hcju/isabelleps huggingface dataset:
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (\<lambda>f2. F (A1.cod f1, f2)) (\<lambda>f2. F (A1.cod f1, f2)) (\<lambda>f2. F (A1.cod f1, f2))"
prems_9: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"
tendstoI: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F x in F. dist (f x) l < e" shows "filterlim f (topological_space.nhds open l) F"
\<R>\<^sub>G_id: fixes s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" shows "\<R>\<^sub>G id id s = s"
a_all_ptrs_in_heap_def: fixes h :: "(_) heap" shows "Shadow_DOM.all_ptrs_in_heap h = (\<forall>host shadow_root_ptr. h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr \<longrightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h)"
vec_last_index: fixes v :: "'b vec"   and n :: "nat"   and m :: "nat"   and i :: "nat" assumes "v \<in> carrier_vec (n + m)"   and "i < m" shows "vec_last v m $ i = v $ (n + i)"
matroid_ax_3_alt: fixes I :: "Points set"   and X :: "Points set"   and Y :: "Points set" assumes "I \<subseteq> X \<inter> Y" shows "rk (X \<union> Y) + rk I \<le> rk X + rk Y"
fps_compose_mult_distrib: fixes c :: "'a fps"   and a :: "'a fps"   and b :: "'a fps" assumes "fps_nth c 0 = (0::'a)" shows "a * b oo c = (a oo c) * (b oo c)"
intros_3: fixes Pa :: "complex mat"   and Q :: "complex mat"   and R :: "complex mat"   and S1 :: "com"   and S2 :: "com" assumes "ps_P.is_quantum_predicate Pa"   and "ps_P.is_quantum_predicate Q"   and "ps_P.is_quantum_predicate R"   and "ps_P.hoare_partial Pa S1 Q"   and "ps_P.hoare_partial Q S2 R" shows "ps_P.hoare_partial Pa (S1;; S2) R"
fls_zero_eqI: fixes f :: "'b fls" assumes "\<And>n. fls_nth f n = (0::'b)" shows "f = 0"
B_2: fixes x :: "'b" shows "norm (h x) \<le> B * norm x"
IH_2: assumes "hs \<noteq> \<langle>\<rangle>" shows "\<Phi> (pass\<^sub>2 hs) - \<Phi> hs \<le> log 2 (real (size hs))"
infinite_cts_filtration_axioms_def: fixes M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure" shows "infinite_cts_filtration_axioms M F \<equiv> F = infinite_coin_toss_space.nat_filtration M"
Compl: fixes a :: "'c set"   and sp :: "'c set"   and A :: "'c set set" assumes "a \<in> sigma_sets sp A" shows "sp - a \<in> sigma_sets sp A"
resolution_decreases_selected_part: fixes Sel :: "'at set"   and P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set"   and A :: "'at"   and n :: "nat" assumes "ordered_sel_resolvent_upon Sel P1 P2 C A"   and "Neg A \<in> P2"   and "finite P1"   and "finite P2"   and "card (selected_part Sel P2) = Suc n" shows "card (selected_part Sel C) = n"
standard_decompI: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "\<And>h U. (h, U) \<in> set (ps\<^sub>+) \<Longrightarrow> k \<le> poly_deg h"   and "\<And>h U d. \<lbrakk>(h, U) \<in> set (ps\<^sub>+); k \<le> d; d \<le> poly_deg h\<rbrakk> \<Longrightarrow> \<exists>h' U'. (h', U') \<in> set ps \<and> poly_deg h' = d \<and> card U \<le> card U'" shows "standard_decomp k ps"
almost_full_on_imp_no_antichain_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and f :: "nat \<Rightarrow> 'a" assumes "almost_full_on P A" shows "\<not> antichain_on P f A"
is_top_sorted_alt: fixes R :: "('b \<times> 'b) set"   and l :: "'b list" shows "is_top_sorted R l = (\<forall>x y. (x, y) \<in> list_before_rel l \<longrightarrow> (y, x) \<notin> R\<^sup>*)"
cancel_less: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "normal F" shows "(F x < F y) = (x < y)"
