hcju/isabelleps huggingface dataset:
cblinfun_of_matrix_apply: fixes a :: "'a \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" shows "cblinfun_apply (cblinfun_of_matrix a) x = (\<Sum>i\<in>CBasis. \<Sum>j\<in>CBasis. (j \<bullet>\<^sub>C x * a i j) *\<^sub>C i)"
bind_gpv_Done: fixes f :: "('d, 'e, 'f) gpv" shows "f \<bind> Generative_Probabilistic_Value.Done = f"
W_4: shows "diff_fun k (charts_submanifold W) f'"
dtor_corec: fixes s :: "'d \<Rightarrow> ('c, ('c coG + 'd) coF) coG0"   and x :: "'d" shows "dtor_coG (dtor_corec_coG s x) = map_pre_coG id (case_sum id (dtor_corec_coG s)) (s x)"
InvariantVarsMAfterBacktrack: fixes M :: "(Literal \<times> bool) list"   and F0 :: "Literal list list"   and Vbl :: "nat set"   and M' :: "(Literal \<times> bool) list" assumes "InvariantVarsM M F0 Vbl"   and "decisions M \<noteq> []"   and "M' = prefixBeforeLastDecision M @ [(opposite (lastDecision M), False)]" shows "InvariantVarsM M' F0 Vbl"
insert_eq: shows "insert x {xa \<in> set vs. xa \<noteq> x} = set vs"
simps_7: fixes x71 :: "nat"   and x72 :: "msg"   and y71 :: "nat"   and y72 :: "msg" shows "(Crypt x71 x72 = Crypt y71 y72) = (x71 = y71 \<and> x72 = y72)"
UN_UN_flatten: fixes C :: "'b \<Rightarrow> 'a set"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "\<Union> (C ` \<Union> (B ` A)) = (\<Union>y\<in>A. \<Union> (C ` B y))"
bind_advantage_def: fixes \<A> :: "'a \<Rightarrow> ('a \<times> nat \<times> nat \<times> nat \<times> nat) spmf" shows "ped_commit.bind_advantage \<A> \<equiv> spmf (ped_commit.bind_game \<A>) True"
bezout_coefficients_def: fixes a :: "'a"   and b :: "'a" shows "bezout_coefficients a b = fst (euclid_ext_aux (1::'a) (0::'a) (0::'a) (1::'a) a b)"
sgnx_at_top_IVT: fixes p :: "real poly"   and a :: "real" assumes "sgnx (poly p) (at_right a) \<noteq> sgnx (poly p) at_top" shows "\<exists>x>a. poly p x = 0"
distinct_109: fixes x101 :: "'a ltln"   and x102 :: "'a ltln"   and x111 :: "'a ltln"   and x112 :: "'a ltln" shows "x101 W\<^sub>n x102 \<noteq> x111 M\<^sub>n x112"
M_perp_hyp2: fixes l :: "proj2_line"   and m :: "proj2_line"   and a :: "proj2"   and b :: "proj2"   and c :: "proj2" assumes "M_perp l m"   and "a \<in> hyp2"   and "proj2_incident a l"   and "b \<in> hyp2"   and "proj2_incident b m"   and "proj2_incident c l"   and "proj2_incident c m" shows "c \<in> hyp2"
fps_compose_mult_distrib_lemma: fixes c :: "'a fps"   and a :: "'a fps"   and b :: "'a fps"   and n :: "nat" assumes "fps_nth c 0 = (0::'a)" shows "fps_nth ((a oo c) * (b oo c)) n = (\<Sum>s = 0..n. \<Sum>i = 0..s. fps_nth a i * fps_nth b (s - i) * fps_nth (c ^ s) n)"
termlemma: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex"   and xa :: "'a"   and val :: "bool" assumes "lowest_tops [i, t, e] = Some xa" shows "(case (restrict_top i xa val, restrict_top t xa val, restrict_top e xa val) of (i, t, e) \<Rightarrow> size i + size t + size e) < (case (i, t, e) of (i, t, e) \<Rightarrow> size i + size t + size e)"
GodExistenceIsValid: shows "\<lfloor>(\<lambda>w. \<exists>x. existsAt x w \<and> (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) w))\<rfloor>"
hyps: fixes \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain" shows "comlist (mk_ps_chain \<Pi> \<pi>') = comlist \<Pi> @ \<pi>'"
pwd_spy_enc_sign_def: fixes n :: "nat" shows "pwd_spy_enc_sign n \<equiv> let (s, S, A, B, C, D) = pwd_spy_enc_mult n; SK = (Some S, {A, B}, {C, D}) in (insert (Spy, Crypt (SesK SK) (Sign n (Auth_PriK n))) s, S, A, B, C, D)"
