hcju/isabelleps huggingface dataset:
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "P T"   and "P F"   and "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
power_0: fixes a :: "'a" shows "a\<^bsup>0\<^esup> = (1::'a)"
absorp1: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x + y) = x"
eq_tr_simps_1: shows "eq\<cdot>TT\<cdot>TT = TT"
det_orthogonal_matrix: fixes Q :: "(('a, 'b) vec, 'b) vec" assumes "orthogonal_matrix Q" shows "det Q = (1::'a) \<or> det Q = - (1::'a)"
inv_identity: shows "inv (\<lambda>a. a) = (\<lambda>a. a)"
h5: assumes "Av = 0"   and "Bv \<le> 0" shows "(Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
assms: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "f y \<le> f x"
proj_append_lfinite: fixes t :: "cnf llist"   and c :: "'id"   and t' :: "cnf llist" assumes "lfinite t" shows "\<pi>\<^bsub>c\<^esub>(t @\<^sub>l t') = \<pi>\<^bsub>c\<^esub>t @\<^sub>l (\<pi>\<^bsub>c\<^esub>t')"
cnf_of_encode_goal_state_set_2: fixes \<Pi> :: "'b strips_problem"   and v :: "'b"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "v \<in> dom (\<Pi>\<^sub>G)" shows "(\<Pi>\<^sub>G) v = Some False \<longrightarrow> (\<exists>!C. C \<in> cnf ((\<Phi>\<^sub>G \<Pi>) t) \<and> C = {(State t (index (\<Pi>\<^sub>\<V>) v))\<inverse>})"
f_shrink_0: fixes xs :: "'b message_af list" shows "xs \<div> 0 = []"
assms: shows "\<lbrace>x\<rbrace>y\<lbrace>\<alpha> z\<rbrace>"
ordLeq_def2: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" shows "(r \<le>o s) = (Well_order r \<and> Well_order s \<and> (\<exists>f. \<forall>a\<in>Field r. f a \<in> Field s \<and> f ` underS r a \<subseteq> underS s (f a)))"
sup_inf_distrib_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
ide_dom: fixes f :: "'a" assumes "Left.arr f" shows "Left.ide (Left.dom f)"
inv_in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VoV.iso f"   and "VoV.in_hom f a b" shows "VoV.in_hom (VoV.inv f) b a"
invertible_right_inverse: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "monoid M composition unit"   and "monoid.invertible M composition unit u"   and "u \<in> M" shows "composition u (monoid.inverse M composition unit u) = unit"
map\<^sub>1_in_vhom: fixes f :: "'a" assumes "C.ide f" shows "\<guillemotleft>\<tau>\<^sub>1 f : G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f) \<Rightarrow>\<^sub>D \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f\<guillemotright>"
simps_189: fixes x121 :: "expr"   and x122 :: "expr"   and x91 :: "expr"   and x92 :: "char list"   and x93 :: "char list list"   and x94 :: "expr" shows "x121;; x122 \<noteq> x91\<bullet>x92{x93} := x94"
subterm_reduction_applicable_aux_def: fixes S :: "'a eclause set"   and t :: "'a trm" shows "subterm_reduction_applicable_aux S t = (is_compound t \<and> (lhs t, t) \<in> trm_ord \<and> (rhs t, t) \<in> trm_ord \<and> (((lhs t, t) \<in> trm_ord \<longrightarrow> trm_rep (lhs t) S \<noteq> lhs t) \<or> ((rhs t, t) \<in> trm_ord \<longrightarrow> trm_rep (rhs t) S \<noteq> rhs t)))"
sgn_divide: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.field_abs_sgn divide inverse times one plus zero minus uminus abs sgn" shows "sgn (divide a b) = divide (sgn a) (sgn b)"
reachable_0: fixes s :: "'c"   and A :: "('c, 'd) ioa" assumes "s \<in> start A" shows "reachable A s"
fA: shows "f (\<Squnion>A) \<in> carrier L"
exhaust: fixes y :: "('e, 'f) splice_state" obtains x11 :: "'e" and x12 :: "'f" where "y = splice_state.Left x11 x12"   | x21 :: "'e" and x22 :: "'f" where "y = splice_state.Right x21 x22"   | x3 :: "'e" where "y = Left_only x3"   | x4 :: "'f" where "y = Right_only x4"
widening_mod_spec_def: shows "widening_mod_spec = SPEC (\<lambda>n. True)"
iteration_refl: fixes x :: "'a" shows "(1::'a) \<le> x\<^sup>\<infinity>"
