hcju/isabelleps huggingface dataset:
equiv_sub_add: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans t u"   and "sub_add t" shows "sub_add u"
qq: fixes i :: "'n set" assumes "i \<in> r" shows "qq i tagged_division_of i \<and> d fine qq i \<and> norm ((\<Sum>(x, K)\<in>qq i. content K *\<^sub>R f x) - integral i f) < k / (real (card r) + 1)"
inext_neq_imp_Max: fixes n :: "nat"   and I :: "nat set" assumes "n \<noteq> inext n I" shows "n < Max I \<or> infinite I"
assms_2: shows "emeasure M (space M) \<noteq> \<infinity>"
add_le_cancel_left: fixes x :: "V"   and y :: "V"   and z :: "V" shows "(x + y \<subseteq>\<^sub>\<circ> x + z) = (y \<subseteq>\<^sub>\<circ> z)"
tp_assoc: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Ring R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "a \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<cdot>\<^sub>r\<^bsub>R\<^esub> c = a \<cdot>\<^sub>r\<^bsub>R\<^esub> (b \<cdot>\<^sub>r\<^bsub>R\<^esub> c)"
maxsimp_map_into_image: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimp X x" shows "SimplicialComplex.maxsimp (f \<turnstile> X) (f ` x)"
simps_3: fixes t :: "'b EV.term"   and u :: "'b EV.term" shows "\<lbrace>t \<^bold>\<star> u\<rbrace> = \<lbrace>t\<rbrace> \<star>\<^sub>D \<lbrace>u\<rbrace>"
q_HPair_def: fixes x :: "hf"   and y :: "hf" shows "q_HPair x y \<equiv> q_Eats (q_Eats 0 (q_Eats (q_Eats 0 y) x)) (q_Eats (q_Eats 0 x) x)"
simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and pat :: "term"   and rhs :: "'c"   and cs :: "(term \<times> 'c) list"   and t :: "'c" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.find_match unapp unconst ((pat, rhs) # cs) t = (case term.match unapp unconst pat t of None \<Rightarrow> term.find_match unapp unconst cs t | Some env \<Rightarrow> Some (env, pat, rhs))"
mid_in_interval: fixes i :: "float interval" shows "mid i \<in>\<^sub>i i"
down_chain_reaches_empty: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "finite UNIV"   and "mono \<tau>"   and "\<forall>i. (\<tau> ^^ (i + 1)) UNIV \<subset> (\<tau> ^^ i) UNIV" shows "\<exists>j. (\<tau> ^^ j) UNIV = {}"
simps_3: fixes x1 :: "'a" shows "nNone \<noteq> nSome x1"
scaleHR_def: shows "scaleHR = *f2* (*\<^sub>R)"
implies_intro'_pre2: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and A :: "Core.term" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "A \<in> \<Gamma>" shows "\<Theta>,\<Gamma> \<turnstile> A \<longmapsto> B"
T_Tinv_of_set: fixes A :: "'a set" assumes "I.invertible_qmpt"   and "A \<in> sets M" shows "Tinv -` (I.Tinv -` A \<inter> space M) \<inter> space M = A"
Var\<^sub>0_power: fixes v :: "'c"   and n :: "nat" shows "Var\<^sub>0 v ^ n = monomial (1::'d) (monomial n v)"
Seq_def: shows "Seq = map (\<^bold>u ys) [0..<a ! k]"
bempty_not_binsert_2: fixes x :: "'c"   and X :: "'c set['d]" shows "binsert x X \<noteq> bempty"
simps_1: fixes F :: "'a formula"   and G :: "'a formula" shows "is_disj (F \<^bold>\<or> G) = (is_lit_plus F \<and> is_disj G)"
iter: fixes P :: "'b \<Rightarrow> bool"   and s :: "'a" assumes "Measurable.pred N P"   and "P \<le> lfp F" shows "emeasure (M s) {x \<in> space N. F P x} = f (\<lambda>s. emeasure (M s) {x \<in> space N. P x}) s"
