hcju/isabelleps huggingface dataset:
ok_mono: fixes m :: "nat"   and mr :: "mregex"   and n :: "nat" assumes "ok m mr"   and "m \<le> n" shows "ok n mr"
in_set_lookup_set_triesD: fixes x :: "'a"   and t :: "('b, 'a list) trie"   and a :: "'b list" assumes "x \<in> set (lookup_tries t a)" shows "x \<in> set_tries t"
winding_number_homotopic_loops: fixes z :: "complex"   and g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex" assumes "homotopic_loops (- {z}) g h" shows "winding_number g z = winding_number h z"
minim_Under: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "minim B \<in> local.Under B"
all_fulfill_def: fixes P :: "'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set" shows "all_fulfill P S = (\<forall>C. C \<in> S \<longrightarrow> P C)"
scale_left_commute: fixes a :: "complex"   and b :: "complex"   and x :: "'d" shows "a *\<^sub>C b *\<^sub>C x = b *\<^sub>C a *\<^sub>C x"
x_eqn: shows "x ^ 3 + a * x\<^sup>2 + b * x + c = 0"
wd_repeat: fixes a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and n :: "nat" assumes "well_def a" shows "well_def (repeat n a)"
coboundedI: fixes A :: "'b word set"   and a :: "'b word" assumes "finite A"   and "a \<in> A" shows "signed.Min A \<le>s a"
abs_pred_tm_def: fixes P :: "('a, 'b) tm \<Rightarrow> bool"   and t :: "('a, 'b) tm" shows "abs_pred P t = True"
permute_cfun_def: fixes p :: "perm"   and f :: "'a \<rightarrow> 'b" shows "p \<bullet> f = (\<Lambda> x. p \<bullet> f\<cdot>(- p \<bullet> x))"
fourier_expansion_periodic_arithmetic_2: fixes k :: "nat"   and f :: "nat \<Rightarrow> complex"   and m :: "nat" assumes "0 < k"   and "periodic_arithmetic f k" shows "f m = (\<Sum>n<k. 1 / of_nat k * (\<Sum>m<k. f m * unity_root k (- int n * int m)) * unity_root k (int m * int n))"
wf_eq_minimal2: fixes r :: "('c \<times> 'c) set" shows "wf r = (\<forall>A. A \<subseteq> Field r \<and> A \<noteq> {} \<longrightarrow> (\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r))"
exists_long_division: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "q \<noteq> []" obtains b :: "'a list list" and r :: "'a list list" where "b \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)" and "r \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)" and "UP.long_divides p q (b, r)"
surj_uni_samp_plus: fixes y :: "nat"   and q :: "nat" assumes "inj_on (\<lambda>b. (y + b) mod q) {..<q}" shows "(\<lambda>b. (y + b) mod q) ` {..<q} = {..<q}"
cmp_simps_2: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "src\<^sub>C (\<Phi>\<^sub>F (f, g)) = src\<^sub>C (F g)"
Nats_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "zero \<in> semiring_1.Nats one plus zero"
gcsrp_i_closed: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> gcsrp_map G N ` set_rcs G N" shows "gcsrp_iop G N a \<in> gcsrp_map G N ` set_rcs G N"
vec_neg: fixes x :: "'a" shows "vec (- x) = - vec x"
that: fixes xs :: "'edge list"   and ys :: "'edge list" assumes "as = xs @ ys"   and "xs \<noteq> []"   and "upd_cs cs xs = []"   and "\<forall>xs' ys'. xs = xs' @ ys' \<and> ys' \<noteq> [] \<longrightarrow> upd_cs cs xs' \<noteq> []"   and "upd_cs [] ys = []" shows "thesis"
pos_zcount_image_zmset: fixes M :: "'a zmultiset"   and t :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>t. 0 \<le> zcount M t"   and "0 < zcount M t" shows "0 < zcount (Auxiliary.image_zmset f M) (f t)"
RModuleEnd_inner_dirsum_el_decomp_nth: fixes Ns :: "'v set list"   and n :: "nat" assumes "\<forall>N\<in>set Ns. RModule FG (\<cdot>) N \<and> N \<subseteq> V"   and "add_independentS Ns"   and "n < length Ns" shows "RModuleEnd FG (\<cdot>) (\<Oplus>N\<leftarrow>Ns. N) (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>Ns. G) then \<Oplus>Ns\<leftarrow> a ! n else (0::'v))"
