hcju/isabelleps huggingface dataset:
elims: fixes x :: "'b"   and xa :: "'b tree"   and xb :: "'b tree"   and y :: "'b tree" assumes "mkt_bal_r x xa xb = y" obtains n :: "'b" and l :: "'b tree" and r :: "'b tree" where "x = n" and "xa = l" and "xb = r" and "y = (if ht r = ht l + 2 then case r of MKT rn rl rr x \<Rightarrow> if ht rr < ht rl then case rl of MKT rln rll rlr x \<Rightarrow> mkt rln (mkt n l rll) (mkt rn rlr rr) else mkt rn (mkt n l rl) rr else mkt n l r)"
hom_closed: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f"   and x :: "'d" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h x \<in> carrier H"
sgn_real_def: fixes a :: "real" shows "sgn a = (if a = 0 then 0 else sgn_bool (0 < a))"
sorted_insort: fixes a :: "'c"   and xs :: "'c list" shows "sorted (insort a xs) = sorted xs"
restriction_consec_clm: fixes v :: "view"   and u :: "view"   and w :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=u--w \<and> consec (lan u) (lan w) \<longrightarrow> restrict u (clm ts) c = bot \<or> restrict w (clm ts) c = bot"
has_field_derivative_ln_Gamma_complex: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "(ln_Gamma has_field_derivative Digamma z) (at z)"
dep2_def: fixes u :: "'a"   and v :: "'a" shows "dep2 u v \<equiv> \<exists>w r s. u = r *\<^sub>R w \<and> v = s *\<^sub>R w"
hom_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<otimes> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
zero_hom_0_axioms_def: fixes hom :: "'a \<Rightarrow> 'b" shows "zero_hom_0_axioms hom \<equiv> \<forall>x. hom x = (0::'b) \<longrightarrow> x = (0::'a)"
induct: fixes P :: "('c \<Rightarrow> 'd \<Rightarrow> assn) \<Rightarrow> 'c list \<Rightarrow> 'd list \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and a1 :: "'c list"   and a2 :: "'d list" assumes "\<And>Pa. P Pa [] []"   and "\<And>Pa a as c cs. P Pa as cs \<Longrightarrow> P Pa (a # as) (c # cs)"   and "\<And>uu_ v va. P uu_ (v # va) []"   and "\<And>uu_ v va. P uu_ [] (v # va)" shows "P a0 a1 a2"
bst_bst_of_list: fixes xs :: "'a list" shows "bst (bst_of_list xs)"
assms_2: shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"
alphaAll_refl: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs" shows "(qGood X \<longrightarrow> X #= X) \<and> (qGoodAbs A \<longrightarrow> A $= A)"
finite_in_arcs: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c" assumes "fin_digraph G" shows "finite (in_arcs G v)"
conjugate_cross_ratio: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo"   and z :: "complex_homo" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3" shows "cross_ratio (conjugate z) (conjugate z1) (conjugate z2) (conjugate z3) = conjugate (cross_ratio z z1 z2 z3)"
not_B: fixes c :: "color" assumes "c \<noteq> B" shows "c = R"
simps_2: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "sp_impl_subj_obj \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = sp_impl_subj_obj"
NC_disjoint_def: fixes v :: "'b V_rec" shows "NC_disjoint v \<equiv> N\<^bsub>v\<^esub> \<inter> C\<^bsub>v\<^esub> = {}"
Pi_split_insert_domain: fixes x :: "'c \<Rightarrow> 'd"   and i :: "'c"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd set" shows "(x \<in> Pi (insert i I) X) = (x \<in> Pi I X \<and> x i \<in> X i)"
conv_contrav: fixes x :: "'a"   and y :: "'a" shows "(x ; y)\<^sup>T = y\<^sup>T ; x\<^sup>T"
to_list_correct: fixes e :: "'a" shows "map_of (Finca_to_list e) = Finca_get e"
ja_less_suc_k: shows "mod_type_class.to_nat ja < Suc k"
