hcju/isabelleps huggingface dataset:
1: shows "pick (insert a I) i = sorted_list_of_set (insert a I) ! i"
not_mayDiverge_Atm: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and atm :: "'c"   and s :: "'b" shows "\<not> PL.mayDiverge tval aval (Atm atm) s"
times_ms_aux_MSLNil_2: fixes F :: "('a \<times> real) msllist" shows "times_ms_aux F MSLNil = MSLNil"
to_list_ga_def: fixes it_init :: "'b \<Rightarrow> 'c Heap"   and it_has_next :: "'c \<Rightarrow> bool Heap"   and it_next :: "'c \<Rightarrow> ('d \<times> 'c) Heap"   and l_empty :: "'a Heap"   and l_prepend :: "'d \<Rightarrow> 'a \<Rightarrow> 'a Heap"   and s :: "'b" shows "to_list_ga it_init it_has_next it_next l_empty l_prepend s \<equiv> it_init s \<bind> (\<lambda>it. l_empty \<bind> (\<lambda>l. to_list_ga_rec it_has_next it_next l_prepend it l \<bind> return))"
mat_rel_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'c mat" shows "mat_rel R A B \<equiv> dim_row A = dim_row B \<and> dim_col A = dim_col B \<and> (\<forall>i j. i < dim_row B \<longrightarrow> j < dim_col B \<longrightarrow> R (A $$ (i, j)) (B $$ (i, j)))"
dvd_prod: fixes A :: "'b set"   and a :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "\<exists>b\<in>A. a dvd f b"   and "finite A" shows "a dvd prod f A"
MutuallyDistinct_def: fixes F :: "('d, 'e, 'f) seqauto set" shows "MutuallyDistinct F = (\<forall>a\<in>F. \<forall>b\<in>F. a \<noteq> b \<longrightarrow> States a \<inter> States b = {})"
stationary_distribution_imp_limit: fixes C :: "'s set"   and N :: "'s pmf"   and y :: "'s" assumes "aperiodic C"   and "essential_class C"   and "countable C"   and "stationary_distribution N"   and "set_pmf N \<subseteq> C"   and "y \<in> C" shows "(\<lambda>n. LINT x|count_space C. \<bar>p y x n - pmf N x\<bar>) \<longlonglongrightarrow> 0"
msg_exhaust: fixes y :: "msg" obtains x :: "agent" where "y = Messages.Agent x__"   | x :: "nat" where "y = Number xa__"   | x :: "nonce_t" where "y = Nonce xb__"   | x :: "ltkey" where "y = LtK xc__"   | x :: "ephkey" where "y = EphK xd__"   | x :: "msg" and x' :: "msg" where "y = \<langle>xe__, x'__\<rangle>"   | x :: "msg" and x' :: "msg" where "y = Enc xf__ x'a__"   | x :: "msg" and x' :: "msg" where "y = Aenc xg__ x'b__"   | x :: "msg" and x' :: "msg" where "y = Sign xh__ x'c__"   | x :: "msg" where "y = Hash xi__"   | x :: "tag" where "y = Tag xj__"   | x :: "msg" and x' :: "msg" where "y = Exp x x'"
edom_ae2ce: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "edom (ae2ce ae) = edom ae"
stream_pred_cases: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" assumes "pred_stream P xs" obtains y :: "'b" and ys :: "'b stream" where "xs = y ## ys" and "P y" and "pred_stream P ys"
f2: fixes x :: "atom" assumes "supp T - set (ba xs) = supp S - set (ba ys)"   and "x \<in> set (ba ys)" shows "x \<sharp> f xs T"
sinks_cons_nonint: fixes u :: "'e"   and D :: "'f \<Rightarrow> 'e"   and x :: "'f"   and I :: "('e \<times> 'e) set"   and xs :: "'f list" assumes "(u, D x) \<notin> I" shows "sinks I D u (x # xs) = sinks I D u xs"
add_0_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.monoid_add plus zero" shows "plus zero a = a"
bot: fixes f :: "'a \<Rightarrow> 'b" shows "hom_graph f {} \<bottom> \<bottom>"
Inr_def: fixes b :: "hf" shows "HF.Inr b \<equiv> \<langle>1, b\<rangle>"
