hcju/isabelleps huggingface dataset:
jordan_nf_order: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and a :: "'a" assumes "jordan_nf A n_as" shows "order a (char_poly A) = sum_list (map fst (filter (\<lambda>na. snd na = a) n_as))"
get_put: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e" assumes "wb_lens x" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
brnL_lsum: fixes cl :: "('test, 'atom, 'choice) cmd list" shows "brnL cl (length cl) = lsum brn cl"
graph_eq_intro: fixes G :: "'c graph"   and G' :: "'c graph" assumes "nodes G = nodes G'"   and "edges G = edges G'" shows "G = G'"
init_simps_6: fixes T :: "Arraytype" shows "init (ArrT T) = nullV"
cl_def: fixes rbt :: "('a, 'b) rbt" shows "cl rbt = (case rbt of Leaf \<Rightarrow> B | Node x21 x22 x23 x24 x25 \<Rightarrow> x22)"
prefix_snoc: fixes xs :: "'b list"   and ys :: "'b list"   and y :: "'b" shows "prefix xs (ys @ [y]) = (xs = ys @ [y] \<or> prefix xs ys)"
2: shows "\<exists>x\<in>X. n_ivl (lookup (S1 \<triangle> S2) x) < n_ivl (lookup S1 x)"
l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
IH: shows "\<forall>ys. length ys < length ps \<longrightarrow> 0 < length ys \<longrightarrow> (\<forall>k. distinct (map (\<lambda>p. p $ k) ys)) \<longrightarrow> balanced (build ks ys)"
quarter_discrim_positive: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "proj2_abs p \<noteq> proj2_abs q"   and "proj2_abs p \<in> hyp2" shows "0 < quarter_discrim p q"
simps_3: fixes l :: "nat" shows "compile l Throw = ([THROW], l)"
assms_6: shows "check_AND (x0, x1) (a0, a1) s' (z0', z1')"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'node SDG_node list"   and x3 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "realizable x1 x2 x3"   and "\<And>n ns n'. matched n ns n' \<Longrightarrow> P n ns n'"   and "\<And>n\<^sub>0 ns n\<^sub>1 p n\<^sub>2 V ns' n\<^sub>3. \<lbrakk>realizable n\<^sub>0 ns n\<^sub>1; P n\<^sub>0 ns n\<^sub>1; n\<^sub>1 -p\<rightarrow>\<^bsub>call\<^esub> n\<^sub>2 \<or> n\<^sub>1 -p:V\<rightarrow>\<^bsub>in\<^esub> n\<^sub>2; matched n\<^sub>2 ns' n\<^sub>3\<rbrakk> \<Longrightarrow> P n\<^sub>0 (ns @ n\<^sub>1 # ns') n\<^sub>3" shows "P x1 x2 x3"
nth_d_is: shows "lnth Ns n = N \<union> {(C0, L)}"
unit_disc_fix_preserve_poincare_collinear3: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "unit_disc_fix M" shows "poincare_collinear {moebius_pt M u, moebius_pt M v, moebius_pt M w} = poincare_collinear {u, v, w}"
zero_sheaf_spec_in_sheaf_spec: fixes U :: "'a set set" assumes "is_zariski_open U" shows "zero_sheaf_spec U \<in> \<O> U"
assert_demonic_comp: fixes p :: "'e \<Rightarrow> bool"   and r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and p' :: "'f \<Rightarrow> bool"   and r' :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "{. p .} \<circ> [: r :] \<circ> {. p' .} \<circ> [: r' :] = {.x. p x \<and> (\<forall>y. r x y \<longrightarrow> p' y).} \<circ> [: r OO r' :]"
rref_suc_k: shows "reduced_row_echelon_form_upt_k (Gauss_Jordan A) (Suc k)"
add_lt_neutral: fixes a :: "'a"   and b :: "'a" assumes "a + b < b" shows "a < (0::'a)"
intros_2: fixes n :: "nat" shows "i_set_indp [\<dots>n]"
apropsOf_L_stateOf: fixes AP' :: "('U, 'C, 'Out) Noninterference.aprop set"   and \<pi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream" assumes "wfp AP' \<pi>" shows "apropsOf \<pi> = L (stateOf \<pi>)"