substitute_cong_induct: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b ttree"   and t :: "'b ttree"   and f' :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> carrier (f x) \<subseteq> A"   and "carrier t \<subseteq> A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "substitute f T t = substitute f' T t"
sub: fixes i :: "nat"   and r :: "nat \<Rightarrow> 'a" shows "\<exists>k. strict_mono k \<and> P i (r \<circ> k)"
case_rews_2: fixes f :: "int \<rightarrow> 'a"   and int :: "int" shows "Integer_case\<cdot>f\<cdot>(MkI\<cdot>int) = f\<cdot>int"
rep_eq: fixes x :: "real"   and xa :: "'a sq_mtx" shows "to_vec (x *\<^sub>R xa) = x *\<^sub>R to_vec xa"
n_def: shows "n = (if J = {} then 0 else Max J)"
coprime_gcd: fixes h :: "'a mod_ring poly"   and c1 :: "'a mod_ring"   and c2 :: "'a mod_ring"   and f :: "'a mod_ring poly" assumes "algebraic_semidom_class.coprime (h - [:c1:]) (h - [:c2:])" shows "algebraic_semidom_class.coprime (gcd f (h - [:c1:])) (gcd f (h - [:c2:]))"
nAct_Null: fixes n :: "enat"   and t :: "cnf llist"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a" assumes "n \<le> llength t"   and "dynamic_component.nAct tCMP active c n t = 0" shows "\<forall>i. enat i < n \<longrightarrow> \<not> active c (lnth t i)"
sel_2: fixes x :: "gauss_int"   and y :: "gauss_int" shows "ImZ (x * y) = ReZ x * ImZ y + ImZ x * ReZ y"
alist_split_set_1: fixes xs :: "('d, 'c) bintree option list" shows "set (vals xs) = fst ` set (elements xs)"
rel_mono_strong: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'a \<times> 'c"   and y :: "'b \<times> 'd"   and R1a :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2a :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_prod R1 R2 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> Basic_BNFs.fsts x; y1 \<in> Basic_BNFs.fsts y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> Basic_BNFs.snds x; y2 \<in> Basic_BNFs.snds y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2" shows "rel_prod R1a R2a x y"
kop_epi: fixes f :: "('d \<Rightarrow> 'c set) \<Rightarrow> 'e"   and g :: "('d \<Rightarrow> 'c set) \<Rightarrow> 'e" assumes "f \<circ> op\<^sub>K = g \<circ> op\<^sub>K" shows "f = g"
coeff_0_reflect_poly_0_iff: fixes p :: "'a poly" shows "(constant_term (reflect_poly p) = (0::'a)) = (p = 0)"
type_less_right_simps_5: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "(\<tau> < Set \<sigma>) = (\<exists>\<phi>. \<tau> = Set \<phi> \<and> \<phi> < \<sigma>)"
backflows_id: fixes E :: "('a \<times> 'a) set" shows "backflows (backflows E) = E"
ln_nth_prime_asymp_equiv: shows "(\<lambda>n. ln (real (nth_prime n))) \<sim>[sequentially] (\<lambda>x. ln (real x))"
cases: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "a \<in> orbit f x" obtains "a = f x"   | y :: "'c" where "a = f y" and "y \<in> orbit f x"
map_iterator_no_cond_rule_insert_P: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option"   and I :: "'b set \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and f :: "'b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "map_iterator it m"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
that: fixes ns' :: "'node list" assumes "g \<turnstile> n-ns'\<rightarrow>n"   and "n' \<in> set ns'"   and "n \<notin> set (tl (butlast ns'))"   and "set ns' \<subseteq> set ns" shows "thesis"
common_abs_def: fixes t :: "'s Trace" shows "common_abs t \<equiv> clock_commonAbs t"
