hcju/isabelleps huggingface dataset:
sig_red_regular_lt: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red (\<prec>\<^sub>t) top_tail F p q" shows "lt q = lt p"
iterate_rule_P: fixes S :: "'a dlist"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "ls.invar S"   and "I (ls.\<alpha> S) \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> ls.\<alpha> S; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" shows "P (ls.iterate S f \<sigma>0)"
S6safe: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S6 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> [UNext memCh mm (resbar rmhist) p]_(rtrner memCh!p, resbar rmhist!p)"
quasi_isometry_on_subset: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and Y :: "'c set" assumes "lambda C -quasi_isometry_on X f"   and "Y \<subseteq> X" shows "lambda C -quasi_isometry_on Y f"
is_element_ptr_kind_obtains: fixes node_ptr :: "(_) node_ptr" assumes "is_element_ptr_kind node_ptr" obtains element_ptr :: "(_) element_ptr" where "node_ptr = cast element_ptr"
mtx_vec_scaleR_commute: fixes A :: "'b sq_mtx"   and c :: "real"   and x :: "(real, 'b) vec" shows "A *\<^sub>V (c *\<^sub>R x) = c *\<^sub>R A *\<^sub>V x"
distinct_1: fixes atom_ordering :: "('a \<times> 'a) set"   and x1 :: "'b"   and x2 :: "'b" assumes "propositional_atoms atom_ordering" shows "Pos x1 \<noteq> Neg x2"
assms_5: shows "v \<in> set (precondition_of op)"
dprod_def: fixes r :: "(('a, 'b) node set \<times> ('a, 'b) node set) set"   and s :: "(('a, 'b) node set \<times> ('a, 'b) node set) set" shows "dprod r s \<equiv> \<Union>(x, x')\<in>r. \<Union>(y, y')\<in>s. {(Scons x y, Scons x' y')}"
eventually_at_open_segment: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> {a<--<b}" shows "\<forall>\<^sub>F y in at x. (y - a) \<bullet> rev.rot (a - b) = 0 \<longrightarrow> y \<in> {a<--<b}"
assms_2: shows "p \<cdot> w \<le> w \<cdot> p \<cdot> w"
dvd_factor_mult_gcd: fixes 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 gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "'a"   and p :: "'a"   and q :: "'a"   and r :: "'a" assumes "class.idom_gcd times one plus zero minus uminus gcd"   and "dvd.dvd times k (times p q)"   and "dvd.dvd times k (times p r)"   and "q \<noteq> zero"   and "r \<noteq> zero" shows "dvd.dvd times k (times p (gcd q r))"
Pi_empty: fixes B :: "'e \<Rightarrow> 'f set" shows "Pi {} B = UNIV"
Fundef_3: shows "list_all2 (\<lambda>x y. x = norm_instr y) xs ys"
measure_of_st_vec'_def: fixes x :: "'b st_vec"   and I :: "'b set" shows "measure_of_st_vec' x I = ennreal (sum (($h) (st_vec x)) I)"
varFalse: fixes b :: "BExpr"   and t :: "Var \<Rightarrow> Val"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "\<not> evalB b t"   and "A s t" shows "(b, A, \<Phi>, s, t) \<in> var"
seq_CSP2: fixes Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "Q is CSP2 healthy" shows "P ;; Q is CSP2 healthy"
triangle_right: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_in_normal_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "V (H g \<epsilon>) (V (\<a> g f g) (H \<eta> g)) = g"
assms_2: fixes b :: "'a" assumes "b \<in> T" shows "\<exists>u. (\<forall>n. u n \<in> S) \<and> u \<longlonglongrightarrow> b \<and> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b"
betw_3n: shows "\<forall>n n' n''. n'' < card X \<and> n < n' \<and> n' < n'' \<longrightarrow> ord (f n) (f n') (f n'')"
standard_uniqueness: fixes W :: "'c set set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a"   and C :: "'c set" assumes "ChamberComplexMorphism W X f"   and "ChamberComplexMorphism W X g"   and "SimplicialComplex.maxsimp W C"   and "fun_eq_on f g C"   and "\<And>Cs. SimplicialComplex.min_maxsimpchain W (C # Cs) \<Longrightarrow> pgallery (f \<Turnstile> (C # Cs))"   and "\<And>Cs. SimplicialComplex.min_maxsimpchain W (C # Cs) \<Longrightarrow> pgallery (g \<Turnstile> (C # Cs))" shows "fun_eq_on f g (\<Union> W)"
