hcju/isabelleps huggingface dataset:
monom_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat" assumes "ring R" shows "ring.monom R a n = a # replicate n \<zero>\<^bsub>R\<^esub>"
matrix_vector_mult_diff_distrib: fixes A :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec"   and y :: "('a, 'c) vec" shows "A *v (x - y) = A *v x - A *v y"
lemma_straddle: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and e :: "real" assumes "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (f has_real_derivative f' x) (at x)"   and "0 < e" shows "\<exists>g. gauge {a..b} g \<and> (\<forall>x u v. a \<le> u \<and> u \<le> x \<and> x \<le> v \<and> v \<le> b \<and> v - u < g x \<longrightarrow> \<bar>f v - f u - f' x * (v - u)\<bar> \<le> e * (v - u))"
axiom_safe: fixes a :: "axiom" shows "fsafe (get_axiom a)"
synIfSC_def: fixes e :: "('c, 'd) Expr"   and c1 :: "(('c, 'd) Expr, 'c) MWLsCom"   and c2 :: "(('c, 'd) Expr, 'c) MWLsCom" shows "synIfSC e c1 c2 \<equiv> \<exists>d. {} \<turnstile>\<^bsub>\<E>\<^esub> e : d \<and> (\<forall>d'. d \<le> d')"
fvi_plus_bound: fixes b :: "nat"   and c :: "nat"   and \<phi> :: "Formula.formula"   and n :: "nat" assumes "\<forall>i\<in>Formula.fvi (b + c) \<phi>. i < n" shows "\<forall>i\<in>Formula.fvi b \<phi>. i < c + n"
tendsto_compose_at: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and F :: "'a filter"   and g :: "'b \<Rightarrow> 'c"   and z :: "'c" assumes "(f \<longlongrightarrow> y) F"   and "g \<midarrow>y\<rightarrow> z"   and "\<forall>\<^sub>F w in F. f w = y \<longrightarrow> g y = z" shows "((g \<circ> f) \<longlongrightarrow> z) F"
min_single_1: fixes x :: "('c, 'd) bintree option"   and a :: "'c" assumes "Binomial_Queue.min [x] = Some a" shows "priority (the x) = a"
degree_to_poly: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "deg Zp (to_polynomial Zp a) = 0"
akra_bazzi_measure_def: fixes x :: "real" shows "akra_bazzi_measure x = nat \<lceil>x\<rceil>"
ps_def: shows "ps = perp_up (Rep_hyp2 c) l"
nth_sorted_list_of_set_greaterThanLessThan: fixes n :: "nat"   and j :: "nat"   and i :: "nat" assumes "n < j - Suc i" shows "list_of {i<..<j} ! n = Suc (i + n)"
rel_frontier_cball: fixes a :: "'a"   and r :: "real" shows "rel_frontier (cball a r) = (if r = 0 then {} else sphere a r)"
intros_2: fixes xps :: "('e \<times> 'f) list"   and uu :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and s :: "'g"   and x :: "'e" assumes "(xps, uu) \<in> c_failures step out s\<^sub>0"   and "s = foldl step s\<^sub>0 (map fst xps)" shows "(xps @ [(x, out s x)], {(y, p). p \<noteq> out (step s x) y}) \<in> c_failures step out s\<^sub>0"
path_swap_ret: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "is_path \<pi>" obtains \<pi>' :: "nat \<Rightarrow> 'n" and n :: "nat" where "is_path \<pi>'" and "\<pi> =\<^bsub>k\<^esub> \<pi>'" and "\<pi>' n = local.return"
wb_lens_weak: fixes x :: "'a \<Longrightarrow> 'b" assumes "wb_lens x" shows "weak_lens x"
open_real: fixes s :: "real set" shows "open s = (\<forall>x\<in>s. \<exists>e>0. \<forall>x'. \<bar>x' - x\<bar> < e \<longrightarrow> x' \<in> s)"
restricted_subformulas_size: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> restricted_subformulas \<phi>" shows "size \<psi> < size \<phi>"
hyps: fixes i :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', \<theta>', sb, \<D>', \<O>', \<R>')]) m = flush_all_until_volatile_write ts m"
gr_le: fixes A :: "'b set set"   and f :: "'b \<Rightarrow> real" assumes "sigma_algebra A"   and "\<forall>a. {w. a < f w} \<in> A" shows "\<forall>a. {w. f w \<le> a} \<in> A"
x_hyp: fixes i :: "'n" shows "\<parallel>x $ i\<parallel> \<le> 1"