2: shows "Arr (rep f) \<and> Arr (rep g) \<and> Arr (rep h) \<and> DOM h = COD g \<and> DOM g = COD f"
Image_singleton_iff: fixes b :: "'b"   and r :: "('c \<times> 'b) set"   and a :: "'c" shows "(b \<in> r `` {a}) = ((a, b) \<in> r)"
that: assumes "P x"   and "r = x"   and "h' = h" shows "thesis"
arr_runit: fixes a :: "'c local.term set" assumes "ide a" shows "arr \<r>[a]"
has_limits_of_shape_def: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and J :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah" assumes "category C" shows "category.has_limits_of_shape C J \<equiv> \<forall>D. diagram J C D \<longrightarrow> (\<exists>a \<chi>. limit_cone J C D a \<chi>)"
pull_up_senior_rank: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n"   and "y \<le> n"   and "semi_mojmir_def.senior \<delta> q\<^sub>0 w x n = semi_mojmir_def.senior \<delta> q\<^sub>0 w y n" shows "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w y n"
exp: assumes "do_log op v e = Some (Exp e)" shows "thesis"
finite_equiv: fixes A :: "'b set" assumes "finite A" shows "finite {R. equiv A R}"
hom_nat_pow: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and x :: "'c"   and n :: "nat" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h (x [^]\<^bsub>G\<^esub> n) = h x [^]\<^bsub>H\<^esub> n"
ex_inj: assumes "class.countable TYPE('c)" shows "\<exists>to_nat. inj to_nat"
basis_def: fixes A :: "'c set" shows "basis A = (lin_indpt A \<and> gen_set A \<and> A \<subseteq> carrier V)"
inj1: shows "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>1)"
Red_F_\<G>_empty_def: fixes N :: "'f set" shows "no_labels.Red_F_\<G>_empty N = (\<Inter>q\<in>Q. no_labels.Red_F_\<G>_empty_q q N)"
lateEarlyInput: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and u :: "name" assumes "P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'" shows "P \<longmapsto>\<^sub>e a<u> \<prec>\<^sub>e P'[x::=u]"
base: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option" shows "visits s e s r []"
sequence_irpt_l: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a \<Rightarrow> ('b \<times> 'a) option"   and M' :: "'b \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'a) option" assumes "\<lbrace>P\<rbrace> M\<dagger>" shows "\<lbrace>P\<rbrace> bind\<^sub>S\<^sub>E M M'\<dagger>"
ord_cong: fixes k1 :: "'a"   and k2 :: "'a"   and n :: "'a" assumes "[k1 = k2] (mod n)" shows "ord n k1 = ord n k2"
minimalEnabled_def: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list"   and msg :: "('a, 'c) message" assumes "execution trans sends start exec trace" shows "execution.minimalEnabled exec trace msg \<equiv> (\<exists>p. isReceiverOf p msg) \<and> enabled (last exec) msg \<and> (\<exists>n<length exec. enabled (exec ! n) msg \<and> (\<forall>n'\<ge>n. n' < length trace \<longrightarrow> msg \<noteq> trace ! n') \<and> (\<forall>n' msg'. (\<exists>p. isReceiverOf p msg') \<and> enabled (last exec) msg' \<and> n' < length trace \<and> enabled (exec ! n') msg' \<and> (\<forall>n''\<ge>n'. n'' < length trace \<longrightarrow> msg' \<noteq> trace ! n'') \<longrightarrow> n \<le> n'))"
case_prodD: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'h"   and b :: "'i" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