monom_pow: fixes a :: "int"   and n :: "nat"   and m :: "nat" assumes "a \<in> carrier INTEG" shows "up_ring.monom (UP INTEG) a n [^]\<^bsub>UP INTEG\<^esub> m = up_ring.monom (UP INTEG) (a [^]\<^bsub>INTEG\<^esub> m) (n * m)"
subst_Pair: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e"   and a :: "'d \<times> 'e" assumes "P x y"   and "a = (x, y)" shows "P (fst a) (snd a)"
valid_expansion: fixes Cn :: "'b set \<Rightarrow> 'b set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "Tarskian_logic Cn"   and "K = Cn A"   and "\<phi> \<in> Cn {}" shows "Cn (K \<union> {\<phi>}) = K"
simply_connected_imp_connected: fixes S :: "'b set" assumes "simply_connected S" shows "connected S"
rel_mono_strong: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c pmf"   and y :: "'d pmf"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_pmf R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_pmf x; yb \<in> set_pmf y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_pmf Ra x y"
E_wf_1: shows "fst ` E \<subseteq> V"
cut_greater_Max_eq: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "I \<down>> t \<noteq> {}" shows "Max (I \<down>> t) = Max I"
length_hideDupsRec: fixes x :: "'a"   and xs :: "'a list" shows "|hideDupsRec x xs| = |xs|"
wf_map_prod_image: fixes r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e" assumes "wf r"   and "inj f" shows "wf (map_prod f f ` r)"
Inf_fold_inf: fixes A :: "'a set" assumes "finite A" shows "\<Sqinter> A = Finite_Set.fold (\<sqinter>) \<top> A"
NewA: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned"   and T :: "ty" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>New T[e]\<rangle>\<guillemotright> A"
rep_of_iff: fixes l :: "nat list"   and i :: "nat" assumes "ufa_invar l"   and "i < length l" shows "rep_of l i = (if l ! i = i then i else rep_of l (l ! i))"
Im_divide_of_nat: fixes z :: "complex"   and n :: "nat" shows "Im (z / of_nat n) = Im z / real n"
nof_vertices_def: shows "nof_vertices = length \<circ> remdups \<circ> concat"
generates_max_idealD_1: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "generates_max_ideal X F" shows "ideal F \<noteq> UNIV"
rtrner_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('d, 'c) chan" shows "rtrner ch \<equiv> \<lambda>s p. (rbit (ch s p), res (ch s p))"
simps_4: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly"   and c' :: "'b poly"   and n' :: "nat"   and p' :: "'b poly" shows "CN c n p *\<^sub>p CN c' n' p' = (if n < n' then CN (c *\<^sub>p CN c' n' p') n (p *\<^sub>p CN c' n' p') else if n' < n then CN (CN c n p *\<^sub>p c') n' (CN c n p *\<^sub>p p') else CN c n p *\<^sub>p c' +\<^sub>p CN 0\<^sub>p n' (CN c n p *\<^sub>p p'))"
sum_negf: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(\<Sum>x\<in>A. - f x) = - sum f A"
app3_list: fixes t1 :: "('a, 'b) FingerTreeStruc"   and xs :: "('a, 'b) Node list"   and t2 :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.toList (FingerTreeStruc.app3 t1 xs t2) = FingerTreeStruc.toList t1 @ FingerTreeStruc.nlistToList xs @ FingerTreeStruc.toList t2"
cball_eq_closed_segment_real: fixes x :: "real"   and e :: "real" shows "cball x e = (if 0 \<le> e then {x - e--x + e} else {})"
z1pdiv2: fixes b :: "int" shows "(2 * b + 1) div 2 = b"
onetwo2: shows "set (labels \<tau>1) \<subseteq> r \<down>s {fst \<alpha>_step} \<and> length (labels \<tau>2) \<le> 1 \<and> set (labels \<tau>2) \<subseteq> {fst \<beta>_step} \<and> set (labels \<tau>3') \<subseteq> r \<down>s {fst \<beta>_step, fst \<alpha>_step} \<and> set (labels \<delta>1''') \<subseteq> r \<down>s {fst \<beta>_step, fst \<alpha>_step} \<and> snd \<alpha>_step = fst \<tau>1 \<and> lst \<tau>1 = fst \<tau>2 \<and> lst \<tau>2 = fst \<tau>3' \<and> lst \<tau>3' = lst \<delta>1''' \<and> fst \<delta>1''' = fst \<delta>3"
type_definition_axioms: shows "type_definition uint word_of_int (uints LENGTH('a))"
