hcju/isabelleps huggingface dataset:
sel_13: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "un_StaticOperationCall2 (StaticOperationCall x61 x62 x63) = x62"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [Local n i vls LI] : ts _> ts'"
monoid_mult_hom_axioms: shows "monoid_mult_hom (map_poly hom)"
dom_lemmas_1: fixes f :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a" assumes "f x = Some y" shows "x \<in> dom f"
eta: shows "s \<rightarrow>\<^sub>\<eta>\<^sup>* t"
pred_inject_2: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and aa :: "'a list" shows "list_all P (a # aa) = (P a \<and> list_all P aa)"
lepoll_times1: fixes B :: "'a set"   and A :: "'b set" assumes "B \<noteq> {}" shows "A \<lesssim> A \<times> B"
case_2: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'd"   and x2 :: "'f" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
lt_row_to_poly_pivot_fun_eqD: fixes S :: "'t set"   and A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and i1 :: "nat"   and i2 :: "nat" assumes "card S = dim_col A"   and "pivot_fun A f (dim_col A)"   and "i1 < dim_row A"   and "i2 < dim_row A"   and "f i1 < dim_col A"   and "f i2 < dim_col A"   and "pps_to_list S ! f i1 = pps_to_list S ! f i2" shows "i1 = i2"
hnorm_scaleHR_less: fixes x :: "real star"   and r :: "real star"   and y :: "'a star"   and s :: "real star" assumes "\<bar>x\<bar> < r"   and "hnorm y < s" shows "hnorm (scaleHR x y) < r * s"
simps_16: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Exp a) vs = approx_floatarith p a vs \<bind> exp_aform_err p"
assms_1: shows "run (p := x) \<sigma> \<sigma>' r"
linear_upper: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and u :: "real"   and x :: "real" assumes "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_real_derivative f' x) (at x within {a..b})"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> f' x \<le> u"   and "x \<in> {a..b}" shows "f x \<le> f a + u * (x - a)"
bij_lens_def: fixes x :: "'d \<Longrightarrow> 'e" shows "bij_lens x \<equiv> weak_lens x \<and> bij_lens_axioms x"
map_fst_pair_pmf: fixes A :: "'b pmf"   and B :: "'c pmf" shows "map_pmf fst (pair_pmf A B) = A"
abs_real_alg_def: fixes x :: "real_alg" shows "\<bar>x\<bar> = (if real_of x < 0 then - x else x)"
mem_rel_interior_cball: fixes x :: "'c"   and S :: "'c set" shows "(x \<in> rel_interior S) = (x \<in> S \<and> (\<exists>e>0. cball x e \<inter> affine hull S \<subseteq> S))"
simps: fixes xs :: "'a"   and x :: "'b"   and ys :: "'a"   and y :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" shows "qAFreshAbs xs x (qAbs ys y X) = ((xs \<noteq> ys \<or> x \<noteq> y) \<and> qAFresh xs x X)"
exec_step_ind_Checkcast_Error: fixes P :: "jvm_prog"   and C :: "char list"   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" assumes "\<not> cast_ok P C h (hd stk)" shows "exec_step_ind (StepI (Checkcast C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt ClassCast\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
merge_union: fixes A :: "'a cset"   and B :: "'a cset"   and f :: "'a \<Rightarrow> 'm" shows "merge (cUn A B) f = bind (merge A f) (\<lambda>A'. bind (merge B f) (\<lambda>B'. return (cUn A' B')))"
NatTransExt': fixes \<eta>1' :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2' :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NTDom \<eta>1' = NTDom \<eta>2'"   and "NTCod \<eta>1' = NTCod \<eta>2'"   and "\<And>X. X \<in> obj\<^bsub>NTCatDom \<eta>1'\<^esub> \<Longrightarrow> \<eta>1' $$ X = \<eta>2' $$ X" shows "MakeNT \<eta>1' = MakeNT \<eta>2'"
