hcju/isabelleps huggingface dataset:
ring_of_sets_def: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "ring_of_sets \<Omega> M \<equiv> semiring_of_sets \<Omega> M \<and> ring_of_sets_axioms M"
sorted_le_last: fixes xs :: "'a list"   and x :: "'a" assumes "sorted xs"   and "x \<in> list.set xs" shows "x \<le> last xs"
inv: shows "oalist_inv_raw (update_by_pair (k, 0::'b) xs)"
super_orientable_def: fixes x :: "'a" shows "super_orientable x \<equiv> \<exists>y. super_orientation x y"
RPO_S_subset_RPO_NS: shows "RPO_S \<subseteq> RPO_NS"
not_in_path_image_join: fixes x :: "'c"   and g1 :: "real \<Rightarrow> 'c"   and g2 :: "real \<Rightarrow> 'c" assumes "x \<notin> path_image g1"   and "x \<notin> path_image g2" shows "x \<notin> path_image (g1 +++ g2)"
simps_66: fixes x1 :: "const" shows "RID\<^sub>V (CV x1) = {}"
cases: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d"   and k :: "'d \<Rightarrow> nat"   and a1 :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)"   and a2 :: "'e"   and a3 :: "('d \<Rightarrow> real) set"   and a4 :: "'e"   and a5 :: "('d \<Rightarrow> real) set" assumes "Regions X v n not_in_X"   and "Regions.step_z_beta X k v n not_in_X a1 a2 a3 a4 a5" obtains A :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)" and l :: "'e" and Z :: "('d \<Rightarrow> real) set" and l' :: "'e" and Z' :: "('d \<Rightarrow> real) set" where "a1 = A" and "a2 = l" and "a3 = Z" and "a4 = l'" and "a5 = Beta_Regions'.Approx\<^sub>\<beta> X k v n not_in_X Z'" and "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"
simps_10: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and f3 :: "('d \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'c"   and x1 :: "'d" shows "(case Const x1 of Const x \<Rightarrow> f1 x | Load x \<Rightarrow> f2 x | Apply x \<Rightarrow> f3 x) = f1 x1"
exec_append: fixes s :: "val option"   and rs :: "(proc \<times> val) list"   and rs' :: "(proc \<times> val) list" shows "(s \<star> rs) \<star> rs' = s \<star> append rs' rs"
le_infI2: fixes b :: "'a multiset"   and x :: "'a multiset"   and a :: "'a multiset" assumes "b \<subseteq># x" shows "a \<inter># b \<subseteq># x"
runit_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<r>\<^sub>1[a] : a \<otimes>\<^sub>1 \<I>\<^sub>1 \<rightarrow> a\<guillemotright>"
normed_vectorspace_def: fixes V :: "'c set"   and norm :: "'c \<Rightarrow> real" shows "normed_vectorspace V norm \<equiv> vectorspace V \<and> Normed_Space.norm V norm"
T_AE_iterates: fixes P :: "'a \<Rightarrow> bool" assumes "almost_everywhere M P" shows "AE x in M. \<forall>n. P ((T ^^ n) x)"
False: shows "\<not> ((fst instr = load_store_type STH \<or> fst instr = load_store_type STHA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s1)))) \<noteq> 0)"
hyps_5: shows "is_lub,P,E \<turnstile> e1' :: U"
distinct_25: fixes x3 :: "('a, 'b, 'c) formula"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) formula" shows "! x3 \<noteq> Exists x51 x52"
element_exists: fixes A :: "'a set" assumes "A \<in> P" shows "\<exists>a\<in>S. a \<in> A"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"   and "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"   and "l_set_disconnected_nodes_get_disconnected_nodes_wf_axioms heap_is_wellformed get_disconnected_nodes set_disconnected_nodes" shows "l_set_disconnected_nodes_get_disconnected_nodes_wf type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
simps: fixes a1 :: "'a itrm"   and a2 :: "'a itrm" shows "idiom_rule a1 a2 = ((\<exists>x. a1 = Pure \<I> \<diamondop> x \<and> a2 = x) \<or> (\<exists>g f x. a1 = Pure \<B> \<diamondop> g \<diamondop> f \<diamondop> x \<and> a2 = g \<diamondop> (f \<diamondop> x)) \<or> (\<exists>f x. a1 = Pure f \<diamondop> Pure x \<and> a2 = Pure (f \<degree> x)) \<or> (\<exists>f x. a1 = f \<diamondop> Pure x \<and> a2 = Pure (\<T> \<degree> x) \<diamondop> f))"