\<delta>_le_\<epsilon>: shows "\<delta> \<le> \<epsilon>"
rep_eq: fixes x :: "('d, 'e) fmap" shows "fset (fmdom x) = dom (fmlookup x)"
distinct_14: fixes ccs1' :: "ccs"   and ccs2' :: "ccs"   and act :: "act"   and ccs :: "ccs" shows "ccs1' \<parallel> ccs2' \<noteq> act.ccs"
lexordp_linorder: shows "class.linorder lexordp_eq ord_class.lexordp"
Dirichlet_kernel_continuous: fixes n :: "nat" shows "continuous_on {- pi..pi} (Dirichlet_kernel n)"
axioms_1: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e" assumes "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G" shows "bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"
rel_funI: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
hyps: fixes t :: "'a llist" assumes "t \<in> UNIV\<^sup>\<omega>" shows "t \<down> i < t"
A: fixes U :: "'b set" shows "filter P (ipurge_tr_aux I D U xs) = ipurge_tr_aux_foldr I D P U xs"
assms: shows "A \<in> sets (lborel_f (Suc n))"
a1: shows "r = memory_write asi addr byte_mask data s \<and> r = Some s' \<and> get_S (cpu_reg_val PSR s) = 0"
dom_dom: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "J.dom (J.dom f) = J.dom f"
sel_1: fixes x1 :: "'b"   and x2 :: "'c" shows "fst (x1, x2) = x1"
equal_ket: fixes A :: "'c ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd"   and B :: "'c ell2 \<Rightarrow>\<^sub>C\<^sub>L 'd" assumes "\<And>x. A *\<^sub>V ket x = B *\<^sub>V ket x" shows "A = B"
second_countable_topology_quotient: fixes p :: "'c \<Rightarrow> 'b" assumes "open = openin (map_topology p euclidean)"   and "surj p"   and "\<And>X. open X \<Longrightarrow> open (p ` X)" shows "class.second_countable_topology open"
simps: fixes a1 :: "pi"   and a2 :: "pi" shows "a1 \<approx> a2 = (\<exists>P Q. a1 = P \<and> a2 = Q \<and> P \<leadsto>\<^sup>^<weakBisim> Q \<and> Q \<approx> P)"
show_law_complex: fixes r :: "complex" shows "show_law showsp_complex r"
in_disconnected_nodes_no_parent: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr"   and owner_document :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None"   and "h \<turnstile> get_owner_document (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 node_ptr) \<rightarrow>\<^sub>r owner_document"   and "h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"   and "known_ptrs h"   and "type_wf h" shows "node_ptr \<in> set disc_nodes"
stalks_are_local: fixes x :: "'b"   and U :: "'b set" assumes "x \<in> U"   and "is_open U" shows "stalk.is_local is_open \<O>\<^sub>X \<rho> add_str mult_str zero_str one_str (neighborhoods x) x U"
app_Var_IT: fixes t :: "dB"   and i :: "nat" assumes "IT t" shows "IT (t \<degree> Var i)"
inf_closed: fixes x :: "'b"   and y :: "'b" shows "brouwer.regular (- x \<sqinter> - y)"
gr: fixes u :: "'b allegorical_term"   and v :: "'b allegorical_term" shows "graph_rule (transl_rule (u \<sqsubseteq> v))"
preserves_arr: fixes f :: "'b" assumes "B.arr f" shows "B.arr (src\<^sub>B f)"
rcm_B: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b l\<rightarrow> c = a l\<rightarrow> b l\<rightarrow> c"
