hcju/isabelleps huggingface dataset:
lb_reflclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and x :: "'b" shows "lb P\<^sup>=\<^sup>= B x = lb P B x"
Hermite_of_upt_row_i_Least: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat" assumes "\<not> is_zero_row i A"   and "echelon_form A"   and "ass_function ass"   and "res_function res"   and "k \<le> nrows A" shows "(LEAST n. Hermite_of_upt_row_i A k ass res $ i $ n \<noteq> (0::'b)) = (LEAST n. A $ i $ n \<noteq> (0::'b))"
coplanar_perm_22: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet D C A B"
map_poly_zero_hom_0_def: fixes hom :: "'a \<Rightarrow> 'b" shows "map_poly_zero_hom_0 hom \<equiv> zero_hom_0 hom"
rawpsubstT_notIn: fixes x :: "'var"   and txs :: "('trm \<times> 'var) list"   and t :: "'trm" assumes "x \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "t \<in> trm"   and "FvarsT t \<inter> snd ` set txs = {}" shows "rawpsubstT t txs = t"
vqj: shows "non_pivot_base A pp qj $ qj = (1::'a)"
collapse: fixes envT :: "('d, 'e) envT" shows "EnvT (run_env envT) = envT"
fimage_frrestriction: fixes r :: "V"   and A :: "V"   and B :: "V" shows "(r \<restriction>\<^sup>r\<^sub>\<bullet> A) `\<^sub>\<bullet> B = A \<inter>\<^sub>\<circ> r `\<^sub>\<bullet> B"
b1: shows "\<exists>M. \<forall>n. norm (x n) < M"
top_on_acom_widen: fixes C1 :: "'a st option acom"   and X :: "char list set"   and C2 :: "'a st option acom" assumes "top_on\<^sub>c C1 X"   and "strip C1 = strip C2"   and "top_on\<^sub>c C2 X" shows "top_on\<^sub>c (C1 \<nabla> C2) X"
finite_imp_null_set_lborel: fixes A :: "'a set" assumes "finite A" shows "A \<in> null_sets lborel"
C2: fixes \<phi> :: "'c"   and Q :: "('a, 'b, 'c) psi" assumes "(\<phi>, Q) mem CsQ" shows "\<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> \<Psi> \<rhd> P \<sim>\<^sub>s Q"
matchC_TMC_def2: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchC_TMC theta d c = (\<forall>s t d' t'. mustT c s \<and> mustT d t \<and> s \<approx> t \<and> (d, t) \<rightarrow>c (d', t') \<longrightarrow> (\<exists>c' s'. (c, s) \<rightarrow>*c (c', s') \<and> s' \<approx> t' \<and> (d', c') \<in> theta))"
ring_inv1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "-\<^sub>a a \<cdot>\<^sub>r b = (-\<^sub>a a) \<cdot>\<^sub>r b \<and> -\<^sub>a a \<cdot>\<^sub>r b = a \<cdot>\<^sub>r (-\<^sub>a b)"
sigma_finite_countable: shows "\<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union> A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)"
lead_coeff_smult: fixes c :: "'c"   and p :: "'c poly" shows "lead_coeff (smult c p) = c * lead_coeff p"
cl_inI: fixes \<E> :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> \<E>"   and "X \<subseteq> \<E>"   and "x \<in> \<E>"   and "\<E>.rank_of (insert x X) = \<E>.rank_of X" shows "x \<in> \<E>.cl_in \<E> X"
cases: fixes x :: "('d \<Rightarrow> 'd \<Rightarrow> bool) \<times> 'd list" obtains P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" where "x = (P__, [])"   | P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" and xa :: "'d" and ys :: "'d list" where "x = (P, xa # ys)"
csiov: shows "s\<inverse> O ov \<subseteq> ov \<union> f\<inverse> \<union> d\<inverse>"
fract_expand_2: shows "1 = Fraction_Field.Fract (1::'a) (1::'a)"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_inverse" shows "a\<inverse>\<^sub>\<real> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_inverse"
phl_skip: fixes x :: "'a ppath set" shows "pp_prod x pp_one \<subseteq> pp_prod pp_one x"