set_comprehension_list_comprehension: fixes f :: "nat \<Rightarrow> 'a"   and x :: "nat"   and a :: "nat" shows "set (map f [x..<a]) = {f i |i. i \<in> {x..<a}}"
arcs_add_arc: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "arcs (pre_digraph.add_arc G a) = insert a (arcs G)"
Min_def: shows "signed.Min \<equiv> semilattice_set.F signed.min"
assms: fixes inp :: "'index \<Rightarrow> ('index, 'bindex) tree option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex) tree option" assumes "liftAll phi inp"   and "liftAll phi binp" shows "phi (Branch inp binp)"
assms: shows "\<forall>G nP i F. wf_graph G \<and> \<not> sinvar G nP \<and> F \<in> set_offending_flows G nP \<and> sinvar (delete_edges G F) nP \<and> (\<not> receiver_violation \<longrightarrow> i \<in> fst ` F \<longrightarrow> \<not> sinvar G (nP(i := otherbot))) \<and> (receiver_violation \<longrightarrow> i \<in> snd ` F \<longrightarrow> \<not> sinvar G (nP(i := otherbot)))"
iszero_numeral: fixes w :: "num" shows "iszero (numeral w) = (numeral w mod int CARD('a bit0) = 0)"
partitions_insert1: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat" assumes "p partitions n"   and "0 < k" shows "p(k := p k + 1) partitions n + k"
minus_r_inv_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A - A = 0\<^sub>m nr nc"
star_n_def: fixes X :: "nat \<Rightarrow> 'a" shows "star_n X = Abs_star (starrel `` {X})"
mono: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "x \<le> y" shows "f x \<le> f y"
RSD_pareto_eqclass_rsd_winners_aux: fixes agents' :: "'agent list" assumes "finite alts"   and "alts \<noteq> {}"   and "set agents' \<subseteq> agents" shows "RSD_pareto_eqclass (set agents') alts R (rsd_winners R alts agents')"
distinct_93: fixes x31 :: "Syntax.var"   and x32 :: "Syntax.term"   and x17 :: "hash" shows "Lam x31 x32 \<noteq> Hash x17"
mult_monom: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a"   and y :: "nat \<Rightarrow>\<^sub>0 nat"   and b :: "'a" shows "MPoly_Type.monom x a * MPoly_Type.monom y b = MPoly_Type.monom (x + y) (a * b)"
mult_isotone: fixes w :: "'a"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "w \<le> y"   and "x \<le> z" shows "w \<sqinter> x \<le> y \<sqinter> z"
fdr_alt_def: fixes n :: "nat" shows "Fast_Dice_Roll.fdr_alt n \<equiv> curry (spmf.fixp_fun (\<lambda>fdr_alt (v, c). Fast_Dice_Roll.fdr_step n v c \<bind> (\<lambda>(v', c'). if c' < n then return_spmf c' else curry fdr_alt (v' - n) (c' - n))))"
welldefined_rs: shows "fBall rs (\<lambda>(uu_, rhs). consts rhs |\<subseteq>| all_consts)"
smallo_imp_smallomega: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "g \<in> \<omega>[F](f)"
path_split2: fixes v :: "'v"   and v' :: "'v"   and pth1 :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list"   and e :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort"   and pth2 :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" shows "path v v' (pth1 @ e # pth2) = (path v (edge_begin e) pth1 \<and> path (edge_begin e) v' (e # pth2))"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 (\<mu> \<star> \<nu>) \<pi>"
Abs_measure_inverse: fixes y :: "'c set \<times> 'c set set \<times> ('c set \<Rightarrow> ennreal)" assumes "y \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>- A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>}" shows "Rep_measure (Abs_measure y) = y"
dvd_mult_cancel_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "(c * a dvd c * b) = (c = (0::'a) \<or> a dvd b)"
prime_factorization_normalize: fixes x :: "'a" shows "prime_factorization (normalize x) = prime_factorization x"
simps_2: fixes v :: "nat"   and va :: "nat"   and vb :: "'b option"   and vc :: "'b tree"   and vd :: "'b tree" shows "combine Leaf (Node v va vb vc vd) = Node v va vb vc vd"
