hcju/isabelleps huggingface dataset:
bindE_cong: fixes M :: "'d \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool"   and M' :: "'d \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool"   and N :: "'f \<Rightarrow> 'd \<Rightarrow> (('e + 'g) \<times> 'd) \<times> bool"   and N' :: "'f \<Rightarrow> 'd \<Rightarrow> (('e + 'g) \<times> 'd) \<times> bool" assumes "M = M'"   and "\<And>v s s'. (Inr v, s') = fst (M' s) \<Longrightarrow> N v s' = N' v s'" shows "M >>=E N = M' >>=E N'"
simps_3: shows "neville_aitken_impl_main [] = 0"
i_Equiv_Exec_Stream_Acc_LocalState_set: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "nat \<Rightarrow> 'h"   and i :: "nat" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "Equiv_Exec_stable_set A equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2"   and "range input \<subseteq> A" shows "equiv_states (i_Exec_Comp_Stream_Acc_LocalState k1 localState1 trans_fun1 (input_fun1 \<circ> input) c1 i) (i_Exec_Comp_Stream_Acc_LocalState k2 localState2 trans_fun2 (input_fun2 \<circ> input) c2 i)"
BfunI: fixes f :: "'e \<Rightarrow> 'f"   and K :: "real"   and F :: "'e filter" assumes "\<forall>\<^sub>F x in F. norm (f x) \<le> K" shows "Bfun f F"
simps_2: fixes k :: "nat" shows "Partition 0 (Suc k) = 0"
prj_emb: fixes x :: "'f" shows "PRJ('e)\<cdot>(EMB('f)\<cdot>x) = COERCE('f, 'e)\<cdot>x"
ring_iso_memE_1: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and x :: "'e" assumes "h \<in> ring_iso R S"   and "x \<in> carrier R" shows "h x \<in> carrier S"
exec1p_exec1_eq: shows "exec1p = (\<lambda>x xa xb xc. (x, xa) \<rightarrow> (xb, xc))"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F (\<Phi>'.Y a') \<Psi>'.map"
R1'_a_def: shows "R1'_a \<equiv> {: p, t, lnk, lbl, mrk \<leadsto> stk, mrk' . (p, t, lnk, lbl, mrk) \<in> Init'' \<and> mrk' = mrk :}"
len_comlist_chainlen: fixes \<Pi> :: "('b, 'a) chain" shows "length (comlist \<Pi>) = chainlen \<Pi>"
xor_def: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c" shows "x \<oplus>\<^bsub>L\<^esub> y = (x \<squnion>\<^bsub>L\<^esub> y) \<sqinter>\<^bsub>L\<^esub> \<^bold>-\<^bsub>L\<^esub> (x \<sqinter>\<^bsub>L\<^esub> y)"
proper_P: fixes ps :: "('a \<times> bool) list" assumes "ps \<in> P" shows "proper_signed_list ps"
bhv2cnf_mono_strict: fixes n :: "nat"   and n' :: "nat"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf" assumes "n < n'"   and "the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t))) - 1 \<le> n" shows "dynamic_component.bhv2cnf tCMP active c t n < dynamic_component.bhv2cnf tCMP active c t n'"
zero_compare_simps_12: fixes a :: "'a" shows "(0::'a) \<le> a\<^sup>2"
min_dist_suc: fixes v :: "nat"   and v' :: "nat"   and d :: "nat" assumes "local.connected v v'"   and "min_dist v v' = Suc d" shows "\<exists>v''. local.connected v v'' \<and> (v'', v') \<in> E \<and> min_dist v v'' = d"
rcs_meet: fixes H :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "G \<guillemotright> H "   and "X \<in> set_rcs G H"   and "Y \<in> set_rcs G H"   and "X \<inter> Y \<noteq> {}" shows "X = Y"
isFDERIV_def: fixes n :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and vs :: "real list" shows "isFDERIV n xs fas vs = ((\<forall>i<n. \<forall>j<n. isDERIV (xs ! i) (fas ! j) vs) \<and> length fas = n \<and> length xs = n)"
Un_eq_UN: fixes A :: "'e set"   and B :: "'e set" shows "A \<union> B = (\<Union>b. if b then A else B)"
ah_leq_il_left: fixes h1 :: "'a \<Rightarrow> 'a set"   and h2 :: "'a \<Rightarrow> 'a set"   and h1' :: "'a \<Rightarrow> 'a set" assumes "h1 [*] h2"   and "h1' \<le> h1" shows "h1' [*] h2"
