and x :: "'a"   and y :: "'a" assumes "a \<le> b"   and "x \<le> y"   and "0 \<le> b"   and "(0::'a) \<le> x" shows "a *\<^sub>R x \<le> b *\<^sub>R y"
diff_bet_ex3: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C" shows "\<exists>D. A \<noteq> D \<and> B \<noteq> D \<and> C \<noteq> D \<and> Col A B D"
cofactor_list_syz_plus: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "cofactor_list_syz n (p + q) = map2 (+) (cofactor_list_syz n p) (cofactor_list_syz n q)"
lebesgue_on_mono: fixes P :: "'a \<Rightarrow> bool"   and S :: "'a set"   and Q :: "'a \<Rightarrow> bool" assumes "almost_everywhere (lebesgue_on S) P"   and "\<And>x. \<lbrakk>P x; x \<in> S\<rbrakk> \<Longrightarrow> Q x" shows "almost_everywhere (lebesgue_on S) Q"
a_group: fixes G :: "('c, 'd) ring_scheme" assumes "abelian_group G" shows "Group.group (add_monoid G)"
simps_3: fixes uv :: "'b" shows "normalized_nnf_match (MatchNot (Match uv)) = True"
r_s_total: fixes x :: "nat" shows "eval r_s [x] \<down>"
ext: fixes g :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr"   and f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.comp g f \<noteq> S.null" shows "S.seq g f"
simps_5: fixes f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
sg_gimg: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "G \<guillemotright> f ` carrier F "
not_\<omega>\<^sub>z_lt_of_nat: fixes n :: "nat" shows "\<not> \<omega>\<^sub>z < of_nat n"
axioms_2: fixes \<eta> :: "'b \<Rightarrow> 'c"   and G :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and G' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "group_homomorphism \<eta> G composition unit G' composition' unit'" shows "Group_Theory.group G composition unit"
abs_zero: 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" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs zero = zero"
QAR_Beta: fixes x :: "name"   and t\<^sub>1 :: "trm"   and t\<^sub>2 :: "trm" shows "App (Lam [x].t\<^sub>1) t\<^sub>2 \<leadsto> t\<^sub>1[x::=t\<^sub>2]"
bind_gpv_Fail: fixes f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "Fail \<bind> f = Fail"
assms_1: shows "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z"
arr_char: fixes f :: "Arr" shows "arr f = (f \<noteq> Null)"
list_of_vec_index: fixes v :: "'b vec"   and j :: "nat" shows "list_of_vec v ! j = v $ j"
constant_imageI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<exists>c. \<forall>x\<in>space M. f x = c" shows "\<forall>x\<in>space M. f x = constant_image f"
exec_instr_IfFalse: fixes i :: "int"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr (IfFalse i) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (let pc' = if hd stk = Bool False then nat (int pc + i) else pc + 1 in (None, h, (tl stk, loc, C\<^sub>0, M\<^sub>0, pc', ics) # frs, sh))"
leg1_simps_3: shows "src (tab\<^sub>1 g) = src (tab\<^sub>0 g)"
rel_spmf_bind_reflI: fixes p :: "'a spmf"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'c spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf P (f x) (g x)" shows "rel_spmf P (p \<bind> f) (p \<bind> g)"
c_defs_2: fixes x :: "'a"   and y :: "'a" shows "cy x y = x + y * ((1::'a) - x\<^sup>2 - y\<^sup>2)"
sel: fixes x :: "'g \<Rightarrow> 'h \<times> 'g" shows "run_state (State x) = x"
graph: fixes \<alpha> :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'e set" assumes "Set_Theory.map \<alpha> S T" shows "\<alpha> \<in> S \<rightarrow>\<^sub>E T"
conjugate_square_positive: fixes conjugate :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.conjugatable_ordered_ring conjugate plus zero minus uminus times less_eq less" shows "less_eq zero (times a (conjugate a))"
simps_5: fixes \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula"   and G :: "'a formula" shows "\<A> \<Turnstile> F \<^bold>\<or> G = (\<A> \<Turnstile> F \<or> \<A> \<Turnstile> G)"
list_lcm: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "x dvd lcm_list xs"