count_roots: fixes ps :: "real poly list"   and p :: "real poly" assumes "sturm_seq ps p"   and "p \<noteq> 0" shows "sign_changes_neg_inf ps - sign_changes_inf ps = card {x. poly p x = 0}"
is_set_category: fixes \<AA> :: "('a \<times> 'a) set" assumes "setcat \<AA>" shows "set_category (concrete_category.COMP {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (\<lambda>C B. compose)) \<AA>"
ioa_triple_proj: fixes x :: "'d set \<times> 'd set \<times> 'd set"   and y :: "'e set"   and z :: "('e \<times> 'd \<times> 'e) set" shows "asig_of (x, y, z) = x \<and> starts_of (x, y, z) = y \<and> trans_of (x, y, z) = z"
is_monic_set_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_monic_set B \<equiv> \<forall>b\<in>B. b \<noteq> 0 \<longrightarrow> lc b = (1::'c)"
lbound_ofI: fixes b :: "'a"   and x :: "'a"   and y :: "'a" assumes "x \<succeq> b"   and "y \<succeq> b" shows "lbound_of x y b"
IH: fixes z :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term"   and yb :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term" assumes "z \<in> set T"   and "yb \<in> set S"   and "timpls_transformable_to TI z yb"   and "wf\<^sub>t\<^sub>r\<^sub>m z"   and "\<forall>f\<in>funs_term z. is_Abs f \<longrightarrow> the_Abs f \<in> A" shows "timpls_transformable_to_pred (A \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) z yb"
cases: fixes a1 :: "'state"   and a2 :: "'state" assumes "scut a1 a2" obtains p :: "'state" and a :: "'action" where "a1 = p" and "a2 = ex a p" and "p \<in> nodes" and "en a p" and "a \<notin> cuts"
intro: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "comm_monoid f z" shows "comm_monoid_set f z"
negligible_poly_times: fixes f :: "nat \<Rightarrow> real"   and n :: "real"   and g :: "nat \<Rightarrow> real" assumes "f \<in> O(\<lambda>x. real x powr n)"   and "negligible g" shows "negligible (\<lambda>x. f x * g x)"
Graph12: fixes a :: "nat set"   and M :: "node list"   and j :: "nat" assumes "a \<subseteq> Blacks M"   and "j < length M" shows "a \<subseteq> Blacks (M[j := Black])"
prime_dvd_multD: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "dvd.dvd times p (times a b)" shows "dvd.dvd times p a \<or> dvd.dvd times p b"
multiset_eqI: fixes A :: "'k multiset"   and B :: "'k multiset" assumes "\<And>x. count A x = count B x" shows "A = B"
bounded_linear_sub: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f x - g x)"
pred_set: fixes P1 :: "'f \<Rightarrow> bool"   and P2 :: "'g \<Rightarrow> bool" shows "Resumption.resumption.pred_resumption P1 P2 = (\<lambda>x. Ball (results x) P1 \<and> Ball (outputs x) P2)"
equiv_neg: fixes P :: "bool"   and Q :: "bool" assumes "P \<Longrightarrow> Q"   and "\<not> P \<Longrightarrow> \<not> Q" shows "P = Q"
simps_9: fixes x3 :: "nat"   and x2 :: "nat" shows "Regions.intv.Greater x3 \<noteq> Regions.intv.Intv x2"
Rf_def: fixes uu :: "'a literal multiset set" shows "Rf uu = {}"
mat_det_adj: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_det (mat_adj A) = cnj (mat_det A)"
strict_trans1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "b < c" shows "a < c"
isDiagonal_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "isDiagonal A = (\<forall>a b. to_nat a \<noteq> to_nat b \<longrightarrow> A $h a $h b = (0::'b))"
simps_82: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and f4 :: "'c"   and f5 :: "char list \<Rightarrow> 'c"   and f6 :: "'c"   and f7 :: "char list \<Rightarrow> 'c"   and f8 :: "'c"   and f9 :: "'c" shows "(case Empty of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2 | Log \<Rightarrow> f3 | Reject \<Rightarrow> f4 | Call x \<Rightarrow> f5 x | Return \<Rightarrow> f6 | Goto x \<Rightarrow> f7 x | Empty \<Rightarrow> f8 | Unknown \<Rightarrow> f9) = f8"