set_take_subset: fixes n :: "nat"   and xs :: "'g list" shows "set (take n xs) \<subseteq> set xs"
axioms_5: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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 get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name" shows "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
one_sheaf_spec_in_sheaf_spec: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and U :: "'d set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U" shows "comm_ring.one_sheaf_spec R addition multiplication zero unit U \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"
CR_divergence_imp_join: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "CR r"   and "(x, y) \<in> r\<^sup>*"   and "(x, z) \<in> r\<^sup>*" shows "(y, z) \<in> r\<^sup>\<down>"
tw_cmpser_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "tw_cmpser G n g = (if n = 0 then td_gchain G 0 g else td_gchain G n g \<and> (\<forall>l\<le>n - 1. \<natural>\<^bsub>G\<^esub>g l \<triangleright> g (Suc l)))"
right_neutral: fixes a :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) ; nil = a"
ta_4: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>a)"
supports_def: fixes S :: "atom set"   and x :: "'e" shows "S supports x \<equiv> \<forall>a b. a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
inject_3: fixes x6 :: "nat"   and y6 :: "nat" shows "(Label x6 = Label y6) = (x6 = y6)"
prob_space_eP: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "prob_space (eP n x)"
mod_0_imp_mod_pred: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "n mod m = 0" shows "(n - Suc 0) mod m = m - Suc 0"
zero_less_two: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (plus one one)"
elims_2: fixes x :: "nat \<Rightarrow> bool"   and xa :: "lit uprod" assumes "sat_clause x xa" obtains \<sigma> :: "nat \<Rightarrow> bool" and l1 :: "lit" and l2 :: "lit" where "x = \<sigma>" and "xa = Upair l1 l2" and "sat_lit \<sigma> l1 \<or> sat_lit \<sigma> l2"
extra_vars_are_not_control_vars: fixes x :: "'Var\<^sub>C" assumes "x \<notin> range var\<^sub>C_of" shows "x \<notin> \<C>\<^sub>C"
from_index_geq0_vector_split_snd: fixes d :: "nat"   and v :: "rat vec"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec"   and m :: "nat"   and i :: "nat" assumes "\<langle>X\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (from_index_geq0_vector d v)"   and "(x, y) = split_n_m_x d m X"   and "i < dim_vec v"   and "i < m" shows "v $ i \<le> y $ i"
h_def: fixes e :: "'v \<times> 'v" shows "h e = (\<Squnion>n. h_plus n e) - (if cycles \<Delta> \<noteq> {} then \<Squnion>n. h_minus n e else 0)"
wf_envs_progress_mono: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and P' :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and db :: "(String.literal, event_data list set list) mapping"   and a :: "nat"   and b :: "nat"   and \<phi> :: "Formula.formula" assumes "wf_envs \<sigma> j P P' V db"   and "a \<le> b" shows "Monitor.progress \<sigma> P \<phi> a \<le> Monitor.progress \<sigma> P' \<phi> b"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_iUntil_finish_event_conv: fixes k :: "nat"   and m :: "'a message_af"   and event :: "nat \<Rightarrow> bool"   and t0 :: "nat"   and t :: "nat"   and s :: "nat \<Rightarrow> 'a message_af"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b" assumes "0 < k"   and "m \<noteq> NoMsg"   and "\<And>t. event t = (t mod k = k - Suc 0)"   and "t0 = t * k"   and "s = output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = ((\<not> event t1. t1 \<U> t2 [0\<dots>] \<oplus> t0. event t2 \<and> s t2 = m) \<or> (\<not> event t1. t1 \<U> t2 [0\<dots>] \<oplus> t0. \<not> event t2 \<and> s t2 = m \<and> (\<circle> t' t2 [0\<dots>]. s t3 = NoMsg. t3 \<U> t4 [0\<dots>] \<oplus> t'. event t4 \<and> s t4 = NoMsg)))"