Sup_eqI: 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 A :: "'b set"   and x :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq y x"   and "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> less_eq z y) \<Longrightarrow> less_eq x y" shows "Sup A = x"
simps: fixes F :: "'a"   and H :: "'l"   and f :: "'m" shows "compile (Prog F H f) = Result.to_option (compile_env (map_option arity \<circ> Finca_get F) load_oracle F \<bind> (\<lambda>F'. Ok (Prog F' H f)))"
RM_subset: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" shows "RM M2 M1 \<Omega> V m i \<subseteq> C M2 M1 \<Omega> V m i"
sub_id_le_nc: fixes x :: "'a" shows "x \<sqinter> 1\<^sub>\<sigma> \<le> nc"
exhaust_sel: fixes state :: "('a, 'b) state" obtains "state = State (\<T> state) (\<B>\<^sub>i\<^sub>l state) (\<B>\<^sub>i\<^sub>u state) (\<V> state) (\<U> state) (\<U>\<^sub>c state)"
fansI: fixes subs :: "'a \<Rightarrow> 'a set" assumes "\<forall>A. finite (subs A)" shows "fans subs"
assms_2: fixes n :: "nat" assumes "e = PExpr1 (PVar n)" shows "P"
Cons_1: fixes ts :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and queue :: "'t list" assumes "thread_oks ts ntas"   and "set queue = dom ts"   and "distinct queue" shows "distinct (enqueue_new_threads queue ntas)"
ofe_fields_def: fixes flow_entry_match :: "('a, 'b) flow_entry_match" shows "ofe_fields flow_entry_match = (case flow_entry_match of OFEntry x1 x2 x3 \<Rightarrow> x2)"
assms: shows "exe_osig_conds (exesorts \<Sigma>)"
ovsmfidiesiOmi: shows "(ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>) O m\<inverse> \<subseteq> d\<inverse> \<union> s\<inverse> \<union> ov\<inverse> \<union> m\<inverse> \<union> f\<inverse> \<union> f \<union> e"
tensor_from_lookup_eqI: fixes ds :: "nat list"   and e\<^sub>1 :: "nat list \<Rightarrow> 'b"   and e\<^sub>2 :: "nat list \<Rightarrow> 'b" assumes "\<And>is. is \<lhd> ds \<Longrightarrow> e\<^sub>1 is = e\<^sub>2 is" shows "tensor_from_lookup ds e\<^sub>1 = tensor_from_lookup ds e\<^sub>2"
fds_deriv_0: shows "fds_deriv 0 = 0"
field_le_epsilon: fixes x :: "'c"   and y :: "'c" assumes "\<And>e. (0::'c) < e \<Longrightarrow> x \<le> y + e" shows "x \<le> y"
principal_dchar_sum_bound': shows "(\<lambda>x. sum_upto (\<lambda>k. \<chi>\<^sub>0 k * mangoldt k / of_nat k) x - Ln (complex_of_real x)) \<in> O(\<lambda>_. 1)"
sorted_del_list: fixes ps :: "('c \<times> 'd) list"   and x :: "'c" assumes "sorted1 ps" shows "sorted1 (del_list x ps)"
wf_cdecl_supD: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and D :: "char list"   and r :: "fdecl list \<times> 'b mdecl list" assumes "wf_cdecl wf_md P (C, D, r)"   and "C \<noteq> Object" shows "is_class P D"
Rineq: fixes R :: "real" assumes "1 \<le> R" shows "R ^ DIM('a) \<le> D * R ^ DIM('b)"
Compl_Domain_subset_wp: fixes act :: "('b \<times> 'b) set"   and B :: "'b set" shows "- Domain act \<subseteq> wp act B"
monad_def: fixes x :: "'b star" shows "monad x = {y. x \<approx> y}"
iso_runit': fixes f :: "'a" assumes "ide f" shows "local.iso \<r>\<^sup>-\<^sup>1[f]"
B_1: shows "upper_asymptotic_density B \<le> (\<Sum>n. upper_asymptotic_density (A n))"
not_distinct_decomp: fixes ws :: "'b list" assumes "\<not> distinct ws" shows "\<exists>xs ys zs y. ws = xs @ [y] @ ys @ [y] @ zs"