assms_4: shows "h \<turnstile> get_root_node_si ptr' \<rightarrow>\<^sub>r root"
disconnected_nodes_eq_h2: fixes doc_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" shows "h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes = h3 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
hmemrec_def: fixes G :: "(hf \<Rightarrow> 'a) \<Rightarrow> hf \<Rightarrow> 'a" shows "hmemrec G \<equiv> wfrec hmem_rel G"
assms: shows "(\<lambda>x. f x - l) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (0::'b)"
add_le_same_cancel2: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a + b \<subseteq># b) = (a \<subseteq># {#})"
Y_ide_is_functor: fixes a :: "'c" assumes "C.ide a" shows "functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a)"
nth_prime_eqI'': fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "\<pi> (real p) = real n + 1" shows "nth_prime n = p"
Sup_closed': shows "\<forall>x\<subseteq>U. \<Squnion>\<^sub>o\<^sub>w x \<in> U"
inheritedInsertEq: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and Gamma :: "'b"   and n :: "nat"   and A :: "(nat \<times> 'b) set" assumes "inherited subs P"   and "\<not> terminal subs Gamma" shows "P (insert (n, Gamma) A) = P A"
set_merge_wrt: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "set (merge_wrt rel xs ys) = set xs \<union> set ys"
simps_3: fixes uu :: "Unboxed.type"   and x :: "'d" shows "generalize_instr (ILoadUbx uu x) = ILoad x"
components_eq_sing_exists: fixes s :: "'c set" shows "(\<exists>a. components s = {a}) = (connected s \<and> s \<noteq> {})"
is_qfN_mult: fixes A :: "int"   and N :: "int"   and B :: "int" assumes "is_qfN A N"   and "is_qfN B N" shows "is_qfN (A * B) N"
sep_conj_impl1: fixes P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "\<And>s. P s \<Longrightarrow> I s"   and "(P ** R) s" shows "(I ** R) s"
lex_pp_lin': fixes t :: "('a, 'b) pp"   and s :: "('a, 'b) pp" assumes "\<not> lex_pp t s" shows "lex_pp s t"
assms: fixes x :: "'a vec" assumes "x \<in> set xs" shows "x \<in> carrier_vec n"
sum_mono: fixes K :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i" shows "sup_monoid.sum f K \<le> sup_monoid.sum g K"
supported_strong_def: fixes a :: "'c zmultiset"   and t :: "'c" shows "supported_strong a t = (\<exists>s<t. zcount a s < 0 \<and> nonpos_upto a s)"
ipurge_tr_rev_t_in_def: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "ipurge_tr_rev_t_in I D u xs \<equiv> \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = []\<rparr>"
oinvariant_def: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<times> 'b \<Rightarrow> bool" shows "(A \<Turnstile> (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. P s)"
number_partition_of_functions_of: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset" assumes "finite A"   and "finite B"   and "number_partition (card A) N"   and "size N \<le> card B" shows "number_partition_of A B (functions_of A B N) = N"
kcomp_prop: fixes f :: "'d \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'c set"   and x :: "'d" shows "(f \<circ>\<^sub>K g) x = \<mu> (\<P> g (f x))"
"1_3": shows "smap (rank A w) (gtrace r v) = sconst k"
and_foldl_sem: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and I :: "('e, 'f, 'd) interp"   and \<Gamma> :: "('e, 'f, 'd) formula list"   and \<phi> :: "('e, 'f, 'd) formula" assumes "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma> TT)"   and "List.member \<Gamma> \<phi>" shows "\<nu> \<in> fml_sem I \<phi>"