InfinitePrime: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> (\<box>\<diamond>P) = (\<box>\<diamond>P$)"
CFG_Construct_def: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and "uses" :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set" shows "CFG_Construct \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses"
y_eq_v1_implies_P_k_neq_P_new: assumes "y = v1" shows "P_k \<noteq> P_new"
finite_set_antichain: fixes A :: "'b antichain" shows "finite (set_antichain A)"
f1: fixes x :: "nat \<Rightarrow> real" assumes "x \<in> topspace (nsphere p)" shows "(\<Sum>j\<le>p. (f x j)\<^sup>2) = 1"
that: fixes b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "b \<in> set bs"   and "is_regular_spair p b"   and "q = Inl (p, b)" shows "thesis"
sim_r_tc: fixes M :: "('e, 'f, 'g) KripkeStructure"   and M' :: "('e, 'f, 'h) KripkeStructure"   and f :: "'g \<Rightarrow> 'h"   and u :: "'g"   and v' :: "'h"   and as :: "'e set" assumes "kripke M"   and "sim M M' f"   and "u \<in> worlds M"   and "(f u, v') \<in> (\<Union> (relations M' ` as))\<^sup>+" obtains v :: "'g" where "f v = v'" and "(u, v) \<in> (\<Union> (relations M ` as))\<^sup>+"
dyad_rec_41_times2: fixes n :: "nat"   and b :: "nat \<Rightarrow> 'b"   and l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and m :: "nat" assumes "0 < n" shows "dyad_rec b l r (2 * ((4 * real m + 1) / 2 ^ Suc n)) = l (dyad_rec b l r (2 * (2 * real m + 1) / 2 ^ n))"
d_x_U: fixes x :: "'a" shows "d (x \<cdot> U) = d x"
skip_inj: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<in> {k. k \<le> n}"   and "j \<in> {k. k \<le> n}"   and "i \<noteq> j" shows "skip k i \<noteq> skip k j"
lb_mem: fixes i :: "nat" shows "lb i \<in> ith (eq_upto (BAD P) lb i) i"
sum_list_simps_2: fixes x :: "'a"   and xs :: "'a list" shows "sum_list (x # xs) = x + sum_list xs"
less_bit1_def: fixes a :: "'a bit1"   and b :: "'a bit1" shows "(a < b) = (Rep_bit1 a < Rep_bit1 b)"
Dom_cod: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Dom (cod f) = Cod f"
im_jointfun: fixes f :: "nat \<Rightarrow> 'h"   and n :: "nat"   and A :: "'h set"   and g :: "nat \<Rightarrow> 'h"   and m :: "nat"   and B :: "'h set" assumes "f \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> m} \<rightarrow> B" shows "jointfun n f m g ` {j. j \<le> Suc (n + m)} = f ` {j. j \<le> n} \<union> g ` {j. j \<le> m}"
Abs: fixes t :: "trm"   and t' :: "trm"   and T :: "ty"   and x :: "name" assumes "t \<longmapsto> t'" shows "(\<lambda>x:T. t) \<longmapsto> (\<lambda>x:T. t')"
1: fixes m :: "int" shows "poly_mod_type TYPE('d) m = (m = int CARD('d))"
paste_def: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" shows "P +< Q = Outside' (Domain Q) P \<union> Q"
info_2_card_2: fixes x :: "real_alg_2"   and y :: "rat" assumes "invariant_2 x"   and "info_2 x = Inl y" shows "real_of_2 x = real_of_rat y"
supp_Conj: fixes xset :: "('a, 'b, 'c) formula set['a]" assumes "finite (supp xset)" shows "supp (Conj xset) = supp xset"
Sup_UNIV: shows "\<Squnion> UNIV = \<top>"
sum_cmul: fixes c :: "'a"   and f :: "'c \<Rightarrow> ('a, 'b) vec"   and S :: "'c set" shows "(\<Sum>x\<in>S. c *s f x) = c *s sum f S"
monad_state_alt_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and alt :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_state_alt return bind get put alt" shows "monad_state_alt (return_writer return) (bind_writer return bind) (get_writer get) (put_writer put) (alt_writer alt)"