independent_length_le_dimension: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "dimension n K E"   and "local.independent K Us"   and "set Us \<subseteq> E" shows "length Us \<le> n"
simps_4: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Inverse a) vs = inverse (interpret_floatarith a vs)"
one_plus_numeral_commute: fixes x :: "num" shows "(1::'a) + numeral x = numeral x + (1::'a)"
factor: fixes x :: "'a"   and y :: "'a" shows "comm_semiring_1_class.factor x y = (\<exists>z. z \<noteq> (0::'a) \<and> x * z = y)"
abs_prts: fixes abs :: "'a \<Rightarrow> '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 inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add_abs abs plus zero minus uminus less_eq less inf sup" shows "abs a = minus (lattice_ab_group_add.pprt zero sup a) (lattice_ab_group_add.nprt zero inf a)"
int_xor_comm: fixes x :: "int"   and y :: "int" shows "x XOR y = y XOR x"
assms: shows "\<lparr>es\<rparr> \<leadsto> \<lparr>es'\<rparr>"
add_mono_thms_linordered_field_5: fixes i :: "'e"   and j :: "'e"   and k :: "'e"   and l :: "'e" assumes "i < j \<and> k < l" shows "i + k < j + l"
zero_le_double_add_iff_zero_le_single_add: fixes a :: "'b" shows "((0::'b) \<le> a + a) = ((0::'b) \<le> a)"
integral_mono: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x" shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
two_lists_cases_snoc_Cons': fixes as :: "'a list"   and bs :: "'b list" obtains "as = []" and "bs = []"   | y :: "'b" and ys :: "'b list" where "as = []" and "bs = y__ # ys__"   | xs :: "'a list" and x :: "'a" where "as = xs__ @ [x__]" and "bs = []"   | xs :: "'a list" and x :: "'a" and y :: "'b" and ys :: "'b list" where "as = xs @ [x]" and "bs = y # ys"
lossless1: shows "lossless_gpv \<I>_full \<A>1"
modRule1Characterise: fixes Ps :: "('a, 'b) sequent list"   and C :: "('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "(Ps, C) \<in> p_e R M N"   and "R \<subseteq> modRules2" shows "\<exists>F Fs \<Gamma> \<Delta> ps r. (Ps, C) = extendRule ( M \<cdot> \<Gamma> \<Rightarrow>* N \<cdot> \<Delta>) r \<and> r \<in> R \<and> (r = (ps,  \<Empt> \<Rightarrow>* \<LM> Modal F Fs  \<RM>) \<or> r = (ps,  \<LM> Modal F Fs  \<RM> \<Rightarrow>* \<Empt>))"
P7: shows "Bet Q' Y C' \<and> Cong Y C' Y C"
eq_fin_range: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "set w\<^sub>1 = set w\<^sub>2"
d_increasing_sub_identity: fixes x :: "'a" assumes "x \<le> top" shows "x \<le> d x"
span_explicit: fixes b :: "'d set" shows "cspan b = {\<Sum>a\<in>t. r a *\<^sub>C a |t r. finite t \<and> t \<subseteq> b}"
stc_HFinite: fixes x :: "complex star" assumes "x \<in> HFinite" shows "stc x \<in> HFinite"
eucl_of_list_prod: fixes xs :: "real list" shows "eucl_of_list xs = (eucl_of_list (take DIM('e) xs), eucl_of_list (drop DIM('e) xs))"
l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes \<equiv> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs \<and> l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"
terminated_imp_maxflow: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and L_left :: "nat list"   and L_right :: "nat list" assumes "rtf_invar c s t f l n L_left L_right"   and "L_right = []" shows "Network.isMaxFlow c s t f"
continuous_within_tendsto_compose': fixes a :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "continuous (at a within s) f"   and "\<And>n. x n \<in> s"   and "(x \<longlongrightarrow> a) F" shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
cUn_commute: fixes A :: "'d cset"   and B :: "'d cset" shows "cUn A B = cUn B A"
R0sk1iar_def: shows "R0sk1iar \<equiv> {(s, s'). signals s = signalsResp s'}"
finite_subset_Union: fixes A :: "'a set"   and \<B> :: "'a set set" assumes "finite A"   and "A \<subseteq> \<Union> \<B>" obtains \<F> :: "'a set set" where "finite \<F>" and "\<F> \<subseteq> \<B>" and "A \<subseteq> \<Union> \<F>"