neg1: shows "Negligible.negligible (\<lambda>n. local.ddh.advantage n (P1_DDH_mal_adv_\<sigma>_true n M z \<A> D))"
preserves_arr: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.VV.arr (C.VV.map f)"
euclidean_size_of_nat: 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 division_segment :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size (semiring_1.of_nat one plus zero n) = n"
reduction_doesnt_break_things_signs: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)" shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set (reduction_signs p qs signs subsets (M_mat signs subsets))"
individually_rational_on_Cd: fixes ds :: "'b set"   and X :: "'a set"   and d :: "'b" assumes "individually_rational_on ds X" shows "Cd d X = dX X d"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_plus" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> vrat_plus). vrat_plus\<lparr>a\<rparr> = b"
assms_4: shows "snd (rep_nat_term u) = snd (rep_nat_term v)"
semi_inverse2: fixes x :: "'b" assumes "x \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "\<pi>\<^sub>*\<^bsub>G\<^esub> x = \<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> x))"
le_iff_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<squnion> y = y)"
jump_poly_1_mult: fixes p1 :: "real poly"   and x :: "real"   and p2 :: "real poly" assumes "poly p1 x \<noteq> 0 \<or> poly p2 x \<noteq> 0" shows "jump_poly 1 (p1 * p2) x = sign (poly p2 x) * jump_poly 1 p1 x + sign (poly p1 x) * jump_poly 1 p2 x"
jkbpCn_length: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and t :: "'e Trace"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview"   and "t \<in> PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n" shows "tLength t = n"
add_commute: fixes V :: "'b set"   and x :: "'b"   and y :: "'b" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "x + y = y + x"
rel_fun_lift: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>x. A (f x) (g x)" shows "((=) ===> A) f g"
nb\<^sub>1: shows "(k, v) \<in> set (List.product ks vs)"
prj_chine_assoc_2: shows "doms.Prj\<^sub>1\<^sub>0 \<cdot> doms.chine_assoc = doms.Prj\<^sub>0\<^sub>1"
le_floor_iff: fixes z :: "int"   and x :: "'c" shows "(z \<le> \<lfloor>x\<rfloor>) = (of_int z \<le> x)"
left_inverse: fixes a :: "'ag" assumes "a \<in> U" shows "-\<^sub>o\<^sub>w a +\<^sub>o\<^sub>w a = 0\<^sub>o\<^sub>w"
falseLiteralsCanBeRemoved: fixes F1 :: "Literal list list"   and v :: "Literal list"   and F2 :: "Literal list list"   and F' :: "Literal list list"   and c :: "Literal list" assumes "equivalentFormulae (F1 @ val2form v @ F2) F'" shows "equivalentFormulae (F1 @ val2form v @ [removeFalseLiterals c v] @ F2) (F' @ [c])"
divide_fract: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "Fraction_Field.Fract a b / Fraction_Field.Fract c d = Fraction_Field.Fract (a * d) (b * c)"
rev_path_def: fixes ps :: "('c \<times> 'd \<times> 'c) list" shows "rev_path ps \<equiv> map (\<lambda>(a, b, c). (c, b, a)) (rev ps)"
emeasure_pair_measure_Times: fixes A :: "'b set"   and N :: "'b measure"   and B :: "(nat \<Rightarrow> 'a) set" assumes "A \<in> sets N"   and "B \<in> P.events" shows "emeasure (N \<Otimes>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M)) (A \<times> B) = emeasure N A * emeasure (Pi\<^sub>M UNIV (\<lambda>i. M)) B"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case SuperM of Static \<Rightarrow> f1 | SuperM \<Rightarrow> f2 | IntVir \<Rightarrow> f3) = f2"
obs_consistent_def: fixes R :: "('a \<times> 'b) set"   and pi :: "'c \<Rightarrow> 'd"   and Sa :: "('a, 'd) spec"   and Sc :: "('b, 'c) spec" shows "obs_consistent R pi Sa Sc \<equiv> \<forall>s t. (s, t) \<in> R \<longrightarrow> pi (obs Sc t) = obs Sa s"
poly_no_roots_less: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x<a. poly p x \<noteq> 0) = (p \<noteq> 0 \<and> count_roots_below p a = (if poly p a = 0 then 1 else 0))"