mheadsD: fixes emh :: "ref_ty \<times> mhead"   and G :: "prog"   and S :: "qtname"   and t :: "ref_ty"   and sig :: "sig" assumes "emh \<in> mheads G S t sig"   and "wf_prog G" shows "(\<exists>C D m. t = ClassT C \<and> declrefT emh = ClassT D \<and> accmethd G S C sig = Some m \<and> declclass m = D \<and> mhead (mthd m) = mhd emh) \<or> (\<exists>I. t = IfaceT I \<and> ((\<exists>im. im \<in> accimethds G (pid S) I sig \<and> mthd im = mhd emh) \<or> (\<exists>m. G \<turnstile> Iface I accessible_in pid S \<and> accmethd G S Object sig = Some m \<and> accmodi m \<noteq> Private \<and> declrefT emh = ClassT Object \<and> mhead (mthd m) = mhd emh))) \<or> (\<exists>T m. t = ArrayT T \<and> G \<turnstile> T.[] accessible_in pid S \<and> accmethd G S Object sig = Some m \<and> accmodi m \<noteq> Private \<and> declrefT emh = ClassT Object \<and> mhead (mthd m) = mhd emh)"
lastxx_com: fixes x :: "nat"   and y :: "nat" shows "lang (L_lasthasxx x y) = lang (L_lasthasxx y x)"
GS_initial_impl_def: fixes I :: "'c \<Rightarrow> node_state option"   and v0 :: "'c"   and succs :: "'c set" shows "GS_initial_impl I v0 succs \<equiv> ([v0], [0], I(v0 \<mapsto> STACK 0), if succs = {} then [] else [(0, succs)])"
let_rsp: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun R1 (rel_fun (rel_fun R1 R2) R2) Let Let"
line_integral_on_path_eq_line_integral_on_equiv_chain_3: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_path \<gamma> one_chain"   and "boundary_chain one_chain"   and "\<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>"   and "finite basis" shows "valid_path \<gamma>"
list_to_poly_def: fixes ts :: "'d list"   and cs :: "'e list" shows "list_to_poly ts cs = Abs_poly_mapping (list_to_fun ts cs)"
less_minus_divide_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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less a (uminus (divide b c)) = (if less zero c then less (times a c) (uminus b) else if less c zero then less (uminus b) (times a c) else less a zero)"
maxne0_mono: fixes y :: "nat list"   and x :: "nat list"   and a :: "nat list" assumes "y \<le>\<^sub>v x" shows "maxne0 y a \<le> maxne0 x a"
older_seniors_cases_subseteq: fixes x :: "nat"   and n :: "nat"   and y :: "nat" obtains "older_seniors x n \<subseteq> older_seniors y n"   | "older_seniors y n \<subseteq> older_seniors x n"
differentiableI: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and net :: "'c filter" assumes "(f has_derivative f') net" shows "f differentiable net"
bind_to_let_conv: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a nres" shows "RETURN x \<bind> f = Let x f"
eq_eqvt: fixes p :: "perm"   and x :: "'c"   and y :: "'c" shows "p \<bullet> (x = y) = (p \<bullet> x = p \<bullet> y)"
converse_Rel_components_1: fixes T :: "V" shows "T\<inverse>\<^sub>R\<^sub>e\<^sub>l\<lparr>ArrVal\<rparr> = (T\<lparr>ArrVal\<rparr>)\<inverse>\<^sub>\<circ>"
v11_app_vconverse_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_uminus" shows "-\<^sub>\<rat> vrat_uminus\<inverse>\<^sub>\<circ>\<lparr>a\<rparr> = a"
intro: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" assumes "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"   and "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"   and "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F"   and "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"   and "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "converse_pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