Ueq': shows "zs' = concat [c', a', d', b'] @ interact as' bs'"
subgraph_strongly_connected_iso: fixes H :: "('a, 'b) pre_digraph" assumes "subgraph H G" shows "strongly_connected (app_iso hom H) = strongly_connected H"
exhaust: fixes y :: "'c extended" obtains "y = \<turnstile>"   | x2 :: "'c" where "y = \<lbrakk>x2\<rbrakk>"   | "y = \<stileturn>"
eq0_iff: fixes x :: "V" shows "(x = 0) = (\<forall>y. y \<notin>\<^sub>\<circ> x)"
chop_concat: fixes P :: "'b list \<Rightarrow> bool"   and splitf :: "'b list \<Rightarrow> 'b list \<times> 'b list"   and xs :: "'b list"   and yss :: "'b list list"   and zs :: "'b list" assumes "is_maxsplitter P splitf"   and "chop splitf xs = (yss, zs)" shows "xs = concat yss @ zs"
\<iota>\<^sub>0_G_Inf: shows "\<iota>\<^sub>0 \<in> G_Inf M"
restrict_on_source: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set" assumes "map f S T" shows "restrict f S = f"
get_scdom_component_def: fixes ptr :: "(_) object_ptr" shows "get_scdom_component ptr = Heap_Error_Monad.bind (get_owner_document ptr) (\<lambda>document. Heap_Error_Monad.bind (get_disconnected_nodes document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (to_tree_order (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document)) (\<lambda>tree_order. Heap_Error_Monad.bind (map_M (to_tree_order \<circ> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r) disc_nodes) (\<lambda>disconnected_tree_orders. return (tree_order @ concat disconnected_tree_orders)))))"
strict_iff_order: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a \<le> b \<and> a \<noteq> b)"
Mem: fixes t :: "tm"   and u :: "tm"   and i :: "name"   and x :: "tm" shows "(t IN u)(i::=x) = subst i x t IN subst i x u"
linorder_on_def: fixes A :: "'b set"   and R :: "('b \<times> 'b) set" shows "linorder_on A R = (refl_on A R \<and> antisym R \<and> trans R \<and> total_on A R)"
assms_1: fixes K :: "'a list set"   and L :: "'a list set" assumes "K \<sim> L" shows "K \<subseteq> lists S \<and> L \<subseteq> lists S"
empty_exposed_face_of: fixes S :: "'b set" shows "{} exposed_face_of S"
surjective: fixes r :: "('a, 'b, 'c, 'd) Measurement_System_scheme" shows "r = \<lparr>mag = mag r, dim = Quantity.dim r, unit_sys = unit_sys r, \<dots> = Measurement_System.more r\<rparr>"
pmf_of_st_vec_inj: fixes x :: "'a st_vec"   and y :: "'a st_vec" shows "(pmf_of_st_vec x = pmf_of_st_vec y) = (x = y)"
a0: fixes x :: "'a" assumes "x \<in> S'" shows "a x = 0"
derivation_const_eq_zero: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a"   and c :: "real" assumes "X \<in> sub.tangent_space p"   and "p \<in> sub.carrier" shows "X (restrict0 sub.carrier (\<lambda>x. c)) = 0"
ground\<^sub>l_subs: fixes l :: "fterm literal"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "ground\<^sub>l l" shows "l \<cdot>\<^sub>l \<sigma> = l"
mcont_monoD: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "mcont luba orda lubb ordb f"   and "orda x y" shows "ordb (f x) (f y)"
algebraic_int_altdef_ipoly: fixes x :: "'b" shows "algebraic_int x = (\<exists>p. ipoly p x = (0::'b) \<and> monic p)"
lalternate_llength: fixes xs :: "'b llist" shows "llength (lalternate xs) * 2 = 1 + llength xs \<or> llength (lalternate xs) * 2 = llength xs"
sum_union: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "\<Sum> (A \<union> B) = \<Sum> A + \<Sum> B"
const_smallo_inverse_powr: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and p :: "real" assumes "filterlim f at_top at_top" shows "((\<lambda>_. c) \<in> o(\<lambda>x. inverse (f x powr p))) = (0 \<le> p \<longrightarrow> c = 0)"
semigroup_axioms: shows "semigroup (\<squnion>1)"
