hcju/isabelleps huggingface dataset:
map_vec_def: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b Matrix.vec" shows "map_vec f v \<equiv> Matrix.vec (dim_vec v) (\<lambda>i. f (v $v i))"
steps_Stuck: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com" shows "\<Gamma> \<turnstile> (c, Stuck) \<rightarrow>\<^sup>* (Skip, Stuck)"
additive_content_division: fixes d :: "'a set set"   and a :: "'a"   and b :: "'a" assumes "d division_of cbox a b" shows "sum content d = content (cbox a b)"
q_1: shows "g \<turnstile> defNode g q-ms\<rightarrow>m"
isoE: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" obtains g :: "'c \<times> 'c" where "C.VV.inverse_arrows f g"
\<I>'2: shows "\<forall>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<longrightarrow> subst_domain \<I>' \<inter> set X = {}"
def_scheme: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "profile_mono_schemeV f g" shows "f x \<equiv> \<lambda>\<tau>. if (\<upsilon> x) \<tau> = true \<tau> then g (x \<tau>) else invalid \<tau>"
gauss_cnj_dvd_iff: fixes a :: "gauss_int"   and b :: "gauss_int" shows "(gauss_cnj a dvd gauss_cnj b) = (a dvd b)"
ideD_3: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "cod a = a"
op_image_fst_colle_def: shows "op_image_fst_colle = (`) fst"
cnf_to_bool_set: fixes f :: "'b \<Rightarrow> bool"   and cnf :: "'b negation_type list" shows "cnf_to_bool f cnf = (\<forall>c\<in>set cnf. case c of Pos a \<Rightarrow> f a | Neg a \<Rightarrow> \<not> f a)"
icard_eq_enat_imp: fixes A :: "'a set"   and n :: "nat" assumes "icard A = enat n" shows "finite A"
map_of_entries: fixes t :: "('a, 'c) mapping_rbt" shows "map_of (entries t) = lookup t"
param_append: fixes R :: "('a \<times> 'b) set" shows "((@), (@)) \<in> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel"
coboundedI: fixes A :: "'c set"   and a :: "'c" assumes "finite A"   and "a \<in> A" shows "Min A \<le> a"
IfFalse: fixes tst :: "'test"   and s :: "'state"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "\<not> tval tst s" shows "(if tst then c1 else c2, s) \<rightarrow>c (c2, s)"
ext_compat_list_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_compat_list_axioms ext \<equiv> \<forall>y x gt xs xs'. y \<noteq> x \<longrightarrow> gt y x \<longrightarrow> ext gt (xs @ y # xs') (xs @ x # xs')"
prefixes_conv_suffixes: fixes xs :: "'c list" shows "prefixes xs = map rev (suffixes (rev xs))"
anonymity_prefs_from_table: fixes xs :: "('agent \<times> 'alt set list) list"   and ys :: "('agent \<times> 'alt set list) list" assumes "prefs_from_table_wf agents alts xs"   and "prefs_from_table_wf agents alts ys"   and "mset (map snd xs) = mset (map snd ys)" shows "sds (prefs_from_table xs) = sds (prefs_from_table ys)"
atLeastLessThan_empty_iff2: fixes a :: "'a \<times> ('a, 'b) operation"   and b :: "'a \<times> ('a, 'b) operation" shows "({} = hb.atLeastLessThan a b) = (\<not> hb a b)"
restrict_nonempty_product: fixes ls :: "'c list"   and l :: "'c"   and k :: "'c"   and ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd"   and m :: "'c"   and ms :: "'c list" assumes "\<not> List.member ls l" shows "(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms) = [k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m] \<oplus> ([k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms) \<oplus> (ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m]) \<oplus> (ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms)"