simps_16: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f4 :: "('b, 'c) Combinators \<Rightarrow> ('b, 'c) Combinators \<Rightarrow> 'a" shows "(case DenyAll of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f1"
borel_compact: fixes A :: "'b set" assumes "compact A" shows "A \<in> sets borel"
function_ring_not_car: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "a \<notin> S"   and "f \<in> carrier F" shows "f a = undefined"
eucl_of_list_eqI: fixes xs :: "real list"   and ys :: "real list" assumes "take DIM('a) (xs @ replicate (DIM('a) - length xs) 0) = take DIM('a) (ys @ replicate (DIM('a) - length ys) 0)" shows "eucl_of_list xs = eucl_of_list ys"
square_norm_one: fixes x :: "'a" assumes "x\<^sup>2 = (1::'a)" shows "norm x = 1"
length_rotate: fixes n :: "nat"   and xs :: "'c list" shows "|rotate n xs| = |xs|"
assms: fixes c :: "('test, 'atom, 'choice) cmd" assumes "c \<in> set cl" shows "discr c"
mult_pos_iff: fixes i :: "int"   and j :: "int" assumes "0 < i"   and "0 \<le> i * j" shows "0 \<le> j"
in_spanE: fixes v :: "'a vec"   and W :: "'a vec set" assumes "v \<in> NC.span W" shows "\<exists>a A. v = NC.lincomb a A \<and> finite A \<and> A \<subseteq> W"
\<F>\<G>_finite: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "finite (\<F>\<G> \<phi> w)"
safety_def: fixes A :: "'b set"   and P :: "'b llist set" shows "safety A P \<equiv> \<forall>t\<in>A\<^sup>\<infinity>. (\<forall>r\<in>finpref A t. \<exists>s\<in>A\<^sup>\<infinity>. r @@ s \<in> P) \<longrightarrow> t \<in> P"
better_OrL_substn: fixes x :: "name"   and y :: "name"   and M :: "trm"   and z :: "name"   and N :: "trm"   and c :: "coname"   and P :: "trm" assumes "x \<sharp> ([y].M, [z].N)" shows "OrL y.M z.N x{x:=<c>.P} = fresh_fun (\<lambda>z'. Cut <c>.P z'.OrL y.M z.N z')"
union_ac_4: fixes x :: "'b"   and y :: "'b"   and M :: "'b multiset" shows "x, y, M = y, x, M"
iiws_imp_iipwn: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration" assumes "dataflow_topology summary results_in"   and "dataflow_topology.inv_imps_work_sum summary results_in c" shows "inv_imp_plus_work_nonneg c"
in_hom_def: fixes f :: "('a, 'a set) Maps.arr"   and a :: "('a, 'a set) Maps.arr"   and b :: "('a, 'a set) Maps.arr" shows "Maps.in_hom f a b \<equiv> a \<in> Maps.domains f \<and> b \<in> Maps.codomains f"
prv_imp_neg_fls: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (imp \<phi> (imp (neg \<phi>) fls))"
gen_cfg_inhabited: fixes ext :: "'d" shows "let g = \<lparr>gen_\<alpha>e = {}, gen_\<alpha>n = [undefined], gen_inEdges = \<lambda>_. [], gen_Entry = undefined, gen_defs = \<lambda>_. {}, gen_uses = \<lambda>_. {}, \<dots> = ext\<rparr> in CFG_wf (\<lambda>_. gen_\<alpha>e g) (\<lambda>_. gen_\<alpha>n g) (\<lambda>_. True) (\<lambda>_. gen_inEdges' g) (\<lambda>_. gen_Entry g) (\<lambda>_. gen_defs g) (\<lambda>_. gen_uses g)"
simps_3: fixes test :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and r :: "'d regex"   and s :: "'d regex" shows "lpd test i (Plus r s) = lpd test i r \<union> lpd test i s"
set_def: fixes r :: "'a ref"   and x :: "'a" shows "Ref.set r x = refs_update (\<lambda>h. h(TYPEREP('a) := (h TYPEREP('a))(addr_of_ref r := to_nat x)))"
degree_poly_lift2: fixes p :: "'d poly" shows "degree (poly_lift2 p) = degree p"
igFreshIGOp_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gFreshGOp MOD" shows "igFreshIGOp (fromMOD MOD)"
dflt_ops_impl: shows "StdMap ahm_basic.dflt_ops"
empty_stack_imp_empty_tj_stack: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s"   and "stack s = []" shows "tj_stack s = []"