set_eqE2: fixes R :: "('a, 'b) eq_object_scheme"   and A :: "'a set"   and B :: "'a set" assumes "A {.=}\<^bsub>R\<^esub> B" obtains "\<forall>a\<in>A. \<exists>b\<in>B. a .=\<^bsub>R\<^esub> b" and "\<forall>b\<in>B. \<exists>a\<in>A. b .=\<^bsub>R\<^esub> a"
ListSum_compl2: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and f :: "'a \<Rightarrow> nat" shows "ListSum (filter P xs) f + ListSum (filter (\<lambda>x. \<not> P x) xs) f = ListSum xs f"
fds_shift_mult: fixes c :: "'b"   and f :: "'b fds"   and g :: "'b fds" shows "fds_shift c (f * g) = fds_shift c f * fds_shift c g"
iso_map\<^sub>1_ide: fixes f :: "'a" assumes "C.ide f" shows "D.iso (\<sigma>.map\<^sub>1 f)"
pre_subdivFace'_Some1': fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and v :: "nat"   and n :: "nat"   and u :: "nat"   and vol :: "nat option list"   and ws :: "nat list"   and f21 :: "face"   and g' :: "graph" assumes "pre_subdivFace' g f v' v n (Some u # vol)"   and "pre_splitFace g v u f ws"   and "f21 = fst (snd (splitFace g v u f ws))"   and "g' = snd (snd (splitFace g v u f ws))" shows "pre_subdivFace' g' f21 v' u 0 vol"
iadd_0_enat: fixes a :: "enat" shows "enat 0 + a = a"
sign_simps_1: fixes ad :: "'a"   and bd :: "'a"   and cd :: "'a" shows "ad + bd + cd = ad + (bd + cd)"
valid_node_Entry_path: fixes prog :: "cmd"   and n :: "w_node" assumes "valid_node prog n" shows "\<exists>as. While_CFG.path prog (_Entry_) as n"
elims_2: fixes x :: "('a, 'b) message" assumes "is_delete x" obtains m :: "'a delete_message" where "x = Delete m"
bernoulli'_fps_altdef: shows "bernoulli'_fps = Abs_fps (\<lambda>n. of_real (bernoulli' n) / fact n)"
finite_insert: fixes a :: "'b"   and A :: "'b set" shows "finite (insert a A) = finite A"
rename_def: fixes h :: "'c \<Rightarrow> 'd" shows "rename h \<equiv> extend (\<lambda>(x, u). h x)"
rel_funD: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and x :: "'d"   and y :: "'e" assumes "(A ===> B) f g"   and "A x y" shows "B (f x) (g y)"
sheaf_spec_on_open_is_comm_ring: fixes U :: "nat set set" assumes "cring0.is_zariski_open U" shows "comm_ring (cring0.sheaf_spec U) (cring0.add_sheaf_spec U) (cring0.mult_sheaf_spec U) (cring0.zero_sheaf_spec U) (cring0.one_sheaf_spec U)"
poly_zero: fixes p :: "'a poly"   and x :: "'a" assumes "poly p x = (0::'a)" shows "(p = 0) = (Polynomial.degree p = 0)"
bot_entails_all: fixes B :: "'a"   and N1 :: "'a set" assumes "B \<in> Bot" shows "{B} \<Turnstile> N1"
restrict_map_UNIV: fixes m :: "'c \<Rightarrow> 'd option" shows "m |` UNIV = m"
noabort: shows "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` F"
bind_gpv_parametric: fixes A :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun (rel_gpv A C) (rel_fun (rel_fun A (rel_gpv B C)) (rel_gpv B C)) (\<bind>) (\<bind>)"
Inf_between_alt: fixes Inf :: "'a inference set"   and N :: "'a set"   and M :: "'a set" shows "inference_system.Inf_between Inf N M = {\<iota> \<in> Inf. \<iota> \<in> inference_system.Inf_from Inf (N \<union> M) \<and> set (prems_of \<iota>) \<inter> M \<noteq> {}}"
in_renamed_in_unlabelled_val_2: fixes \<beta> :: "'d \<Rightarrow> 'd"   and l :: "'d"   and \<alpha> :: "'e \<Rightarrow> 'e"   and v :: "('e, 'd, 'f) val" assumes "bij \<beta>" shows "(\<beta> l \<in> LID\<^sub>V (\<R>\<^sub>V \<alpha> \<beta> v)) = (l \<in> LID\<^sub>V v)"
EM_remainder_cong: fixes a :: "int"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat"   and a' :: "int" assumes "\<And>x. real_of_int a \<le> x \<Longrightarrow> f x = g x"   and "n = n'"   and "a = a'" shows "EM_remainder n f a = EM_remainder n' g a'"