simps_1: fixes i :: "int"   and ks :: "int list" shows "divisor (Le i ks) = 1"
arc_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_stone_relation_algebra.arc f" shows "\<exists>e. f e = top \<and> (\<forall>d. e \<noteq> d \<longrightarrow> f d = bot)"
invert_side_of_triangle_1: fixes h :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.arr h"   and "VV.comp f g = h"   and "VV.iso f" shows "VV.seq (VV.inv f) h \<and> g = VV.comp (VV.inv f) h"
Int_Un_distrib2: fixes B :: "'a set"   and C :: "'a set"   and A :: "'a set" shows "(B \<union> C) \<inter> A = B \<inter> A \<union> C \<inter> A"
outer_measure_close: fixes f :: "'a set \<Rightarrow> ennreal"   and X :: "'a set"   and e :: "ennreal" assumes "outer_measure M f X < e" shows "\<exists>A. range A \<subseteq> M \<and> disjoint_family A \<and> X \<subseteq> \<Union> (range A) \<and> (\<Sum>i. f (A i)) < e"
skl1_inv1I: fixes x :: "skl1_state" assumes "\<And>Ra A B gny Nr. 0 < signalsInit x (Commit A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) \<Longrightarrow> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress x Ra = Some {xnx, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Ra xnr = Some Nr \<and> sklvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))" shows "x \<in> skl1_inv1"
flub_def: fixes A :: "'b fset set"   and t :: "'b fset" shows "flub A t = (if \<forall>a\<in>A. a |\<subseteq>| t then Abs_fset (\<Union> (fset ` A)) else t)"
mC_ZOC_sym: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "sym theta"   and "mC_ZOC theta c d s t I0 P F" shows "mC_ZOC theta d c t s (F I0) (F ` P) (inv_into P F)"
vec_def: fixes n :: "nat"   and x :: "'b list" shows "vec n x = (length x = n)"
drop_tape_of_cons: fixes q :: "nat"   and lm :: "nat list"   and x :: "nat" assumes "Suc q < length lm"   and "x = lm ! q" shows "<drop q lm> = Oc # Oc \<up> x @ Bk # <drop (Suc q) lm>"
l2_inv6D: fixes Ra :: "fid_t"   and x :: "l2_state"   and K :: "msg" assumes "role (guessed_runs Ra) = Init"   and "run_ended (progress x Ra)"   and "guessed_frame Ra xsk = Some K"   and "x \<in> l2_inv6" shows "Aenc K (epubKF (Ra $ kE)) \<in> extr (bad x) (ik x) (chan x)"
degree_aform_err_min_range_antimono: fixes p :: "nat"   and F :: "floatarith \<Rightarrow> floatarith"   and D :: "floatarith \<Rightarrow> floatarith"   and l :: "float"   and u :: "float"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "min_range_antimono p F D l u X = Some Y"   and "degree_aform_err X \<le> n" shows "degree_aform_err Y \<le> n"
toplevel_summands_lang: fixes r :: "'b rexp"   and s :: "'b rexp"   and n :: "nat" assumes "r \<in> toplevel_summands s" shows "lang n r \<subseteq> lang n s"
Grp_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "BNF_Def.Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
atlas_def: shows "l.atlas = {c. domain c \<subseteq> carrier \<and> (\<forall>c'\<in>charts. l-smooth_compat c c')}"
intro: fixes x :: "'b \<Longrightarrow> 'c" assumes "weak_lens x"   and "mwb_lens_axioms x" shows "mwb_lens x"
tensor_preserves_ide: fixes a :: "'d"   and b :: "'d" assumes "ide a"   and "ide b" shows "ide (a \<otimes> b)"