mycasexx_def: fixes x :: "'a"   and y :: "'a" shows "mycasexx x y = Plus (seq [Star (Plus (Atom x) (Atom y)), Atom x, Atom x]) (seq [Star (Plus (Atom x) (Atom y)), Atom y, Atom y])"
ii_not_nonpos_Reals: shows "\<i> \<notin> \<real>\<^sub>\<le>\<^sub>0"
actrans_def: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "_actrans P v \<equiv> P \<or> Unchanged v"
a_inv: fixes h :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set" assumes "h \<in> generate_ring R H" shows "\<ominus>\<^bsub>R\<^esub> h \<in> generate_ring R H"
ASSUME_simps_1: shows "aASSUME True = areturn ()"
not_zero_y: shows "\<not> is_zero_row_upt_k y (Suc k) (Gauss_Jordan_in_ij A (0::'rows) (mod_type_class.from_nat k))"
invertible_Hermite_of: fixes ass :: "'a \<Rightarrow> 'a"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'c) vec, 'b) vec"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "ass_function ass"   and "is_bezout_ext bezout" shows "\<exists>P. invertible P \<and> Hermite_of A ass res bezout = P ** A"
irr: shows "as \<subseteq> {q. irreducible q \<and> monic q}"
strict_subtuple_mono: fixes xm :: "'e \<rightharpoonup>\<^sub>f 'd"   and ym :: "'e \<rightharpoonup>\<^sub>f 'd"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> fmran' xm; y \<in> fmran' ym\<rbrakk> \<Longrightarrow> f x y \<longrightarrow> g x y" shows "strict_subtuple f xm ym \<longrightarrow> strict_subtuple g xm ym"
set_bset_inverse: fixes x :: "'c set['d]" shows "Abs_bset (set_bset x) = x"
raw_converter_invariant_run_converter: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>" shows "raw_converter_invariant \<I> \<I>' run_converter (WT_converter \<I> \<I>')"
Abs_poly_inverse: fixes y :: "nat \<Rightarrow> 'b" assumes "y \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'b)}" shows "coeff (Abs_poly y) = y"
conga__lea: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C CongA D E F" shows "A B C LeA D E F"
part_def: fixes f :: "'c \<Rightarrow> 'a"   and pivot :: "'a"   and xs :: "'c list" shows "part f pivot xs = (filter (\<lambda>x. f x < pivot) xs, filter (\<lambda>x. f x = pivot) xs, filter (\<lambda>x. pivot < f x) xs)"
nxtActLe: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "n \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
mult_float_mono1: fixes a :: "real"   and b :: "real"   and ab :: "real"   and bb :: "real"   and aa :: "real"   and ba :: "real"   and ac :: "real"   and bc :: "real"   and prec :: "nat" assumes "a \<le> b"   and "ab \<le> bb"   and "aa \<le> a"   and "b \<le> ba"   and "ac \<le> ab"   and "bb \<le> bc" shows "plus_down prec (nprt aa * pprt bc) (plus_down prec (nprt ba * nprt bc) (plus_down prec (pprt aa * pprt ac) (pprt ba * nprt ac))) \<le> plus_down prec (nprt a * pprt bb) (plus_down prec (nprt b * nprt bb) (plus_down prec (pprt a * pprt ab) (pprt b * nprt ab)))"
SD_pref_profile: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and q :: "'b pmf"   and p :: "'b pmf" assumes "pref_profile_wf agents alts R"   and "i \<in> agents" shows "SD (R i) q p = (p \<in> lotteries_on alts \<and> q \<in> lotteries_on alts \<and> (\<forall>x\<in>alts. measure_pmf.prob q (preferred_alts (R i) x) \<le> measure_pmf.prob p (preferred_alts (R i) x)))"
single_valued_lfp_Mon2Rel: fixes c :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> (unit \<times> 'a) option" shows "single_valued (lfp (\<Gamma> c (Mon2Rel B)))"
head_isnpolyh: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "isnpolyh (head p) n0"
thms_I: fixes H :: "'b pl set"   and p :: "'b pl" shows "H |- p -> p"
pcr_cr_eq: shows "pcr_ocircline = cr_ocircline"