QR_lemma_05: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "card ({(int p - 1) div 2<..int p - 1} \<times> {0<..(int q - 1) div 2}) = QR.r p q"
antisym_conv: fixes y :: "'d list"   and x :: "'d list" assumes "prefix y x" shows "prefix x y = (x = y)"
redT_updTs_None: fixes ts :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option"   and tas :: "('c, 'a, 'd) new_thread_action list"   and t :: "'c" assumes "redT_updTs ts tas t = None" shows "ts t = None"
cases: fixes a1 :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and a2 :: "typ" assumes "wf_type a1 a2" obtains \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" and \<kappa> :: "String.literal" and Ts :: "typ list" where "a1 = \<Sigma>__" and "a2 = Ty \<kappa> Ts" and "type_arity \<Sigma>__ \<kappa> = Some (length Ts)" and "\<forall>T\<in>set Ts. wf_type \<Sigma>__ T"   | \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" and S :: "String.literal set" and a :: "variable" where "a1 = \<Sigma>" and "a2 = Tv a S" and "wf_sort (subclass (osig \<Sigma>)) S"
ord_FO_\<Gamma>_def: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset" shows "ord_FO_\<Gamma> S = {uu_. \<exists>CAs DA AAs As \<sigma> E. uu_ = Infer (mset CAs) DA E \<and> ord_resolve_rename S CAs DA AAs As \<sigma> E}"
is_tm_tdghm_axioms: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
kruskal_spanning_invariant_def: fixes f :: "'a"   and g :: "'a"   and h :: "'a" shows "kruskal_spanning_invariant f g h \<equiv> symmetric g \<and> h = h\<^sup>T \<and> g \<sqinter> - - h = h \<and> spanning_forest f (- h \<sqinter> g)"
fls_mult_const_nth_2: fixes f :: "'a fls"   and x :: "'a"   and n :: "int" shows "fls_nth (f * fls_const x) n = fls_nth f n * x"
finite_measure_UNION: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets M"   and "disjoint_family A" shows "(\<lambda>i. Sigma_Algebra.measure M (A i)) sums Sigma_Algebra.measure M (\<Union> (range A))"
someI: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f" assumes "P x" shows "P (Eps P)"
inf_mono: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<le> c"   and "b \<le> d" shows "a \<cdot> b \<le> c \<cdot> d"
simps_2: shows "cast_and_box Ubx2 = map_option box_ubx2 \<circ> cast_Ubx2"
natural_isomorphism_axioms_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" shows "natural_isomorphism_axioms A B \<tau> \<equiv> \<forall>a. partial_magma.ide A a \<longrightarrow> category.iso B (\<tau> a)"
IH: fixes env :: "v sem_env"   and s :: "'a state" shows "\<exists>s' r. evaluate_match True env s v pes v' (s', r)"
delete_edge_length: fixes v :: "'b"   and v' :: "'b"   and G :: "'b list_graph" shows "length (edgesL (FiniteListGraph.delete_edge v v' G)) \<le> length (edgesL G)"
equivp_evquivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "equivp (equivclp r)"
path_join: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) = (path g1 \<and> path g2)"
chain_reparam_weak_path_def: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "chain_reparam_weak_path \<gamma> one_chain \<equiv> \<exists>l. set l = one_chain \<and> distinct l \<and> reparam \<gamma> (rec_join l) \<and> valid_chain_list l \<and> l \<noteq> []"
right_diff_distrib_numeral: fixes v :: "num"   and b :: "'c"   and c :: "'c" shows "numeral v * (b - c) = numeral v * b - numeral v * c"
is_pole_inverse_power: fixes n :: "nat"   and a :: "complex" assumes "0 < n" shows "is_pole (\<lambda>z. 1 / (z - a) ^ n) a"
abs_eq: fixes xb :: "'d"   and xa :: "'c"   and x :: "('a, 'b, 'c, 'd) FL_Formula.Tree" shows "FL_Formula.Act\<^sub>\<alpha> xb xa (FL_Formula.abs_Tree\<^sub>\<alpha> x) = FL_Formula.abs_Tree\<^sub>\<alpha> (FL_Formula.Tree.tAct xb xa x)"
mset_neg_after_summary: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'c zmultiset"   and S :: "'b antichain" assumes "dataflow_topology summary results_in"   and "mset_neg M = {#}" shows "mset_neg (dataflow_topology.after_summary results_in M S) = {#}"