span_eq: fixes Sa :: "'a set"   and T :: "'a set" assumes "Sa \<subseteq> S"   and "T \<subseteq> S" shows "(span Sa = span T) = (Sa \<subseteq> span T \<and> T \<subseteq> span Sa)"
smult_r_distr: fixes a :: "'a"   and x :: "'e"   and y :: "'e" assumes "a \<in> carrier R"   and "x \<in> carrier M2"   and "y \<in> carrier M2" shows "a \<odot>\<^bsub>M2\<^esub> (x \<oplus>\<^bsub>M2\<^esub> y) = a \<odot>\<^bsub>M2\<^esub> x \<oplus>\<^bsub>M2\<^esub> a \<odot>\<^bsub>M2\<^esub> y"
crel_vs1_executeD: fixes P :: "heap \<Rightarrow> bool"   and update :: "'g \<Rightarrow> 'h \<Rightarrow> unit Heap"   and lookup :: "'g \<Rightarrow> 'h option Heap"   and dp :: "'g \<Rightarrow> 'h"   and R :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and a :: "'i"   and b :: "'j Heap"   and heap :: "heap" assumes "dp_consistency_heap P update lookup"   and "dp_consistency_heap.crel_vs1 P lookup dp R a b"   and "P heap"   and "dp_consistency.cmem (heap_mem_defs.lookup' lookup) dp heap" obtains x :: "'j" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "dp_consistency.cmem (heap_mem_defs.lookup' lookup) dp heap'" and "R a x"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case LT of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f1"
rref_condition4_explicit: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'d" assumes "reduced_row_echelon_form A"   and "\<not> is_zero_row i A"   and "i \<noteq> j" shows "A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = (0::'b)"
sum_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and r :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.semiring_0 plus zero times" shows "times r (comm_monoid_add.sum plus zero f A) = comm_monoid_add.sum plus zero (\<lambda>n. times r (f n)) A"
path_obs_subset: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b"   and S :: "'b set" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "\<forall>n'\<in>set (CFG.sourcenodes sourcenode as). n' \<notin> S" shows "CFG.obs sourcenode targetnode valid_edge n' S \<subseteq> CFG.obs sourcenode targetnode valid_edge n S"
assms_2: shows "\<exists>k<e_length e. eval r_psi [i, k] \<up>"
b: fixes x :: "'a"   and y :: "'a"   and init :: "'a list"   and qs :: "'a list" assumes "x \<noteq> y"   and "init \<in> {[x, y], [y, x]}"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])" shows "T\<^sub>p_on_rand' BIT (type0 init x y) qs = 15 / 10"
CEApp_no_restr: fixes e :: "exp"   and x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and r :: "C" shows "(\<N>\<lbrakk> App e x \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r \<sqsubseteq> ((\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r \<down>CFn \<rho> x)\<cdot>r"
closed_Collect_conj: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.topological_space open"   and "topological_space.closed open {x. P x}"   and "topological_space.closed open {x. Q x}" shows "topological_space.closed open {x. P x \<and> Q x}"
ma_of_rat: fixes r :: "rat" shows "real_of_rat r = real_of (ma_of_rat r)"
orth_at_symmetry: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and V :: "'p" assumes "X OrthAt A B C U V" shows "X OrthAt A B C V U"
str_del_point_blocks_def: fixes p :: "'a" shows "des.str_del_point_blocks p \<equiv> filter_mset ((\<notin>) p) (str_del_point_blocks p)"
PrfP_imp_OrdP: fixes s :: "tm"   and k :: "tm"   and t :: "tm" shows "{PrfP s k t} \<turnstile> OrdP k"
Un: fixes a :: "'a set"   and A :: "'a set set"   and t :: "'a set" assumes "a \<in> A"   and "t \<in> tiling A"   and "a \<subseteq> - t" shows "a \<union> t \<in> tiling A"
nodouble_def: fixes x :: "'a"   and y :: "'a" shows "nodouble x y = Plus (seq [question (Atom x), Star (Times (Atom y) (Atom x)), Atom y]) (seq [question (Atom y), Star (Times (Atom x) (Atom y)), Atom x])"