filtercomap_inf: fixes f :: "'a \<Rightarrow> 'b"   and F1 :: "'b filter"   and F2 :: "'b filter" shows "filtercomap f (inf F1 F2) = inf (filtercomap f F1) (filtercomap f F2)"
raw_term_induct: fixes y :: "('d, 'e) trm"   and P :: "('d, 'e) trm \<Rightarrow> bool" assumes "y \<in> {\<theta>. dfree \<theta>}"   and "\<And>x. P (raw_term x)" shows "P y"
UN_I: fixes a :: "'b"   and A :: "'b set"   and b :: "'c"   and B :: "'b \<Rightarrow> 'c set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
degree_le: fixes n :: "nat"   and p :: "'d poly" assumes "\<forall>i>n. coeff p i = (0::'d)" shows "degree p \<le> n"
pnormal_length: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a list" assumes "class.semiring_0 plus zero times"   and "semiring_0.pnormal zero p" shows "0 < length p"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "\<lbrace>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<rbrace> = \<a>\<^sub>D[\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>]"
cases: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and a1 :: "('n, 'p, 'r, 't, 'v, 'x) U"   and a2 :: "('o, 'q, 's, 'u, 'w, 'x) U" assumes "rel_U L1 Co1 Co2 Contra1 Contra2 a1 a2" obtains x :: "('n, 'p, 'r, 't, 'v, 'x) U" and y :: "('o, 'q, 's, 'u, 'w, 'x) U" where "a1 = x" and "a2 = y" and "rel_G (rel_U L1 Co1 Co2 Contra1 Contra2) L1 Co1 Co2 Contra1 Contra2 (D_U x) (D_U y)"
nonzero_mult_closed: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a" assumes "domain R"   and "a \<in> nonzero R"   and "b \<in> nonzero R" shows "a \<otimes>\<^bsub>R\<^esub> b \<in> nonzero R"
at_lst_asgd_not_ge: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "rational_preference carrier relation"   and "carrier \<noteq> {}"   and "x \<in> carrier"   and "y \<in> carrier"   and "x \<notin> at_least_as_good y carrier relation" shows "(x, y) \<notin> relation"
disjoint_insert_1: fixes B :: "'a set"   and a :: "'a"   and A :: "'a set" shows "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
intro_elim_6_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<equiv> \<psi> in v]"   and "[\<phi> in v]" shows "[\<psi> in v]"
assms_2: shows "z + 1 / 2 \<notin> \<int>\<^sub>\<le>\<^sub>0"
add3_self: fixes p :: "'a"   and q :: "'a" shows "ccw (0::'a) p (p + q) = ccw (0::'a) p q"
prod_ennreal: fixes A :: "'d set"   and f :: "'d \<Rightarrow> real" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i" shows "(\<Prod>i\<in>A. ennreal (f i)) = ennreal (prod f A)"
proj_set_mono_2: fixes A :: "('a strand_label \<times> 'b) list"   and B :: "('a strand_label \<times> 'b) list"   and n :: "'a" assumes "set A \<subseteq> set B" shows "set (proj_unl n A) \<subseteq> set (proj_unl n B)"
Fun: fixes x2a :: "('a, 'c) Term.term" assumes "x2a \<in> set ts" shows "scf_term scf (x2a \<cdot> \<sigma>) = scf_term scf x2a \<cdot> (\<lambda>x. scf_term scf (\<sigma> x))"
has_integral_cexp_minus_to_infinity: fixes a :: "complex"   and c :: "real" assumes "0 < Re a" shows "((\<lambda>x. exp (x *\<^sub>R - a)) has_integral exp (c *\<^sub>R - a) / a) {c..}"
case2: shows "direction lineB = 0 \<longrightarrow> lineB \<parallel>  lineA"
simps_7: fixes vf :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" shows "[] \<cong> {[], [], vf} = True"
\<G>\<F>_\<F>_subset: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> \<phi> w \<subseteq> \<F> \<phi> w"
Uim: fixes I :: "ZF"   and u :: "ZF" assumes "Elem I U"   and "Elem u (Fun I U)" shows "Elem (HOLZF.Sum (HOLZF.Range u)) U"