assert_bound_cases: fixes a :: "'a atom"   and s :: "('i, 'a) state"   and P' :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> (('i, 'a) state \<Rightarrow> (nat, 'i \<times> 'a) mapping) \<Rightarrow> (('i, 'a) state \<Rightarrow> (nat, 'i \<times> 'a) mapping) \<Rightarrow> (('i, 'a) state \<Rightarrow> nat \<Rightarrow> 'a option) \<Rightarrow> (('i, 'a) state \<Rightarrow> nat \<Rightarrow> 'a option) \<Rightarrow> (((nat, 'i \<times> 'a) mapping \<Rightarrow> (nat, 'i \<times> 'a) mapping) \<Rightarrow> ('i, 'a) state \<Rightarrow> ('i, 'a) state) \<Rightarrow> (('i, 'a) state \<Rightarrow> nat \<Rightarrow> 'i) \<Rightarrow> (('i, 'a) state \<Rightarrow> nat \<Rightarrow> 'i) \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a atom) \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a atom) \<Rightarrow> ('i, 'a) state \<Rightarrow> bool"   and i :: "'i"   and P :: "('i, 'a) state \<Rightarrow> bool" assumes "\<And>c x dir. \<lbrakk>dir = Positive \<or> dir = Negative; a = LE dir x c; \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)\<rbrakk> \<Longrightarrow> P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) s"   and "\<And>c x dir. \<lbrakk>dir = Positive \<or> dir = Negative; a = LE dir x c; \<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x); \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)\<rbrakk> \<Longrightarrow> P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (set_unsat [i, LI dir s x] (update\<B>\<I> (UBI_upd dir) i x c s))"   and "\<And>c x dir. \<lbrakk>dir = Positive \<or> dir = Negative; a = LE dir x c; x \<notin> lvars (\<T> s); lt dir c (\<langle>\<V> s\<rangle> x); \<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x); \<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)\<rbrakk> \<Longrightarrow> P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (update x c (update\<B>\<I> (UBI_upd dir) i x c s))"   and "\<And>c x dir. \<lbrakk>dir = Positive \<or> dir = Negative; a = LE dir x c; x \<in> lvars (\<T> s); \<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x); \<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)\<rbrakk> \<Longrightarrow> P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (update\<B>\<I> (UBI_upd dir) i x c s)"   and "\<And>c x dir. \<lbrakk>dir = Positive \<or> dir = Negative; a = LE dir x c; \<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x); \<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x); \<not> lt dir c (\<langle>\<V> s\<rangle> x)\<rbrakk> \<Longrightarrow> P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (update\<B>\<I> (UBI_upd dir) i x c s)"   and "\<And>s. P s = P' (\<lambda>x y. y < x) \<B>\<^sub>i\<^sub>l \<B>\<^sub>i\<^sub>u \<B>\<^sub>l \<B>\<^sub>u \<B>\<^sub>i\<^sub>l_update \<I>\<^sub>l \<I>\<^sub>u Geq Leq s"   and "\<And>s. P s = P' (<) \<B>\<^sub>i\<^sub>u \<B>\<^sub>i\<^sub>l \<B>\<^sub>u \<B>\<^sub>l \<B>\<^sub>i\<^sub>u_update \<I>\<^sub>u \<I>\<^sub>l Leq Geq s" shows "P (assert_bound (i, a) s)"
emptyseq_collapse_eq: fixes s :: "nat \<Rightarrow> 'b" assumes "emptyseq s" shows "\<natural> s = s"
\<K>_\<S>_closed': fixes s :: "'s \<times> ('c \<Rightarrow> real) set"   and D :: "('s \<times> ('c \<Rightarrow> real) set) pmf"   and t :: "'s \<times> ('c \<Rightarrow> real) set" assumes "s \<in> \<S>"   and "D \<in> \<K> s"   and "t \<in> set_pmf D" shows "t \<in> \<S>"
forall_spec: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and a :: "'e" assumes "\<forall>b. P b \<longrightarrow> Q b"   and "P a" shows "Q a"