simps_1: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'b" shows "shd (sscan f xs a) = f (shd xs) a"
le_cInf_iff: fixes S :: "'b multiset set"   and a :: "'b multiset" assumes "S \<noteq> {}"   and "subset_mset.bdd_below S" shows "(a \<subseteq># Inf S) = (\<forall>x\<in>S. a \<subseteq># x)"
space_closed: fixes \<Omega> :: "'b set"   and M :: "'b set set" assumes "subset_class \<Omega> M" shows "M \<subseteq> Pow \<Omega>"
converseI: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
Sup1: fixes x :: "'b Assertion"   and A :: "'b Assertion set" assumes "x \<in> A" shows "x \<le> \<Squnion>A"
normalize_idem: fixes a :: "'a"   and b :: "'a" shows "normalize (gcd a b) = gcd a b"
measurable_envelope_def: fixes M :: "'b measure"   and A :: "'b set"   and E :: "'b set" shows "measurable_envelope M A E = (A \<subseteq> E \<and> E \<in> sets M \<and> (\<forall>F\<in>sets M. emeasure M (F \<inter> E) = outer_measure_of M (F \<inter> A)))"
Inf_action: fixes S :: "('a, 'b) action set" shows "Inf S \<equiv> if S = {} then top else action_of (\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> action.relation_of ` S})"
Above_Field: fixes x :: "'d"   and r :: "('d \<times> 'd) set"   and X :: "'d set" assumes "x \<in> Above r X" shows "x \<in> Field r"
f: shows "(\<lambda>(x, y). f x y) \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
\<pi>_inverse_exists: fixes m :: "nat" shows "\<exists>n. primes_pi (real n) = real m"
hom_zero: shows "map_poly to_fract 0 = 0"
hseqI: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and a :: "'c"   and b :: "'c"   and \<nu> :: "'c"   and c :: "'c" assumes "horizontal_composition V H src trg"   and "horizontal_homs.in_hhom V src trg \<mu> a b"   and "horizontal_homs.in_hhom V src trg \<nu> b c" shows "partial_magma.arr V (H \<nu> \<mu>)"
prefix_iff: fixes xs :: "'a list"   and zs :: "'a list" shows "(xs \<le> zs) = (\<exists>ys. zs = xs @ ys)"
hoare_mono: fixes S :: "'a \<Rightarrow> 'b"   and Q :: "'a"   and R :: "'a"   and P :: "'b" assumes "mono S"   and "Q \<sqsubseteq> R"   and "\<Turnstile> P{| S |}Q" shows "\<Turnstile> P{| S |}R"
right_neutral: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and a :: "'e" assumes "monoid f z" shows "f a z = a"
GrpE: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" assumes "Grp A f x y" obtains "f x = y" and "x \<in> A"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b rexp"   and b :: "'c rexp" assumes "rel_rexp R a b" obtains "a = Zero" and "b = Zero"   | "a = One" and "b = One"   | x :: "'b" and y :: "'c" where "a = Atom x" and "b = Atom y" and "R x y"   | x1 :: "'b rexp" and x2 :: "'b rexp" and y1 :: "'c rexp" and y2 :: "'c rexp" where "a = Plus x1 x2" and "b = Plus y1 y2" and "rel_rexp R x1 y1" and "rel_rexp R x2 y2"   | x1a :: "'b rexp" and x2a :: "'b rexp" and y1a :: "'c rexp" and y2a :: "'c rexp" where "a = Times x1a x2a" and "b = Times y1a y2a" and "rel_rexp R x1a y1a" and "rel_rexp R x2a y2a"   | xa :: "'b rexp" and ya :: "'c rexp" where "a = Star xa" and "b = Star ya" and "rel_rexp R xa ya"
LeftDerivationIntrosAt_early: fixes index :: "nat" assumes "1 \<le> index \<and> index + 1 < length L" shows "LeftDerivationIntrosAt \<alpha> E L' index"
insert_Diff_single: fixes a :: "'b"   and A :: "'b set" shows "(a::'b) \<triangleright> (A::'b set) - {a} = a \<triangleright> A"
chainE_length: fixes w :: "nat \<Rightarrow> 'a list"   and k :: "nat" assumes "chain w" obtains l :: "nat" where "k < length (w l)"
rt_fresh_asD1: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