basic_invar_def: fixes s :: "('v, 'a) state_scheme" shows "basic_invar s \<equiv> set (stack s) = dom (discovered s) - dom (finished s) \<and> distinct (stack s) \<and> (stack s \<noteq> [] \<longrightarrow> last (stack s) \<in> V0) \<and> dom (finished s) \<subseteq> dom (discovered s) \<and> Domain (pending s) \<subseteq> dom (discovered s) - dom (finished s) \<and> pending s \<subseteq> E"
abs_convergent_euler_product': fixes f :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>n. norm (f n))"   and "completely_multiplicative_function f" shows "abs_convergent_prod (\<lambda>p. if prime p then inverse ((1::'a) - f p) else (1::'a))"
interp_ins_find_ref: fixes xs :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a"   and ys :: "('a \<times> 'a option) list" assumes "insert_ops (xs @ [(oid, Some ref)] @ ys)"   and "ref \<in> set (interp_ins (xs @ [(oid, Some ref)] @ ys))" shows "\<exists>r. (ref, r) \<in> set xs"
converse_mono: fixes r :: "('c \<times> 'a) set"   and s :: "('c \<times> 'a) set" shows "(r\<inverse> \<subseteq> s\<inverse>) = (r \<subseteq> s)"
pref_in_at_least_as: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq> y" shows "x \<in> at_least_as_good y carrier relation"
varnum_def: fixes X :: "'b set"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "varnum X t = (if keys t - X = {} then 0 else Suc (Max (elem_index ` (keys t - X))))"
wlconf_upd: fixes G :: "prog"   and s :: "st"   and l :: "'a \<Rightarrow> val option"   and L :: "'a \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and vn :: "'a" assumes "G,s\<turnstile>l[\<sim>\<Colon>\<preceq>]L"   and "G,s\<turnstile>v\<Colon>\<preceq>T"   and "L vn = Some T" shows "G,s\<turnstile>l(vn \<mapsto> v)[\<sim>\<Colon>\<preceq>]L"
borel_measurable_subalgebra: fixes N :: "'a measure"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "sets N \<subseteq> sets M"   and "space N = space M"   and "f \<in> borel_measurable N" shows "f \<in> borel_measurable M"
path_cons_elim: fixes a :: "'action"   and r :: "'action list"   and p :: "'state" assumes "path (a # r) p" obtains "en a p" and "path r (ex a p)"
equiv_utransD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "equiv_utrans t u"   and "unitary P" shows "t P = u P"
invar_C_ss_V: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat"   and u :: "nat" assumes "nf_invar c src dst PRED C N d"   and "u \<in> C" shows "u \<in> Graph.V c"
iso_def: shows "local.iso \<equiv> index_f \<circ> (\<lambda>q. eq_app_right language `` {path_to q})"
poincare_line_inversion_full: fixes u :: "complex_homo"   and v :: "complex_homo"   and x :: "complex_homo" assumes "u \<noteq> v"   and "on_circline (poincare_line u v) x" shows "on_circline (poincare_line u v) (inversion x)"
not_None_eq: fixes x :: "\<langle>'d\<rangle>\<^sub>\<bottom>" shows "(x \<noteq> None) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
sparse_mono: fixes \<Delta> :: "real"   and P :: "(int \<times> int) set"   and \<delta> :: "real" assumes "sparse \<Delta> P"   and "\<delta> \<le> \<Delta>" shows "sparse \<delta> P"
inj_idom_hom_axioms: shows "inj_idom_hom of_rat"
sc_a_0: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and a :: "'i" assumes "R module M"   and "a \<in> carrier R" shows "a \<cdot>\<^sub>s\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>M\<^esub>"
not_None_eq: fixes x :: "'c option" shows "(x \<noteq> None) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
