hcju/isabelleps huggingface dataset:
BinOp: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp"   and v\<^sub>2 :: "val"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and bop :: "bop"   and v :: "val" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>"   and "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>"
rev_predicate1D: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "P x"   and "P \<le> Q" shows "Q x"
bounded_linear_linepath': fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "bounded_linear f" shows "f \<circ> linepath a b = linepath (f a) (f b)"
simps_1: fixes \<A> :: "'c ltl set" shows "\<A> \<Turnstile>\<^sub>P true = True"
lex_square_def: shows "lex_square \<equiv> less_than <*lex*> Abacus.lex_triple"
sorted_nprt_spvec: fixes v :: "(nat \<times> 'a) list" assumes "sorted_spvec v" shows "sorted_spvec (nprt_spvec v)"
divisors_pos_fun_def: fixes df :: "'a \<Rightarrow> 'a list" shows "divisors_pos_fun df \<equiv> (\<forall>x. x \<noteq> (0::'a) \<longrightarrow> set (df x) = {d. d dvd x \<and> (0::'a) < d}) \<and> (\<forall>x. distinct (df x))"
map_def: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set" shows "map \<alpha> S T \<equiv> \<alpha> \<in> S \<rightarrow>\<^sub>E T"
intros_2: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" assumes "P \<turnstile> C \<preceq>\<^sup>* D" shows "P \<turnstile> Class C \<le> Class D"
preserves_reflects_arr: fixes f :: "'j" shows "C.arr (A.map f) = J.arr f"
HOLCF_trans_rules_1: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
llength_ltake: fixes n :: "enat"   and xs :: "'a llist" shows "llength (ltake n xs) = min n (llength xs)"
plus_monotone_strict: fixes s :: "'a"   and t :: "'a"   and u :: "'a" assumes "s \<prec> t" shows "s + u \<prec> t + u"
hom_mult_eq_zero: fixes x :: "'a poly"   and y :: "'a poly" assumes "x * y = 0" shows "coeff x 0 * coeff y 0 = (0::'a)"
cod_char: fixes f :: "'c \<times> 'c" shows "C.VxV.cod f = (if C.arr (fst f) \<and> C.arr (snd f) then (C.cod (fst f), C.cod (snd f)) else (C.null, C.null))"
HAStates_def: fixes HA :: "('d, 'e, 'f) hierauto" shows "HAStates HA = \<Union> (States ` SAs HA)"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "hom.in_hom f a b = (hom.arr a \<and> hom.arr b \<and> hom.arr f \<and> \<guillemotleft>f : a \<Rightarrow> b\<guillemotright>)"
xml_of_term_def: fixes uu :: "term" shows "xml_of_term uu = [XML_Tree]"
JVM_typesafe_axioms: shows "JVM_typesafe addr2thread_id thread_id2addr empty_heap allocate typeof_addr heap_read heap_write hconf P"
dropWhile_not_vol_write_empty: fixes sb :: "'b memref list" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb = []"
self_factor_iso: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "(\<lambda>X. the_elem ((\<lambda>x. \<one>\<^bsub>G\<^esub>) ` X)) \<in> Group.iso (G Mod carrier G) (G\<lparr>carrier := {\<one>\<^bsub>G\<^esub>}\<rparr>)"
surjective: fixes r :: "('c, 'd) Graph_scheme" shows "r = \<lparr>verts = V\<^bsub>r\<^esub>, arcs = E\<^bsub>r\<^esub>, \<dots> = more r\<rparr>"
gbm_from_list_impl: shows "graph_from_list gbm_\<alpha> gbm_invar gbm_from_list"
IF2map_simps: fixes f :: "'e \<Rightarrow> 'd"   and v :: "('e, 'e IF1, 'e IF2) F2" shows "IF2map f (ctor2 v) = ctor2 (F2map f (IF1map f) (IF2map f) v)"
simps: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option"   and \<tau> :: "'d \<Rightarrow> ('e, 'd, 'f) val option"   and e :: "('e, 'd, 'f) expr" shows "(\<sigma>, \<tau>, e)\<^sub>\<sigma> = \<sigma>"