fpxs_nth_add: fixes f :: "'b fpxs"   and g :: "'b fpxs"   and r :: "rat" shows "fpxs_nth (f + g) r = fpxs_nth f r + fpxs_nth g r"
timpl_closure'_inv''_3: fixes f :: "('a, 'b, 'c) prot_fun"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and g :: "('a, 'b, 'c) prot_fun"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and TI :: "('c set \<times> 'c set) set" assumes "(Fun f S, Fun g T) \<in> timpl_closure' TI"   and "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> TI\<^sup>+"
null: fixes x :: "'a" assumes "x \<in> space M" shows "{x} \<in> null_sets M"
closedin_topspace: fixes U :: "'b topology" shows "closedin U (topspace U)"
fin_2: fixes n :: "'b" assumes "n \<in> N" shows "finite (B1 n) \<and> finite (B2 n)"
appliedLearn_def: fixes stateA :: "State"   and stateB :: "State" shows "appliedLearn stateA stateB \<equiv> getConflictFlag stateA = True \<and> \<not> getC stateA el getF stateA \<and> getF stateB = getF stateA @ [getC stateA] \<and> getM stateB = getM stateA \<and> getConflictFlag stateB = True \<and> getC stateB = getC stateA"
rep_eq: fixes x :: "('c, 'd) lifted_pair"   and xa :: "('c, 'd) lifted_pair" shows "Rep_lifted_pair (x \<squnion> xa) = (\<lambda>f. triple.pairs_sup (Rep_lifted_pair x f) (Rep_lifted_pair xa f))"
lookup_def: fixes x :: "'v" shows "lookup x = ask (\<lambda>s. case s x of None \<Rightarrow> fail | Some y \<Rightarrow> return y)"
count_greater_eq_Suc_zero_iff: fixes M :: "'b multiset"   and x :: "'b" shows "(Suc 0 \<le> count M x) = (x \<in># M)"
sup2_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion>2 (b \<squnion>2 c) = a \<squnion>2 b \<squnion>2 c"
tObsC_abs_jview_eq: fixes a :: "'a"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" assumes "spr_jview a t' = spr_jview a t" shows "tObsC_abs t = tObsC_abs t'"
match_matches: fixes t :: "('e, 'd) Term.term"   and p :: "('e, 'd) Term.term"   and \<sigma> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "match t p = Some \<sigma>" shows "p \<cdot> \<sigma> = t"
canonical_retraction_simplex_retraction2: fixes A :: "'a set set"   and C :: "'a set"   and a :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "C \<in> A"   and "a \<in> A" shows "canonical_retraction A C ` a = a"
smaller_than_small: fixes A :: "'b set"   and B :: "'b set" assumes "small A"   and "B \<subseteq> A" shows "small B"
star_mod_def: shows "(mod) \<equiv> *f2* (mod)"
E_of_succ_def: fixes succ :: "'b \<Rightarrow> 'c set" shows "E_of_succ succ \<equiv> {(u, v). v \<in> succ u}"
diff: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "additive f" shows "f (x - y) = f x - f y"
shift_def: fixes lab :: "'b list \<Rightarrow> 'c"   and k :: "'b" shows "BNF_Greatest_Fixpoint.shift lab k = (\<lambda>kl. lab (k # kl))"
ccSUP_upper: fixes A :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "countable A"   and "i \<in> A" shows "f i \<le> Sup (f ` A)"
5: shows "bprv (imp (subst \<chi> \<langle>\<chi>\<rangle> xx) (subst \<phi> \<langle>\<psi>\<rangle> xx))"
meets_wd: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a" assumes "class.interval meets \<I>"   and "meets p q" shows "\<I> p \<and> \<I> q"
add_neg_numeral_simps_3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus (numeral.numeral one plus m)) (uminus (numeral.numeral one plus n)) = uminus (plus (numeral.numeral one plus m) (numeral.numeral one plus n))"
t_stop_def: shows "local.t_stop = ego2.t_stop + \<delta>"