Uniq_I: fixes P :: "'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> y = x" shows "Uniq P"
simps: fixes G :: "'a pair_pre_digraph"   and u :: "'a"   and v :: "'a"   and w :: "'a" shows "subdivide G (u, v) w = \<lparr>pverts = pverts G \<union> {w}, parcs = parcs G - {(u, v), (v, u)} \<union> {(u, w), (w, u), (w, v), (v, w)}\<rparr>"
zero_flow_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> E" shows "f (u, v) = (0::'capacity)"
nn_integral_measure_pmf: fixes p :: "'c pmf"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf p) f = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
SK: fixes K :: "'a set" assumes "K \<in> \<D>" shows "\<exists>x. x \<in> S \<inter> K \<and> K \<subseteq> g x"
r_not_DA_in_tl: fixes p :: "('b, 'c) Combinators list"   and a :: "('b, 'c) Combinators" assumes "wellformed_policy1_strong p"   and "a \<in> set p"   and "a \<noteq> DenyAll" shows "a \<in> set (tl p)"
assms_4: fixes ko :: "'o" shows "h ([(k, v)], ko) = map_val_raw (\<lambda>k. f k (0::'b)) ([(k, v)], ko)"
assms_4: shows "h \<turnstile> get_elements_by_class_name ptr idd \<rightarrow>\<^sub>r results"
alw_Stream: fixes P :: "'a stream \<Rightarrow> bool"   and x :: "'a"   and s :: "'a stream" shows "alw P (x ## s) = (P (x ## s) \<and> alw P s)"
add_increasing: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq b c" shows "less_eq b (plus a c)"
stream_times: fixes t :: "'a tree"   and t' :: "'a tree" shows "stream (t * t') = stream t * stream t'"
lm071: shows "injectionsUniverse = runiqs \<inter> converse ` runiqs"
bot_entails_all: fixes B :: "'f"   and N1 :: "'f set" assumes "B \<in> Bot" shows "{B} \<Turnstile>Q N1"
index_zero_implies_same_group_unique: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "\<B> index ps = 0" shows "\<exists>!G. G \<in> \<G> \<and> ps \<subseteq> G"
run_bind_nondet: fixes bind :: "'d \<Rightarrow> ('f \<Rightarrow> 'd) \<Rightarrow> 'd"   and merge :: "'f \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd"   and m :: "('e, 'd) nondetT"   and f :: "'e \<Rightarrow> ('e, 'd) nondetT" shows "run_nondet (nondetM_base.bind_nondet bind merge m f) = bind (run_nondet m) (\<lambda>A. merge A (run_nondet \<circ> f))"
NewA: fixes E :: "env"   and T :: "ty"   and dt :: "loc \<Rightarrow> ty option"   and i :: "expr" assumes "is_acc_type (prg E) (pkg E) T"   and "E,dt\<Turnstile>i\<Colon>-PrimT Integer" shows "E,dt\<Turnstile>New T[i]\<Colon>-T.[]"
intro: fixes dom :: "'b set"   and carrier :: "'c set"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "dom \<noteq> {}"   and "\<And>i. i \<in> dom \<Longrightarrow> Order_Predicates.preorder_on carrier (R i)"   and "\<And>i x y. i \<notin> dom \<Longrightarrow> \<not> R i x y" shows "preorder_family dom carrier R"
relcompI: fixes a :: "'a"   and b :: "'c"   and r :: "('a \<times> 'c) set"   and c :: "'d"   and s :: "('c \<times> 'd) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
that: fixes k :: "nat"   and k' :: "nat" assumes "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"   and "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> (Suc k) \<noteq> \<pi>' (Suc k')" shows "thesis"
rep_eq: fixes x :: "'b multiset"   and xa :: "'b multiset" shows "count (x - xa) = (\<lambda>a. count x a - count xa a)"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> (a \<sqinter> b) = a \<sqinter> b"
Minimal_Solutions_imp_Solutions: fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b" shows "(x, y) \<in> Linear_Diophantine_Equations.Solutions a b"