CIModel: fixes T :: "(ZF, ZF) Axioms" assumes "ZFAxioms T" shows "Model (CI T) T"
Cons_1: fixes t1 :: "'a event list"   and t2 :: "'a event list"   and u :: "'a event list" assumes "s setinterleaves ((t1 @ t2, u), C)" shows "set t2 \<subseteq> set (drop (length t1) s)"
f5: fixes r :: "real"   and ra :: "real" shows "r + ra = ra + r"
mk_poincare_line_cmat_hermitean: fixes A :: "real"   and B :: "complex" shows "hermitean (mk_poincare_line_cmat A B)"
steps_plusI: fixes p :: "bool list"   and q :: "bool list"   and A :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" assumes "(p, q) \<in> steps A w" shows "(p, q) \<in> steps (RegExp2NA.plus A) w"
assms_1: fixes n :: "nat" assumes "n \<le> m" shows "psi n \<le> c * ln 2 * real n"
advantage_def: fixes \<A> :: "(('plain \<times> 'plain) \<times> 'b, 'plain, 'cipher) gpv \<times> ('cipher \<Rightarrow> 'b \<Rightarrow> (bool, 'plain, 'cipher) gpv)" shows "advantage \<A> = \<bar>spmf (ind_cpa \<A>) True - 1 / 2\<bar>"
csin_ccos_squared_add: fixes a :: "real" shows "complex_of_real (cos a) * complex_of_real (cos a) + complex_of_real (sin a) * complex_of_real (sin a) = 1"
insert_dom: fixes f :: "'d \<Rightarrow> 'c option"   and x :: "'d"   and y :: "'c" assumes "f x = Some y" shows "insert x (dom f) = dom f"
inj_enum: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "inj_on (kuhn_simplex.enum base upd) {..n}"
commD: fixes x :: "'b list"   and t :: "'b list"   and y :: "'b list" assumes "x \<in> t*"   and "y \<in> t*" shows "x \<cdot> y = y \<cdot> x"
cltn2_abs_mult: fixes k :: "real"   and A :: "((real, 3) vec, 3) vec" assumes "k \<noteq> 0"   and "invertible A" shows "cltn2_abs (k *\<^sub>R A) = cltn2_abs A"
unique_Na: fixes A :: "agent"   and B :: "agent"   and M :: "nat"   and Na :: "nat"   and evs :: "event list"   and B' :: "agent"   and M' :: "nat" assumes "Says A B \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce Na, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"   and "Says A B' \<lbrace>Nonce M', Agent A, Agent B', Crypt (shrK A) \<lbrace>Nonce Na, Nonce M', Agent A, Agent B'\<rbrace>\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> orb" shows "B = B' \<and> M = M'"
hseqE: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (H \<nu> \<mu>)" obtains "partial_magma.arr V \<mu>" and "partial_magma.arr V \<nu>" and "src \<nu> = trg \<mu>"
connected_component_disjoint: fixes S :: "'c set"   and a :: "'c"   and b :: "'c" shows "(connected_component_set S a \<inter> connected_component_set S b = {}) = (a \<notin> connected_component_set S b)"
pointed_iff_empty_complete: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "extremed A less_eq = {{}}-complete A (\<lambda>x y. less_eq y x)"
Lim_def: fixes A :: "'d filter"   and f :: "'d \<Rightarrow> 'c" shows "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
interaction_bounded_byI: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat" assumes "interaction_bound consider gpv \<le> n" shows "interaction_bounded_by consider gpv n"
cod_in_codomains: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.codomains f \<noteq> {}" shows "A1xA2.cod f \<in> A1xA2.codomains f"
mult_mat_vec_def: fixes A :: "'c mat"   and v :: "'c vec" shows "A *\<^sub>v v \<equiv> vec (dim_row A) (\<lambda>i. row A i \<bullet> v)"
det_row_add: fixes k :: "'b"   and a :: "'b \<Rightarrow> ('a, 'b) vec"   and b :: "'b \<Rightarrow> ('a, 'b) vec"   and c :: "'b \<Rightarrow> ('a, 'b) vec" shows "det (\<chi>i. if i = k then a i + b i else c i) = det (\<chi>i. if i = k then a i else c i) + det (\<chi>i. if i = k then b i else c i)"