ae: shows "AE x in S. AE y in T. Pxy (x, y) = Px x * Py y"
doubleForall: fixes A :: "atom fm"   and L :: "real list" shows "eval (AllN 2 A) L = eval (AllQ (AllQ A)) L"
inject_3: fixes x3 :: "(Term_Class.term \<times> sterm) list"   and y3 :: "(Term_Class.term \<times> sterm) list" shows "(Sabs x3 = Sabs y3) = (x3 = y3)"
mem_S: fixes N :: "nat"   and x :: "real" assumes "\<forall>n\<ge>N. x < X n" shows "x \<in> S"
finite_SequentialRacing: fixes US :: "'d pupdate set" assumes "finite US" shows "(SOME u. u \<in> SequentialRacing US) \<in> SequentialRacing US"
feasible_Sup_trans: fixes S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set" assumes "\<forall>t\<in>S. feasible t" shows "feasible (Sup_trans S)"
tail_in_verts: fixes G :: "nat graph_rec"   and e :: "nat \<times> nat" assumes "cfg_doms G"   and "e \<in> g_E G" shows "tail G e \<in> set (g_V G)"
eq_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "ordering less_eq less" shows "(a = b) = (less_eq a b \<and> less_eq b a)"
BMax_rec_eq1: fixes R :: "nat \<Rightarrow> bool"   and x :: "nat" shows "BMax_rec R x = (GREATEST z. R z \<and> z \<le> x \<or> z = 0)"
ds2: fixes k :: "nat" assumes "k < n" shows "matrix_sum d (\<lambda>k. adjoint (M k) * M k * A) k \<in> carrier_mat d d"
mod_0_imp_mod_mult_right_0: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n mod m = 0" shows "n * k mod m = 0"
oneTS_steps_3: fixes x :: "'a"   and y :: "'a"   and hs :: "'a list" assumes "x \<noteq> y" shows "TS_step_d ([x, y], hs) x = ((0, []), x # hs)"
deleteAround_eq: fixes g :: "graph"   and v :: "nat"   and ps :: "(nat \<times> nat) list" shows "deleteAround g v ps = deleteAround' g v ps"
classical_operator_exists_def: fixes \<pi> :: "'c \<Rightarrow> 'd option" shows "classical_operator_exists \<pi> = cblinfun_extension_exists (range ket) (\<lambda>t. case \<pi> (inv ket t) of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
pnorm_alt: fixes r :: "'a rexp" shows "pnorm r = pnorm_alt r"
subset_ordLeq_diff_infinite: fixes B :: "'d set"   and A :: "'d set" assumes "infinite B"   and "A \<subseteq> B"   and "|A| <o |B|" shows "infinite (B - A)"
type_aexpr: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and e :: "aexp" shows "type_aexpr \<Gamma> e (\<Union> (to_total \<Gamma> ` aexp_vars e))"
stirling_sum'_def: fixes j :: "nat"   and m :: "nat"   and x :: "'a" shows "stirling_sum' j m x = (- (1::'a)) ^ Suc j * (\<Sum>k\<le>m. of_real (bernoulli' k) * pochhammer (of_nat (Suc k)) (j - 1) * inverse x ^ (k + j))"
dg_prod_ArrI: fixes f :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "vsv f"   and "\<D>\<^sub>\<circ> f = I"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Arr\<rparr>" shows "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"
ss_inA: shows "\<forall>\<tau>\<in>set ss. \<tau> \<in> A"
assms_4: shows "k \<in> dom (map_of (concat l))"
fdia_fbox_de_morgan_2: fixes x :: "'a"   and y :: "'a" shows "ad ( |x\<rangle> y) =  |x] ad y"
cod_simp: fixes f :: "'c local.term set \<times> 'c local.term set" assumes "CC.arr f" shows "CC.cod f = (cod (fst f), cod (snd f))"
simps_2: fixes v1 :: "'c \<times> nat"   and e :: "('c \<times> nat) edge"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and sub :: "('c \<times> nat) \<times> 'c \<times> nat" shows "uses_sub v1 (e # es) v2 sub = (v1 \<noteq> src e \<and> sub = (v1, src e) \<or> uses_sub (tgt e) es v2 sub)"