is_applicable_then_enabled_2: fixes s :: "nat \<Rightarrow> nat option"   and ast_op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat"   and name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat" assumes "s \<in> valid_states"   and "ast_op \<in> set ast\<delta>"   and "is_operator_applicable_in s (abs_ast_operator ast_op)"   and "lookup_operator ((fst \<circ> the \<circ> lookup_action \<circ> abs_ast_operator) ast_op) = Some (name, pres, effs, layer)" shows "s \<in> subsuming_states (map_of pres)"
signed_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus 1 = (if LENGTH('c) = 1 then uminus one else one)"
failures_def: fixes P :: "'b process" shows "failures P \<equiv> fst (Rep_process P)"
ex_un_\<epsilon>: shows "\<exists>!\<epsilon>. \<guillemotleft>\<epsilon> : f \<star> g \<Rightarrow> trg f\<guillemotright> \<and> S.UP \<epsilon> = UP.unit (trg f) \<cdot>\<^sub>S \<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g))"
effect_def: fixes c :: "'c Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'c" shows "effect c h h' r = (execute c h = Some (r, h'))"
Act_Basic_5: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<A>(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> (\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<A>\<psi>) in v]"
quicksort_correct_case5: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and xs0 :: "'c list"   and lo :: "nat"   and hi :: "nat"   and xs :: "'c list"   and xs' :: "'c list"   and p :: "nat"   and xs'' :: "'c list" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. x \<noteq> y \<Longrightarrow> R (h x) (h y) \<or> R (h y) (h x)"   and "quicksort_pre R h xs0 lo hi xs"   and "partition_spec R h xs lo hi xs' p"   and "\<not> p - Suc 0 \<le> lo"   and "hi \<le> Suc p"   and "quicksort_post R h lo (p - Suc 0) xs' xs''" shows "quicksort_post R h lo hi xs xs''"
388: fixes x :: "'a"   and y :: "'a" shows "- - (x \<sqinter> y) \<sqinter> - x = bot"
continuous_map_open: fixes T1 :: "'b topology"   and T2 :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and U :: "'c set" assumes "continuous_map T1 T2 f"   and "openin T2 U" shows "openin T1 (f -` U \<inter> topspace T1)"
si_empty3: fixes x :: "'c event"   and s :: "'c event list"   and X :: "'c event set" shows "setinterleaving (x # s, X, []) = (if x \<in> X then {} else {x # u |u. u setinterleaves ((s, []), X)})"
continuous_map_in_subtopology: fixes X :: "'c topology"   and X' :: "'d topology"   and S :: "'d set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X (subtopology X' S) f = (continuous_map X X' f \<and> f ` topspace X \<subseteq> S)"
find_indices_empty: fixes x :: "'b"   and Vs :: "'b list" shows "(find_indices x Vs = []) = (x \<notin> set Vs)"
Postdomination_axioms: shows "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"
s_2: shows "s \<le> sum_weights (map (\<lambda>(s, l, t). l) xs)"
scale: fixes r :: "complex"   and b :: "'a" shows "f (r *\<^sub>C b) = r *\<^sub>C f b"
prj_chine_assoc'_1: shows "fgh.Prj\<^sub>1\<^sub>1 \<cdot> fgh.chine_assoc' = fgh.Prj\<^sub>1"
Ball_set_replicate: fixes n :: "nat"   and a :: "'g"   and P :: "'g \<Rightarrow> bool" shows "(\<forall>x\<in>set (replicate n a). P x) = (P a \<or> n = 0)"
\<theta>_def: shows "\<theta> \<equiv> (#>\<^bsub>relcycle_group (p - Suc 0) (subtopology X S) {}\<^esub>) (singular_relboundary_set (p - Suc 0) (subtopology X S) {}) \<circ> chain_boundary p"
while_eq_nfoldli: fixes c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and l :: "'d list"   and \<sigma> :: "'c" shows "WHILE\<^sub>T (FOREACH_cond c) (FOREACH_body f) (l, \<sigma>) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>) = nfoldli l c f \<sigma>"
simps_1: fixes idx :: "idx"   and m :: "nat" shows "wf0 idx (Fo m) = LESS FO m idx"