filt_equiv_filtration: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure" assumes "filt_equiv F M N" shows "Filtration.filtration N F"
mod: fixes a :: "nat" shows "\<Phi> (a mod d) = \<Phi> a"
a_get_root_node_locs_def: shows "Shadow_DOM.get_root_node_locs = Shadow_DOM.get_ancestors_locs"
NR\<^sub>N_recurrence': fixes n :: "nat"   and u :: "nat"   and l :: "nat" assumes "0 < n"   and "p \<in> {0<..<1}" shows "NR\<^sub>N n u l = (ennreal (p * real l) + ennreal p * NR\<^sub>N (n - 1) u l + ennreal (q * real u) + ennreal q * (\<Sum>k<n - 1. NR\<^sub>N (k + 1) u l * ennreal (pmf (binomial_pmf (n - 1) q) k))) / ennreal (1 - q ^ n)"
ortho: fixes i :: "nat"   and j :: "nat" assumes "i < length gs"   and "j < length gs" shows "(gs ! i \<bullet> gs ! j = (0::'a)) = (i \<noteq> j)"
natLeq_cinfinite: shows "BNF_Cardinal_Arithmetic.cinfinite natLeq"
Neg: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "fv \<phi>' = {}" shows "wf_mformula \<sigma> j P V n R (MNeg \<phi>) (formula.Neg \<phi>')"
that: fixes u :: "real"   and v :: "real" assumes "a *\<^sub>R x + b *\<^sub>R y + c *\<^sub>R z = x + u *\<^sub>R (y - x) + v *\<^sub>R (z - x)"   and "0 \<le> u"   and "0 \<le> v"   and "u + v \<le> 1" shows "thesis"
of_bool_half_eq_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "bool" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (zero_neq_one.of_bool one zero b) (numeral.numeral one plus (num.Bit0 num.One)) = zero"
assms_3: shows "[A*\<^sub>vx]\<le>b"
fmrestrict_set_empty: fixes A :: "'d set" shows "fmrestrict_set A fmempty = fmempty"
cases: fixes a1 :: "sequent list \<times> sequent"   and a2 :: "form" assumes "leftPrincipal a1 a2" obtains C :: "sequent" and A :: "form" and Ps :: "sequent list" where "a1 = (Ps, C)" and "a2 = A" and "C = ( \<LM> A  \<RM> \<Rightarrow>* \<Empt>)" and "A \<noteq> ff"
assms_2: shows "set_mset Q \<subseteq> r \<down>m N - r \<down>m M"
mult_monomials_dir_one: fixes p :: "'a mpoly"   and q :: "'a mpoly" shows "monomials (p * q) \<subseteq> {a + b |a b. a \<in> monomials p \<and> b \<in> monomials q}"
cptn_onlyif_cptn_mod_aux: fixes a :: "'b com"   and s :: "'b"   and Q :: "'b com option"   and t :: "'b"   and xs :: "('b com option \<times> 'b) list" assumes "(Some a, s) -c\<rightarrow> (Q, t)"   and "(Q, t) # xs \<in> cptn_mod" shows "(Some a, s) # (Q, t) # xs \<in> cptn_mod"
compress_top_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "compress_top w1 w2" shows "kauff_mat w1 = kauff_mat w2"
sig_regular_top_reduced_lc_unique: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "is_sig_GB_upt d G (lt q)"   and "p \<in> dgrad_sig_set' (length fs) d"   and "q \<in> dgrad_sig_set' (length fs) d"   and "lt p = lt q"   and "lc p = lc q"   and "\<not> is_sig_red (\<prec>\<^sub>t) (=) G p"   and "\<not> is_sig_red (\<prec>\<^sub>t) (=) G q" shows "punit.lc (rep_list p) = punit.lc (rep_list q)"
CLDomain: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "ZFAxioms T" shows "CLDomain T [A,e,B]\<^bsub>T\<^esub> = A"
composite_cell_in_hom: fixes w :: "'a"   and u :: "'a"   and \<theta> :: "'a" assumes "C.ide w"   and "\<guillemotleft>w : src\<^sub>C u \<rightarrow>\<^sub>C src\<^sub>C f\<guillemotright>"   and "\<guillemotleft>\<theta> : f \<star>\<^sub>C w \<Rightarrow>\<^sub>C u\<guillemotright>" shows "\<guillemotleft>(r \<star>\<^sub>C \<theta>) \<cdot>\<^sub>C \<a>\<^sub>C[r, f, w] \<cdot>\<^sub>C (\<rho> \<star>\<^sub>C w) : g \<star>\<^sub>C w \<Rightarrow>\<^sub>C r \<star>\<^sub>C u\<guillemotright>"