extended_distance_pos: fixes a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" assumes "a \<in> extended_distance_set"   and "b \<in> extended_distance_set" shows "0 \<le> extended_distance a b"
l: shows "\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e"
fmdrop_fset_single: fixes a :: "'c"   and m :: "('c, 'd) fmap" shows "fmdrop_fset {|a|} m = fmdrop a m"
Class_def: shows "orbit.Class = (\<lambda>a\<in>S. {b \<in> S. (b, a) \<in> Orbit_Relation})"
DomainI: fixes a :: "'h"   and b :: "'i"   and r :: "('h \<times> 'i) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<preceq> y"   and "y \<preceq> x" shows "x = y"
perfect_sec_P2_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool \<times> bool" shows "and_secret_sharing.perfect_sec_P2 m1 m2 \<equiv> and_R2 m1 m2 = and_funct m1 m2 \<bind> (\<lambda>(s1, s2). and_secret_sharing.Ideal2 m2 m1 s2)"
test_comp_test_top: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "y \<sqinter> (x \<sqinter> (1::'a)) * top * (z \<sqinter> (1::'a)) = (x \<sqinter> (1::'a)) * y * (z \<sqinter> (1::'a))"
prems_2: shows "form_of_lit A \<^bold>\<or> disj_of_clause (removeAll A (f # F)), \<Gamma> \<Rightarrow>\<^sub>n"
absorb2: fixes b :: "'f"   and a :: "'f" assumes "b \<le> a" shows "min a b = b"
has_integral_component_ubound: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and a :: "'c"   and b :: "'c"   and k :: "'d"   and B :: "real" assumes "(f has_integral i) (cbox a b)"   and "\<forall>x\<in>cbox a b. f x \<bullet> k \<le> B"   and "k \<in> Basis" shows "i \<bullet> k \<le> B * content (cbox a b)"
stmt_step_distinct_load_tmps: fixes \<theta> :: "nat \<Rightarrow> nat option"   and s :: "stmt"   and t :: "nat"   and s' :: "stmt"   and t' :: "nat"   and "is" :: "instr list" assumes "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is)" shows "distinct_load_tmps is"
assms_1: shows "\<forall>\<^sub>F x in sequentially. \<forall>y\<in>A. f x y = g x y"
isoI: fixes h :: "'f \<Rightarrow> 'g"   and G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" assumes "h \<in> hom G H"   and "bij_betw h (carrier G) (carrier H)" shows "h \<in> Group.iso G H"
iwlsFSb_iterAll_termFSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Iteration.model" assumes "iwlsFSb MOD" shows "termFSbImorph (iter MOD) (iterAbs MOD) MOD"
r_into_trancl: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> r" shows "(a, b) \<in> r\<^sup>+"
path_component_of: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "path_component_of X x y = (\<exists>T. path_connectedin X T \<and> x \<in> T \<and> y \<in> T)"
simps_8: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case EQ of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f2"
trans_def: fixes r :: "('b \<times> 'b) set" shows "trans r = (\<forall>x y z. x \<succeq>[r] y \<longrightarrow> y \<succeq>[r] z \<longrightarrow> x \<succeq>[r] z)"
decompose_com_basic_com: fixes c' :: "('g, 'h, 'i, 'j) com"   and ctxts :: "(('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com) \<times> (('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list)"   and c :: "('g, 'h, 'i, 'j) com" assumes "(c', ctxts) \<in> decompose_com c" shows "basic_com c'"
simps: fixes v :: "nat"   and p :: "real mpoly" shows "decreasePower v p = (case chop (to_list v p) of [] \<Rightarrow> (p, 0) | (p, i) # L \<Rightarrow> (\<Sum>(term, x)\<leftarrow>(p, i) # L. term * Var v ^ (x - i), i))"
