hcju/isabelleps huggingface dataset:
finite_reach_product: fixes \<iota>\<^sub>m :: "'b \<Rightarrow> 'c option"   and \<Sigma> :: "'d set"   and \<delta>\<^sub>m :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c" assumes "finite (dom \<iota>\<^sub>m)"   and "\<And>x. x \<in> dom \<iota>\<^sub>m \<Longrightarrow> finite (reach \<Sigma> (\<delta>\<^sub>m x) (the (\<iota>\<^sub>m x)))" shows "finite (reach \<Sigma> (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m)"
it_step_insert_iff: fixes it :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "it \<subseteq> S"   and "x \<in> it" shows "S - (it - {x}) = insert x (S - it)"
xor_bot: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<oplus> \<bottom> = x"
nontriv: assumes "class.nontriv TYPE('b)" shows "1 < CARD('b)"
fps_left_inverse': fixes x :: "'a"   and f :: "'a fps"   and y :: "'a" assumes "x * fps_nth f 0 = (1::'a)"   and "fps_nth f 0 * y = (1::'a)" shows "fps_right_inverse f y * f = 1"
set_4: fixes x2 :: "'b" shows "vars_hd (Sym x2) = {}"
elims: fixes x :: "'b list"   and xa :: "'b list"   and y :: "'b list" assumes "splice x xa = y" obtains ys :: "'b list" where "x = []" and "xa = ys__" and "y = ys__"   | xb :: "'b" and xs :: "'b list" and ys :: "'b list" where "x = xb # xs" and "xa = ys" and "y = xb # splice ys xs"
ldropn_Suc_conv_ldropn: fixes n :: "nat"   and xs :: "'g llist" assumes "enat n < llength xs" shows "LCons (lnth xs n) (ldropn (Suc n) xs) = ldropn n xs"
altc_sample_option_def: fixes altc_sample :: "'f \<Rightarrow> ('g \<Rightarrow> 'e) \<Rightarrow> 'e"   and p :: "'f"   and f :: "'g \<Rightarrow> ('d, 'e) optionT" shows "altc_sample_option altc_sample p f = OptionT (altc_sample p (\<lambda>x. run_option (f x)))"
mult_mat_of_col: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "A * mat_of_col v = mat_of_col (A *\<^sub>v v)"
drefl: shows "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x"
intro: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme" assumes "cring R"   and "abelian_group M"   and "module_axioms R M" shows "module R M"
abs_Re_le_cmod: fixes x :: "complex" shows "\<bar>Re x\<bar> \<le> cmod x"
h3d: assumes "MPoly_Type.degree p2 var = 0" shows "B = 0"
"4_2": shows "v =\<^sub>F s\<^sub>1 @ s\<^sub>3"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> \<BB>\<lparr>Dom\<rparr>"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> \<BB>\<lparr>Dom\<rparr>" shows "b = c"
asym: fixes a :: "'a list"   and b :: "'a list" assumes "a <lex b"   and "b <lex a" shows "False"
is_tdghmD_5: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vcard \<NN> = 5\<^sub>\<nat>"
0: shows "skel (X #[x' \<and> x]_xs) = skel X"
prefix_length_le: fixes xs :: "'c list"   and ys :: "'c list" assumes "prefix xs ys" shows "length xs \<le> length ys"
commute: fixes a :: "'c"   and b :: "'c" shows "a \<squnion> b = b \<squnion> a"
3: fixes x :: "'a" shows "(\<lambda>k. indicat_real (\<Union> (S ` {..k})) x) \<longlonglongrightarrow> indicat_real (\<Union> (range S)) x"
locally_full: fixes f :: "'a"   and f' :: "'a"   and \<nu> :: "'b" assumes "C.ide f"   and "C.ide f'"   and "src\<^sub>C f = src\<^sub>C f'"   and "trg\<^sub>C f = trg\<^sub>C f'"   and "\<guillemotleft>\<nu> : F f \<Rightarrow>\<^sub>D F f'\<guillemotright>" shows "\<exists>\<mu>. \<guillemotleft>\<mu> : f \<rightarrow>\<^sub>C f'\<guillemotright> \<and> F \<mu> = \<nu>"
assms_3: shows "(i1, Insert e1 n1) \<in> set (node_deliver_messages xs)"
es_inputs_outputs_disjoint_def: fixes ES :: "'c ES_rec" shows "es_inputs_outputs_disjoint ES \<equiv> I\<^bsub>ES\<^esub> \<inter> O\<^bsub>ES\<^esub> = {}"
ConjE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>& \<psi> in v]" shows "[\<phi> in v] \<and> [\<psi> in v]"
subst_atms_def: fixes AA :: "('a, 'b) Term.term set"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subst_atms AA \<sigma> = AA \<cdot>\<^sub>s\<^sub>e\<^sub>t \<sigma>"
prems_5: shows "a # slist \<in> state_set (state_list s as)"
square_mat: fixes n :: "nat"   and A :: "complex Matrix.mat" assumes "gate n A" shows "square_mat A"