prec_eq_None_or_equal: fixes s1 :: "val option"   and s2 :: "val option" assumes "s1 \<preceq> s2" shows "s1 = \<bottom> \<or> s1 = s2"
assms: fixes d :: "'a \<Rightarrow> nat"   and hs :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) set"   and m :: "nat"   and p :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and q :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) \<times> ('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and data :: "nat \<times> 'b" assumes "dickson_grading d"   and "set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "unique_idx (p # q # hs @ map (fst \<circ> snd) ps @ map (snd \<circ> snd) ps) data"   and "\<And>p' q'. \<lbrakk>(p', q') \<in>\<^sub>p snd ` set ps; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "p \<in> B"   and "q \<in> B"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "crit data hs ps p q" shows "crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q)"
ffieldD: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<F>\<^sub>\<bullet> r" shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r \<union>\<^sub>\<circ> \<R>\<^sub>\<bullet> r"
isZFfunE: fixes f :: "ZF" assumes "isZFfun f" obtains "isOpair f" and "isOpair (Fst f)" and "isFun (Snd f)" and "Snd f |\<subseteq>| HOLZF.Domain (Snd f) |\<times>| HOLZF.Range (Snd f)" and "HOLZF.Domain (Snd f) = |dom|f \<and> HOLZF.Range (Snd f) |\<subseteq>| |cod|f"
qSwap_twice_preserves_qGood: fixes zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm" shows "qGood (X #[[x \<and> y]]_zs #[[x' \<and> y']]_zs') = qGood X"
filter_compliant_stateful_ACS_accu_induction_maximal: fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and edgesList :: "('b \<times> 'b) list"   and accu :: "('b \<times> 'b) list"   and stateful :: "('b \<times> 'b) set" assumes "valid_reqs (get_ACS M)"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "set edgesList \<subseteq> E"   and "set accu \<subseteq> E"   and "stateful = set (filter_compliant_stateful_ACS_accu \<lparr>nodes = V, edges = E\<rparr> M accu edgesList)"   and "\<forall>e\<in>E - (set edgesList \<union> set accu \<union> {e \<in> E. e \<in> backflows E}). \<not> \<Union> (get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = set accu \<union> {e}\<rparr>)) \<subseteq> backflows (filternew_flows_state \<lparr>hosts = V, flows_fix = E, flows_state = set accu \<union> {e}\<rparr>)" shows "\<forall>e\<in>E - (stateful \<union> {e \<in> E. e \<in> backflows E}). \<not> \<Union> (get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = stateful \<union> {e}\<rparr>)) \<subseteq> backflows (filternew_flows_state \<lparr>hosts = V, flows_fix = E, flows_state = stateful \<union> {e}\<rparr>)"
add_Pair: fixes a :: "'b"   and b :: "'c"   and c :: "'b"   and d :: "'c" shows "(a, b) + (c, d) = (a + c, b + d)"
netglobalE: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool"   and s :: "'s net_state"   and Q :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool" assumes "netglobal P s"   and "\<And>\<sigma>. \<lbrakk>P \<sigma>; fst (initmissing (netgmap sr s)) = \<sigma>\<rbrakk> \<Longrightarrow> Q \<sigma>" shows "netglobal Q s"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a AND b) AND c = a AND b AND c"
mUnionM_return: fixes A :: "'a multiset" shows "mUnionM {#return {#x#}. x \<in># A#} = return A"