endpoints_refine: fixes xi :: "'cedge"   and x :: "'edge" assumes "\<alpha> xi = x" shows "endpoints xi \<le> \<Down> Id (a_endpoints x)"
swap_swap: fixes p :: "'g \<times> 'h" shows "prod.swap (prod.swap p) = p"
right_unique_rel_gpv'': fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool" assumes "right_unique A"   and "right_unique C"   and "right_total R" shows "right_unique (rel_gpv'' A C R)"
map_value_image_filter_correct_1: fixes m :: "'m1"   and f :: "'u \<Rightarrow> 'v1 \<Rightarrow> 'v2 option" assumes "invar1 m" shows "invar2 (map_value_image_filter f m)"
sel_4: fixes x11 :: "'a"   and x12 :: "'a"   and x13 :: "'a"   and x14 :: "'b" shows "pac_src2 (Add x11 x12 x13 x14) = x12"
smult_as_map_poly: fixes a :: "'c" shows "smult a = map_poly ((*) a)"
map_bset_permute: fixes p :: "perm"   and B :: "'e set['f]" shows "p \<bullet> B = map_bset (permute p) B"
pp_sup_p: fixes x :: "'a" shows "dense (- (x \<sqinter> - x))"
exp_conj_top_left: fixes P :: "'a \<Rightarrow> real" assumes "sound P" shows "\<guillemotleft> \<lambda>_. True \<guillemotright> && P = P"
strongly_connected_imp_induce_subgraph_strongly_connected: fixes H :: "('a, 'b) pre_digraph" assumes "subgraph H (G \<restriction> S)"   and "strongly_connected H" shows "strongly_connected ((G \<restriction> S) \<restriction> verts H)"
inf_set_def: fixes A :: "'d set"   and B :: "'d set" shows "A \<inter> B = Collect (inf (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
enat_ord_simps_3: fixes q :: "enat" shows "q \<le> \<infinity>"
Inf_set_fold: fixes xs :: "'a list" shows "\<Sqinter> (set xs) = fold (\<sqinter>) xs \<top>"
rel_compp: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_option (R OO S) = rel_option R OO rel_option S"
bind_return_distr': fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "M \<bind> (\<lambda>x. return N (f x)) = distr M N f"
prod_power_distrib: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and n :: "nat" shows "prod f A ^ n = (\<Prod>x\<in>A. f x ^ n)"
ibind_strict_1: fixes f :: "unit \<Rightarrow> 'ma" shows "bind bot f = bot"
accn_def: fixes i :: "nat" shows "accn i \<equiv> {q. i \<in> acc q}"
filter_cong: fixes xs :: "'b list"   and ys :: "'b list"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> ran\<^sub>l ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
swap_simp: fixes x :: "'h"   and y :: "'g" shows "prod.swap (x, y) = (y, x)"
f'_def: shows "f' = (\<lambda>x. 1 - (1 - q powr x) ^ n)"
divide_cancel_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "(c / a = c / b) = (c = (0::'a) \<or> a = b)"
assms: shows "vcard (\<R>\<^sub>\<circ> xs) = vcard xs"
sup_cancel_left2: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "- x \<squnion> a \<squnion> (x \<squnion> b) = \<top>"
interp_rga_ids: fixes xs :: "('a \<times> 'a option) list" assumes "rga_ops xs" shows "set (interp_rga xs) = set (map fst xs)"
zippers_rose_tree_blinding_of: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and z :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list \<times> ('a, 'b) rose_tree\<^sub>m"   and e :: "'c \<Rightarrow> 'a"   and p :: "('c \<times> 'c rose_tree list \<times> 'c rose_tree list) list"   and t :: "'c rose_tree" assumes "blinding_of_on UNIV h bo"   and "z \<in> set (zippers_rose_tree e h (p, t))" shows "blinding_of_tree h bo (tree_of_zipper\<^sub>m z) (tree_of_zipper\<^sub>m (blind_path e h p, embed_source_tree e t))"