\<alpha>nodes_finite: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool"   and g :: "'a" assumes "wgraph_by_map M_empty M_update M_delete M_lookup M_invar"   and "wgraph_by_map.invar M_lookup M_invar g" shows "finite (wgraph_by_map.\<alpha>nodes_aux M_lookup g)"
xy: shows "projr (while isl f (Inl (Suc 0, Suc D))) = (x, y)"
sup_state_opt_any_Some: fixes P :: "'b prog"   and s :: "ty list \<times> ty err list"   and X :: "(ty list \<times> ty err list) option" shows "P \<turnstile> \<lfloor>s\<rfloor> \<le>' X = (\<exists>s'. X = \<lfloor>s'\<rfloor> \<and> P \<turnstile> s \<le>\<^sub>i s')"
inv_start: shows "inv_locate_a (as, lm) (n, l, r) ires"
Elem_explode_in: fixes a :: "ZF"   and A :: "ZF"   and B :: "ZF set" assumes "Elem a A"   and "explode A \<subseteq> B" shows "a \<in> B"
semigroup_add_ow_Ball_def: fixes S :: "'c set"   and pls :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "semigroup_add_ow S pls = ((\<forall>a\<in>S. \<forall>b\<in>S. \<forall>c\<in>S. pls (pls a b) c = pls a (pls b c)) \<and> (\<forall>a\<in>S. \<forall>b\<in>S. pls a b \<in> S))"
subset_shift_var_set: fixes A :: "nat set"   and V :: "nat set" assumes "Suc -` A \<subseteq> V" shows "A \<subseteq> shift_var_set V"
de_morgan_3: fixes x :: "'a"   and y :: "'a" shows "- (- x \<cdot> - y) = x + y"
transfer_operations_rl_7: shows "(RETURN o2 appr_inf_inner ops optns, Inf_inners) \<in> appr_rell \<rightarrow> rl_rel \<rightarrow> \<langle>rnv_rel\<rangle>nres_rel"
matcheq_matchAny: fixes m :: "'a match_expr"   and \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and a :: "action"   and p :: "'b" assumes "\<not> has_primitive m" shows "matcheq_matchAny m = matches \<gamma> m a p"
xor_left_inverse: fixes x :: "'e"   and y :: "'e" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<oplus> x \<oplus> y = y"
double_dual_iso_def: fixes x :: "'a \<Rightarrow> complex" shows "G'.double_dual_iso x = (\<lambda>\<chi>. if character (Characters G) \<chi> then \<chi> x else 0)"
elims: fixes x :: "bool list"   and xa :: "tree"   and y :: "tree" assumes "delete x xa = y" obtains T :: "tree" where "x = []" and "xa = T" and "y = Leaf"   | ds :: "bool list" and T\<^sub>1 :: "tree" and T\<^sub>2 :: "tree" where "x = True # ds__" and "xa = Branching T\<^sub>1__ T\<^sub>2__" and "y = Branching (delete ds__ T\<^sub>1__) T\<^sub>2__"   | ds :: "bool list" and T\<^sub>1 :: "tree" and T\<^sub>2 :: "tree" where "x = False # dsa__" and "xa = Branching T\<^sub>1 T\<^sub>2" and "y = Branching T\<^sub>1 (delete dsa__ T\<^sub>2)"   | a :: "bool" and ds :: "bool list" where "x = a # ds" and "xa = Leaf" and "y = Leaf"
Inst: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and v :: "val"   and s1 :: "state"   and b :: "bool"   and T :: "ref_ty" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"   and "b = (v \<noteq> Null \<and> G,snd s1\<turnstile>v fits RefT T)" shows "G\<turnstile>Norm s0 \<midarrow>e InstOf T-\<succ>Bool b\<midarrow>n\<rightarrow> s1"
case_cong_weak: fixes sum :: "'a + 'b"   and sum' :: "'a + 'b"   and f1 :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c" assumes "sum = sum'" shows "(case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (case sum' of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x)"