simps_2: fixes x :: "'b"   and n :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" shows "AVL.insert x (MKT n l r h) = (if x = n then MKT n l r h else if x < n then mkt_bal_l n (AVL.insert x l) r else mkt_bal_r n l (AVL.insert x r))"
False: shows "\<not> (x = (0::'a) \<or> y = (0::'a))"
actual_desc_1: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<equiv> (\<^bold>\<exists>!x. \<^bold>\<A>\<phi> x) in v]"
ring_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "ring_ow U plus zero minus uminus times \<equiv> semiring_ow U plus times \<and> ab_group_add_ow U plus zero minus uminus"
borel_measurable_Min: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M" shows "(\<lambda>x. MIN i\<in>I. f i x) \<in> borel_measurable M"
Let_3: shows "\<Delta> @ \<Gamma> : body \<Down>\<^bsub>L\<^esub> \<Theta> : z"
hull_rel_def: shows "hull_rel \<equiv> {(xa, x). hull_relp xa x}"
chamber_distance_le: fixes C :: "'b set"   and Cs :: "'b set list"   and D :: "'b set" assumes "codomain.gallery (C # Cs @ [D])" shows "order.greater_eq (Suc (length Cs)) (codomain.chamber_distance C D)"
tr: assumes "fst s < fst r"   and "fst t = fst s" shows "snd t \<le> snd r"
matchC_ZO_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchC_ZO theta c d \<equiv> \<forall>s t c' s'. s \<approx> t \<and> (c, s) \<rightarrow>c (c', s') \<longrightarrow> s' \<approx> t \<and> (c', d) \<in> theta \<or> (\<exists>d' t'. (d, t) \<rightarrow>c (d', t') \<and> s' \<approx> t' \<and> (c', d') \<in> theta) \<or> (\<exists>t'. (d, t) \<rightarrow>t t' \<and> s' \<approx> t' \<and> discr c')"
addrow_mat: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat"   and l :: "nat"   and a :: "'a"   and k :: "nat" assumes "A \<in> carrier_mat n nc"   and "l < n" shows "addrow a k l A = addrow_mat n a k l * A"
simps_1: fixes uu :: "('a \<times> 'a) set"   and uv :: "'b \<Rightarrow> 'a"   and U :: "'a set" shows "sources_aux uu uv U [] = U"
real_integrable_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" shows "integrable M f = (f \<in> borel_measurable M \<and> \<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity> \<and> \<integral>\<^sup>+ x. ennreal (- f x) \<partial>M \<noteq> \<infinity>)"
apex_composite: shows "f_src.apex = f.leg0 \<down>\<down> src.leg1"
cones_map_mapsto: fixes f :: "'s" assumes "S.arr f" shows "cones_map f \<in> extensional (cones (S.cod f)) \<inter> (cones (S.cod f) \<rightarrow> cones (S.dom f))"
eval_on_carrier: fixes p :: "nat \<Rightarrow> 'a"   and S :: "('e, 'f) ring_scheme"   and phi :: "'a \<Rightarrow> 'e"   and s :: "'e" assumes "p \<in> carrier P" shows "eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub>i\<in>{..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
to_e_list_1: fixes a :: "b_e" shows "[$a] = $* [a]"
preorder_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "preorder_on A r \<equiv> refl_on A r \<and> trans r"
ipurge_ref_aux_subset: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and xs :: "'e list"   and X :: "'e set" shows "ipurge_ref_aux I D U xs X \<subseteq> X"
wf_graph_add_delete_edge_simp: fixes Y :: "('v \<times> 'v) set" shows "add_edge a1 a2 (delete_edges G Y) = delete_edges G (Y - {(a1, a2)})"
periodic_orbit_def: fixes x :: "'a" shows "periodic_orbit x = (closed_orbit x \<and> 0 < period x)"
simps_4: fixes p :: "('c, 'b) fm"   and q :: "('c, 'b) fm" shows "nominals (p \<^bold>\<or> q) = nominals p \<union> nominals q"
ts_ord_anti: fixes x :: "'a"   and y :: "'a" assumes "n x \<sqsubseteq> n y" shows "t y \<sqsubseteq> t x"