square_integrable_imp_lspace: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f square_integrable S" shows "f \<in> lspace (lebesgue_on S) 2"
rep_eq: fixes x :: "nat"   and xa :: "'b pdevs"   and xb :: "'b pdevs" shows "pdevs_apply (msum_pdevs x xa xb) = msum_pdevs_raw x (pdevs_apply xa) (pdevs_apply xb)"
xMS: shows "{y \<in> S. x \<le> y} \<subseteq> S"
differentiable_iff_scaleR: fixes f :: "real \<Rightarrow> 'a"   and F :: "real filter" shows "(f differentiable F) = (\<exists>d. (f has_derivative (\<lambda>x. x *\<^sub>R d)) F)"
simps_4: fixes x3 :: "nat \<times> nat" shows "RELABEL' \<noteq> SAT_PUSH' x3"
map_comp: fixes g :: "'f \<Rightarrow> 'e"   and f :: "'g \<Rightarrow> 'f"   and v :: "'g pmf" shows "map_pmf g (map_pmf f v) = map_pmf (g \<circ> f) v"
fin_string_upto_length: fixes n :: "nat"   and L :: "fin_string" assumes "List.member (fin_string_upto n) L" shows "ilength L \<le> n"
finite_field_factorization_int: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly list" assumes "square_free_m f"   and "finite_field_factorization_int p f = (c, fs)" shows "unique_factorization_m f (c, mset fs) \<and> c \<in> {0..<p} \<and> (\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p})"
ide_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o \<AA>" shows "ide (mkIde A)"
snth_sset: fixes s :: "'a stream"   and n :: "nat" shows "s !! n \<in> sset s"
errMOD_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" shows "errMOD MOD \<equiv> \<lparr>igWls = \<lambda>s eX. case eX of ERR \<Rightarrow> False | OK X \<Rightarrow> igWls MOD s X, igWlsAbs = \<lambda>(us, s) eA. case eA of ERR \<Rightarrow> False | OK A \<Rightarrow> igWlsAbs MOD (us, s) A, igVar = \<lambda>xs x. OK (igVar MOD xs x), igAbs = \<lambda>xs x eX. if eX \<noteq> ERR \<and> (\<exists>s. isInBar (xs, s) \<and> igWls MOD s (check eX)) then OK (igAbs MOD xs x (check eX)) else ERR, igOp = \<lambda>delta einp ebinp. if liftAll (\<lambda>X. X \<noteq> ERR) einp \<and> liftAll (\<lambda>A. A \<noteq> ERR) ebinp \<and> igWlsInp MOD delta (checkI einp) \<and> igWlsBinp MOD delta (checkI ebinp) then OK (igOp MOD delta (checkI einp) (checkI ebinp)) else ERR, igFresh = \<lambda>ys y eX. if eX \<noteq> ERR \<and> (\<exists>s. igWls MOD s (check eX)) then igFresh MOD ys y (check eX) else True, igFreshAbs = \<lambda>ys y eA. if eA \<noteq> ERR \<and> (\<exists>us s. igWlsAbs MOD (us, s) (check eA)) then igFreshAbs MOD ys y (check eA) else True, igSwap = \<lambda>zs z1 z2 eX. if eX \<noteq> ERR \<and> (\<exists>s. igWls MOD s (check eX)) then OK (igSwap MOD zs z1 z2 (check eX)) else ERR, igSwapAbs = \<lambda>zs z1 z2 eA. if eA \<noteq> ERR \<and> (\<exists>us s. igWlsAbs MOD (us, s) (check eA)) then OK (igSwapAbs MOD zs z1 z2 (check eA)) else ERR, igSubst = \<lambda>ys eY y eX. if eY \<noteq> ERR \<and> igWls MOD (asSort ys) (check eY) \<and> eX \<noteq> ERR \<and> (\<exists>s. igWls MOD s (check eX)) then OK (igSubst MOD ys (check eY) y (check eX)) else ERR, igSubstAbs = \<lambda>ys eY y eA. if eY \<noteq> ERR \<and> igWls MOD (asSort ys) (check eY) \<and> eA \<noteq> ERR \<and> (\<exists>us s. igWlsAbs MOD (us, s) (check eA)) then OK (igSubstAbs MOD ys (check eY) y (check eA)) else ERR\<rparr>"
homeomorphic_map_id: fixes X :: "'c topology"   and Y :: "'c topology" shows "homeomorphic_map X Y id = (Y = X)"
remove_invalid_point_block: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and p :: "'b"   and bl :: "'b set" assumes "incidence_system point_set block_collection"   and "p \<notin> point_set"   and "bl \<in># block_collection" shows "bl - {p} = bl"