vimage_VLambda_vrange: fixes A :: "V"   and f :: "V \<Rightarrow> V"   and B :: "V" shows "VLambda A f `\<^sub>\<circ> B = \<R>\<^sub>\<circ> (VLambda (A \<inter>\<^sub>\<circ> B) f)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>zs ys xs gt. \<lbrakk>\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x; ext gt zs ys; ext gt ys xs\<rbrakk> \<Longrightarrow> ext gt zs xs" shows "ext_irrefl_trans_strong_axioms ext"
exhaust_sel: fixes converter :: "('a, 'b, 'c, 'd) converter" obtains "converter = Converter (run_converter converter)"
square_free_m_smultD: fixes c :: "int"   and f :: "int poly" assumes "square_free_m (smult c f)" shows "square_free_m f"
abs_eq_iff: fixes x :: "'a ltln"   and y :: "'a ltln" shows "(abs_ltln\<^sub>P x = abs_ltln\<^sub>P y) = x \<sim>\<^sub>P y"
fmrel_on_fset_fmdom: fixes ym :: "'e \<rightharpoonup>\<^sub>f 'g"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and xm :: "'e \<rightharpoonup>\<^sub>f 'f"   and k :: "'e" assumes "fmrel_on_fset (fmdom ym) f xm ym"   and "k |\<in>| fmdom ym" shows "k |\<in>| fmdom xm"
assms_3: shows "(y, z) \<in> r\<^sup>*"
realpow_pos_nth2: fixes a :: "real"   and n :: "nat" assumes "0 < a" shows "\<exists>r>0. r ^ Suc n = a"
eqvtY: shows "eqvt (bisim O bangRel weakBisim O bisim)"
elims: fixes x :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and xa :: "'at Literal set set"   and xb :: "nat"   and y :: "'at Literal set set" assumes "inferred_clause_sets x xa xb = y" obtains R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" and S :: "'at Literal set set" where "x = Ra__" and "xa = Sa__" and "xb = 0" and "y = simplify Sa__"   | R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" and S :: "'at Literal set set" and N :: "nat" where "x = R" and "xa = S" and "xb = Suc N" and "y = simplify (add_all_deducible_clauses R (inferred_clause_sets R S N))"
that: fixes h' :: "complex \<Rightarrow> complex" assumes "continuous_on UNIV h'"   and "\<And>z. (h has_field_derivative h' z) (at z)" shows "thesis"
F0: fixes i :: "nat" assumes "i \<noteq> 0" shows "(local.F i = 0) = (local.k < i)"
ln_upper_15_neg: fixes x :: "real" assumes "0 < x"   and "x \<le> 1" shows "ln x \<le> ln_upper_15 x"
assms: shows "low_equal s1 s2 \<and> t1 = in_reg_mod w curr_win rd s1 \<and> t2 = in_reg_mod w curr_win rd s2"
subject_reduction: 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 e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P,E,hp s \<turnstile> e : T" shows "\<exists>T'. P,E,hp s' \<turnstile> e' : T' \<and> P \<turnstile> T' \<le> T"
not_trivial_limit_within: fixes x :: "'a"   and S :: "'a set" shows "(at x within S = bot) \<noteq> (x \<in> closure (S - {x}))"
1: shows "U \<subseteq> \<Union> (insert (- K) (F ` \<C>))"
nzero: shows "\<forall>x. d1 < x \<and> x < c \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)"
fresh_string: fixes a :: "'a"   and s :: "char list" shows "a \<sharp> s"
a: shows "A \<union> {.\<not> p} \<turnstile> q"
split_Nil_iff: fixes xs :: "'b list"   and ys :: "'b list" shows "(splice xs ys = []) = (xs = [] \<and> ys = [])"
pick_asTerm: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood qX" shows "pick (asTerm qX) #= qX"
eq: shows "adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) * (tensor_P (1\<^sub>m N) (mat_incr K) * Q2 * adjoint (tensor_P (1\<^sub>m N) (mat_incr K))) * tensor_P (1\<^sub>m N) (mat_incr K) = Q2"
not_eq_order_implies_strict: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<noteq> b"   and "a \<preceq> b" shows "a \<prec> b"