cases: fixes a :: "('fun, 'var) extstrand_step list" assumes "well_analyzed a" obtains "a = []"   | A :: "('fun, 'var) extstrand_step list" and x :: "('fun, 'var) strand_step" where "a = A__ @ [Step x]" and "well_analyzed A__"   | A :: "('fun, 'var) extstrand_step list" and t :: "('fun, 'var) Term.term" where "a = A @ [Decomp t]" and "well_analyzed A" and "t \<in> subterms\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) - range Var"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (a = b\<midarrow>b)"
conjugate_exponent_ennreal_iff: fixes p :: "ennreal"   and q :: "ennreal" assumes "1 \<le> p" shows "(q = conjugate_exponent p) = (1 / p + 1 / q = 1)"
M_def: shows "M = mat_mult (mult.row_length (mat_to_cols_list A)) (mat_to_cols_list A) (mat_to_cols_list B)"
list_conj_def: fixes fs :: "'a fm list" shows "list_conj fs = foldr and fs TrueF"
wf_digraph_axioms: shows "wf_digraph (G \<restriction> {w. u \<rightarrow>\<^sup>* w})"
exists_next_marker_free_state: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and cid :: "nat"   and p :: "'d"   and q :: "'d"   and t :: "('d, 'e, 'f) event list"   and i :: "nat" assumes "computation channel trans send recv init final"   and "channel cid = Some (p, q)"   and "distributed_system.trace channel trans send recv init t final" shows "\<exists>!j. computation.next_marker_free_state channel trans send recv init t i cid = j \<and> i \<le> j \<and> Marker \<notin> set (msgs (distributed_system.s channel trans send recv init t j) cid)"
l2_inv6I: fixes x :: "l2_state" assumes "\<And>Ra A B gny. \<lbrakk>guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; in_progress (progress x Ra) xsk; sklvl1.guessed_frame Ra xgny = Some gny\<rbrakk> \<Longrightarrow> sklvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))" shows "x \<in> l2_inv6"
freeliftid_Abs_freeword_conv_sum_list: fixes S :: "'b set"   and xs :: "('b \<times> bool) list" assumes "PreCoxeterSystem S"   and "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "freeword_funlift id (Abs_freeword xs) = sum_list (map fst xs)"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> (a \<squnion> b) = a \<squnion> b"
case_prodI: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
st_Infinitesimal: fixes x :: "real star" assumes "x \<in> Infinitesimal" shows "st x = 0"
H_consr: fixes q' :: "'a"   and q :: "'a"   and p :: "'a"   and x :: "'a" assumes "\<tt>\<tt> q' \<le> \<tt>\<tt> q"   and "H p x q'" shows "H p x q"
set_mset_diff: fixes M :: "'b multiset"   and N :: "'b multiset" shows "set_mset (M - N) = {a. count N a < count M a}"
simps: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and a :: "'a stream" shows "(\<phi> suntil \<psi>) a = ((\<exists>\<omega>. a = \<omega> \<and> \<psi> \<omega>) \<or> (\<exists>\<omega>. a = \<omega> \<and> \<phi> \<omega> \<and> (\<phi> suntil \<psi>) (stl \<omega>)))"
nonneg_Reals_pow_I: fixes a :: "'a"   and n :: "nat" assumes "a \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "a ^ n \<in> \<real>\<^sub>\<ge>\<^sub>0"
hcomp_in_vhom: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr"   and h :: "('a S.E.term, 'a) S.arr"   and k :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>S g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow>\<^sub>S k\<guillemotright>"   and "S.src h = S.trg f" shows "\<guillemotleft>\<nu> \<star>\<^sub>S \<mu> : h \<star>\<^sub>S f \<Rightarrow>\<^sub>S k \<star>\<^sub>S g\<guillemotright>"
subgroup_in_rcosets: assumes "Group.group (add_monoid G)" shows "H \<in> rcosets\<^bsub>add_monoid G\<^esub> H"
assms: shows "adjoint_equivalence (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<eta> \<epsilon>"
strict_trans2: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "b \<prec> a"   and "c \<preceq> b" shows "c \<prec> a"