agg_sum_bot: shows "aggregation.sum_0 (\<lambda>k. bot) {k. True} = zero"
sum_list_map_remove1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'b"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'a" assumes "class.comm_monoid_add plus zero"   and "x \<in> set xs" shows "monoid_add.sum_list plus zero (map f xs) = plus (f x) (monoid_add.sum_list plus zero (map f (remove1 x xs)))"
wlt_increasing: fixes B :: "'c set"   and F :: "'c program" shows "B \<subseteq> wlt F B"
remove_child_writes: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" shows "writes (remove_child_locs ptr |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r) (remove_child ptr child) h h'"
list_combE_2: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "simple_syntactic_and P"   and "P (list_comb f xs)"   and "x \<in> set xs" shows "P x"
cod_simps_3: shows "fg.cod.leg1 = fg.leg1"
eq_axs_def_7: shows "eq_abstract_rule_ax \<equiv> Ct STR ''Pure.all'' ((Core.aT \<rightarrow> propT) \<rightarrow> propT) $ Abs Core.aT (mk_eq' Core.bT (Core.f $ Bv 0) (Core.g $ Bv 0)) \<longmapsto> mk_eq (Abs Core.aT (Core.f $ Bv 0)) (Abs Core.aT (Core.g $ Bv 0))"
intro: fixes f :: "'b \<Rightarrow> 'c" assumes "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" shows "bounded_linear_axioms f"
normalized_n_primitive_alist_and'_append: fixes disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> bool"   and ls1 :: "'b negation_type list"   and ls2 :: "'b negation_type list" shows "normalized_n_primitive (disc, sel) f (alist_and' (ls1 @ ls2)) = (normalized_n_primitive (disc, sel) f (alist_and' ls1) \<and> normalized_n_primitive (disc, sel) f (alist_and' ls2))"
27: shows "minimum_spanning_tree w g r \<and> t \<le> w"
le_2: shows "\<I>2 \<le> \<I>2'"
vars_mset_union: fixes A :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) multiset"   and B :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) multiset" shows "vars_mset (A + B) = vars_mset A \<union> vars_mset B"
int2: shows "integrable (measure_spmf (pair_spmf coin_spmf coin_spmf)) (\<lambda>x. spmf (case x of (m0, m1) \<Rightarrow> S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A1 view (m0, m1))) True)"
simps: fixes a :: "('f + 'g + 'h, 'i) Old_Datatype.node set" shows "(a \<in> resumption) = ((\<exists>aa. a = TERMINAL aa) \<or> (\<exists>r b. a = LINEAR b r \<and> r \<in> resumption) \<or> (\<exists>rs c. a = BRANCH c rs \<and> (\<forall>x. rs x \<in> resumption)))"
absorb_iff2: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (inf a b = b)"
inv_Mp2_def: fixes m :: "int"   and m2 :: "int" shows "poly_mod.inv_Mp2 m m2 = map_poly (poly_mod.inv_M2 m m2)"
expunge_store_ids_imply_messages_same: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i :: "'a"   and e1 :: "'b"   and mo :: "'a"   and r :: "'a"   and e2 :: "'b" assumes "xs prefix of j"   and "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"   and "(r, Expunge e2 i r) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
closedE: fixes s :: "'a set" assumes "s closed" obtains "carrier - s open"
lim_ln_1_plus_x_over_x_at_0: shows "(\<lambda>x. ln (1 + x) / x) \<midarrow>0\<rightarrow> 1"
Frame: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} C { Q}" shows "\<turnstile>\<^sub>3\<^sub>a {P \<and>* F} C { Q \<and>* F}"
prl_def: shows "prl \<equiv> \<lambda>(f, n). n = 0 \<and> least f"
X_poly_plus_nat_pow_closed: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "X_poly_plus R a [^]\<^bsub>UP R\<^esub> n \<in> carrier (UP R)"
conv_times: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>T = x\<^sup>T \<cdot> y\<^sup>T"
dvd_eq_mod_eq_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "dvd.dvd times a b = (modulo b a = zero)"
ast\<delta>_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.ast\<delta> problem \<equiv> case problem of (D, I, G, \<delta>) \<Rightarrow> \<delta>"
sstart_sets: fixes S :: "'c set"   and xs :: "'c list" shows "sstart S xs \<in> sets (stream_space (count_space UNIV))"