mono_sqrt': fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "local.sqrt m \<le> local.sqrt n"
FDTs: shows "P \<turnstile> C has_fields FDTs"
remdups_adj_append_dropWhile: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "remdups_adj (xs @ y # ys) = remdups_adj (xs @ [y]) @ remdups_adj (dropWhile (\<lambda>x. x = y) ys)"
suffix_def: fixes xs :: "'d list"   and ys :: "'d list" shows "suffix xs ys = (\<exists>zs. ys = zs @ xs)"
filter_min_def: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "General.filter_min rel xs = General.filter_min_aux rel xs []"
word_log_esimps_3: fixes x :: "'a word" shows "x || 0 = x"
fst_conv: fixes x1 :: "'a"   and x2 :: "'b" shows "fst (x1, x2) = x1"
x_prop: shows "x \<in> set (xs @ [(oid, Some ref)] @ ys) \<and> fst x = ref"
cross7_components_2: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "(x \<times>\<^sub>7 y) $ 2 = x $ 4 * y $ 1 - x $ 1 * y $ 4 + x $ 3 * y $ 5 - x $ 5 * y $ 3 + x $ 6 * y $ 7 - x $ 7 * y $ 6"
iso_point_in_block_img_iff: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and p :: "'c"   and bl :: "'c set" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "p \<in> \<V>"   and "bl \<in># \<B>" shows "(p \<in> bl) = (bij_map p \<in> bij_map ` bl)"
real_root_pow_pos: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 < x" shows "root n x ^ n = x"
OK_None_bot: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b option err" shows "OK None \<sqsubseteq>\<^bsub>Err.le (Opt.le r)\<^esub> x"
poly_mod: fixes q :: "'a poly"   and x :: "'a"   and p :: "'a poly" assumes "poly q x = (0::'a)" shows "poly (p mod q) x = poly p x"
monoid_axioms: shows "Group_Theory.monoid R (+) \<zero>"
bounded_pos: fixes S :: "'a set" shows "bounded S = (\<exists>b>0. \<forall>x\<in>S. norm x \<le> b)"
has_contour_integral_subpath_refl: fixes f :: "complex \<Rightarrow> complex"   and u :: "real"   and g :: "real \<Rightarrow> complex" shows "(f has_contour_integral 0) (subpath u u g)"
line_assm_Cons: fixes \<A> :: "'b \<Rightarrow> bool"   and k :: "'b"   and ks :: "'b set" shows "line_assm \<A> (k \<triangleright> ks) = (if \<A> k then Atom k else \<^bold>\<not> (Atom k)) \<triangleright> line_assm \<A> ks"
sum_list_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero \<equiv> monoid_list.F plus zero"
lattice_of_eq_set: fixes fs :: "'a vec list"   and gs :: "'a vec list" assumes "set fs = set gs"   and "set fs \<subseteq> carrier_vec n" shows "lattice_of fs = lattice_of gs"
imirror_imirror_ident: fixes I :: "nat set" assumes "finite I" shows "imirror (imirror I) = I"
H2_1: shows "u2 \<in> borel_measurable M"
ccField_ccBindsExtra: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ccField (ccBindsExtra \<Gamma>\<cdot>(ae, G)) \<subseteq> fv \<Gamma> \<union> ccField G"
Inf_insert: fixes a :: "'b"   and A :: "'b set" shows "Inf (insert a A) = inf a (Inf A)"
gcsort_stab_input: fixes key :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "stab_inv (\<lambda>k. filter (\<lambda>x. key x = k) xs) key (0, [length xs], xs)"
ty_ex_defined: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and fs :: "local_state \<times> global_state"   and a :: "action" assumes "ty_fs \<Gamma> fs"   and "ty_la \<Gamma> a"   and "la_en fs a = Some True" shows "la_ex fs a \<noteq> None"
cases: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b list"   and a2 :: "'b list" assumes "lexp order a1 a2" obtains x :: "'b" and y :: "'b" and xs :: "'b list" and ys :: "'b list" where "a1 = x__ # xs__" and "a2 = y # ys__" and "order x__ y" and "length xs__ = length ys__"   | xs :: "'b list" and ys :: "'b list" and x :: "'b" where "a1 = x # xs" and "a2 = x # ys" and "lexp order xs ys"