add_sign_intros_1: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "{#} \<subset># a"   and "{#} \<subseteq># b" shows "{#} \<subset># a + b"
check_eqv_sound: fixes \<AA> :: "interp"   and idx :: "idx"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and \<psi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" assumes "#\<^sub>V \<AA> = idx"   and "check_eqv idx \<phi> \<psi>" shows "WS1S.sat \<AA> \<phi> = WS1S.sat \<AA> \<psi>"
Reals_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>"   and "b \<in> \<real>" shows "a * b \<in> \<real>"
IH: shows "\<forall>v\<in>subst_domain \<theta>. subst_elim \<theta> v"
indep_in_def: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set" assumes "indep_system carrier indep" shows "indep_system.indep_in indep \<E> X = (X \<subseteq> \<E> \<and> indep X)"
dim_zero_eq: fixes A :: "'b set" assumes "local.dim A = 0" shows "A = {} \<or> A = {0::'b}"
b: shows "rec_exec rec_F [code tp, bl2wc (<lm>)] = rs - Suc 0"
s'_eq_s: fixes x :: "'a"   and i :: "nat" assumes "x \<in> space M" shows "s' i x = s i x"
lcm_dvdD1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times (lcm a b) c" shows "dvd.dvd times a c"
ide_map\<^sub>0_obj: fixes a :: "'c" assumes "C.obj a" shows "D.ide (\<sigma>'.map\<^sub>0 a)"
example_fg_def: shows "example_fg \<equiv> \<lparr>edges = {((0, 0), Call 1, 0, 1), ((1, 0), Spawn 0, 1, 0), ((1, 0), Call 0, 1, 0)}, main = 0, entry = \<lambda>p. (p, 0), return = \<lambda>p. (p, 1), mon = \<lambda>p. if p = 1 then {0::'d} else {}, proc_of = \<lambda>(p, x). p\<rparr>"
1: shows "\<forall>x\<in>set (a # args). is_closed x"
cases: fixes slices :: "'b \<Rightarrow> 'b set set"   and W :: "'b set"   and p :: "'b"   and R :: "'b set"   and a :: "'b" assumes "stellar slices W"   and "stellar.not_blocked slices W p R a" obtains Sl :: "'b set" and q :: "'b" where "a = q__" and "Sl__ \<in> slices p" and "\<forall>q\<in>Sl__ \<inter> W. q \<notin> R \<and> \<not> stellar.blocking_min slices W R q" and "q__ \<in> Sl__"   | p' :: "'b" and Sl :: "'b set" and q :: "'b" where "a = q" and "stellar.not_blocked slices W p R p'" and "p' \<in> W" and "Sl \<in> slices p'" and "\<forall>q\<in>Sl \<inter> W. q \<notin> R \<and> \<not> stellar.blocking_min slices W R q" and "q \<in> Sl"
isCont_tan: fixes x :: "'b" assumes "cos x \<noteq> (0::'b)" shows "isCont tan x"
is_poincare_line_0_imag_is_y_axis: fixes l :: "circline"   and y :: "complex_homo" assumes "is_poincare_line l"   and "0\<^sub>h \<in> circline_set l"   and "y \<in> circline_set l \<inter> circline_set y_axis"   and "y \<noteq> 0\<^sub>h"   and "y \<noteq> \<infinity>\<^sub>h" shows "l = y_axis"
at_within_def: fixes a :: "'b"   and s :: "'b set" shows "at a within s = inf (nhds a) (principal (s - {a}))"
that: fixes B :: "'a set"   and N :: "'a set" assumes "B \<in> sets borel"   and "negligible N"   and "B \<union> N = S" shows "thesis"
all_union_of: fixes P :: "'b set set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool"   and R :: "'b set \<Rightarrow> bool" shows "(\<forall>S. (P union_of Q) S \<longrightarrow> R S) = (\<forall>T. P T \<and> T \<subseteq> Collect Q \<longrightarrow> R (\<Union> T))"
set_sort: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "set (FWNormalisationCore.sort xs l) = set xs"
assms_3: fixes k :: "'a" assumes "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)" shows "P k (lookup_raw xs k) (lookup_raw ys k)"
lessThan_iff: fixes i :: "'a"   and k :: "'a" shows "(i \<in> {..<k}) = (i < k)"