Map_ide_2: fixes a :: "('a E.term, 'a) arr" assumes "ide a" shows "Map a = EVAL (Cod a)"
assms_1: shows "rt (\<sigma> i) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip)"
infinite_Icc_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less" shows "infinite (ord.atLeastAtMost less_eq a b) = less a b"
SDG_def: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list" shows "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses \<equiv> CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses \<and> Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"
assms: shows "exact_seq (G # l, q)  \<longlongrightarrow>\<^bsub>g\<^esub> H"
max_id: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "max x y = x"
gen_lossless_gpv_Done: fixes co :: "bool"   and \<I> :: "('d, 'e) \<I>"   and x :: "'f" shows "gen_lossless_gpv co \<I> (Generative_Probabilistic_Value.Done x)"
continuous_on_def: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> G"   and "y \<in> G" shows "compose G (x)\<^sub>L (y)\<^sub>L = (x \<cdot> y)\<^sub>L"
pappus12: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and Q :: "Points"   and R :: "Points" assumes "is_pappus1 A B C A' B' C' P Q R" shows "is_pappus2 A B C A' B' C' P Q R"
dgrad_set_le_def: fixes d :: "'d \<Rightarrow> nat"   and S :: "'d set"   and T :: "'d set" shows "dgrad_set_le d S T = (\<forall>s\<in>S. \<exists>t\<in>T. d s \<le> d t)"
bin_nth_sc: fixes n :: "nat"   and b :: "bool"   and w :: "int" shows "bit (bin_sc n b w) n = b"
assms_2: assumes "\<bar>a\<bar> = \<bar>b\<bar>" shows "thesis"
independent_directional_derivative: fixes k :: "enat"   and a :: "'c" assumes "k \<noteq> 0" shows "independent (directional_derivative k a ` Basis)"
path2_not_Nil2: fixes g :: "'g"   and n :: "'node"   and m :: "'node" shows "\<not> g \<turnstile> n-[]\<rightarrow>m"
inB: fixes x :: "nat \<times> nat"   and y :: "nat" assumes "P x y" shows "y \<in> B"
surjective: fixes r :: "('b, 'c, 'd, 'e) BEState_scheme" shows "r = \<lparr>es = es r, ps = ps r, \<dots> = BEState.more r\<rparr>"
markov_inequality: fixes M :: "'a measure"   and X :: "'a \<Rightarrow> real"   and t :: "real" assumes "prob_space M"   and "\<And>a. 0 \<le> X a"   and "integrable M X"   and "0 < t" shows "\<P>(a in M. t \<le> X a) \<le> integral\<^sup>L M X / t"
bdd_below_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "preorder.bdd_below less_eq {}"
continuous_pre_zeta: fixes a :: "real"   and s :: "complex"   and A :: "complex set" assumes "0 < a" shows "continuous (at s within A) (pre_zeta a)"
ccNeighbors_lub: fixes x :: "Vars.var"   and Gs :: "CoCalls set" shows "ccNeighbors x (lub Gs) = lub (ccNeighbors x ` Gs)"
not_in_Gromov_boundary': fixes x :: "'b" shows "to_Gromov_completion x \<notin> Gromov_boundary"
truncate_up_switch_sign_mono: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "x \<le> 0"   and "0 \<le> y" shows "truncate_up prec x \<le> truncate_up prec y"
card_of_mono1: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "|A| \<le>o |B|"
null_setsI: fixes M :: "'c measure"   and A :: "'c set" assumes "emeasure M A = 0"   and "A \<in> sets M" shows "A \<in> null_sets M"
assms_1: shows "map_poly complex_of_real q = r * map_poly complex_of_real p"
Pxxy_eq: fixes x :: "'b"   and y :: "'c" shows "Pxxy (x, x, y) = Pxy (x, y)"
inf_sup_distrib1: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