decompose_Some: fixes f :: "'c"   and ss :: "('c, 'd) Term.term list"   and g :: "'c"   and ts :: "('c, 'e) Term.term list"   and E :: "(('c, 'd) Term.term \<times> ('c, 'e) Term.term) list" assumes "decompose (Fun f ss) (Fun g ts) = Some E" shows "f = g \<and> length ss = length ts \<and> E = zip ss ts"
last_message_append_replicate_NoMsg: fixes xs :: "'d message_af list"   and n :: "nat" shows "last_message (xs @ NoMsg\<^bsup>n\<^esup>) = last_message xs"
select_convs_2: fixes carrier :: "'d set"   and eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and le :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and more :: "'c" shows "gorder.more \<lparr>carrier = carrier, eq = eq, le = le, \<dots> = more\<rparr> = more"
arc_to_block_to_ends: fixes G :: "('a, 'b) pre_digraph"   and e :: "'b" assumes "graph G" shows "{fst (arc_to_ends G e), snd (arc_to_ends G e)} = graph.arc_to_block G e"
normalize_poly_def: fixes p :: "(char list multiset \<times> int) multiset" shows "normalize_poly p = fold_mset add_to_coefficient {#} p"
UnI1: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A" shows "c \<in> A \<union> B"
properfactor_cong_l: fixes x' :: "'a list"   and x :: "'a list"   and y :: "'a list" assumes "x' \<sim>\<^bsub>K [X]\<^esub> x"   and "properfactor (K [X]) x y"   and "x \<in> carrier (K [X])"   and "x' \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "properfactor (K [X]) x' y"
fds_left_inverse: fixes f :: "'a fds" assumes "fds_nth f 1 \<noteq> (0::'a)" shows "inverse f * f = 1"
fresh_distinct_ups: fixes V :: "Vars.var set"   and S :: "stack_elem list" assumes "atom ` V \<sharp>* S" shows "V \<inter> upds S = {}"
linordered_field_no_ub: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less x y"
not_isFault_iff: fixes t :: "('a, 'b) xstate" shows "(\<not> isFault t) = (\<forall>f. t \<noteq> Fault f)"
Rep_triangle_inverse: fixes x :: "triangle" shows "Abs_triangle (Rep_triangle x) = x"
iso_iff2: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and> (\<forall>a\<in>Field r. \<forall>b\<in>Field r. ((a, b) \<in> r) = ((f a, f b) \<in> r')))"
assms_2: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>'"
UN_Un: fixes M :: "'c \<Rightarrow> 'a set"   and A :: "'c set"   and B :: "'c set" shows "\<Union> (M ` (A \<union> B)) = \<Union> (M ` A) \<union> \<Union> (M ` B)"
Cast: fixes A :: "triple set"   and P :: "assn"   and e :: "expr"   and C :: "cname"   and Q :: "vassn" assumes "A \<turnstile>\<^sub>e {P} e {\<lambda>v s. (case v of Null \<Rightarrow> True | Addr a \<Rightarrow> obj_class s a \<preceq>C C) \<longrightarrow> Q v s}" shows "A \<turnstile>\<^sub>e {P} Cast C e {Q}"
cases: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b llist" assumes "full_chain R a" obtains x :: "'b" where "a = LCons x__ LNil" and "\<forall>y. \<not> R x__ y"   | xs :: "'b llist" and x :: "'b" where "a = LCons x xs" and "full_chain R xs" and "R x (lhd xs)"
main: fixes pa :: "real poly"   and i :: "nat" assumes "(pa, i) \<in> set pis" shows "set (roots_of_real_main pa) = {x. poly pa x = 0}"
exec_If\<^sub>C': fixes \<sigma> :: "'a control_state_scheme"   and P :: "'a control_state_scheme \<Rightarrow> bool"   and B\<^sub>1 :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and B\<^sub>2 :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> _if_SECLEAN P B\<^sub>1 B\<^sub>2 ;- M) = (\<sigma> \<Turnstile> M)"
sublinear_sub_conj: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sublinear t" shows "sub_conj t"