hyps_8: shows "y\<^sub>3 = - y\<^sub>1 - l * (x\<^sub>3 - x\<^sub>1)"
lcf_mem: fixes p :: "'a" assumes "p \<in> carrier R" shows "lcf R S X p \<in> carrier S"
release_propag_on_create_graph: fixes \<mu> :: "'a ltlr"   and \<eta> :: "'a ltlr" shows "create_graph \<phi> \<le> SPEC (\<lambda>nds. \<forall>n\<in>nds. \<forall>n'\<in>nds. \<mu> R\<^sub>r \<eta> \<in> old n \<and> name n \<in> incoming n' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old n \<or> \<eta> \<in> old n \<and> \<mu> R\<^sub>r \<eta> \<in> old n')"
continuous_cross: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> (real, 3) vec"   and g :: "'a \<Rightarrow> (real, 3) vec" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. f x \<times> g x)"
boffa: fixes x :: "'a" assumes "x ; x = x" shows "x\<^sup>\<star> = 1' + x"
simps_3: fixes l :: "bdt"   and v :: "nat"   and r :: "bdt"   and env :: "bool list" shows "eval (Bdt_Node l v r) env = (if env ! v then eval r env else eval l env)"
correctCompositionDiffLevelsA93: shows "correctCompositionDiffLevels sA93"
it_to_sorted_list_weaken: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and R' :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and s :: "'f set" assumes "R \<le> R'" shows "it_to_sorted_list R s \<le> it_to_sorted_list R' s"
assms_3: shows "(a + b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
rcos_i: shows "J +> i = \<one>\<^bsub>R Quot J\<^esub>"
runit_in_hom_1: fixes f :: "'a" assumes "ide f" shows "arr (PB.runit f) \<and> src f \<in> sources (PB.runit f) \<and> trg f \<in> targets (PB.runit f)"
Times_Int_Times: fixes A :: "'e set"   and B :: "'f set"   and C :: "'e set"   and D :: "'f set" shows "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
Var: fixes rs :: "lam list"   and rs' :: "lam list"   and x :: "name" assumes "rs [\<rightarrow>\<^sub>l] rs'" shows "Var x \<degree>\<degree> rs \<rightarrow>\<^sub>l Var x \<degree>\<degree> rs'"
comp_assoc_assoc'_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "src f = trg g"   and "src g = trg h" shows "V (\<a> f g h) (bicategory.\<a>' V H \<a> src trg f g h) = H f (H g h)"
refl: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and \<R> :: "('e \<Rightarrow> real) set set"   and l :: "'d"   and R :: "('e \<Rightarrow> real) set" shows "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l, R\<rangle>"
left_total_Grp: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "left_total (BNF_Def.Grp A f) = (A = UNIV)"
bisimE_2: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "Q \<sim> P"
fequalityE: fixes A :: "'a fset"   and B :: "'a fset" assumes "A = B" obtains "A |\<subseteq>| B" and "B |\<subseteq>| A"
eq_FnI: fixes n :: "Arity"   and f1 :: "Value \<rightarrow> Value"   and f2 :: "Value \<rightarrow> Value" assumes "\<And>v. eq (pred\<cdot>n) (f1\<cdot>v) (f2\<cdot>v)" shows "eq n (Fn\<cdot>f1) (Fn\<cdot>f2)"
put_put: fixes s :: "'s"   and s' :: "'s"   and m :: "'m" shows "put s (put s' m) = put s' m"
c2: fixes n :: "nat" assumes "x = 2 * n + 1" shows "P"
total_well_order_extension: fixes p :: "('a \<times> 'a) set" assumes "wf p" shows "\<exists>w. p \<subseteq> w \<and> Well_order w \<and> Field w = UNIV"
is_alt_factorization_of_def: fixes fctrs :: "'a \<times> ('a \<times> nat) list"   and p :: "'a poly" shows "is_alt_factorization_of fctrs p = (interp_alt_factorization fctrs = p \<and> (0::'a) \<notin> set (map fst (snd fctrs)) \<and> distinct (map fst (snd fctrs)))"