locally_negligible_alt: fixes S :: "'c set" shows "negligible S = (\<forall>x\<in>S. \<exists>U. openin (top_of_set S) U \<and> x \<in> U \<and> negligible U)"
intros_2: fixes x :: "name"   and ss :: "lam list"   and ss' :: "lam list"   and r :: "lam"   and r' :: "lam" assumes "x \<sharp> (ss, ss')"   and "r \<rightarrow>\<^sub>s r'"   and "ss [\<rightarrow>\<^sub>s] ss'" shows "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s (Lam [x].r') \<degree>\<degree> ss'"
addfunset_def: fixes A :: "'c set"   and M :: "'d set" shows "addfunset A M \<equiv> {f. supp f \<subseteq> A \<and> range f \<subseteq> M \<and> (\<forall>x\<in>A. \<forall>y\<in>A. f (x + y) = f x + f y)}"
access_matrix_pretty_ipv6_def: fixes c :: "parts_connection"   and rs :: "128 simple_rule list" shows "access_matrix_pretty_ipv6 c rs \<equiv> if \<not> simple_firewall_without_interfaces rs then undefined else let (V, E) = access_matrix c rs; formatted_nodes = map (\<lambda>(v_repr, v_range). (ipv6addr_toString v_repr, ipv6addr_wordinterval_toString v_range)) V in Let (map (\<lambda>(s, d). (ipv6addr_toString s, ipv6addr_toString d)) E) (Pair formatted_nodes)"
Sup_least: fixes A :: "('d \<times> 'e) set"   and z :: "'d \<times> 'e" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod x z" shows "less_eq_prod (Sup_prod A) z"
eval: shows "G\<turnstile>s0 \<midarrow>Body D c-\<succ>v\<rightarrow> s1"
c: shows "abc_steps_l (0, lm) (A @ B) na = (sa, lma)"
better_AndL1_substn: fixes y :: "name"   and x :: "name"   and M :: "trm"   and c :: "coname"   and P :: "trm" assumes "y \<sharp> [x].M" shows "AndL1 x.M y{y:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P z'.AndL1 x.M z')"
fid_on_vdiff: fixes A :: "V"   and B :: "V" shows "fid_on (A -\<^sub>\<circ> B) = fid_on A -\<^sub>\<circ> fid_on B"
elims_2: fixes x :: "'a list"   and xa :: "('a \<times> 'a) list" assumes "wf_list_graph_impl x xa" obtains V :: "'a list" where "x = V__" and "xa = []"   | V :: "'a list" and v1 :: "'a" and v2 :: "'a" and Es :: "('a \<times> 'a) list" where "x = V" and "xa = (v1, v2) # Es" and "v1 \<in> set V \<and> v2 \<in> set V \<and> wf_list_graph_impl V Es"
rel_fun_Sup_distl_weak: fixes G :: "('c \<Rightarrow> 'd) set"   and \<rho> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "G \<noteq> {}" shows "rel_magma.times_rel_fun \<rho> f (\<Squnion>G) = \<Squnion>{rel_magma.times_rel_fun \<rho> f g |g. g \<in> G}"
1: shows "enat i < llength (lproject A (ltake (enat (nth_least (lift (liset A w)) k)) w))"
simps_1: fixes F :: "'d frame" shows "\<lparr>\<nu>*[]\<rparr>F = F"
homeomorphism_of_subsets: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and S' :: "'c set"   and T'' :: "'d set"   and T' :: "'d set" assumes "homeomorphism S T f g"   and "S' \<subseteq> S"   and "T'' \<subseteq> T"   and "f ` S' = T'" shows "homeomorphism S' T' f g"
awalkI_apath: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and p :: "'c list"   and v :: "'b" assumes "pre_digraph.apath G u p v" shows "pre_digraph.awalk G u p v"
pw_econc_iff_1: fixes E :: "('c \<times> 'e) set"   and R :: "('d \<times> 'f) set"   and m :: "('e + 'f) nres"   and ei :: "'c" shows "inres (\<Down>\<^sub>E E R m) (Inl ei) = (nofail m \<longrightarrow> (\<exists>e. inres m (Inl e) \<and> (ei, e) \<in> E))"