derivative_on_prod_subset: fixes X' :: "'a set" assumes "X' \<subseteq> X'" shows "derivative_on_prod T X' f f'"
ccINF_greatest: fixes A :: "'b set"   and u :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> u \<le> f i" shows "u \<le> Inf (f ` A)"
assms_1: shows "x \<in>\<^sub>\<circ> vid_on A"
subst_atmss_comp_subst: fixes AAA :: "'a set set"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "AAA \<cdot>ass (\<tau> \<odot> \<sigma>) = AAA \<cdot>ass \<tau> \<cdot>ass \<sigma>"
mgu_img_composed_var_term: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and f :: "'b"   and Z :: "'a list" assumes "mgu s t = Some \<delta>"   and "Fun f (map Var Z) \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t subst_range \<delta>" shows "\<exists>Z'. map \<delta> Z' = map Var Z \<and> Fun f (map Var Z') \<in> subterms s \<union> subterms t"
oassert_def: fixes \<Phi> :: "bool" shows "oassert \<Phi> \<equiv> if \<Phi> then Some () else None"
relChain_def: fixes r :: "('b \<times> 'b) set"   and As :: "'b \<Rightarrow> 'c" shows "relChain r As \<equiv> \<forall>i j. (i, j) \<in> r \<longrightarrow> As i \<le> As j"
cases: fixes biG :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)"   and a :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)" assumes "subdivision biG a" obtains "a = biG" and "bidirected_digraph (fst biG) (snd biG)"   | biI :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)" and biH :: "('a, 'b) pre_digraph \<times> ('b \<Rightarrow> 'b)" and u :: "'a" and v :: "'a" and w :: "'a" and uv :: "'b" and uw :: "'b" and vw :: "'b" where "a = biH" and "subdivision biG biI" and "subdivision_step (fst biI) (snd biI) (fst biH) (snd biH) (u, v, w) (uv, uw, vw)"
0: shows "local.dom \<tau> \<star> local.dom \<mu> \<noteq> null \<and> local.dom \<mu> \<star> local.dom \<nu> \<noteq> null \<and> cod \<tau> \<star> cod \<mu> \<noteq> null \<and> cod \<mu> \<star> cod \<nu> \<noteq> null"
weight_spmf_of_set_finite: fixes A :: "'c set" assumes "finite A"   and "A \<noteq> {}" shows "weight_spmf (spmf_of_set A) = 1"
E: shows "(Asset n, Crypt (Auth_ShaKey n) (PriKey S)) \<in> s \<longrightarrow> PriKey S \<notin> spied s"
csol_4: shows "(csol t0 x0 t solves_ode f) {t0--t} X"
fp: shows "finprod G g S = (\<Otimes>x\<in>S. h (j x))"
refine_init_using_invariants: fixes Tc :: "('e, 'f) TS_scheme"   and R :: "('g \<times> 'e) set"   and Ta :: "('g, 'h) TS_scheme"   and I :: "'g set"   and J :: "'e set" assumes "init Tc \<subseteq> R `` init Ta"   and "init Ta \<subseteq> I"   and "init Tc \<subseteq> J" shows "init Tc \<subseteq> (R \<inter> I \<times> J) `` init Ta"
well_related_iff_neg_well_founded: fixes A :: "'a set" shows "well_related_set A (\<sqsubseteq>) = well_founded A (\<lambda>x y. \<not> y \<sqsubseteq> x)"
Sup_upper2: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and u :: "'c"   and A :: "'c set"   and v :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "u \<in> A"   and "less_eq v u" shows "less_eq v (Sup A)"
ennreal_tendsto_0_iff: fixes f :: "nat \<Rightarrow> real" assumes "\<And>n. 0 \<le> f n" shows "(\<lambda>n. ennreal (f n)) \<longlonglongrightarrow> 0 = f \<longlonglongrightarrow> 0"
extremum_strict: fixes a :: "'a list" shows "\<not> strict_prefix a []"
m_closed: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)" shows "x \<otimes>\<^bsub>poly_ring R\<^esub> y \<in> carrier (poly_ring R)"