simps_4: fixes p :: "pf"   and q :: "pf"   and xs :: "nat list" shows "eval_pf (Or p q) xs = (eval_pf p xs \<or> eval_pf q xs)"
fps_to_fls_nonzeroI: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "fps_to_fls f \<noteq> 0"
assms_2: shows "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"
select_convs_3: fixes types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "global \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = global"
same_append_eq: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs \<cdot> ys = xs \<cdot> zs) = (ys = zs)"
e_1: shows "e \<in> UNIV \<rightarrow> {- 1..1}"
antipar_1: shows "trg (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)\<^sup>* = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)"
continuous: shows "continuous_on (UNIV \<times> X) (\<lambda>(t, x). f x)"
induct: fixes P :: "nat list \<Rightarrow> nat \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b) \<Rightarrow> (('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec \<times> (('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec \<times> (('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "nat"   and a2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a3 :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec \<times> (('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec \<times> (('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "\<And>i bezout P A Q. P [] i bezout (P, A, Q)"   and "\<And>j xs i bezout P A Q. \<lbrakk>A $h from_nat i $h from_nat i dvd A $h from_nat j $h from_nat j \<Longrightarrow> P xs i bezout (P, A, Q); \<And>x xa y xb ya xc yb xd yc xe xf xg yd. \<lbrakk>\<not> A $h from_nat i $h from_nat i dvd A $h from_nat j $h from_nat j; x = bezout (A $h from_nat i $h from_nat i) (A $h from_nat j $h from_nat j); (xa, y) = x; (xb, ya) = y; (xc, yb) = ya; (xd, yc) = yb; xe = diagonal_step A i j yc xd; xf = diagonal_step_PQ A i j bezout; (xg, yd) = xf\<rbrakk> \<Longrightarrow> P xs i bezout (xg ** P, xe, Q ** yd)\<rbrakk> \<Longrightarrow> P (j # xs) i bezout (P, A, Q)" shows "P a0 a1 a2 a3"
\<sigma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_not_empty: shows "\<sigma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G \<noteq> {}"
sl2: shows "abscissa A - abscissa B \<in> radical_sqrt"
homotopic_with_imp_property: fixes P :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "homotopic_with P X Y f g" shows "P f \<and> P g"
odd_of_distinct: fixes p :: "nat \<Rightarrow> nat"   and i :: "nat"   and n :: "nat" assumes "odd_of_distinct p i \<noteq> 0"   and "\<And>i. p i \<noteq> 0 \<Longrightarrow> i \<le> n" shows "1 \<le> i \<and> i \<le> n"
simps_7: fixes x :: "nat"   and P :: "pol"   and Q :: "pol"   and y :: "nat"   and R :: "pol" shows "Pinj x P \<langle>+\<rangle> PX Q y R = (if x = 0 then P \<langle>+\<rangle> PX Q y R else if x = 1 then PX Q y (R \<langle>+\<rangle> P) else PX Q y (R \<langle>+\<rangle> Pinj (x - 1) P))"
set_sel_2: fixes a :: "'c tree" assumes "a \<noteq> \<langle>\<rangle>" shows "value a \<in> set_tree a"
cobounded2: fixes b :: "'b upred"   and a :: "'b upred" shows "(a \<or> b) \<sqsubseteq> b"
lc: fixes s :: "'a"   and t :: "'a"   and a :: "'b"   and u :: "'a"   and b :: "'b" assumes "(s, t) \<in> L a"   and "(s, u) \<in> E b" shows "\<exists>xs. local.lcliff r a b xs \<and> (t, u) \<in> local.lconv xs"
gseqE: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and A :: "'b set" assumes "(f, g) \<in> mbs.gseq A" obtains i :: "nat" where "\<forall>i. f i \<in> A" and "\<forall>i. g i \<in> A" and "size (g i) < size (f i)" and "\<forall>j<i. f j = g j"
nonpos_Ints_Int: fixes x :: "'a" assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0" shows "x \<in> \<int>"
bangRightSC: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<parallel> !P \<leadsto>[Rel] !P"