ordinal_less_plusR: fixes y :: "ordinal"   and x :: "ordinal" assumes "0 < y" shows "x < x + y"
assms_3: shows "0 \<notin> rep_list ` range seq"
that: fixes i :: "nat" assumes "k \<le> i"   and "\<phi> i = \<psi> i" shows "thesis"
sub_ci: fixes l :: "'a list"   and P :: "pol"   and Q :: "pol" assumes "in_carrier l" shows "Ipol l (P \<langle>-\<rangle> Q) = Ipol l P \<ominus> Ipol l Q"
simps_4: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c" shows "(case LNil of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f1"
indRelRTPO_impl_TRel_is_strong_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "(\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q')) \<and> (\<forall>P Q Q'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q'))" shows "strong_reduction_simulation (TRel\<^sup>+) Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel\<^sup>+ \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel\<^sup>+))"
assms_5: shows "sKS s \<notin> LocalSecrets P"
div_mult_mod_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (times (divide a b) b) (modulo a b) = a"
simps_17: fixes Vs :: "char list list"   and e :: "char list exp" shows "compE\<^sub>1 Vs (throw e) = throw (compE\<^sub>1 Vs e)"
rep_eq: shows "lookup 0 = (\<lambda>k. 0::'e)"
redex_Catch_False: fixes c :: "('a, 'b, 'c) com"   and c'' :: "('a, 'b, 'c) com"   and c' :: "('a, 'b, 'c) com" shows "(redex c = Catch c'' c') = False"
intro: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi bprv"   and "Deduct2_axioms fmla Fvars prv bprv" shows "Deduct2 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv"
distinct_1: fixes x2 :: "qtname" shows "NullT \<noteq> IfaceT x2"
prems: fixes a :: "'a"   and b :: "'a" assumes "a \<in> insert a__ A"   and "b \<in> insert a__ A"   and "a \<noteq> b" shows "coprime a b"
intros_1: shows "OclVoid \<sqsubset>\<^sub>B Boolean"
ag_mOp_closed: fixes A :: "('h, 'i) aGroup_scheme"   and x :: "'h" assumes "aGroup A"   and "x \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> x \<in> carrier A"
variation_mult_pos_2: fixes c :: "real"   and x :: "real"   and y :: "real" assumes "0 < c" shows "variation x (c * y) = variation x y"
mono_sup: fixes f :: "'g multiset \<Rightarrow> 'h"   and A :: "'g multiset"   and B :: "'g multiset" assumes "subset_mset.mono f" shows "sup (f A) (f B) \<le> f (A \<union># B)"
fin: shows "finite {p. prime p \<and> p dvd n \<and> \<not> p dvd a}"
case_4: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a" shows "(case NT of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f4"
that_2: assumes "y \<in> S"   and "y \<noteq> x" shows "thesis"
assms_3: shows "h = g \<oplus>\<^bsub>UP R\<^esub> monom (UP R) a 0"
l7_15: fixes A :: "'p"   and P :: "'p"   and P' :: "'p"   and Q :: "'p"   and Q' :: "'p"   and R :: "'p"   and R' :: "'p" assumes "A Midpoint P P'"   and "A Midpoint Q Q'"   and "A Midpoint R R'"   and "Bet P Q R" shows "Bet P' Q' R'"
rel_restrict_trancl_mem: fixes a :: "'a"   and b :: "'a"   and A :: "('a \<times> 'a) set"   and R :: "'a set" assumes "(a, b) \<in> (rel_restrict A R)\<^sup>+" shows "(a, b) \<in> rel_restrict (A\<^sup>+) R"
map_M_pure_E: fixes h :: "'a"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and xs :: "'d list"   and ys :: "'c list"   and x :: "'d" assumes "h \<turnstile> map_M g xs \<rightarrow>\<^sub>r ys"   and "x \<in> set xs"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (g x) h" obtains y :: "'c" where "h \<turnstile> g x \<rightarrow>\<^sub>r y" and "y \<in> set ys"