cod_simps_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "identity_arrow_of_spans C \<mu>" shows "span_in_category.apex C (Cod \<mu>) = span_in_category.apex C (Dom \<mu>)"
helper_defined_single: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and m :: "'c match_expr"   and a :: "action"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state" assumes "wf (called_by_chain \<Gamma>)"   and "\<forall>rsg\<in>ran \<Gamma> \<union> {[Rule m a]}. wf_chain \<Gamma> rsg"   and "\<forall>rsg\<in>ran \<Gamma> \<union> {[Rule m a]}. \<forall>r\<in>set rsg. (\<nexists>chain. get_action r = Goto chain) \<and> get_action r \<noteq> Unknown"   and "a \<noteq> Return" shows "\<exists>t. \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], s\<rangle> \<Rightarrow> t"
add_poly_spec_def: fixes p :: "int mpoly"   and q :: "int mpoly" shows "add_poly_spec p q = SPEC (\<lambda>r. p + q - r \<in> More_Modules.ideal polynomial_bool)"
tt_monomial_plus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c" assumes "p \<noteq> 0"   and "lt p \<prec>\<^sub>t v" shows "tt (monomial c v + p) = tt p"
exhaust: fixes y :: "dbfm" obtains x11 :: "dbtm" and x12 :: "dbtm" where "y = DBMem x11 x12"   | x21 :: "dbtm" and x22 :: "dbtm" where "y = DBEq x21 x22"   | x31 :: "dbfm" and x32 :: "dbfm" where "y = DBDisj x31 x32"   | x4 :: "dbfm" where "y = DBNeg x4"   | x5 :: "dbfm" where "y = DBEx x5"
fresh_at_path_def: fixes t :: "('form, 'rule, 'c, 'd) itnode rose_tree"   and "is" :: "nat list" shows "fresh_at_path t is = \<Union> (fresh_at t ` set (prefixes is))"
ereal_divide_le_posI: fixes x :: "ereal"   and z :: "ereal"   and y :: "ereal" assumes "0 < x"   and "z \<noteq> - \<infinity>"   and "z \<le> x * y" shows "z / x \<le> y"
idealp_filterp_dual: shows "idealp = filterp \<circ> (`) \<partial>"
cases: fixes a1 :: "nat \<times> nat \<times> thread"   and a2 :: "nat \<times> nat \<times> thread" assumes "bstep_sng a1 a2" obtains c :: "nat" and n :: "nat" where "a1 = (c__, n__, INIT)" and "a2 = (c__, (n__ + 1) mod (k * N), WAIT (n__ mod N))"   | c :: "nat" and tk :: "nat" and n :: "nat" where "a1 = (ca__, na__, WAIT tk__)" and "a2 = (ca__, na__, WAIT tk__)" and "ca__ \<noteq> tk__"   | c :: "nat" and n :: "nat" where "a1 = (cb__, nb__, WAIT cb__)" and "a2 = (cb__, nb__, HOLD cb__)"   | c :: "nat" and n :: "nat" and tk :: "nat" where "a1 = (cc__, nc__, HOLD tka__)" and "a2 = (cc__, nc__, REL tka__)"   | c :: "nat" and n :: "nat" and tk :: "nat" where "a1 = (c, n, REL tk)" and "a2 = ((tk + 1) mod N, n, INIT)"
mk_linear_orders_induct: fixes r :: "'a list"   and C :: "'a set"   and B :: "'a set"   and P :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a list \<Rightarrow> bool" assumes "r \<in> local.mk_linear_orders C B"   and "B \<subseteq> A"   and "\<And>C B. \<lbrakk>B \<subseteq> A; f B = {}\<rbrakk> \<Longrightarrow> P C B []"   and "\<And>C B b cs. \<lbrakk>B \<subseteq> A; cs \<in> local.mk_linear_orders {} (B - {b}); b \<in> f B; f B \<subseteq> C; P {} (B - {b}) cs\<rbrakk> \<Longrightarrow> P C B (b # cs)"   and "\<And>C B b cs. \<lbrakk>B \<subseteq> A; cs \<in> local.mk_linear_orders C (B - {SOME x. x \<in> f B - C}); b \<in> f B; b \<notin> C; P C (B - {SOME x. x \<in> f B - C}) cs\<rbrakk> \<Longrightarrow> P C B ((SOME x. x \<in> f B - C) # cs)" shows "P C B r"