Keys_not_empty: fixes a :: "'b \<Rightarrow>\<^sub>0 'c"   and A :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "a \<in> A"   and "a \<noteq> 0" shows "Keys A \<noteq> {}"
random_serial_dictatorship_altdef: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite agents" shows "random_serial_dictatorship agents alts R = pmf_of_set (permutations_of_set agents) \<bind> (\<lambda>agents'. pmf_of_set (rsd_winners R alts agents'))"
bounded_clinear_cinner_right_comp: fixes g :: "'b \<Rightarrow> 'c"   and a :: "'c" assumes "bounded_clinear g" shows "bounded_clinear (\<lambda>x. \<langle>a, g x\<rangle>)"
sorted_wrt_mergesort_by_rel: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "\<And>x y. R x y \<or> R y x"   and "\<And>x y z. \<lbrakk>R x y; R y z\<rbrakk> \<Longrightarrow> R x z" shows "sorted_wrt R (mergesort_by_rel R xs)"
step: fixes x :: "'b node \<times> 'b node set" assumes "(\<mu> or\<^sub>r \<eta> \<in> old (fst x) \<longrightarrow> {\<mu>, \<eta>} \<inter> (old (fst x) \<union> new (fst x)) \<noteq> {}) \<and> (\<forall>nd\<in>snd x. \<mu> or\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<inter> old nd \<noteq> {})" shows "f x \<le> SPEC (\<lambda>x'. \<forall>nd\<in>snd x'. \<mu> or\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<inter> old nd \<noteq> {})"
ennreal_suminf_neq_top: fixes f :: "nat \<Rightarrow> real" assumes "summable f"   and "\<And>i. 0 \<le> f i" shows "(\<Sum>i. ennreal (f i)) \<noteq> \<top>"
axioms_1: fixes R :: "('e, 'f) ring_scheme" assumes "field R" shows "domain R"
mbisim_final_thread_preserve2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "s1 \<approx>m s2"   and "r2.final_thread s2 t" shows "r1.final_thread s1 t"
get_dom_component_to_tree_order_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and c :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c" shows "set nodes \<subseteq> set c"
IH: fixes y :: "nat" assumes "y < n"   and "3 < y"   and "\<not> prime y" shows "\<exists>qs. prime_factors y = set qs \<and> prod_list qs = y \<and> 2 \<le> length qs"
shift_cfs: fixes p :: "nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" assumes "p \<in> carrier P" shows "shift k p n = p (k + n)"
simple_distributed: fixes M :: "'d measure"   and X :: "'d \<Rightarrow> 'e"   and Px :: "'e \<Rightarrow> real" assumes "simple_distributed M X Px" shows "distributed M (count_space (X ` space M)) X (\<lambda>x. ennreal (Px x))"
comp_char: fixes g :: "'a"   and f :: "'a" shows "hom'.comp g f = (if hom'.arr f \<and> hom'.arr g \<and> seq g f then g \<cdot> f else null)"
word_and_le1: fixes y :: "'b word"   and a :: "'b word" shows "y AND a \<le> a"
SB: fixes S :: "'a set" assumes "S \<in> B" shows "open S \<and> connected S \<and> S \<noteq> {}"
antisym: fixes b :: "'g"   and a :: "'g" assumes "b \<le> a"   and "a \<le> b" shows "a = b"
val_def: fixes SkewBinomialTree :: "('c, 'd) SkewBinomialTree" shows "val SkewBinomialTree = (case SkewBinomialTree of Node x1 x2 x3 x4 \<Rightarrow> x1)"
cInf_restrict_nat: fixes x :: "nat"   and A :: "nat set" assumes "x \<in> A" shows "Inf A = Inf {y \<in> A. y \<le> x}"
rel_interior_eq_empty: fixes S :: "'c set" assumes "convex S" shows "(rel_interior S = {}) = (S = {})"
m_zmax_gt_each: fixes n :: "nat"   and f :: "nat \<Rightarrow> int" assumes "\<forall>j\<le>n. f j \<in> Zset" shows "\<forall>j\<le>n. f j \<le> m_zmax n f"
space_count_space: fixes \<Omega> :: "'k set" shows "space (count_space \<Omega>) = \<Omega>"
prod_def: fixes A :: "'a set"   and B :: "'b set" shows "Nitpick.prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
effect_heapE: fixes f :: "heap \<Rightarrow> 'a \<times> heap"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "effect (heap f) h h' r" obtains "h' = snd (f h)" and "r = fst (f h)"
congruent_triangle_sas_6: fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'b"   and b2 :: "'b"   and c1 :: "'a"   and c2 :: "'b" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "angle a1 b1 c1 = angle a2 b2 c2" shows "dist c1 b1 = dist b2 c2"