remove_vertex_path_from_to: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and x :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H\<^esub> w"   and "x \<in> V\<^bsub>H\<^esub>"   and "x \<notin> set xs" shows "v \<leadsto>xs\<leadsto>\<^bsub>H.remove_vertex x\<^esub> w"
cases: fixes x :: "('a, 'b, 'varSort, 'var, 'c) qAbs" obtains xs :: "'varSort" and x :: "'var" and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm" where "xa__ = qAbs xs x X"
times_pratt1_def: fixes x :: "pratt1"   and y :: "pratt1" shows "x \<cdot> y \<equiv> x +\<^sub>1 y"
collapse: fixes gauss_int :: "gauss_int" shows "Gauss_Int (ReZ gauss_int) (ImZ gauss_int) = gauss_int"
mset_le_incr_right1: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "a \<subseteq># b + c"
composite_functor_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'b \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'e" shows "composite_functor A B C F G \<equiv> functor A B F \<and> functor B C G"
Mismatch_1: fixes P :: "pi"   and c :: "name"   and x :: "name"   and P' :: "pi"   and a :: "name"   and b :: "name" assumes "P \<Longrightarrow>c<\<nu>x> \<prec> P'"   and "a \<noteq> b" shows "[a\<noteq>b]P \<Longrightarrow>c<\<nu>x> \<prec> P'"
nonfinal_empty_mrexp: fixes r :: "'b rexp" shows "\<not> final (empty_mrexp r)"
hyps_1: shows "\<forall>mthd Cs'. \<not> P \<turnstile> C has least M = mthd via Cs'"
assms_2: shows "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = A"
perm_string: fixes pi :: "('a \<times> 'a) list"   and s :: "char list" shows "pi \<bullet> s = s"
nominals_mapi_branch_bridge: fixes j :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and k :: "'c"   and xs :: "(nat \<times> nat) set" assumes "Nom k at j in' set branch" shows "branch_nominals (mapi_branch (bridge k j xs) branch) = branch_nominals branch"
mem_interior: fixes x :: "'b"   and S :: "'b set" shows "(x \<in> interior S) = (\<exists>e>0. ball x e \<subseteq> S)"
card_bit0: shows "CARD('a bit0) = 2 * CARD('a)"
tendsto_ine\<^sub>N_iff_limitin_ge1: fixes p :: "ennreal"   and M :: "'a measure"   and u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a \<Rightarrow> real" assumes "1 \<le> p" shows "tendsto_ine\<^sub>N (\<LL> p M) u x = limitin (topology\<^sub>N (\<LL> p M)) u x sequentially"
nonPrincipalInvertRight: fixes R1 :: "(sequent list \<times> sequent) set"   and R2 :: "(sequent list \<times> sequent) set"   and R3 :: "(sequent list \<times> sequent) set"   and R :: "(sequent list \<times> sequent) set"   and r :: "sequent list \<times> sequent"   and ps :: "sequent list"   and c :: "sequent"   and n :: "nat"   and F :: "char list"   and x :: "NominalSequents.var"   and A :: "form"   and \<Gamma>' :: "form multiset"   and \<Delta>' :: "form multiset"   and \<Gamma> :: "form multiset"   and \<Delta> :: "form multiset"   and S :: "sequent"   and Ps :: "sequent list"   and n' :: "nat" assumes "R1 \<subseteq> upRules"   and "R2 \<subseteq> nprovRules"   and "R3 \<subseteq> provRules"   and "R = Ax \<union> R1 \<union> R2 \<union> R3"   and "r \<in> R1 \<or> r \<in> R2 \<or> r \<in> R3"   and "r = (ps, c)"   and "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> F \<nabla> [x].A, m) \<in> derivable  R* \<longrightarrow> (\<forall>r'\<in>R. rightPrincipal r' (F \<nabla> [x].A) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> \<not> multSubst \<Gamma>' \<and> \<not> multSubst \<Delta>' \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable  R*)"   and "( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> F \<nabla> [x].A, n) \<in> derivable  R*"   and "\<forall>r'\<in>R. rightPrincipal r' (F \<nabla> [x].A) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"   and "\<not> multSubst \<Gamma>' \<and> \<not> multSubst \<Delta>'"   and "\<not> rightPrincipal r (F \<nabla> [x].A)"   and "extendRule S r = (Ps,  \<Gamma> \<Rightarrow>* \<Delta> \<oplus> F \<nabla> [x].A)"   and "n = n' + 1"   and "\<forall>p\<in>set Ps. \<exists>n\<le>n'. (p, n) \<in> derivable  R*"   and "Ps \<noteq> []" shows "\<exists>m\<le>n. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable  R*"
num_ivl_def: fixes n :: "int" shows "num_ivl n = {n\<dots>n}"
opt_no_defection_def': fixes s :: "opt_v_state"   and round_votes :: "process \<Rightarrow> val option" shows "opt_no_defection s round_votes \<equiv> \<forall>v Q. quorum_for Q v (last_vote s) \<longrightarrow> round_votes ` Q \<subseteq> {None, Some v}"
vimage_empty: fixes f :: "'d \<Rightarrow> 'e" shows "f -` {} = {}"
prim_sem_env_def: shows "CakeML_Code.prim_sem_env = snd prim_sem_res"
C_pos: shows "\<forall>\<^sub>F x in at_top. 0 < eval C x"
simps_1: fixes uu :: "char list" shows "(Iface [] \<le> Iface uu) = True"
nob: shows "\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). real_of_int x = b + real_of_int j)"
compositionality: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "map_option f (map_option g option) = map_option (f \<circ> g) option"
less_eq_prod_def: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "(x \<le> y) = (fst x \<le> fst y \<and> snd x \<le> snd y)"
fds_abs_converges_power: fixes f :: "'a fds"   and s :: "'a"   and n :: "nat" assumes "fds_abs_converges f s" shows "fds_abs_converges (f ^ n) s"
edgesT_diff_sng_inv_eq: fixes T :: "'b ugraph"   and x :: "'b"   and y :: "'b" shows "(edges T - {(x, y), (y, x)})\<inverse> = edges T - {(x, y), (y, x)}"
P13: shows "Col T2 C D \<and> Bet A T2 B"
redundant_def: fixes g :: "'g" shows "redundant g \<equiv> \<exists>v\<in>allVars g. trivial g v"
iprev_induct: fixes P :: "nat \<Rightarrow> bool"   and I :: "nat set"   and n :: "nat" assumes "P (Max I)"   and "\<And>n. \<lbrakk>n \<in> I; P n\<rbrakk> \<Longrightarrow> P (iprev n I)"   and "finite I"   and "n \<in> I" shows "P n"
assms_2: shows "\<forall>x. List.member S x \<longrightarrow> x < length subsets"
isnpolyh_Suc_const: fixes p :: "'b poly"   and n :: "nat" assumes "isnpolyh p (Suc n)" shows "isconstant p"
sym_sum_inter_num_to_zero: fixes b1 :: "'a set" assumes "b1 \<in># \<B>" shows "(\<Sum>bl\<in>#remove1_mset b1 \<B>. (b1 |\<inter>| bl - \<Lambda>)\<^sup>2) = 0"
listset_appendE: fixes ys :: "'b list"   and xs1 :: "'b set list"   and xs2 :: "'b set list" assumes "ys \<in> listset (xs1 @ xs2)" obtains ys1 :: "'b list" and ys2 :: "'b list" where "ys1 \<in> listset xs1" and "ys2 \<in> listset xs2" and "ys = ys1 @ ys2"
JF_cind2: fixes R1 :: "('d JF1 \<times> 'd JF1) set"   and R2 :: "('d JF2 \<times> 'd JF2) set" assumes "sbis UNIV UNIV dtor1 dtor2 R1 R2" shows "R2 \<subseteq> Id"
times_invertible_eq_zero: fixes A :: "((real, 'a) vec, 'a) vec"   and x :: "(real, 'a) vec" assumes "invertible A"   and "x v* A = 0" shows "x = 0"
adj_cancel_eq2: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "g \<circ> f \<circ> g = g"
gen_span_closed: fixes gens :: "'a set" assumes "gens \<subseteq> carrier G" shows "\<langle>gens\<rangle>\<^bsub>G\<^esub> \<subseteq> carrier G"
image_is_empty: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "(f ` A = {}) = (A = {})"
wt_e: shows "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-Class statC"