cnf_form_of_is: fixes F :: "'b formula" assumes "is_nnf F" shows "is_cnf (cnf_form_of F)"
assms_5: shows "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"
tt_eqI: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p"   and "\<And>u. u \<in> keys p \<Longrightarrow> v \<preceq>\<^sub>t u" shows "tt p = v"
A: shows "(G_impl, G) \<in> igbg_impl_rel_ext Re node_rel"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "\<pi> f = C.null"
less_option_def: fixes x :: "'a option"   and y :: "'a option" shows "(x < y) = (case y of None \<Rightarrow> False | Some y \<Rightarrow> case x of None \<Rightarrow> True | Some x \<Rightarrow> x < y)"
ExI: fixes A :: "fm"   and i :: "name" assumes "ss_fm A" shows "ss_fm (SyntaxN.Ex i A)"
ContractsWithIRC_axioms: shows "ContractsWithIRC Xd Xh Pd Ch"
poly_induct6: fixes p :: "nat \<Rightarrow> 'a"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "p \<in> carrier P"   and "\<And>a n. a \<in> carrier R \<Longrightarrow> Q (monom P a 0)"   and "\<And>a n p. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>; p \<in> carrier P; deg R p < n; Q p\<rbrakk> \<Longrightarrow> Q (p \<oplus>\<^bsub>P\<^esub> monom P a n)" shows "Q p"
poss1: shows "squash (rpoly q x1) = - 1 \<and> squash (rpoly q x2) = 1 \<longrightarrow> (\<exists>r<x1. x2 \<le> r \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
fnt_carrier_fnt_nbt: shows "\<forall>x\<in>carrier. finite (no_better_than x carrier relation)"
IH: shows "higher_differentiable_on {..<0} f k \<and> (\<forall>t<0. nth_derivative k f t 1 = 0)"
\<beta>t: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau>' t $ Fv x \<tau>') (subst_bv (Fv x \<tau>') (bind_fv (x, \<tau>') t))"
nn_integral_divide: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x / c \<partial>M = integral\<^sup>N M f / c"
assms_1: shows "q \<in> dterms \<Gamma> p"
terminating_path_root_end_connected: fixes r :: "'a"   and x :: "'a"   and e :: "'a" assumes "terminating_path_root_end r x e" shows "x ; 1 \<le> x\<^sup>+ ; e"
integral_std_normal_moment_odd: fixes k :: "nat" shows "LBINT x. std_normal_density x * x ^ (2 * k + 1) = 0"
cmod_mult: fixes a :: "complex"   and b :: "complex" shows "cmod (a * b) = cmod a * cmod b"
smult_closed: fixes H :: "'h set"   and R :: "('i, 'j) ring_scheme"   and M :: "('i, 'h, 'k) module_scheme"   and a :: "'i"   and x :: "'h" assumes "Module.submodule H R M"   and "a \<in> carrier R"   and "x \<in> H" shows "a \<odot>\<^bsub>M\<^esub> x \<in> H"
expand_set: fixes g :: "'d \<Rightarrow> 'c" shows "Sum_any g = sum g {a. g a \<noteq> (0::'c)}"
unrest_closure: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred" shows "x \<sharp> [P]\<^sub>u"
set_mult_inclusion: fixes H :: "'a set"   and P :: "'a set" assumes "subgroup H H'"   and "P \<subseteq> carrier H'"   and "H <#>\<^bsub>H'\<^esub> P \<subseteq> H" shows "P \<subseteq> H"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
lift_streams_def: fixes xs :: "'d stream list" shows "lift_streams xs = foldr (smap2 (#)) xs (pure [])"
Mp_conv: fixes x :: "int poly" shows "q.Mp x =m x"
assms_1: fixes p :: "'a" assumes "p \<in># A" shows "prime p"
abs_summable_on_empty: fixes f :: "'c \<Rightarrow> 'd" shows "f abs_summable_on {}"
rem_first: shows "arr_mset arr (i + 1) r = arr_mset arr i r - {#arr i#}"
pfsubst_box_simp: fixes p :: "('a, 'b + 'd, 'c) hp"   and q :: "('a, 'b + 'd, 'c) formula"   and \<sigma> :: "'d \<Rightarrow> ('a, 'b, 'c) formula" shows "PFsubst ([[p]]q) \<sigma> = ([[PPsubst p \<sigma>]]PFsubst q \<sigma>)"
cong: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i"   and x :: "'h"   and y :: "'h" assumes "f = g"   and "x = y" shows "f x = g y"