collection_ternop_type_det: fixes op :: "collection_ternop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<upsilon>\<^sub>1 :: "'a OCL_Types.type"   and \<upsilon>\<^sub>2 :: "'a OCL_Types.type" assumes "collection_ternop_type op \<tau> \<sigma> \<rho> \<upsilon>\<^sub>1"   and "collection_ternop_type op \<tau> \<sigma> \<rho> \<upsilon>\<^sub>2" shows "\<upsilon>\<^sub>1 = \<upsilon>\<^sub>2"
ide_REP: fixes A :: "('a, 'a set) arr" assumes "arr A" shows "B.ide (REP A)"
expr_encode_inj: fixes x :: "expr"   and y :: "expr" assumes "expr_encode x = expr_encode y" shows "x = y"
assms_15: shows "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"
select_convs_1: fixes verts :: "'c set"   and arcs :: "'d set"   and tail :: "'d \<Rightarrow> 'c"   and head :: "'d \<Rightarrow> 'c"   and more :: "'e" shows "pre_digraph.verts \<lparr>pre_digraph.verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = verts"
sequence_to_failure_length: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM" assumes "well_formed M1"   and "well_formed M2"   and "observable M1"   and "observable M2"   and "\<not> M1 \<preceq> M2" shows "\<exists>xs. sequence_to_failure M1 M2 xs \<and> length xs \<le> |M2| * |M1|"
boundedI: fixes A :: "'a DBMEntry set"   and x :: "'a DBMEntry" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> a \<preceq> x" shows "linordered_monoid.Max A \<preceq> x"
0: shows "deg R (p \<oplus>\<^bsub>P\<^esub> monom P a n) = n"
Arrow: fixes F :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> (alt \<times> alt) set" assumes "F \<in> SWF"   and "unanimity F"   and "IIA F" shows "\<exists>i. dictator F i"
funion_iff: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" shows "(c |\<in>| A |\<union>| B) = (c |\<in>| A \<or> c |\<in>| B)"
lc_call: fixes t :: "sterm"   and l :: "Label"   and s :: "sterm" shows "lc (Call t l s) = (lc t \<and> lc s)"
ucast_up_eq: fixes x :: "'c word"   and y :: "'d word" assumes "UCAST('c \<rightarrow> 'b) x = UCAST('d \<rightarrow> 'b) y"   and "LENGTH('e) \<le> LENGTH('b)" shows "UCAST('c \<rightarrow> 'e) x = UCAST('d \<rightarrow> 'e) y"
awalk_cyc_decomp_def: fixes p :: "'b list" shows "awalk_cyc_decomp p = (SOME qrs. is_awalk_cyc_decomp p qrs)"
wf_converse_trancl: fixes r :: "('a \<times> 'a) set" assumes "wf (r\<inverse>)" shows "wf ((r\<^sup>+)\<inverse>)"
Act\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and f :: "'d"   and \<alpha> :: "'c" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha>)"
lookup_single_pp: fixes x :: "'b"   and e :: "'a"   and y :: "'b" shows "lookup_pp (single_pp x e) y = (e when x = y)"
unfold_congs_1: fixes r :: "('c, 'd, 'e) configuration_scheme"   and r' :: "('c, 'd, 'e) configuration_scheme"   and v' :: "'d zmultiset"   and f :: "'d zmultiset \<Rightarrow> 'd zmultiset"   and f' :: "'d zmultiset \<Rightarrow> 'd zmultiset" assumes "r = r'"   and "c_records r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "c_records_update f r = c_records_update f' r'"
diff_Un_incl_triv: fixes A :: "'b set"   and D :: "'b set"   and C :: "'b set"   and E :: "'b set"   and B :: "'b set" assumes "A \<subseteq> D"   and "C \<subseteq> E" shows "A - B \<union> C \<subseteq> D \<union> E"
orelse_thms_5: fixes y :: "tr" shows "(y orelse y) = y"
mat_multI_def: fixes ze :: "'b"   and pl :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ti :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and nr :: "nat"   and m1 :: "'b list list"   and m2 :: "'b list list" shows "mat_multI ze pl ti nr m1 m2 \<equiv> map (matT_vec_multI ze pl ti (Matrix_Legacy.transpose nr m1)) m2"