eq_init_forall_le: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat"   and g :: "nat \<Rightarrow> nat option"   and m :: "nat" assumes "f \<triangleright> n = g \<triangleright> n"   and "m \<le> n" shows "f \<triangleright> m = g \<triangleright> m"
type_less_x_OrderedSet_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = OrderedSet \<rho>"   and "\<rho> < \<sigma>" shows "\<tau> < OrderedSet \<sigma>"
distinct_25: fixes x5 :: "String.literal" shows "NT \<noteq> Class x5"
bij_betw_comp_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and f' :: "'b \<Rightarrow> 'c"   and A'' :: "'c set" assumes "bij_betw f A A'" shows "bij_betw f' A' A'' = bij_betw (f' \<circ> f) A A''"
Suc_to_right: fixes n :: "nat"   and m :: "nat" assumes "Suc n = m" shows "n = m - Suc 0"
svar_value: fixes name :: "name" shows "\<not> is_value (Svar name)"
match_list_singleton: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "match_list \<gamma> [m] a p = matches \<gamma> m a p"
det_sq_matrix_eq: fixes A :: "(('a, 'b) vec, 'b) vec" shows "Square_Matrix.det (from_vec A) = det A"
NN0: fixes e :: "real" assumes "0 < e" shows "(1 / (real k * \<delta>)) ^ NN e < e"
minus_1: fixes x :: "real" shows "fds_perzeta (x - 1) = fds_perzeta x"
uconnected_refcl: fixes E :: "'b uprod set" shows "(uconnected E)\<^sup>* = (uconnected E)\<^sup>="
1: assumes "P \<turnstile> typeof\<^bsub>h(a \<mapsto> (C, S'))\<^esub> v = \<lfloor>Class C'\<rfloor>" shows "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Class C'\<rfloor>"
skl1_inv4_trans: shows "{skl1_inv4} TS.trans skl1 {> skl1_inv4}"
norm_less_rminus: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "x < y - n" shows "x + (n + 1) \<le> y"
elimVar_atom_single: fixes var :: "nat"   and A :: "atom"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and A' :: "atomUni"   and L2 :: "atom list"   and L2' :: "atomUni list" assumes "convert_atom var A (xs' @ x # xs) = Some A'"   and "convert_atom_list var L2 (xs' @ x # xs) = Some L2'"   and "length xs' = var" shows "eval (elimVar var L2 [] A) (xs' @ x # xs) = evalUni (elimVarUni_atom L2' A') x"
assms_1: shows "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'"
cases: fixes a1 :: "'c"   and a2 :: "'c multiset"   and a3 :: "'d el_step"   and a4 :: "'c"   and a5 :: "'c multiset"   and S :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'c \<times> 'c multiset) set" assumes "((a1, a2), a3, a4, a5) \<in> gtrp S" obtains s :: "'c" and c :: "'c multiset" and e :: "'d" and s' :: "'c" and c' :: "'c multiset" where "a1 = sa__" and "a2 = ca__" and "a3 = LOC e__" and "a4 = s'a__" and "a5 = c'a__" and "((sa__, ca__), e__, s'a__, c'a__) \<in> S"   | s :: "'c" and sl :: "'c" and c :: "'c multiset" and e :: "'d" and s' :: "'c" and c' :: "'c multiset" where "a1 = sl" and "a2 = add_mset s c" and "a3 = ENV e" and "a4 = sl" and "a5 = add_mset s' c'" and "((s, add_mset sl c), e, s', add_mset sl c') \<in> S"
prfOf_prv_Pf: fixes n :: "'b"   and \<phi> :: "'c" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prfOf n \<phi>" shows "bprv (psubst Pf [(n, yy), (\<langle>\<phi>\<rangle>, xx)])"
intros: fixes x :: "('a \<times> 'b) set['c]"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "set_bset x \<subseteq> {(x, y). R x y}" shows "rel_bset R (map_bset fst x) (map_bset snd x)"