qAFresh_qSwap_exchange2: fixes zs :: "'d"   and z2 :: "'e"   and z1 :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "qAFresh zs z2 (X #[[z2 \<and> z1]]_zs) = qAFresh zs z1 X"
simps_10: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b pred \<Rightarrow> 'a"   and f3 :: "'b pred \<Rightarrow> 'b seq \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b pred" shows "(case Insert x21 x22 of Empty \<Rightarrow> f1 | Insert x xa \<Rightarrow> f2 x xa | Join x xa \<Rightarrow> f3 x xa) = f2 x21 x22"
2: shows "H * e * (top * H * e)\<^sup>\<star> \<le> H * e"
assms_1: shows "P \<Longrightarrow>\<^sup>^a[b] \<prec> P'"
mult_cong: fixes m :: "int"   and x :: "int"   and y :: "int" assumes "residues m" shows "x mod m \<otimes>\<^bsub>residue_ring m\<^esub> (y mod m) = x * y mod m"
static_reduce': fixes p :: "'a pfm"   and r :: "'a pfm" assumes "static p"   and "static r" shows "static (reduce' r p)"
update_element_type_det: fixes \<tau> :: "'b OCL_Types.type"   and \<sigma> :: "'c OCL_Types.type"   and \<rho>\<^sub>1 :: "'c OCL_Types.type"   and \<rho>\<^sub>2 :: "'c OCL_Types.type" assumes "update_element_type \<tau> \<sigma> \<rho>\<^sub>1"   and "update_element_type \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
ActualNotE: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<A>\<^bold>\<not>\<phi> in v]" shows "[\<^bold>\<not>\<^bold>\<A>\<phi> in v]"
fcompose_0: fixes q :: "'a poly"   and r :: "'a poly" shows "fcompose 0 q r = 0"
right_inverse: fixes a :: "'a perm" shows "a * inverse a = 1"
singleActionChain: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<longmapsto> \<alpha> \<prec> P'" shows "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"
plus_1: fixes x :: "'a" shows "f (g1 x) = f x"
nprv_clear5_1: fixes \<phi>2 :: "'fmla"   and \<phi>3 :: "'fmla"   and \<phi>4 :: "'fmla"   and \<phi>5 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>1 :: "'fmla" assumes "nprv {\<phi>2, \<phi>3, \<phi>4, \<phi>5} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<phi>5 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2, \<phi>3, \<phi>4, \<phi>5} \<psi>"
emeasure_eq_0_AE: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "AE x in M. \<not> P x" shows "emeasure M {x \<in> space M. P x} = 0"
simps_2: fixes m :: "n"   and n :: "n" shows "mul2 (S m) n = add2 n (mul2 m n)"
edges_Gu: fixes G :: "nat set \<times> nat set set"   and u :: "nat" shows "uedges (G -- u) \<subseteq> uedges G"
lcdk: shows "l - j cd\<^bsup>\<pi> \<guillemotleft> j\<^esup>\<rightarrow> k"
Done_Pause: fixes out :: "'g"   and c :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "resumption_ord ABORT (Resumption.resumption.Pause out c)"
Rep_prod_inverse: fixes x :: "'a \<times> 'b" shows "Abs_prod (Rep_prod x) = x"
vimage_right_total_transfer: fixes B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "bi_unique B"   and "right_total A" shows "((A ===> B) ===> rel_set B ===> rel_set A) (\<lambda>f X. f -` X \<inter> Collect (Domainp A)) (-`)"
convex_bound_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and a :: "'a"   and y :: "'a"   and u :: "'a"   and v :: "'a" assumes "class.linordered_semiring_1 plus minus less_eq less zero times one"   and "less_eq x a"   and "less_eq y a"   and "less_eq zero u"   and "less_eq zero v"   and "plus u v = one" shows "less_eq (plus (times u x) (times v y)) a"
that: fixes y :: "'c" assumes "y \<in> indets p"   and "x \<in> indets (f y)" shows "thesis"