RedT_preserves_ts_inv_ok: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ttas :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and I :: "'t \<Rightarrow> 'a option"   and Q :: "'a \<Rightarrow> 't \<Rightarrow> 'x \<Rightarrow> 'm \<Rightarrow> bool" assumes "s -\<triangleright>ttas\<rightarrow>* s'"   and "ts_inv_ok (thr s) I" shows "ts_inv_ok (thr s') (upd_invs I Q (concat (map (thr_a \<circ> snd) ttas)))"
select_result_I: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog"   and P :: "'z \<Rightarrow> bool" assumes "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> P x" shows "P |h \<turnstile> f|\<^sub>r"
oalist_eq_rawD: fixes ys :: "('a \<times> 'c) list \<times> 'o"   and xs :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_inv ys"   and "oalist_eq_raw xs ys" shows "lookup_raw xs = lookup_raw ys"
on_curvep_imp_in_carrierp: fixes a :: "int"   and b :: "int"   and p :: "int \<times> int \<times> int" assumes "on_curvep a b p" shows "in_carrierp p"
TNil: fixes b :: "'c"   and b' :: "'d" assumes "xs = TNil b"   and "ys = TNil b'"   and "Q b b'" shows "thesis"
vector_2_2: fixes x :: "'b"   and y :: "'b" shows "vector [x, y] $ 2 = y"
as_irred: fixes a :: "'a poly"   and i :: "nat" assumes "(a, i) \<in> as" shows "irreducible\<^sub>d a"
union_of_def: fixes P :: "'d set set \<Rightarrow> bool"   and Q :: "'d set \<Rightarrow> bool" shows "P union_of Q \<equiv> \<lambda>S. \<exists>\<U>. P \<U> \<and> \<U> \<subseteq> Collect Q \<and> \<Union> \<U> = S"
LLL_d_Suc: fixes fs :: "int vec list"   and k :: "nat" assumes "LLL_invariant_weak fs"   and "k < m" shows "rat_of_int (d fs (Suc k)) = \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) k\<parallel>\<^sup>2 * rat_of_int (d fs k)"
rbt_comp_ins: shows "local.rbt_comp_ins = ord.rbt_ins (lt_of_comp c)"
ThrowThrow: fixes 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 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" shows "P \<turnstile> \<langle>throw (throw e),s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
distinct_51: fixes x4 :: "MIR.num"   and x91 :: "int"   and x92 :: "MIR.num"   and x93 :: "MIR.num" shows "Neg x4 \<noteq> CF x91 x92 x93"
psubstBinp_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option"   and binp :: "'h \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option" assumes "goodEnv rho"   and "goodBinp binp" shows "goodBinp binp %%[rho]"
"2_1": shows "x \<in> lang (subst (w2rexp xs) \<sigma>) @@ lang (subst (w2rexp ys) \<sigma>)"
BIT_absch_le: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list" assumes "BIT_Off acts qs init"   and "init \<noteq> []" shows "T_on_rand' BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs \<le> 7 / 4 * real (T init qs acts) - 3 / 4 * real (length qs)"
cone_determines_arrow_1: fixes d :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "'c"   and \<chi> :: "'j \<Rightarrow> 'c" assumes "J_C.ide d"   and "Limit.cone (\<cdot>\<^sub>J) (\<cdot>) (\<chi>.\<Phi>.Ya.Cop_S.Map d) a \<chi>" shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map a d (\<chi>.\<Phi>.Ya.Cop_S.MkArr (constant_functor.map (\<cdot>\<^sub>J) (\<cdot>) a) (\<chi>.\<Phi>.Ya.Cop_S.Map d) \<chi>)"
f: shows "f \<in> lspace (lebesgue_on S) (ennreal p)"
simps: fixes a1 :: "'c itself"   and a2 :: "'d itself" shows "rel_itself a1 a2 = (a1 = TYPE('c) \<and> a2 = TYPE('d))"
xor_list_inverse: fixes xs :: "'b list" shows "xs [\<oplus>] xs = replicate (length xs) \<bottom>"