intros: fixes X :: "'b nmultiset"   and M :: "'b nmultiset multiset" assumes "X \<in># M" shows "sub_nmsetp X (MSet M)"
comm_monoid_axioms: shows "comm_monoid (*) (1::'a)"
par_nonabort: fixes i :: "'a::refinement_lattice"   and j :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (i::'a::refinement_lattice)"   and "chaos \<sqsubseteq> (j::'a::refinement_lattice)" shows "chaos \<sqsubseteq> (i::'a::refinement_lattice) \<parallel> (j::'a::refinement_lattice)"
unat_minus_one_word: shows "unat (- 1) = 2 ^ LENGTH('a) - 1"
closed_Union: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set set" assumes "class.topological_space open"   and "finite S"   and "\<forall>T\<in>S. topological_space.closed open T" shows "topological_space.closed open (\<Union> S)"
FreeGroup_0_closed: fixes S :: "'a set" shows "0 \<in> FreeGroup S"
distinct_69: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi' :: "pi" shows "[name1\<frown>name2]pi \<noteq> !pi'"
vdomain_vunion: fixes A :: "V"   and B :: "V" shows "\<D>\<^sub>\<circ> (A \<union>\<^sub>\<circ> B) = \<D>\<^sub>\<circ> A \<union>\<^sub>\<circ> \<D>\<^sub>\<circ> B"
WTrtBinOpEq: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E,h \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E,h \<turnstile> e\<^sub>2 : T\<^sub>2" shows "P,E,h \<turnstile> e\<^sub>1 \<guillemotleft>Eq\<guillemotright> e\<^sub>2 : Boolean"
hyps_1: shows "a \<noteq> (0::'a) \<or> p \<noteq> 0"
inf_exists: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "\<exists>i. is_glb L i A"
has_codomain_iff_arr: fixes f :: "'a \<times> 'a" shows "(VV.codomains f \<noteq> {}) = VV.arr f"
eqpoll_imp_lepoll: fixes A :: "'a set"   and B :: "'b set" assumes "A \<approx> B" shows "A \<lesssim> B"
get_host_locs_impl: shows "get_host_locs = local.a_get_host_locs"
intro: fixes xs :: "V" assumes "vsv xs"   and "vfsequence_axioms xs" shows "vfsequence xs"
element_ptr_kinds_commutes: fixes element_ptr :: "(_) element_ptr"   and h :: "(_) heap" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr |\<in>| node_ptr_kinds h) = (element_ptr |\<in>| element_ptr_kinds h)"
simps_2: fixes a :: "'a"   and b :: "'a"   and bs :: "'a list" shows "hideDupsRec a (b # bs) = (if a = b then None # hideDupsRec b bs else Some b # hideDupsRec b bs)"
cases: fixes x :: "('c, 'd) chain" obtains uu_ :: "'c" where "x = \<lbrace> uu_ \<rbrace>"   | uv_ :: "'c" and uw_ :: "'d" and \<Pi> :: "('c, 'd) chain" where "x = \<lbrace> uv_ \<rbrace> \<cdot> uw_ \<cdot> \<Pi>"
add_mono_Fin_le: fixes y1 :: "'a"   and x1 :: "'a extended"   and y2 :: "'a"   and x2 :: "'a extended" assumes "Fin y1 \<le> x1"   and "Fin y2 \<le> x2" shows "Fin (y1 + y2) \<le> x1 + x2"
prime_factors_gauss_int_of_nat_def: fixes n :: "nat" shows "prime_factors_gauss_int_of_nat n = (if n = 0 then {} else \<Union>p\<in>prime_factors n. set (factor_gauss_int_prime_nat p))"
quasi_isometry_betweenD_4: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "lambda C -quasi_isometry_between X Y f"   and "x \<in> X"   and "y \<in> X" shows "dist (f x) (f y) \<le> lambda * dist x y + C"