minus_unique: fixes y :: "'c"   and x :: "'c"   and y' :: "'c" assumes "y \<oplus>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"   and "x \<oplus>\<^bsub>M\<^esub> y' = \<zero>\<^bsub>M\<^esub>"   and "x \<in> carrier M"   and "y \<in> carrier M"   and "y' \<in> carrier M" shows "y = y'"
prj_simps_12: shows "C.seq fg.prj\<^sub>0 \<p>\<^sub>0[g.leg0, f.leg1 \<cdot> fg.prj\<^sub>1]"
fv_not_fresh: fixes x :: "'b"   and e :: "'a" shows "atom x \<sharp> e = (x \<notin> fv e)"
sum_associativity: fixes x :: "i"   and y :: "i"   and z :: "i" shows "x \<oplus> (y \<oplus> z) = (x \<oplus> y) \<oplus> z"
map_of_asetI1: fixes S :: "('b \<times> 'c) set"   and a :: "'b"   and b :: "'c" assumes "unique_keys_set S"   and "(a, b) \<in> S" shows "map_of_aset S\<langle>a\<rangle> = Some b"
prod_of_cont_is_cont: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and g :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_continuous f"   and "is_Zp_continuous g" shows "is_Zp_continuous (f \<otimes>\<^bsub>function_ring (carrier Zp) Zp\<^esub> g)"
abc_abd_bcdbdc: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "MinkowskiSpacetime \<E> \<P> betw"   and "betw a b c"   and "betw a b d"   and "c \<noteq> d" shows "betw b c d \<or> betw b d c"
simps_1: shows "BinomialHeapStruc.queue_to_multiset [] = {#}"
MS_step: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and l1 :: "int"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and l2 :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and m :: "nat" assumes "(M, l, s, n, l1, r) \<in> Step"   and "(M, l1, r, k, l2, t) \<in> MStep"   and "m = Suc k + n" shows "(M, l, s, m, l2, t) \<in> MStep"
primes_upto_def: fixes n :: "nat" shows "primes_upto n = sorted_list_of_set {m. m \<le> n \<and> prime m}"
outstanding_non_volatile_refs_owned_or_read_only: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "non_volatile_owned_or_read_only False \<S>\<^sub>s\<^sub>b \<O> sb"
unit_simps_5: shows "trg \<eta>\<^sub>0 = src e\<^sub>0"
cComm2: fixes a :: "name"   and b :: "name"   and P' :: "pi"   and x :: "name"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a[b] \<prec> P'"   and "!P \<longmapsto> a<x> \<prec> P''"   and "x \<sharp> C"   and "\<And>C. F C (!P) (a<x> \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> P' \<parallel> P''[x::=b])"
circline_eq_cmat_def: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "circline_eq_cmat A B = (\<exists>k. k \<noteq> 0 \<and> B = cor k *\<^sub>s\<^sub>m A)"
self_codomain_iff_ide: fixes a :: "(bool, bool list) concrete_category.arr" shows "(a \<in> codomains a) = ide a"
simps_25: fixes v :: "formula list" shows "is_constraint (Ands v) = False"
lpCons: fixes a :: "'a"   and p :: "'a poly" assumes "a \<noteq> (0::'a) \<or> p \<noteq> 0" shows "P (pCons a p) 0"
Recv_ev': shows "ev' = Recv i'' (occurs_on ev') s u'' u''' m'"
G_def: fixes z :: "complex" shows "G z = (if z = 0 then 1 else sin (a * z) / (a * z))"
lconf_upd: fixes P :: "'b prog"   and h :: "'heap"   and l :: "String.literal \<Rightarrow> 'addr val option"   and E :: "String.literal \<Rightarrow> ty option"   and v :: "'addr val"   and T :: "ty"   and V :: "String.literal" assumes "P,h \<turnstile> l (:\<le>) E"   and "P,h \<turnstile> v :\<le> T"   and "E V = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> l(V \<mapsto> v) (:\<le>) E"
psize_def: fixes p :: "'b poly" shows "psize p = (if p = 0 then 0 else Suc (degree p))"