hom_nat_pow: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c"   and x :: "'a"   and n :: "nat" assumes "ring_hom_ring R S h"   and "x \<in> carrier R" shows "h (x [^]\<^bsub>R\<^esub> n) = h x [^]\<^bsub>S\<^esub> n"
power_0_left: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero n = (if n = 0 then one else zero)"
RPDs_aux_def: shows "RPDs_aux = saturate (\<lambda>S. S \<union> \<Union> (RPD ` S))"
pow_generator_eq_iff_cong: fixes G :: "('d, 'e) cyclic_group_scheme"   and x :: "nat"   and y :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "finite (carrier G)" shows "(\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> x = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> y) = [x = y] (mod order G)"
eps_induct: fixes P :: "V \<Rightarrow> bool"   and a :: "V" assumes "\<And>x. (\<And>y. y \<in> elts x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
ptensor_mat_def: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "d.ptensor_mat m1 m2 = tensor_mat d.dims0 d.vars1' m1 m2"
eq_refl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
conjugate_cancel_iff: fixes a :: "'a"   and b :: "'a" shows "(conjugate a = conjugate b) = (a = b)"
simps: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b" shows "((a1, a2) \<in> encoding.indRelRPO Enc) = ((\<exists>S. a1 = SourceTerm S \<and> a2 = TargetTerm (Enc S)) \<or> (\<exists>S. a1 = SourceTerm S \<and> a2 = SourceTerm S) \<or> (\<exists>T. a1 = TargetTerm T \<and> a2 = TargetTerm T) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> (P, Q) \<in> encoding.indRelRPO Enc \<and> (Q, R) \<in> encoding.indRelRPO Enc))"
ge: shows "\<forall>\<^sub>F x in F. g x < ereal C"
size_4: fixes x21 :: "'c"   and x22 :: "('c, 'd) Term.term list" shows "size (Fun x21 x22) = size_list size x22 + Suc 0"
cols_transpose: fixes A :: "'b mat" shows "cols A\<^sup>T = rows A"
sort_is_sortedQ: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "all_in_list p l"   and "singleCombinators p" shows "FWNormalisationCore.sorted (qsort p l) l"
WT_elim_cases_2: 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 E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and e :: "expr"   and T :: "ty" assumes "P,E \<turnstile> Cast C e :: T" obtains D :: "char list" where "T = Class C" and "P,E \<turnstile> e :: Class D" and "is_class P C" and "P \<turnstile> Path D to C unique \<or> (\<forall>Cs. \<not> P \<turnstile> Path D to C via Cs )"
assms: shows "g \<in> \<Theta>(\<lambda>x. real x powr p / ln (real x))"
minus_iarray_def: fixes A :: "'c iarray"   and B :: "'c iarray" shows "A - B = IArray.of_fun (\<lambda>n. A !! n - B !! n) (IArray.length A)"
last_drop: fixes n :: "nat"   and xs :: "'c list" assumes "n < |xs|" shows "last (drop n xs) = last xs"
addSubsetSet_empty: fixes s :: "'a set"   and ts :: "'a set set" shows "addSubsetSet s (ts - {{}}) - {{}} = addSubsetSet s ts - {{}}"
pickQFreshEnv: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option) set"   and xs :: "'varSort" assumes "|V| <o |UNIV| \<or> finite V"   and "|XS| <o |UNIV| \<or> finite XS"   and "Ball XS qGood"   and "|Rho| <o |UNIV| \<or> finite Rho"   and "Ball Rho qGoodEnv" shows "pickQFreshEnv xs V XS Rho \<notin> V \<and> (\<forall>X\<in>XS. qFresh xs (pickQFreshEnv xs V XS Rho) X) \<and> (\<forall>rho\<in>Rho. qFreshEnv xs (pickQFreshEnv xs V XS Rho) rho)"
open_segment_eq_real_ivl: fixes a :: "real"   and b :: "real" shows "{a<--<b} = (if a \<le> b then {a<..<b} else {b<..<a})"