i_set_cont_min_eq: shows "i_set_cont = i_set_cont_min"
simple_cfg_on: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and ct :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" assumes "Finite_Markov_Decision_Process K S"   and "ct \<in> Pi S K" shows "Finite_Markov_Decision_Process.simple K S ct s \<in> Markov_Decision_Process.cfg_on K s"
get_shadow_root_eq_h2: fixes ptr' :: "(_) element_ptr"   and shadow_root_opt :: "(_) shadow_root_ptr option" shows "h2 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_opt = h' \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_opt"
adjacent_half_chamber_system_image_fg: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>"   and "D \<in> g \<turnstile> folding_g.\<C>"   and "C \<sim> D" shows "f ` D = C"
no_grey_refsD_2: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state"   and r :: "'g"   and p :: "'d process_name" assumes "no_grey_refs s" shows "r \<notin> WL p s"
sup_absorb2: fixes x :: "'a multiset"   and y :: "'a multiset" assumes "x \<subseteq># y" shows "x \<union># y = y"
not_Limit_if_in_Limit_plus_omega: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Limit \<alpha>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>"   and "\<beta> \<in>\<^sub>\<circ> \<alpha> + \<omega>" shows "\<not> Limit \<beta>"
pgallery: fixes xs :: "'a set list" assumes "pgallery xs" shows "gallery xs"
xs2'_def: shows "xs2' = take (Suc hi - lo) (drop lo xs')"
incl_HK: fixes J :: "'a set" assumes "subring J R"   and "I \<subseteq> J" shows "generate_ring (R\<lparr>carrier := J\<rparr>) I \<subseteq> J"
7: shows "w * r \<sqinter> w\<^sup>T\<^sup>+ * r = bot"
float_defs_3: fixes a :: "('c, 'd) IEEE.float" shows "is_zero a = (IEEE.exponent a = 0 \<and> fraction a = 0)"
IH2: shows "phis'_aux g v {m} (phis'_aux g v (set ms) phis) = snd (readVariableRecursive g v m (phis'_aux g v (set ms) phis))"
quorum_def: fixes Q :: "'a set" shows "orig.quorum Q \<equiv> \<forall>p\<in>Q \<inter> W. \<exists>Sl\<in>slices p. Sl \<subseteq> Q"
affine_dependent_choose: fixes S :: "'a set"   and a :: "'a" assumes "\<not> affine_dependent S" shows "affine_dependent (insert a S) = (a \<notin> S \<and> a \<in> affine hull S)"
neutr_eq_iff: fixes a :: "'c"   and b :: "'c" shows "(bot = a \<squnion> b) = (a = bot \<and> b = bot)"
possible_many_calls: fixes x :: "'b"   and x' :: "'b" shows "possible (many_calls x) x' = (x = x')"
group_l_invI: assumes "\<And>x. x \<in> carrier (DirProds G I) \<Longrightarrow> \<exists>y\<in>carrier (DirProds G I). y \<otimes>\<^bsub>DirProds G I\<^esub> x = \<one>\<^bsub>DirProds G I\<^esub>" shows "Group.group (DirProds G I)"
ci_min: shows "\<forall>(D, j)\<in>#mset (map (apfst mset) P). weight (mset C, i) \<le> weight (D, j)"
assms: shows "s \<rightarrow>\<^bsub>\<alpha>\<^esub> s'"
adopt_node_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> adopt_node_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M parent owner_document document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
nlcons_list: fixes a :: "('c, 'd) Node"   and t :: "('c, 'd) FingerTreeStruc" shows "toList (nlcons a t) = nodeToList a @ toList t"
hyps: shows "applied_rule_rev C x (removeShadowRules2 ys) = applied_rule_rev C x ys"
InvariantUniq_def: fixes M :: "(Literal \<times> bool) list" shows "InvariantUniq M \<equiv> uniq (elements M)"
subst_simp_var1: fixes x :: "'a"   and M :: "'a trm" shows "Var x[x ::= M] = M"