I_2: fixes i :: "nat" assumes "Suc i < n" shows "I i < n"
list_of_oalist_shift_keys: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and cmp_term :: "'b nat_term_order"   and t :: "'c"   and f :: "'e \<Rightarrow> 'e"   and xs :: "('b, 'e) oalist_ntm" assumes "gd_nat_term pair_of_term term_of_pair cmp_term" shows "list_of_oalist_ntm (gd_nat_term.shift_map_keys pair_of_term term_of_pair t f xs) = map_raw (\<lambda>kv. (term_powerprod.splus pair_of_term term_of_pair t (fst kv), f (snd kv))) (list_of_oalist_ntm xs)"
comm_groupE_4: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and y :: "'b" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
that: assumes "box a b \<subseteq> A"   and "x \<in> box a b"   and "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i" shows "thesis"
Lambert_W'_lessI: fixes y :: "real"   and x :: "real" assumes "y * exp y < x"   and "x < 0" shows "Lambert_W' x < y"
ub_sum: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and f :: "'d \<Rightarrow> 'c"   and i :: "'d" assumes "class.idempotent_semiring one times sup less_eq less bot" shows "less_eq (f i) (comm_monoid_add.sum sup bot f {k. True})"
p_\<alpha>: shows "\<alpha>' = p \<bullet> \<alpha>"
eq: shows "(\<lambda>x. EM_remainder' n (\<lambda>x. of_real (f x)) (real_of_int a) (real_of_int x)) = (\<lambda>x. of_real (EM_remainder' n f (real_of_int a) (real_of_int x)))"
a_closed: fixes G :: "('d, 'e) ring_scheme"   and x :: "'d"   and y :: "'d" assumes "abelian_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> y \<in> carrier G"
simps_10: fixes C :: "char list"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" shows "fv (C\<bullet>\<^sub>sF{D} := e\<^sub>2) = fv e\<^sub>2"
R_consts_1: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n true\<^sub>n"
simps_4: fixes v :: "ty"   and va :: "ty list" shows "conformable [] (v # va) = False"
elims: fixes x :: "bool \<times> nat \<times> int vec list"   and y :: "int vec list" assumes "basis_reduction_main x = y" obtains upw :: "bool" and i :: "nat" and fs :: "int vec list" where "x = (upw, i, fs)" and "y = (if i < m \<and> LLL_invariant upw i fs then basis_reduction_main (basis_reduction_step upw i fs) else fs)"
sorted_on_inv_2: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and xs :: "'a list" assumes "sorted_on r (x # xs)" obtains "x \<in> Field r" and "\<forall>y\<in>set xs. (x, y) \<in> r" and "sorted_on r xs"
le_zero_eq: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
eq_name_eq_effs': fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and name' :: "char list"   and pres' :: "(nat \<times> nat) list"   and effs' :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer' :: "nat"   and s :: "nat \<Rightarrow> nat option" assumes "lookup_operator ((fst \<circ> the \<circ> lookup_action \<circ> abs_ast_operator) (name, pres, effs, layer)) = Some (name', pres', effs', layer')"   and "is_standard_operator' (name, pres, effs, layer)"   and "(name, pres, effs, layer) \<in> set ast\<delta>"   and "s \<in> valid_states" shows "effs = effs'"
spaceN_iff: fixes x :: "'c"   and N :: "'c quasinorm" shows "(x \<in> space\<^sub>N N) = (eNorm N x < \<infinity>)"
lookup_monom_mult_plus: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" shows "lookup (local.punit.monom_mult c t p) (local.punit.splus t v) = c * lookup p v"