high_scores: fixes u :: "nat \<Rightarrow> real"   and i :: "nat" assumes "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> \<infinity>" shows "\<exists>n\<ge>i. \<forall>l\<le>n. u l \<le> u n"
deriv_boundI: fixes n :: "nat"   and a :: "'a"   and r :: "('a \<times> 'a) set" assumes "\<And>b m. \<lbrakk>n < m; (a, b) \<in> r ^^ m\<rbrakk> \<Longrightarrow> False" shows "deriv_bound r a n"
split_time_empty: fixes i :: "nat"   and n :: "nat"   and s :: "nat \<Rightarrow> 'a list"   and t :: "nat" assumes "i < n"   and "0 < i" shows "split_time s n (n * t + i) = []"
simps: fixes q\<^sub>0 :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<phi> :: "'b ltl" shows "ltl_to_rabin_base_def.initial q\<^sub>0 q\<^sub>0\<^sub>M \<phi> = (q\<^sub>0 \<phi>, \<iota>\<^sub>\<times> (\<^bold>G \<phi>) (semi_mojmir_def.initial \<circ> q\<^sub>0\<^sub>M \<circ> theG))"
GroupHomSet_def: fixes G :: "'a set"   and H :: "'b set" shows "GroupHomSet G H \<equiv> {T. GroupHom G T} \<inter> {T. T ` G \<subseteq> H}"
iff_simps_1: fixes p :: "fm" shows "Ferrack.iff p p = T"
bl_to_bin_rep_False: fixes n :: "nat" shows "bl_to_bin (replicate n False) = 0"
assms: shows "rt = update rt dip (dsn, dsk, flag, hops, nhip, pre)"
root_test_divergence: fixes f :: "nat \<Rightarrow> 'b" assumes "1 < limsup (\<lambda>n. ereal (root n (norm (f n))))" shows "\<not> summable f"
presWlsAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'c, 'i) Recursion.model_scheme" shows "presWlsAbs hA MOD \<equiv> \<forall>us s A. wlsAbs (us, s) A \<longrightarrow> gWlsAbs MOD (us, s) (hA A)"
add_diff_cancel: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a +\<^sub>o\<^sub>w b -\<^sub>o\<^sub>w b = a"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "sinits\<cdot>(x :# xs) = [::] :# smap\<cdot>(x:#)\<cdot>(sinits\<cdot>xs)"
assms_1: shows "nonneg_cexpr (shift_var_set V) (case_nat t2 \<Gamma>) e"
fmadd_assoc: fixes m :: "('d, 'e) fmap"   and n :: "('d, 'e) fmap"   and p :: "('d, 'e) fmap" shows "m ++\<^sub>f (n ++\<^sub>f p) = m ++\<^sub>f n ++\<^sub>f p"
intro: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "('a, 'b) insert_message" assumes "integrate_insert_commute M a s"   and "integrate_insert_commute_insert_axioms M a m" shows "integrate_insert_commute_insert M a s m"
simps_10: fixes v :: "'b acom"   and va :: "'b acom"   and vb :: "'b" shows "(v;; va \<sqsubseteq> SKIP {vb}) = False"
preserves_reflects_arr: fixes f :: "'a" shows "arr (L f) = arr f"
quot_cond_function: fixes P :: "'d quot \<Rightarrow> 'e quot \<Rightarrow> bool"   and f :: "'d quot \<Rightarrow> 'e quot \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "'d"   and b :: "'e" assumes "\<And>X Y. P X Y \<Longrightarrow> f X Y \<equiv> g (pick X) (pick Y)"   and "\<And>x x' y y'. \<lbrakk>\<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor>; \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor>; P \<lfloor>x\<rfloor> \<lfloor>y\<rfloor>; P \<lfloor>x'\<rfloor> \<lfloor>y'\<rfloor>\<rbrakk> \<Longrightarrow> g x y = g x' y'"   and "P \<lfloor>a\<rfloor> \<lfloor>b\<rfloor>" shows "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
atLeastLessThan_inj_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "{a..<b} = {c..<d}"   and "a < b"   and "c < d" shows "a = c"
elims: fixes x :: "'a \<times> 'b Filters.filter"   and y :: "'a \<times> 'b Filters.filter" assumes "pairs_uminus x = y" obtains xa :: "'a" and ya :: "'b Filters.filter" where "x = (xa, ya)" and "y = (- xa, phi xa)"
closed_segment: fixes a :: "'b"   and b :: "'b" shows "closed {a--b}"
x2: shows "x\<^sub>2 [^] 3 = \<ominus> (a \<otimes> x\<^sub>2 \<oplus> b)"
repr_independence: fixes y :: "'a"   and H :: "'a set"   and x :: "'a" assumes "y \<in> H #> x"   and "x \<in> carrier G"   and "subgroup H G" shows "H #> x = H #> y"
fm_measurable: fixes N :: "'b set set"   and M :: "'d measure" assumes "f ` J \<in> N" shows "fm \<in> Pi\<^sub>M J (\<lambda>_. M) \<rightarrow>\<^sub>M Pi\<^sub>F N (\<lambda>_. M)"