inject_2: fixes b :: "'h"   and b' :: "'h" shows "(Inr b = Inr b') = (b = b')"
isPartition_map_trans: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and hi :: "nat"   and xs :: "'c list"   and lo :: "nat"   and p :: "nat" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "hi < length xs"   and "isPartition_map R h xs lo hi p" shows "\<forall>i j. lo \<le> i \<and> i < p \<and> p < j \<and> j \<le> hi \<longrightarrow> R (h (xs ! i)) (h (xs ! j))"
emeasure_eq_measure: fixes A :: "'a set" shows "emeasure M A = ennreal (M.prob A)"
restrict_subseteq: fixes v :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "restrict v f c \<le> f c"
map_iterator_dom_correct: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option" assumes "map_iterator it m" shows "set_iterator (map_iterator_dom it) (dom m)"
ufact: shows "unique_factorization_m (q.Mp (Polynomial.smult ilc f)) (a, mset b)"
presT: shows "rel_preserves_pred (_Collect (P, Q) (\<exists>SP SQ. \<lbrakk>SP\<rbrakk> \<in>T P \<and> \<lbrakk>SQ\<rbrakk> \<in>T Q \<and> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel)) Pred"
Dom_HcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Dom (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Dom t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom u"
iTILL_div_Max: fixes n :: "nat"   and k :: "nat" shows "Max ([\<dots>n] \<oslash> k) = n div k"
bitwise_inner_prod_def: fixes n :: "nat"   and i :: "nat"   and j :: "nat" shows "i \<cdot>\<^bsub>n\<^esub>  j = (\<Sum>k = 0..<n. bin_rep n i ! k * bin_rep n j ! k)"
ssum_map_sinl': fixes f :: "'f \<rightarrow> 'e"   and g :: "'h \<rightarrow> 'g"   and x :: "'f" assumes "f\<cdot>\<bottom> = \<bottom>" shows "ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)"
nb\<^sub>2: shows "\<forall>(v, a)\<in>dom (\<phi>\<^sub>S \<Psi> s). \<forall>(v, a')\<in>dom (\<phi>\<^sub>S \<Psi> s). (\<phi>\<^sub>S \<Psi> s) (v, a) = Some True \<and> (\<phi>\<^sub>S \<Psi> s) (v, a') = Some True \<longrightarrow> (v, a) = (v, a')"
obtainit: fixes r :: "nat list"   and rs :: "nat rexp list" assumes "r \<in> lang (verund rs)" shows "\<exists>x\<in>set rs. r \<in> lang x"
inject: fixes x1 :: "'b trm"   and x2 :: "'b trm"   and y1 :: "'b trm"   and y2 :: "'b trm" shows "(Eq x1 x2 = Eq y1 y2) = (x1 = y1 \<and> x2 = y2)"
supp_Pred\<^sub>\<alpha>: fixes f :: "'d"   and \<phi> :: "'b" shows "supp (Pred\<^sub>\<alpha> f \<phi>) = supp f \<union> supp \<phi>"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "b \<le> a" shows "b < a"
known_ptrs_known_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "DocumentClass.known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "DocumentClass.known_ptr ptr"
assms_1: shows "WT_core \<I>_adv_core \<I>_usr_core I_core core"
nat_pow_eone: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "x [^] 1 = x"
nxt_singles: fixes S :: "'b set"   and x' :: "'b" shows "nxt (singles S) x' = (if x' \<in> S then without x' (singles S) else singles S)"
pick_reduce_set: fixes i :: "nat"   and m :: "nat"   and I :: "nat set" assumes "i < card {a. a < m \<and> a \<in> I}" shows "pick I i = pick {a. a < m \<and> a \<in> I} i"
poss2: shows "squash (rpoly q x1) = 1 \<and> squash (rpoly q x2) = - 1 \<longrightarrow> (\<exists>r>x1. r \<le> x2 \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
phl_skip: fixes x :: "'a" shows "x \<parallel> 1\<^sub>\<pi> \<le> 1\<^sub>\<pi> \<parallel> x"
wp2WhileTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "bval b s" shows "wp\<^sub>2 c (wp\<^sub>2 (WHILE b DO c) Q) s + 1 \<le> wp\<^sub>2 (WHILE b DO c) Q s"