basic_monos_1: fixes A :: "'h set" shows "A \<subseteq> A"
DirProds_iso: fixes f :: "'e \<Rightarrow> 'f"   and I :: "'e set"   and J :: "'f set"   and Gs :: "'e \<Rightarrow> ('g, 'h) monoid_scheme"   and Hs :: "'f \<Rightarrow> ('i, 'j) monoid_scheme" assumes "bij_betw f I J"   and "\<And>i. i \<in> I \<Longrightarrow> Gs i \<cong> Hs (f i)"   and "\<And>i. i \<in> I \<Longrightarrow> Group.group (Gs i)"   and "\<And>j. j \<in> J \<Longrightarrow> Group.group (Hs j)" shows "DirProds Gs I \<cong> DirProds Hs J"
addK_def: shows "addK \<equiv> \<Lambda> x y. bindK\<cdot>x\<cdot>(\<Lambda> xv. bindK\<cdot>y\<cdot>(\<Lambda> yv. returnK\<cdot>(xv + yv)))"
dominates_unsnoc: fixes g :: "'g"   and n :: "'node"   and m :: "'node"   and m' :: "'node" assumes "invar g"   and "dominates g n m"   and "m' \<in> set (predecessors g m)"   and "n \<noteq> m" shows "dominates g n m'"
oldUses_def: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "oldUses g n = var' g ` uses' g n"
assms: fixes x :: "'a"   and z :: "'a" shows "f x (f x z) = f (f x x) z"
convex_hull_exchange_Union: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> convex hull S" shows "convex hull S = (\<Union>b\<in>S. convex hull insert a (S - {b}))"
simps: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" shows "coeff_cube_to_path (k, \<gamma>) = (if k = 1 then \<gamma> else reversepath \<gamma>)"
inv_upto_def: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m :: "nat" shows "local.inv_upto p A m \<equiv> \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> j < m \<longrightarrow> p A i j"
ag_eq_sol1: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and x :: "'e"   and b :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "x \<in> carrier A"   and "b \<in> carrier A"   and "a \<plusminus>\<^bsub>A\<^esub> x = b" shows "x = -\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> b"
Partial_order_restrict_rel: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" assumes "Partial_order R" shows "Partial_order (restrict_rel A R)"
standard_cone_decomp_snd_split_3: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "F \<subseteq> P[X]"   and "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "g \<in> punit.reduced_GB F" shows "poly_deg g \<le> Suc (Max (poly_deg ` fst ` set (snd (split 0 X (lpp ` punit.reduced_GB F)))))"
extended_simps_1: fixes x :: "'a extended" shows "(\<turnstile> < x) = (x \<noteq> \<turnstile>)"
Max_wrt_of_weak_ranking_Cons: fixes x :: "'b set"   and xs :: "'b set list" assumes "is_weak_ranking (x # xs)" shows "Max_wrt (of_weak_ranking (x # xs)) = x"
axioms_1: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" assumes "retract G" shows "galois_connection G"
trivial_normal_series: shows "normal_series (G\<lparr>carrier := H\<rparr>) [{\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>}, carrier (G\<lparr>carrier := H\<rparr>)]"
integrable_on_little_subintervals: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<forall>x\<in>cbox a b. \<exists>d>0. \<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow> f integrable_on cbox u v" shows "f integrable_on cbox a b"
suminf_binaryset_eq: fixes f :: "'c set \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" assumes "f {} = (0::'b)" shows "(\<Sum>n. f (binaryset A B n)) = f A + f B"
fresh_at_base_permI: fixes a :: "'a"   and p :: "perm" assumes "atom a \<sharp> p" shows "p \<bullet> a = a"
bind_returns_result_E: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and y :: "'c" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y" obtains x :: "'d" and h' :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>r y"
n_def: shows "n = nat \<lceil>log c K\<rceil>"
index_mat_four_block_3: fixes A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat"   and D :: "'b mat" shows "dim_col (four_block_mat A B C D) = dim_col A + dim_col D"