countable_Union: fixes X :: "'a set set" assumes "countable X"   and "X \<subseteq> M" shows "\<Union> X \<in> M"
inject_2: fixes x21 :: "freeRes"   and x22 :: "pi"   and y21 :: "freeRes"   and y22 :: "pi" shows "(FreeR_Rep x21 x22 = FreeR_Rep y21 y22) = (x21 = y21 \<and> x22 = y22)"
wellformed_defP: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp" shows "wellformed \<Gamma> = wfP (\<lambda>q p. p \<leadsto>\<^bsub>\<Gamma>\<^esub> q)"
E6: shows "(\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. \<Sum>i | i < length qs \<and> (qs ! i = y \<or> qs ! i = x). ALG' A qs init i y + ALG' A qs init i x) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALGxy A qs init x y)"
face_cycle_set_self: fixes a :: "'b" shows "a \<in> H.face_cycle_set a"
zero_b_k: shows "is_zero_row_upt_k b k A"
domain_eq: shows "Domainp pcr_complex_homo_coords = (\<lambda>v. v \<noteq> vec_zero)"
\<tau>3'': shows "\<tau>3'' \<in> seq ars"
n: shows "(c12', s') \<rightarrow>(n) (c', t)"
inf_absorb1: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y" shows "inf x y = x"
oLimit: fixes F :: "ordinal \<Rightarrow> ordinal"   and f :: "nat \<Rightarrow> ordinal" assumes "normal F" shows "F (oLimit f) = oLimit (\<lambda>n. F (f n))"
2: shows "ennreal (1 / q * EH\<^sub>N n) = NR\<^sub>N n 1 1"
hyps_5: shows "(a, y) \<in> rtrancl_on F (r\<inverse>)"
assms_2: shows "r \<in> {- int k<..<int k}"
differing_vars_neg: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and x :: "'b"   and mem1 :: "'b \<Rightarrow> 'c"   and mem2 :: "'b \<Rightarrow> 'c"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list"   and i :: "nat" assumes "sifum_security stop eval some_val some_val'"   and "x \<notin> sifum_security.differing_vars_lists mem1 mem2 mems i" shows "fst (mems ! i) x = mem1 x \<and> snd (mems ! i) x = mem2 x"
Rcd_type2': fixes \<Gamma> :: "binding list"   and fs :: "(char list \<times> trm) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list" assumes "\<Gamma> \<turnstile> Rcd fs : RcdT fTs" shows "\<Gamma> \<turnstile> map (\<lambda>(l, T). (l, the (fs\<langle>l\<rangle>\<^sub>?))) fTs [:] fTs"
rec: fixes f :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"   and x :: "'b \<Rightarrow> 'c" shows "rec_T3 f (T3.E x) = f x"
is_ub_upward: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "class.po below"   and "po.is_ub below S x"   and "below x y" shows "po.is_ub below S y"
basis_subset_eq: fixes B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set" assumes "basis B\<^sub>1"   and "basis B\<^sub>2"   and "B\<^sub>1 \<subseteq> B\<^sub>2" shows "B\<^sub>1 = B\<^sub>2"
Call_lemma': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and \<sigma> :: "'e"   and p :: "'d"   and Z :: "'e"   and c :: "('e, 'd, 'f) com" assumes "\<forall>q\<in>dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call q,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>Call q \<down> Normal s \<and> ((s, q), \<sigma>, p) \<in> termi_call_steps \<Gamma>} Call q {t. \<Gamma>\<turnstile> \<langle>Call q,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call q,Normal Z\<rangle> \<Rightarrow> Abrupt t}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>Call p \<down> Normal \<sigma> \<and> (\<exists>c'. \<Gamma>\<turnstile> (Call p, Normal \<sigma>) \<rightarrow>\<^sup>+ (c', Normal s) \<and> c \<in> redexes c')} c {t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
closed_Int: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "topological_space.closed open T" shows "topological_space.closed open (S \<inter> T)"
exists_eq_simple_sort: fixes s :: "atom_sort" shows "\<exists>a. a \<in> {a. sort_of a = s}"
simps_1: fixes Vs :: "char list list"   and C :: "char list" shows "compE\<^sub>1 Vs (new C) = new C"
globs_agree: shows "list_all2 glob_agree (s.globs s) (s_globs \<S>)"
pathCards_noneD: fixes ps :: "Vars.var list set"   and x :: "Vars.var" assumes "pathsCard ps x = none" shows "x \<notin> \<Union> (set ` ps)"