fv_supp_exp: fixes e :: "exp" shows "supp e = atom ` fv e"
insert_before_locs_def: shows "Shadow_DOM.insert_before_locs \<equiv> l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before_locs Shadow_DOM.set_child_nodes_locs Shadow_DOM.adopt_node_locs set_disconnected_nodes_locs"
derivable_clauses_are_finite: fixes C :: "'a eclause"   and P :: "'a eclause set"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "derivable C P S \<sigma> k C'"   and "\<forall>x\<in>P. finite (cl_ecl x)" shows "finite (cl_ecl C) \<and> finite C'"
decrpoly_zero: fixes p :: "poly" shows "(decrpoly p = 0\<^sub>p) = (p = 0\<^sub>p)"
simps_6: fixes M :: "('b, 'c) model"   and g :: "'d \<Rightarrow> 'b"   and uw :: "'b"   and i :: "'d"   and p :: "('c, 'd) fm" shows "(M, g, uw \<Turnstile> \<^bold>@ i p) = (M, g, g i \<Turnstile> p)"
sep_disj_prod_commute_2: fixes n :: "'b"   and ps :: "'a" shows "(0::'a, n) ## (ps, 0::'b)"
ndet_def: fixes P :: "'b process"   and Q :: "'b process" shows "P |-| Q \<equiv> Abs_process (F P \<union> F Q, D P \<union> D Q)"
abs_summable_on_Sigma_project1': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "(\<lambda>(x, y). f x y) abs_summable_on Sigma A B"   and "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)" shows "(\<lambda>x. infsetsum (f x) (B x)) abs_summable_on A"
LT_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and r :: "('b \<times> 'b) set"   and t :: "('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> (bool \<times> 'b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> (bool \<times> 'b \<times> 'a) list) \<times> ('a \<times> (bool \<times> 'b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> 'a \<times> (bool \<times> 'b \<times> 'a) list" shows "LT ars r t = (let (\<beta>, \<alpha>, \<sigma>1, \<sigma>2, \<sigma>3, \<tau>1, \<tau>2, \<tau>3) = t in LT1 ars r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<and> LT1 ars r \<alpha> \<beta> \<tau>1 \<tau>2 \<tau>3 \<and> lst_conv \<sigma>3 = lst_conv \<tau>3)"
is_valid_problem_strips_operator_variable_sets_3: fixes \<Pi> :: "'b strips_problem"   and op :: "'b strips_operator" assumes "is_valid_problem_strips \<Pi>"   and "op \<in> set (\<Pi>\<^sub>\<O>)" shows "set (delete_effects_of op) \<subseteq> set (\<Pi>\<^sub>\<V>)"
LIMSEQ_unique: fixes "open" :: "'e set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.t2_space open"   and "filterlim X (topological_space.nhds open a) sequentially"   and "filterlim X (topological_space.nhds open b) sequentially" shows "a = b"
smult_sg_nsg: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "G \<guillemotright> H \<diamondop>\<^bsub>G\<^esub> N "
set_rel_br: fixes a :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" shows "\<langle>br a I\<rangle>set_rel = br ((`) a) (\<lambda>X. Ball X I)"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom (src f))"
Iio_eq_empty_iff: fixes n :: "'b" shows "({..<n} = {}) = (n = bot)"
gb_sig_z_def: fixes rword_strict :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and fs0 :: "('a \<Rightarrow>\<^sub>0 'c) list" shows "gb_sig_z rword_strict fs0 = (let fs = rev (remdups (rev (removeAll 0 fs0))); res = rb_spp_aux fs rword_strict (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), 0) in (fst (fst res), snd res))"
subst_subtype: fixes \<Delta> :: "binding list"   and V :: "POPLmark.type"   and \<Gamma> :: "binding list"   and T :: "POPLmark.type"   and U :: "POPLmark.type" assumes "\<Delta> @ VarB V \<Colon> \<Gamma> \<turnstile> T <: U" shows "\<down>\<^sub>e 1 0 \<Delta> @ \<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> T <: \<down>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> U"
puree_def: fixes P :: "bool" shows "\<up> P \<equiv> \<lambda>h. h = (0::'c) \<and> P"