card_atLeastZeroLessThan_int: fixes u :: "int" shows "card {0..<u} = nat u"
strict_border_take_\<ff>: fixes i :: "nat"   and s :: "'a list" assumes "0 < i"   and "i \<le> length s" shows "strict_border (take (\<ff> s i - 1) s) (take i s)"
fn_norm_ub: fixes f :: "'a \<Rightarrow> real"   and b :: "real" assumes "Function_Norm.continuous V f norm"   and "b \<in> B V f" shows "b \<le> \<parallel>f\<parallel>\<hyphen>V"
LFP_least_fixed_point: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "Mono\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L"   and "x \<in> fps L f" shows "LFP\<^bsub>L\<^esub> f \<sqsubseteq>\<^bsub>L\<^esub> x"
zeta_partial_sum_le_pos: fixes s :: "real" assumes "0 < s"   and "s \<noteq> 1" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>x. real x powr - s) x - (x powr (1 - s) / (1 - s) + Re (zeta (complex_of_real s)))\<bar> \<le> c * x powr - s"
DomFunctor: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "dom\<^bsub>CatCod F\<^esub> (F ## f) = F @@ dom\<^bsub>CatDom F\<^esub> f"
pure_returns_heap_eq: fixes h :: "'e"   and f :: "('e, 'f, 'g) prog"   and h' :: "'e" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "pure f h" shows "h = h'"
tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "'c" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "dom \<theta> \<union> load_tmps is = {i. i < t}"
Agree: fixes cfg :: "('p, 'v, 's) configuration" shows "reachable c cfg \<longrightarrow> agreement cfg"
univEq: shows "UNIV = {uu_. \<exists>x y. uu_ = x \<and> x = Abs_fin_string y \<and> y \<in> {s. length s \<le> MAX_STR}}"
flush_all_until_volatile_write_append_unflushed: fixes sb :: "'b memref list"   and i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and p' :: "'b"   and is' :: "instr list"   and sbx :: "'b memref list"   and \<D>' :: "'c"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}"   and "i < length ts"   and "ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', xs, sb @ sbx, \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
Graph_Invariants_axioms: shows "Graph_Invariants B PB QB"
Ord_IN_Ord: fixes H :: "fm set"   and l :: "tm"   and k :: "tm" assumes "H \<turnstile> l IN k"   and "H \<turnstile> OrdP k" shows "H \<turnstile> OrdP l"
Fun_lam: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : hfsetcat.prod a b \<rightarrow> c\<guillemotright>" shows "Fun (hfsetcat.\<Lambda> a b c f) = (\<lambda>x\<^bold>\<in>local.set a. UP (happ (hlam (ide_to_hf a) (ide_to_hf b) (ide_to_hf c) (arr_to_hfun f)) (DOWN x)))"
c: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < length fs" shows "fs ! j $ i = of_int (c i j)"
simps_7: fixes f :: "'f"   and p :: "'e" shows "mark_guards f (Call p) = Call p"
adopt_node_child_in_heap: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and child :: "(_) node_ptr" assumes "h \<turnstile> ok Shadow_DOM.adopt_node document_ptr child" shows "child |\<in>| node_ptr_kinds h"
poly_mapping_eqI_proj: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>k. proj_poly k p = proj_poly k q" shows "p = q"
beukers_integral3_conv_nn_integral: fixes n :: "nat" shows "beukers_integral3 n = enn2real (beukers_nn_integral3 n)"
assms: fixes a :: "'a"   and b :: "'b" assumes "(a, b) \<in> fset cs" shows "f a b = b"
hyps: assumes "\<exists>a. trace c t'' a"   and "prefix t' t''" shows "\<exists>a. trace c t' a"
LPDi_MSkip: fixes i :: "nat"   and n :: "nat" shows "LPDi i (MSkip n) \<subseteq> MSkip ` {i. i \<le> n}"
collapse: fixes stream :: "'c stream" shows "shd stream ## stl stream = stream"