hom_decompose_prod_root_unity_main: fixes p :: "'a poly"   and k :: "nat" shows "decompose_prod_root_unity_main (map_poly hom p) k = map_prod id (map_poly hom) (decompose_prod_root_unity_main p k)"
fun_of_def: fixes S :: "('b \<times> 'c) set" shows "fun_of S \<equiv> \<lambda>x y. (x, y) \<in> S"
frgv_impl_rel_ext_internal_def: fixes Rm :: "('b \<times> 'd) set"   and Rv :: "('a \<times> 'c) set" shows "frgv_impl_rel_ext Rm Rv \<equiv> \<langle>Rm, \<langle>Rv\<rangle>list_set_rel, \<langle>Rv\<rangle>slg_rel, \<langle>Rv\<rangle>list_set_rel\<rangle>gen_g_impl_rel_ext"
a: shows "\<nexists>l. \<forall>l'\<ge>l. f l' = f (Suc l')"
valid_conj_PL: fixes p :: "'a"   and q :: "'a" shows "\<tturnstile> (p .\<and>. q) = (\<tturnstile> p \<and> \<tturnstile> q)"
n_eq: shows "enat n = (\<Sum>i<m. llength (lnth (LCons xs' xss'') i)) + enat n'"
elims: fixes x :: "game option"   and xa :: "game option"   and y :: "game option" assumes "Choiceo x xa = y" obtains \<alpha> :: "game" and \<beta> :: "game" where "x = Agame \<alpha>__" and "xa = Agame \<beta>" and "y = Agame (\<alpha>__ \<union>\<union> \<beta>)"   | \<alpha> :: "game option" where "x = \<alpha>" and "xa = undefg" and "y = undefg"   | v :: "game" where "x = undefg" and "xa = Agame v" and "y = undefg"
measurable_compose_countable': fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and g :: "'d \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> f i \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M count_space I"   and "countable I" shows "(\<lambda>x. f (g x) x) \<in> M \<rightarrow>\<^sub>M N"
points_of_path_iff_1: fixes x :: "'a" shows "(x + x\<^sup>T) ; 1 = x\<^sup>T ; 1 + start_points x"
upd_cs_snoc_Call: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and cs :: "'a list"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "CFG.upd_cs kind cs (as @ [a]) = a # CFG.upd_cs kind cs as"
isomorphic_dags_eq_sym: fixes st\<^sub>1 :: "dag"   and st\<^sub>2 :: "dag"   and var :: "ref \<Rightarrow> nat" shows "isomorphic_dags_eq st\<^sub>1 st\<^sub>2 var = isomorphic_dags_eq st\<^sub>2 st\<^sub>1 var"
sequentially_imp_eventually_at_left: fixes b :: "'b"   and a :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "b < a"   and "\<And>f. \<lbrakk>\<And>n. b < f n; \<And>n. f n < a; incseq f; f \<longlonglongrightarrow> a\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F n in sequentially. P (f n)" shows "eventually P (at_left a)"
cl_type_is: shows "cl_type cl = t1s _> t2s"
B1: fixes x :: "'b" assumes "x \<in> B" shows "norm x = 1"
ordering_oLimit: fixes A :: "ordinal set"   and f :: "nat \<Rightarrow> ordinal" assumes "normal_set A" shows "OrdinalVeblen.ordering A (oLimit f) = oLimit (\<lambda>n. OrdinalVeblen.ordering A (f n))"
treewidth_def: fixes G :: "('a, 'b) Graph_scheme" assumes "Graph G" shows "Graph.treewidth G \<equiv> Min {TreeDecomposition.width T bag |T bag. TreeDecomposition G T bag}"
rank_invar_cons_up: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list"   and t' :: "('c, 'd) SkewBinomialTree" assumes "rank_invar (t # bq)"   and "rank t' < rank t" shows "rank_invar (t' # t # bq)"
restrict_map_to_empty: fixes m :: "'a \<Rightarrow> 'b option" shows "m |` {} = Map.empty"
weak_reduction_correspondence_simulation_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_correspondence_simulation Rel Cal" shows "weak_reduction_correspondence_simulation (Rel\<^sup>=) Cal"