valid_regions_distinct: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> intv"   and J :: "'a \<Rightarrow> 'a \<Rightarrow> intv'"   and r :: "('a \<times> 'a) set"   and I' :: "'a \<Rightarrow> intv"   and J' :: "'a \<Rightarrow> 'a \<Rightarrow> intv'"   and r' :: "('a \<times> 'a) set"   and v :: "'a \<Rightarrow> real" assumes "valid_region X k I J r"   and "valid_region X k I' J' r'"   and "v \<in> region X I J r"   and "v \<in> region X I' J' r'" shows "region X I J r = region X I' J' r'"
simps_4: fixes v :: "('a, 'b) Combinators"   and va :: "('a, 'b) Combinators"   and y :: "('a, 'b) Combinators list" shows "removeShadowRules2 (v \<oplus> va # y) = v \<oplus> va # removeShadowRules2 y"
simps_24: fixes vc :: "bexp"   and vd :: "'a acom"   and ve :: "'a acom"   and vf :: "'a"   and v :: "char list"   and va :: "aexp"   and vb :: "'a" shows "(IF vc THEN vd ELSE ve {vf} \<le> v ::= va {vb}) = False"
hoare_r_weaken_pre_1: fixes p :: "'a upred"   and Q :: "'a hrel"   and r :: "'a upred"   and q :: "'a upred" assumes "\<lbrace>p\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u" shows "\<lbrace>p \<and> q\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u"
UnionE: fixes A :: "'b"   and C :: "'b set set" assumes "A \<in> \<Union> C" obtains X :: "'b set" where "A \<in> X" and "X \<in> C"
Dup_omit_branch_mem_same_opening: fixes p :: "('c, 'd) fm"   and i :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and xs :: "(nat \<times> nat) set" assumes "Dup p i branch xs"   and "p at i in branch" shows "p at i in omit_branch xs branch"
exec1_drop_suffix: fixes cs1 :: "com list"   and cs2 :: "com list"   and s :: "state"   and cs1' :: "com list"   and s' :: "state" assumes "(cs1 @ cs2, s) \<rightarrow> (cs1' @ cs2, s')"   and "cs1 \<noteq> []" shows "(cs1, s) \<rightarrow> (cs1', s')"
density_cong': fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x" shows "density M f = density M g"
intros: fixes a :: "'g"   and b :: "'h"   and r :: "('g \<times> 'h) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
prime_factors_list_prime: fixes n :: "nat" assumes "prime n" shows "\<exists>qs. prime_factors n = set qs \<and> prod_list qs = n \<and> length qs = 1"
vs: shows "vs \<in> F \<rightarrow> carrier_vec n"
rel_setI: fixes A :: "'d set"   and B :: "'e set"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" shows "rel_set R A B"
simps_2: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" shows "one_call_in_path x (y # xs) = (if x = y then no_call_in_path x xs else one_call_in_path x xs)"
cmp_in_hom: 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"   and \<mu>\<nu> :: "'c \<times> 'c" 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"   and "partial_magma.arr (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>" shows "partial_magma.in_hom V\<^sub>D (composite_pseudofunctor.cmp V\<^sub>B H\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<mu>\<nu>) (((\<lambda>\<mu>\<nu>. H\<^sub>D (fst \<mu>\<nu>) (snd \<mu>\<nu>)) \<circ> weak_arrow_of_homs.FF V\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>D src\<^sub>D trg\<^sub>D (G \<circ> F)) (partial_magma.dom (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>)) ((G \<circ> F \<circ> (\<lambda>\<mu>\<nu>. H\<^sub>B (fst \<mu>\<nu>) (snd \<mu>\<nu>))) (partial_magma.cod (subcategory.comp (product_category.comp V\<^sub>B V\<^sub>B) (\<lambda>\<mu>\<nu>. partial_magma.arr V\<^sub>B (fst \<mu>\<nu>) \<and> partial_magma.arr V\<^sub>B (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))) \<mu>\<nu>))"
f_def: shows "f = (\<lambda>x xa. if xa \<in> {a..real x} then xa powr e else 0)"
Pi_mem: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set"   and x :: "'g" assumes "f \<in> Pi A B"   and "x \<in> A" shows "f x \<in> B x"
induct: fixes P :: "('c, 'd) cconstraint \<Rightarrow> bool"   and a0 :: "('c, 'd) cconstraint" assumes "\<And>cc1 cc2. \<lbrakk>P cc1; P cc2\<rbrakk> \<Longrightarrow> P (AND cc1 cc2)"   and "\<And>c uu_. P (LT c uu_)"   and "\<And>c uv_. P (LE c uv_)"   and "\<And>c uw_. P (EQ c uw_)"   and "\<And>c ux_. P (GE c ux_)"   and "\<And>c uy_. P (GT c uy_)" shows "P a0"