hd_dropWhile: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" assumes "dropWhile P xs \<noteq> []" shows "\<not> P (hd (dropWhile P xs))"
dehomogenize_ideal_subset: fixes x :: "'b"   and F :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "dehomogenize x ` ideal F \<subseteq> ideal (dehomogenize x ` F)"
intro: fixes s :: "nat \<Rightarrow> nat option" assumes "s \<in> \<R>" shows "total_cons s"
hyps_2: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> x"   and "\<And>n. \<lbrakk>n \<in>\<^sub>\<circ> \<omega>; P n\<rbrakk> \<Longrightarrow> P (ZFC_in_HOL.succ n)" shows "P y"
g: fixes r :: "nat" shows "eval g [n, r, b, j, v] \<down>= e_snoc r (if n = j then v else e_nth b n)"
Z: fixes w :: "int"   and bl :: "bool list" shows "bin_to_bl_aux 0 w bl = bl"
A: shows "t_find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (p\<^sub>2 # ps) \<le> 8"
dw_closed_rtrancl: fixes A :: "'a set"   and a :: "'a"   and b :: "'a" assumes "dw_closed A"   and "a \<in> A"   and "a \<rightarrow>\<^sup>* b" shows "b \<in> A"
inj_on_Int: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj_on f A \<or> inj_on f B" shows "inj_on f (A \<inter> B)"
oalist_inv_rawD2: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw x xs" shows "sorted_wrt (lt x) (map fst xs)"
RH1: shows "relH {a. a < lim h \<and> a \<notin> as1} h (the_state \<sigma>)"
hyps_2: shows "P \<turnstile> \<langle>e_,s\<^sub>0_\<rangle> \<rightarrow>* \<langle>throw e'_,s\<^sub>1_\<rangle>"
simps_1: fixes sr :: "'e \<Rightarrow> 'c \<times> 'd"   and i :: "nat"   and s :: "'e"   and R :: "nat set" shows "netgmap sr (NodeS i s R) = ([i \<mapsto> fst (sr s)], NodeS i (snd (sr s)) R)"
dsr2: fixes x :: "'a"   and y :: "'a" shows "ad (ad (x \<cdot> ad (ad y))) = ad (ad (x \<cdot> y))"
cost_nonneg: fixes w :: "nat \<Rightarrow> real"   and m :: "nat"   and S :: "nat \<Rightarrow> 'b set"   and R :: "'b set"   and i :: "nat" assumes "Set_Cover w m S" shows "0 \<le> Set_Cover.cost w S R i"
simps: fixes i :: "'a"   and j :: "nat" shows "is_valid_state_id (i, j) = (j \<le> length (events i))"
sup_eq_bot_iff: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<oplus> y = mbot) = (x = mbot \<and> y = mbot)"
assms_5: shows "Ana_invar_subst (ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A)"
assms_1: shows "\<I> \<turnstile>g gpv \<surd>"
assms_5: shows "outs_gpv \<I> gpv \<subseteq> A"
singletonize_L4Ports_dst: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and pts :: "ipt_l4_ports"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "match_list (\<beta>, \<alpha>) (map (Match \<circ> Dst_Ports) (singletonize_L4Ports pts)) a p = matches (\<beta>, \<alpha>) (Match (Dst_Ports pts)) a p"
Infty_neq_0_1: shows "\<infinity> \<noteq> 0"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<squnion> (a \<squnion> c) = a \<squnion> (b \<squnion> c)"
assms_1: shows "vs @ xs \<in> C M2 M1 \<Omega> V m i"
bang_eq: fixes x :: "'b word"   and y :: "'b word" shows "(x = y) = (\<forall>n. x !! n = y !! n)"
inf1: fixes A :: "'e"   and k :: "'e"   and a :: "'e"   and b :: "'e" assumes "A \<equiv> inf k a" shows "inf A b \<equiv> inf k (inf a b)"
less_is_non_zero_p1: fixes a :: "'b word"   and k :: "'b word" assumes "a < k" shows "a + 1 \<noteq> 0"
units_of_def: fixes G :: "('a, 'b) monoid_scheme" shows "units_of G = \<lparr>carrier = Units G, monoid.mult = (\<otimes>\<^bsub>G\<^esub>), one = \<one>\<^bsub>G\<^esub>\<rparr>"
restrict_relation_carrier: shows "restrict_relation carrier (of_weak_ranking (weak_ranking_aux carrier)) = of_weak_ranking (weak_ranking_aux carrier)"
convert_None: fixes t1 :: "t"   and v :: "v"   and t2 :: "t"   and sx :: "sx option" assumes "types_agree t1 v"   and "cvt t2 sx v = None" shows "\<lparr>[$C v, $Cvtop t2 Convert t1 sx]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
Cons_3: shows "v \<in> set (Koszul_syz_sigs_aux (b # bs) k)"