path_conv_loc_imps_fw: fixes loc1 :: "'loc"   and loc2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and c :: "('loc, 't) configuration" assumes "path loc1 loc2 xs"   and "distinct xs" shows "\<exists>M. loc_imps_fw c loc1 loc2 M xs"
agree_comm: fixes A :: "(real, 'c) vec \<times> (real, 'c) vec"   and B :: "(real, 'c) vec \<times> (real, 'c) vec"   and V :: "('c + 'c) set" assumes "Vagree A B V" shows "Vagree B A V"
many_strongly_connected_implies_10_eq: fixes x :: "'a" assumes "many_strongly_connected x" shows "x ; x\<^sup>T ; x\<^sup>\<star> = x\<^sup>+"
l_set_mode_get_tag_name_axioms_def: fixes set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_tag_name_axioms set_mode_locs get_tag_name_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
vwalk_arcs_tl: fixes xs :: "'a list" shows "vwalk_arcs (tl xs) = tl (vwalk_arcs xs)"
measure_pmf_eq_density: fixes p :: "'a pmf" shows "measure_pmf p = density (count_space UNIV) (\<lambda>x. ennreal (pmf p x))"
mahler_landau_graeffe_approximation_core: fixes g :: "int poly"   and f :: "int poly"   and k :: "nat" assumes "g = graeffe_poly_impl f k" shows "mahler_measure f \<le> root (2 ^ Suc k) (real_of_int (\<Sum>a\<leftarrow>coeffs g. a * a))"
\<alpha>_def: shows "\<alpha> \<equiv> \<lambda>p. var_rename (max_var_set (fv (pair p)))"
SUP_insert: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` (a \<triangleright> A)) = sup (f a) (Sup (f ` A))"
intros_25: fixes n :: "i32"   and v1 :: "v"   and v2 :: "v" assumes "int_eq n 0" shows "\<lparr>[$C v1, $C v2, $C ConstInt32 n, $Select]\<rparr> \<leadsto> \<lparr>[$C v2]\<rparr>"
total: fixes x :: "'alt"   and y :: "'alt" assumes "x \<in> alts'"   and "y \<in> alts'" shows "Ri' x y \<or> Ri' y x"
substEnv_def: fixes xs :: "'varSort"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and x :: "'var" shows "substEnv xs X x \<equiv> psubstEnv (idEnv [x \<leftarrow> X]_xs)"
proj_append_1: fixes l :: "'a"   and A' :: "('a strand_label \<times> 'b) list"   and B' :: "('a strand_label \<times> 'b) list" shows "proj l (A' @ B') = proj l A' @ proj l B'"
keys_simps_2: fixes c :: "color"   and l :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and r :: "('b, 'c) RBT_Impl.rbt" shows "RBT_Impl.keys (Branch c l k v r) = RBT_Impl.keys l @ k # RBT_Impl.keys r"
ord_spmf_eq_leD: fixes p :: "'c spmf"   and q :: "'c spmf"   and x :: "'c" assumes "ord_spmf (=) p q" shows "spmf p x \<le> spmf q x"
abs_eq: fixes xa :: "nat \<Rightarrow>\<^sub>0 nat"   and x :: "'a" shows "MPoly_Type.monom xa x = MPoly (Poly_Mapping.single xa x)"
w_in_hom_2: shows "\<guillemotleft>\<chi>.chine : \<chi>.chine \<Rightarrow> \<chi>.chine\<guillemotright>"
Basic: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and f :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "\<Gamma>\<turnstile>Basic f \<down> Normal s"
hyps_3: assumes "\<And>x. x \<in> S \<Longrightarrow> set (coeffs (f x)) \<subseteq> \<real>" shows "set (coeffs (prod f S)) \<subseteq> \<real>"
Qg_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "G / H = \<lparr>carrier = set_rcs G H, Group.top = c_top G H, iop = c_iop G H, one = H\<rparr>"
CL_axioms_def: fixes cl :: "'b potype" shows "CL_axioms cl \<equiv> cl \<in> CompleteLattice"
sign_img_def: shows "sign_img = \<lparr>carrier = {- 1, 1}, monoid.mult = (*), one = 1\<rparr>"
"3_4": shows "oalist_inv_raw ((kx, vx) # xs)"
intrinsic_border_max: fixes x :: "'a list"   and ys :: "'a list" assumes "strict_border x ys" shows "length x \<le> length (intrinsic_border ys)"