ih: shows "\<forall>q. (q, x) \<in> measure FOL_Axiomatic.size_fm \<longrightarrow> (q \<in> H \<longrightarrow> \<lbrakk>\<^bold>#, \<^bold>\<dagger>, \<lambda>P ts. \<^bold>\<ddagger> P ts \<in> H\<rbrakk> q) \<and> ((\<^bold>\<not> q) \<in> H \<longrightarrow> \<not> \<lbrakk>\<^bold>#, \<^bold>\<dagger>, \<lambda>P ts. \<^bold>\<ddagger> P ts \<in> H\<rbrakk> q)"
fls_divide_shift_numer: fixes m :: "int"   and f :: "'a fls"   and g :: "'a fls" shows "fls_shift m f / g = fls_shift m (f / g)"
induct: fixes P :: "int \<times> int \<Rightarrow> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "int \<times> int"   and a1 :: "int"   and a2 :: "(int \<times> int) list" assumes "\<And>uu_ uv_. P uu_ uv_ []"   and "\<And>p uw_ p\<^sub>0. P p uw_ [p\<^sub>0]"   and "\<And>p \<delta> p\<^sub>0 v va. (\<And>x. \<lbrakk>x = dist_code p p\<^sub>0; \<not> \<delta> \<le> (snd p\<^sub>0 - snd p)\<^sup>2\<rbrakk> \<Longrightarrow> P p (min \<delta> x) (v # va)) \<Longrightarrow> P p \<delta> (p\<^sub>0 # v # va)" shows "P a0 a1 a2"
lexp_head: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "order x y"   and "length xs = length ys" shows "lexp order (x # xs) (y # ys)"
kk_def: shows "kk = Polynomial.order ev (char_poly A)"
sum_power2_le_zero_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (plus (power.power one times x 2) (power.power one times y 2)) zero = (x = zero \<and> y = zero)"
Dom_dom: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "local.Dom (local.dom f) = local.Dom f"
le_phi': fixes \<phi> :: "real" assumes "\<phi> \<le> \<phi>'"   and "0 \<le> \<phi>" shows "sin \<phi> \<le> d / l"
PseudoTermination: fixes cc :: "('p, 'v, 's) configuration"   and Q :: "'p set" shows "terminationPseudo 1 cc Q"
fsmultD: fixes a :: "'f"   and v :: "('f, 'g) aezfun \<Rightarrow> 'v" shows "a \<currency>\<currency> v = a \<delta>\<delta> (0::'g) \<currency> v"
ideD_2: fixes a :: "'a" assumes "A.ide a" shows "A.dom a = a"
Union_empty_conv: fixes A :: "'a set set" shows "(\<Union> A = {}) = (\<forall>x\<in>A. x = {})"
to_poly_is_ring_hom: shows "to_polynomial R \<in> ring_hom R P"
reds_isValI: fixes z :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and L :: "Vars.var list" assumes "isVal z" shows "\<Gamma> : z \<Down>\<^bsub>L\<^esub> \<Gamma> : z"
in_angle_one_side: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "\<not> Col A B C"   and "\<not> Col B A P"   and "P InAngle A B C" shows "A B OS P C"
max_subgraph_def: fixes P :: "('c, 'd) pre_digraph \<Rightarrow> bool"   and H :: "('c, 'd) pre_digraph" shows "aG.max_subgraph P H \<equiv> subgraph H (app_iso hom G) \<and> P H \<and> (\<forall>H'. H' \<noteq> H \<and> subgraph H H' \<longrightarrow> \<not> (subgraph H' (app_iso hom G) \<and> P H'))"
permute_stack_elem_def: fixes \<pi> :: "perm"   and x :: "stack_elem" shows "\<pi> \<bullet> x = (case x of Alts e1 e2 \<Rightarrow> Alts (\<pi> \<bullet> e1) (\<pi> \<bullet> e2) | Arg v \<Rightarrow> Arg (\<pi> \<bullet> v) | Upd v \<Rightarrow> Upd (\<pi> \<bullet> v) | Dummy v \<Rightarrow> Dummy (\<pi> \<bullet> v))"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_snoc_axioms ext" shows "ext_snoc ext"
higher_pderiv_mult: fixes n :: "nat"   and p :: "'a poly"   and q :: "'a poly" shows "(pderiv ^^ n) (p * q) = (\<Sum>k\<le>n. Polynomial.smult (of_nat (n choose k)) ((pderiv ^^ k) p * (pderiv ^^ (n - k)) q))"
fork_3: shows "r' \<notin> RID\<^sub>G s"