map_is_simplicial_morph: fixes f :: "'a \<Rightarrow> 'b" shows "SimplicialComplex (f \<turnstile> X)"
attract: fixes y :: "'a set"   and x :: "'a set"   and z :: "'a set" assumes "y \<sqsubseteq>\<^sup>s x"   and "x \<sqsubseteq>\<^sup>s y"   and "z \<sqsubseteq>\<^sup>s y"   and "x \<in> Q"   and "y \<in> Q"   and "z \<in> Q" shows "z \<sqsubseteq>\<^sup>s x"
exhaust: fixes y :: "dynamic" obtains "y = DNil"   | x2 :: "bool" where "y = DBool x2"   | x3 :: "nat" where "y = DNum x3"
a_coset_add_assoc: fixes G :: "('a, 'b) ring_scheme"   and M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "abelian_group G"   and "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "M +>\<^bsub>G\<^esub> g +>\<^bsub>G\<^esub> h = M +>\<^bsub>G\<^esub> g \<oplus>\<^bsub>G\<^esub> h"
a_heap_is_wellformed_def: fixes h :: "(_) heap" shows "local.a_heap_is_wellformed h = (local.a_acyclic_heap h \<and> local.a_all_ptrs_in_heap h \<and> local.a_distinct_lists h \<and> local.a_owner_document_valid h)"
ord_rec_succ: fixes k :: "hf"   and T :: "hf"   and G :: "hf \<Rightarrow> hf"   and H :: "hf \<Rightarrow> hf" assumes "Ord k" shows "ord_rec T G H (succ k) = G (ord_rec T G H k)"
assms: shows "M \<le> SPEC (\<lambda>c. (c, a) \<in> R)"
rel_interior_Epigraph: fixes f :: "'b \<Rightarrow> ereal"   and x :: "'b"   and z :: "real" assumes "Lower_Semicontinuous.convex_on UNIV f" shows "((x, z) \<in> rel_interior (Epigraph UNIV f)) = (x \<in> rel_interior (domain f) \<and> f x < ereal z)"
agree_on_def: fixes G :: "('d, 'e) labeled_graph"   and f\<^sub>1 :: "('e \<times> 'f) set"   and f\<^sub>2 :: "('e \<times> 'f) set" shows "agree_on G f\<^sub>1 f\<^sub>2 \<equiv> \<forall>v\<in>vertices G. f\<^sub>1 `` {v} = f\<^sub>2 `` {v}"
advantage_def: fixes \<A> :: "('y, 'z) distinguisher"   and res1 :: "('y, 'z) resource"   and res2 :: "('y, 'z) resource" shows "advantage \<A> res1 res2 = \<bar>spmf (connect \<A> res1) True - spmf (connect \<A> res2) True\<bar>"
rev_nth: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "rev xs ! n = xs ! (length xs - Suc n)"
strict_mono_Suc_iff: fixes f :: "nat \<Rightarrow> 'c" shows "strict_mono f = (\<forall>n. f n < f (Suc n))"
steps_appendI: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "G'.steps (xs @ [x])"   and "x \<rightarrow> y" shows "G'.steps (xs @ [x, y])"
wf_nFAND: fixes idx :: "'n"   and \<phi>s :: "('a, 'k) aformula list" shows "local.wf idx (nFAND \<phi>s) = (\<forall>\<phi>\<in>set \<phi>s. local.wf idx \<phi>)"
base: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and input :: "'f"   and \<I> :: "('e, 'f) \<I>"   and x :: "('d, 'e, 'f) gpv" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "x = c input" shows "sub_gpvsp \<I> x gpv"
map_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'a" assumes "constant_transformation A B g" shows "constant_transformation.map A B g f \<equiv> if partial_magma.arr A f then g else partial_magma.null B"
index_col: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "col A j $v i = A $$ (i, j)"
img_ord_def: fixes f :: "'b \<Rightarrow> 'd"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> 'c" shows "img_ord f ord = (\<lambda>x y. ord (f x) (f y))"
R: fixes x :: "'a" assumes "x \<in> A" shows "R x (f x)"
vc: shows "vector \<equiv> \<lambda>A B. (fst A - fst B, snd A - snd B)"