dec_clocked_unclocked_equiv: fixes mn :: "char list list"   and env :: "v sem_env"   and s1 :: "'b state"   and d :: "dec"   and s2 :: "'b state"   and r :: "(v sem_env, v) result" shows "evaluate_dec False mn env s1 d (s2, r) = (\<exists>c. evaluate_dec True mn env (update_clock (\<lambda>_. c) s1) d (update_clock (\<lambda>_. 0) s2, r) \<and> r \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s1 = clock s2)"
assms: shows "x \<le> 319 / 10\<^sup>2"
cases: fixes a :: "'b \<times> 'b"   and w :: "'b list" assumes "a \<in> rotate_rel w" obtains n :: "nat" where "a = mismatch_pair w (rotate n w)" and "0 < n" and "n < \<^bold>|w\<^bold>|"
root_periodN: fixes w :: "'b list"   and r :: "'b list" assumes "w \<noteq> \<epsilon>"   and "w \<le>p r\<^sup>\<omega>" shows "periodN w \<^bold>|r\<^bold>|"
simps: 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 a1 :: "char list"   and a2 :: "char list list" shows "Subobjs\<^sub>R P a1 a2 = ((\<exists>C. a1 = C \<and> a2 = [C] \<and> is_class P C) \<or> (\<exists>C D Cs. a1 = C \<and> a2 = C # Cs \<and> P \<turnstile> C \<prec>\<^sub>R D \<and> Subobjs\<^sub>R P D Cs))"
n_def: shows "n = max (max \<bar>lead_coeff f\<bar> (int (degree f))) (separable_bound f)"
zero_less_\<alpha>: fixes G :: "nat set \<times> nat set set" assumes "uverts G \<noteq> {}" shows "0 < \<alpha> G"
d_cex: shows "\<not> INTERP N \<Turnstile> DA"
GodDefsAreEquivalent: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<forall>xa. \<P> xa w \<longrightarrow> xa x) = (\<forall>xa. \<P> xa w = xa x))\<rfloor>"
tensor_P_left_right_partial2: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and m3 :: "'a mat"   and m4 :: "'a mat" assumes "m1 \<in> carrier_mat K K"   and "m2 \<in> carrier_mat K K"   and "m3 \<in> carrier_mat N N"   and "m4 \<in> carrier_mat K K" shows "tensor_P (1\<^sub>m N) m1 * tensor_P m3 m2 * tensor_P (1\<^sub>m N) m4 = tensor_P m3 (m1 * m2 * m4)"
I: fixes n :: "'a" shows "(f n has_integral I n) (cbox a b)"
is_singletonI': fixes A :: "'e set" assumes "A \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "is_singleton A"
image_mult_atLeastAtMost_if: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 c :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "times c ` ord.atLeastAtMost less_eq x y = (if less zero c then ord.atLeastAtMost less_eq (times c x) (times c y) else if less_eq x y then ord.atLeastAtMost less_eq (times c y) (times c x) else {})"
split_pairD: fixes x :: "'b"   and y :: "'c"   and p :: "'b \<times> 'c" assumes "(x, y) = p" shows "x = fst p \<and> y = snd p"
naturality: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.\<ll>' (\<F>C.cod f) \<cdot> \<F>C.map f = \<F>C.tensor \<F>C.unity f \<cdot> \<F>C.\<ll>' (\<F>C.dom f)"
top_le_conv: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b"   and x :: "'b" assumes "order r"   and "Semilat.top r T" shows "(T \<sqsubseteq>\<^bsub>r\<^esub> x) = (x = T)"
sup_commute: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> y = y \<oplus> x"
fin: shows "finite {(R, f). R \<in> Rs \<and> Domain f = vertices (fst R) \<and> univalent f \<and> nextMax (Range f) \<le> N}"
set_zip_map2: fixes a :: "'a"   and z :: "'b"   and xs :: "'a list"   and f :: "'c \<Rightarrow> 'b"   and ys :: "'c list" assumes "(a, z) \<in> set (zip xs (map f ys))" shows "\<exists>b. (a, b) \<in> set (zip xs ys) \<and> z = f b"
less_le_not_le: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" shows "(x \<sqsubset> y) = (x \<sqsubseteq> y \<and> y \<notin> subterms x)"
diff_tr_def: fixes s1 :: "('c, 'd) alpha_rp_scheme"   and s2 :: "('c, 'e) alpha_rp_scheme" shows "diff_tr s1 s2 = tr s1 - tr s2"
wtL_gL: fixes l :: "('fsym, 'psym) lit" assumes "Ik.wtL l" shows "GE.wtL (gL l)"