le_iff_diff_le_0: fixes 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 a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a b = less_eq (minus a b) zero"
nprv_cut: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F \<phi>"   and "nprv (insert \<phi> F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F \<psi>"
wb_lens_axioms_def: fixes x :: "'d \<Longrightarrow> 'e" shows "wb_lens_axioms x \<equiv> \<forall>\<sigma>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
elims: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and x :: "('a \<Rightarrow> 'd) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'd)"   and xa :: "'a \<times> 'd"   and xb :: "'b list"   and y :: "nat" assumes "On_Off.T_on' step t x xa xb = y" obtains A :: "('a \<Rightarrow> 'd) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'd)" and s :: "'a \<times> 'd" where "x = A__" and "xa = s__" and "xb = []" and "y = 0"   | A :: "('a \<Rightarrow> 'd) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'd)" and s :: "'a \<times> 'd" and r :: "'b" and rs :: "'b list" where "x = A" and "xa = s" and "xb = r # rs" and "y = t (fst s) r (fst (snd A s r)) + On_Off.T_on' step t A (On_Off.Step step A s r) rs"
prv_lcnj_mono_imp: fixes \<phi>1s :: "'fmla list"   and \<phi>2s :: "'fmla list" assumes "set \<phi>1s \<subseteq> fmla"   and "set \<phi>2s \<subseteq> fmla"   and "\<forall>\<phi>2\<in>set \<phi>2s. \<exists>\<phi>1\<in>set \<phi>1s. prv (imp \<phi>1 \<phi>2)" shows "prv (imp (lcnj \<phi>1s) (lcnj \<phi>2s))"
map_closed: fixes a :: "'a" assumes "a \<in> G" shows "Class a \<in> Partition"
adjacent_nodes_finite: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" assumes "Finite_Graph c" shows "finite (Graph.adjacent_nodes c u)"
sel_1: fixes l :: "'a \<Rightarrow> 'a"   and r :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "root (tree_iterate l r s) = s"
adjointFO_safe: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "\<And>i. dsafe (\<sigma> i)" shows "is_interp (adjointFO I \<sigma> \<nu>)"
mult': fixes c :: "ennreal"   and u :: "'a \<Rightarrow> ennreal" assumes "c < top"   and "u \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> u x < top"   and "P u" shows "P (\<lambda>x. c * u x)"
until_monoL: fixes \<phi>1 :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<phi>2 :: "'a stream \<Rightarrow> bool" assumes "(\<phi>1 until \<psi>) xs"   and "\<And>xs. \<phi>1 xs \<Longrightarrow> \<phi>2 xs" shows "(\<phi>2 until \<psi>) xs"
borel_measurable_cos: shows "cos \<in> borel_measurable borel"
TensorDiag_term_Unity: fixes t :: "'b local.term" shows "t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<I> = t"
REC_rule: fixes body :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and pre :: "'c \<Rightarrow> bool"   and x :: "'c"   and M :: "'c \<Rightarrow> 'd" assumes "trimono body"   and "pre x"   and "\<And>f x. \<lbrakk>\<And>x. pre x \<Longrightarrow> f x \<le> M x; pre x; f \<le> REC body\<rbrakk> \<Longrightarrow> body f x \<le> M x" shows "REC body x \<le> M x"
equiv_iff_eq_norm: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and u :: "'b local.term" assumes "free_monoidal_category C" shows "((monoidal_language.Arr C t \<and> monoidal_language.Arr C u \<and> monoidal_language.Dom C t = monoidal_language.Dom C u \<and> monoidal_language.Cod C t = monoidal_language.Cod C u) \<and> monoidal_language.Diagonalize C t = monoidal_language.Diagonalize C u) = (monoidal_language.Arr C t \<and> monoidal_language.Arr C u \<and> free_monoidal_category.norm C t = free_monoidal_category.norm C u)"
univ_poly_zero: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" shows "\<zero>\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> = []"
is_ground_lit_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and L :: "'b literal" shows "substitution_ops.is_ground_lit subst_atm L = substitution_ops.is_ground_atm subst_atm (atm_of L)"
remove: fixes A :: "'b set"   and x :: "'b"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "x \<in> A" shows "sum g A = g x + sum g (A - {x})"