rspp_Actual_out_in_slice: fixes x :: "nat"   and a :: "'edge"   and xs :: "'var list"   and S :: "'node SDG_node set"   and f :: "'var \<Rightarrow> 'val option"   and g :: "'var \<Rightarrow> 'val option" assumes "x < length (ParamDefs (targetnode a))"   and "valid_edge a"   and "length (ParamDefs (targetnode a)) = length xs"   and "Actual_out (targetnode a, x) \<in> S" shows "rspp (targetnode a) S xs f g (ParamDefs (targetnode a) ! x) = g (xs ! x)"
diff_cs: shows "fst (cs (S t 0) cid) \<noteq> fst (cs (S t i) cid)"
unit_factor_0: shows "unit_factor (0::'a) = (0::'a)"
strictly_generalizes_trans: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and E :: "'a literal multiset" assumes "C \<prec>\<cdot> D"   and "D \<prec>\<cdot> E" shows "C \<prec>\<cdot> E"
crename_ImpR_aux: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and c :: "coname"   and M :: "trm"   and e :: "coname" assumes "R[a\<turnstile>c>b] = ImpR x.<c>.M e" shows "a = e \<and> a = b \<or> a \<noteq> e"
closed_in_D_1: fixes A :: "'a set" assumes "closed A" shows "emeasure M A = Sup (emeasure M ` {K. K \<subseteq> A \<and> compact K})"
word_le_make_less: fixes y :: "'a Word.word"   and x :: "'a Word.word" assumes "y \<noteq> - 1" shows "(x \<le> y) = (x < y + 1)"
outs_converter_def: fixes \<I> :: "('b, 'c) \<I>"   and \<I>' :: "('a, 'd) \<I>"   and conv :: "('b, 'c, 'a, 'd) converter" shows "outs_converter \<I> \<I>' conv \<equiv> {x. outsp_converter \<I> \<I>' x conv}"
ipurge_tr_aux_idem: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list" shows "ipurge_tr_aux I D U (ipurge_tr_aux I D U xs) = ipurge_tr_aux I D U xs"
reguler_refinementE: fixes \<X> :: "'c \<Rightarrow> 'a set"   and I :: "'c set" assumes "carrier \<subseteq> \<Union> (\<X> ` I)"   and "\<And>i. i \<in> I \<Longrightarrow> open (\<X> i)" obtains N :: "nat set" and \<psi> :: "nat \<Rightarrow> ('a, 'b) chart" where "\<And>i. i \<in> N \<Longrightarrow> \<psi> i \<in> atlas" and "(domain \<circ> \<psi>) ` N refines \<X> ` I" and "regular_cover N \<psi>"
hoare_partial_correct_def: fixes dims :: "nat list"   and P :: "complex mat"   and S :: "com"   and Q :: "complex mat" shows "state_sig.hoare_partial_correct dims P S Q = (\<forall>\<rho>\<in>state_sig.density_states dims. trace (P * \<rho>) \<le> trace (Q * state_sig.denote dims S \<rho>) + (trace \<rho> - trace (state_sig.denote dims S \<rho>)))"
join_below_iff: fixes x :: "'a"   and y :: "'a"   and a :: "'a" assumes "compatible x y" shows "(x \<squnion> y \<sqsubseteq> a) = (x \<sqsubseteq> a \<and> y \<sqsubseteq> a)"
convexD: fixes s :: "'e set"   and x :: "'e"   and y :: "'e"   and u :: "real"   and v :: "real" assumes "convex s"   and "x \<in> s"   and "y \<in> s"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
Longest_common_prefix_unique: fixes L :: "'c list set" assumes "L \<noteq> {}" shows "\<exists>!ps. (\<forall>xs\<in>L. prefix ps xs) \<and> (\<forall>qs. (\<forall>xs\<in>L. prefix qs xs) \<longrightarrow> length qs \<le> length ps)"
qPsubstAbs_preserves_alphaAbs1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and B :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodEnv rho"   and "qGoodAbs A \<or> qGoodAbs B"   and "A $= B" shows "A $[[rho]] $= B $[[rho]]"
subst_Bool_ign_res_eq: fixes P :: "'b assn"   and b :: "bool"   and x :: "vals"   and Y :: "vals"   and s :: "state"   and Z :: "'b" shows "(P\<leftarrow>=b\<down>=x) Y s Z = ((P\<leftarrow>=b) Y s Z \<and> Y = x)"