disc_2: fixes x21 :: "'g"   and x22 :: "('g, 'h) tllist" shows "\<not> is_TNil (TCons x21 x22)"
a_def: shows "a \<equiv> ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi"
ffd_fbd_conjugation: fixes f :: "'c \<Rightarrow> 'd set"   and X :: "'d set"   and Y :: "'c set" shows "(fd\<^sub>\<F> f X \<inter> Y = {}) = (X \<inter> bd\<^sub>\<F> f Y = {})"
Integer_arith_simps_1: fixes a :: "int"   and b :: "int" shows "MkI\<cdot>a + MkI\<cdot>b = MkI\<cdot>(a + b)"
type3_def: fixes init :: "'a list"   and x :: "'a"   and y :: "'a" shows "type3 init x y = bernoulli_pmf (5 / 10) \<bind> (\<lambda>a. bernoulli_pmf (5 / 10) \<bind> (\<lambda>b. return_pmf (if [a, b] ! index init x \<and> \<not> [a, b] ! index init y then ([x, y], [a, b], init) else ([y, x], [a, b], init))))"
check_plan'_correct: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list" assumes "wf_ast_problem problem" shows "check_plan' problem \<pi>s = ast_problem.check_plan problem \<pi>s"
I_Lam: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list"   and i :: "nat" assumes "spectral_radius_1_jnf_max A n m lam n_as"   and "i \<in> spectral_radius_1_jnf_max.I n m n_as" shows "(m, spectral_radius_1_jnf_max.Lam n_as i) \<in> set n_as"
assms: shows "pseudo_proj_True n w = pseudo_proj_True n x"
cfun_below_iff: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'b" shows "(f \<sqsubseteq> g) = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)"
is_SNF_Smith_mxn: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m n" shows "is_SNF A (Smith_mxn A)"
Pxz_nn: fixes x :: "'b"   and z :: "'d" assumes "x \<in> space S"   and "z \<in> space P" shows "0 \<le> Pxz (x, z)"
collapse: fixes interval :: "'a interval" shows "Interval (low interval) (high interval) = interval"
is_unit_left_imp_coprime: fixes a :: "'a"   and b :: "'a" assumes "is_unit a" shows "coprime a b"
concat_in_star: fixes ws :: "'b list list"   and A :: "'b list set" assumes "set ws \<subseteq> A" shows "concat ws \<in> star A"
list_all2_Nil: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and ys :: "'d list" shows "list_all2 P [] ys = (ys = [])"
i_less_filter: shows "i < length (filter (\<lambda>ys. j < length ys) xs)"
add: fixes b1 :: "'m"   and b2 :: "'m" shows "g' (b1 + b2) = g' b1 + g' b2"
mem_case_prodE: fixes z :: "'a"   and c :: "'b \<Rightarrow> 'c \<Rightarrow> 'a set"   and p :: "'b \<times> 'c" assumes "z \<in> (case p of (x, xa) \<Rightarrow> c x xa)" obtains x :: "'b" and y :: "'c" where "p = (x, y)" and "z \<in> c x y"
NF: shows "nofail (m' \<bind> WHILE c' f')"
singleton_set_union: fixes X :: "'c set" shows "(\<Union>Y\<in>singleton_set X. Y) = X"
assms: fixes \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* \<Psi>"   and "distinct (bn \<alpha>)" shows "\<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel"
1: fixes a :: "real"   and c :: "'a"   and r :: "real" assumes "a \<noteq> 0" shows "(*\<^sub>R) a ` cball c r \<subseteq> cball (a *\<^sub>R c) (\<bar>a\<bar> * r)"
Suc_length_conv: fixes n :: "nat"   and xs :: "'b list" shows "(Suc n = \<^bold>|xs\<^bold>|) = (\<exists>y ys. xs = y # ys \<and> \<^bold>|ys\<^bold>| = n)"
drop_all_iff: fixes n :: "nat"   and xs :: "'b list" shows "(drop n xs = []) = (length xs \<le> n)"
hom_0: fixes x :: "'a" assumes "[:x:] = 0" shows "x = (0::'a)"
Beta: fixes r :: "dB"   and s :: "dB"   and ss :: "dB list"   and t :: "dB" assumes "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>s t" shows "Abs r \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
starr_applied_def: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and F :: "'a \<Rightarrow> bool" shows "starr_applied h1 h2 h0 F \<equiv> (h1,h2\<triangleright>h0) \<and> \<not> F h0"