dfs_invariant: fixes ks :: "int list"   and l :: "int"   and S :: "nat option list \<times> int list"   and xs :: "int list"   and I :: "nat option list \<times> int list \<Rightarrow> bool"   and n :: "nat" assumes "dioph_invariant ks l S"   and "list_all (dioph_is_node ks l) xs"   and "I S"   and "\<And>S x. \<lbrakk>\<not> dioph_memb x S; dioph_is_node ks l x; dioph_invariant ks l S; I S\<rbrakk> \<Longrightarrow> I (dioph_ins x S)" shows "I (dioph_dfs.dfs n ks S xs)"
lifting_inv: shows "lifting_inv init_fin_final init_fin (\<lambda>_. init_fin_lift P)"
residual_network_sel_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" assumes "countable_network \<Delta>" shows "capacity (countable_network.residual_network \<Delta> f) (x, y) = (if edge \<Delta> x y then capacity \<Delta> (x, y) - f (x, y) else if y = source \<Delta> then 0 else f (y, x))"
typeII_chain_line_integral_exists_boundary'': shows "\<forall>(k, \<gamma>)\<in>two_chain_horizontal_boundary two_chain. line_integral_exists F {j} \<gamma>"
intFI_def: shows "intFI \<equiv> MonotProblem.intFI TE_wtFsym wtPsym TE_arOf TE_resOf parOf TE_\<Phi>"
fresh_6: fixes a :: "W.tvar"   and x1 :: "W.var"   and x2 :: "trm" shows "a \<sharp> Lam [x1].x2 = a \<sharp> [x1].x2"
WTrt_elim_cases_9: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and bop :: "bop"   and e\<^sub>2 :: "char list exp"   and T :: "ty" assumes "P,E,h \<turnstile> e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2 : T" obtains T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" where "bop = Eq" and "T = Boolean" and "P,E,h \<turnstile> e\<^sub>1 : T\<^sub>1" and "P,E,h \<turnstile> e\<^sub>2 : T\<^sub>2"   | "bop = Add" and "T = Integer" and "P,E,h \<turnstile> e\<^sub>1 : Integer" and "P,E,h \<turnstile> e\<^sub>2 : Integer"
Adec: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "Aenc X Y \<in> parts H" shows "X \<in> parts H"
length_mtf2: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "length (mtf2 n x xs) = length xs"
distinct1_rotate: fixes xs :: "'a list" shows "distinct (rotate1 xs) = distinct xs"
induced_automorphism_bij: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "bij (OpposedThinChamberComplexFoldings.induced_automorphism X f g)"
\<Delta>\<Phi>_del_min: fixes h :: "'b heap" assumes "hps h \<noteq> []"   and "no_Empty h" shows "\<Phi> (del_min h) - \<Phi> h \<le> 3 * log 2 (real (size_hps (hps h))) - real (length (hps h)) + 2"
SFAssInitRed: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and v :: "val"   and h :: "heap"   and l :: "sfields" assumes "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh D = \<lfloor>(sfs, Done)\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := Val v,(h, l, sh),False\<rangle> \<rightarrow> \<langle>INIT D ([D],False) \<leftarrow> C\<bullet>\<^sub>sF{D} := Val v,(h, l, sh),False\<rangle>"
trans: fixes b :: "'f word"   and a :: "'f word"   and c :: "'f word" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
similar_matrix_simm: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" assumes "similar_matrix A B" shows "similar_matrix B A"
sorted_list_of_set_length: fixes A :: "'a set" shows "length (sorted_list_of_set A) = card A"
ne1: fixes x :: "'a" assumes "x \<in> T"   and "b \<bullet> x = 0" shows "b \<bullet> g x \<noteq> 1"