eventually_ae_filter: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" shows "almost_everywhere M P = (\<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
Sup_atLeastAtMost: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq x y" shows "Sup (ord.atLeastAtMost less_eq x y) = y"
finite_verts: fixes p :: "real"   and n :: "nat" assumes "edge_space p" shows "finite (edge_space.S_verts n)"
add_neg_numeral_special_8: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) one = zero"
val_ab: shows "valof (a / b) = valof (zerosign (of_bool (sign a \<noteq> sign b)) (IEEE.round To_nearest (valof a / valof b)))"
limpt_of_limpts: fixes x :: "'b"   and S :: "'b set" assumes "x islimpt {y. y islimpt S}" shows "x islimpt S"
ccINF_greatest: 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 A :: "'d set"   and u :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq u (f i)" shows "less_eq u (Inf (f ` A))"
swap_simps_2: fixes a :: "name"   and b :: "name" shows "[(a, b)] \<bullet> b = a"
dw_closedI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; r a b\<rbrakk> \<Longrightarrow> b \<in> A" shows "relation.dw_closed r A"
TC_INDUCT: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x y. R x y \<longrightarrow> P x y"   and "\<forall>x y z. P x y \<and> P y z \<longrightarrow> P x z" shows "\<forall>u v. TC R u v \<longrightarrow> P u v"
bisim: shows "(\<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'))"
COPY_rec: shows "COPY = (left`?`x \<rightarrow> right`!`x \<rightarrow> COPY)"
PPc_nonempt: fixes c :: "('a, 'b) MWLsCom" shows "PPc c \<noteq> []"
lrev_LCons: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "lrev (a ## r) = lrev r @@ a ## LNil"
iia_onD: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set"   and B :: "'d set"   and C :: "'d set"   and a :: "'d" assumes "iia_on A f"   and "B \<subseteq> A"   and "C \<subseteq> B"   and "a \<in> C"   and "a \<in> f B" shows "a \<in> f C"
ax2: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "\<turnstile> \<box>F \<longrightarrow> \<box>[\<box>F]_v"
assms_1: shows "ct_suffix [TSome T_i32] cts"
parent_edge: fixes G :: "('c, 'd) Graph_scheme"   and root :: "'c"   and v :: "'c" assumes "RootedTree G root"   and "v \<in> V\<^bsub>G\<^esub>"   and "v \<noteq> root" shows "v \<rightarrow>\<^bsub>G\<^esub> RootedTree.parent G root v"
DynPDG_path_CFG_path: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>d* n'" shows "n -as\<rightarrow>* n'"
bound_compare_defs_13: fixes c :: "'a"   and b :: "'a option" shows "c \<le>\<^sub>l\<^sub>b b = \<unlhd>\<^sub>l\<^sub>b (<) c b"
sup_top_right: fixes x :: "'b" shows "x \<squnion> \<top> = \<top>"
targets_iso_closed: fixes a :: "'a"   and \<mu> :: "'a"   and a' :: "'a" assumes "a \<in> targets \<mu>"   and "a \<cong> a'" shows "a' \<in> targets \<mu>"
Assign: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and a :: "aexp" shows "\<turnstile>\<^sub>1 {\<lambda>l s. P l (s[a/x])} x ::= a { \<lambda>s. Suc 0 \<Down> P}"
wf_sort_code: fixes sub :: "(String.literal \<times> String.literal) list"   and S :: "String.literal set" assumes "exe_class_conds sub" shows "wf_sort (set sub) S = exewf_sort sub S"
substitution_def: fixes g :: "'g" shows "substitution g \<equiv> THE v'. isTrivialPhi g (chooseNext g) v'"
i_shrink_last_assoc: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat" shows "f \<div>\<^sub>l a \<div>\<^sub>l b = f \<div>\<^sub>l (a * b)"
B_invariant: fixes a :: "'a"   and b :: "'a" assumes "PA a"   and "A a b" shows "QA b"
cut_bound'_mono: fixes a :: "nat"   and b :: "nat"   and F :: "'b formula" assumes "a \<le> b" shows "cut_bound' a F \<le> cut_bound' b F"
chine_simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "partial_magma.cod C (Chn \<mu>) = span_in_category.apex C (Cod \<mu>)"