trace_square_less_square_trace: fixes B :: "complex mat"   and n :: "nat" assumes "B \<in> carrier_mat n n"   and "diagonal_mat B"   and "\<And>i. i < n \<Longrightarrow> 0 \<le> B $$ (i, i)" shows "trace (B * B) \<le> (trace B)\<^sup>2"
weak_ring_morphism_is_iso: fixes f :: "'a \<Rightarrow> 'd"   and I :: "'a set" assumes "weak_ring_morphism f I R" shows "(\<lambda>x. the_elem (f ` x)) \<in> ring_iso (R Quot I) (image_ring f R)"
equalityD1: fixes A :: "'l set"   and B :: "'l set" assumes "A = B" shows "A \<subseteq> B"
exp_pi_i': shows "exp (\<i> * cor pi) = - 1"
simps: fixes k :: "nat"   and xs :: "'b list" shows "T_brauns k xs = (if xs = [] then 0 else let ys = take (2 ^ k) xs; zs = drop (2 ^ k) xs; ts = brauns (k + 1) zs in 4 * min (2 ^ k) (length xs) + T_brauns (k + 1) zs)"
top_right_normalize: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and bot :: "'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "bounded_quasi_semilattice f top bot normalize" shows "f a top = normalize a"
intros_15: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val"   and T' :: "ty"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and e\<^sub>2 :: "expr"   and T\<^sub>2 :: "ty" assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor>"   and "P,E(V \<mapsto> T),h \<turnstile> e\<^sub>2 :' T\<^sub>2"   and "P \<turnstile> T' \<le> T"   and "is_type P T" shows "P,E,h \<turnstile> {V:T; V:=Val v;; e\<^sub>2} :' T\<^sub>2"
fsafe_Diamond_simps: fixes a :: "('a, 'b, 'c) hp"   and p :: "('a, 'b, 'c) formula" shows "fsafe (\<langle> a \<rangle> p) = (hpsafe a \<and> fsafe p)"
len: shows "length (coeffs G2) \<le> length (coeffs G1)"
dg_GRPH_Cod_app: fixes x :: "V"   and \<alpha> :: "V" assumes "x \<in>\<^sub>\<circ> all_dghms \<alpha>" shows "dg_GRPH \<alpha>\<lparr>Cod\<rparr>\<lparr>x\<rparr> = x\<lparr>HomCod\<rparr>"
sstar_increasing: fixes y :: "'a" assumes "has_least_prefixpoint (Sf y)" shows "y \<le> sstar y"
diff_fun_def: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'e" shows "diff_fun k charts f \<equiv> diff k charts charts_eucl f"
one_le_floor: fixes x :: "'b" shows "(1 \<le> \<lfloor>x\<rfloor>) = ((1::'b) \<le> x)"
continuous_transform_within: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and d :: "real"   and g :: "'a \<Rightarrow> 'b" assumes "continuous (at x within s) f"   and "0 < d"   and "x \<in> s"   and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "continuous (at x within s) g"
idiom_comp: fixes g :: "'a itrm"   and f :: "'a itrm"   and x :: "'a itrm" shows "idiom_rule (Pure \<B> \<diamondop> g \<diamondop> f \<diamondop> x) (g \<diamondop> (f \<diamondop> x))"
steps_Stuck_prop: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate"   and c' :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate" assumes "\<Gamma> \<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"   and "s = Stuck" shows "s' = Stuck"
fempty_ffilter: fixes A :: "'b fset" shows "ffilter (\<lambda>_. False) A = {||}"
subst_simp_unit: fixes x :: "'a"   and M :: "'a trm" shows "Unit[x ::= M] = Unit"
rep_eq: fixes x :: "'x"   and xa :: "'x fset" shows "fset (finsert x xa) = insert x (fset xa)"
extra_axiom_imp_Pf: fixes e :: "(name, hf) finfun" shows "Pf \<lbrakk>\<guillemotleft>extra_axiom\<guillemotright>\<rbrakk>e"
Liminf_at: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "Liminf (at x) f = (SUP e\<in>{0<..}. Inf (f ` (ball x e - {x})))"