Ccompt: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and X :: "'a"   and Y :: "'a"   and g :: "'b"   and Z :: "'a" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y"   and "g maps\<^bsub>C\<^esub> Y to Z" shows "f ;;\<^bsub>C\<^esub> g maps\<^bsub>C\<^esub> X to Z"
zero_matrix_def: shows "mzero = (\<lambda>e. bot + bot)"
bl_to_bin_aux_cat: fixes bs :: "bool list"   and nv :: "nat"   and v :: "int"   and w :: "int" shows "bl_to_bin_aux bs (concat_bit nv v w) = concat_bit (nv + length bs) (bl_to_bin_aux bs v) w"
same_pmod_iff_pdivides: fixes K :: "('c multiset \<Rightarrow> 'a) set"   and a :: "('c multiset \<Rightarrow> 'a) list"   and b :: "('c multiset \<Rightarrow> 'a) list"   and q :: "('c multiset \<Rightarrow> 'a) list" assumes "subfield K L"   and "a \<in> carrier (K [X]\<^bsub>L\<^esub>)"   and "b \<in> carrier (K [X]\<^bsub>L\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>L\<^esub>)" shows "(L.pmod a q = L.pmod b q) = q pdivides\<^bsub>L\<^esub> (a \<ominus>\<^bsub>K [X]\<^bsub>L\<^esub>\<^esub> b)"
mul_ext_stri_imp_nstri: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and as :: "'a list"   and bs :: "'a list" assumes "fst (mul_ext f as bs)" shows "snd (mul_ext f as bs)"
simps_5: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> and\<^sub>c \<psi> = (\<xi> \<Turnstile>\<^sub>c \<phi> \<and> \<xi> \<Turnstile>\<^sub>c \<psi>)"
delete_min_nonempty: fixes xs :: "('c, 'd) bintree option list"   and t :: "('c, 'd) bintree" assumes "normalized xs"   and "xs \<noteq> []"   and "find_min xs = Some t" shows "delete_min xs = Binomial_Queue.normalize (meld (map Some (rev (children t))) (map (match (priority t)) xs))"
closed_of_int_image: fixes A :: "int set" shows "closed (of_int ` A)"
valid_W_invD0_3: fixes r :: "'d"   and s :: "'f process_name \<Rightarrow> ('e, 'f, 'g, 'd) Model.local_state"   and p :: "'f process_name"   and q :: "'f process_name" assumes "r \<in> ghost_honorary_grey (s p)"   and "valid_W_inv s" shows "r \<notin> W (s q)"
is_map_to_list_def: fixes Rk :: "('h \<times> 'i) set"   and Rv :: "('j \<times> 'k) set"   and Rm :: "('h \<times> 'i) set \<Rightarrow> ('j \<times> 'k) set \<Rightarrow> ('l \<times> ('i \<Rightarrow> 'k option)) set"   and tsl :: "'l \<Rightarrow> ('h \<times> 'j) list" shows "is_map_to_list Rk Rv Rm tsl \<equiv> is_map_to_sorted_list (\<lambda>_ _. True) Rk Rv Rm tsl"
exhaust_sel: fixes tag_const :: "'b tag_const" obtains "tag_const = \<tau>\<^sub>c\<^sub>s\<^sub>t (the_tag_const tag_const)"
invertible_inverse_closed: fixes u :: "'a \<Rightarrow> 'a" assumes "symmetric.sub.invertible u"   and "u \<in> G" shows "symmetric.sub.inverse u \<in> G"
safe_step_preserves_read_only_unowned: fixes ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "(ts, m, \<S>) \<Rightarrow>\<^sub>d (ts', m', \<S>')"   and "safe_delayed (ts, m, \<S>)"   and "simple_ownership_distinct ts"   and "read_only_unowned \<S> ts" shows "read_only_unowned \<S>' ts'"
tame10_def: fixes g :: "graph" shows "tame10 g = (let n = countVertices g in 13 \<le> n \<and> n \<le> 15)"
simps_2: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and x :: "'f"   and xs :: "'f list" shows "c_sources I D u (x # xs) = (if \<exists>v\<in>c_sources I D u xs. (D x, v) \<in> I then insert (D x) (c_sources I D u xs) else c_sources I D u xs)"
cases: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and a1 :: "ty"   and a2 :: "ty" assumes "P \<turnstile> a1 \<le> a2" obtains T :: "ty" where "a1 = T" and "a2 = T"   | C :: "char list" and D :: "char list" where "a1 = Class C__" and "a2 = Class D" and "P \<turnstile> C__ \<preceq>\<^sup>* D"   | C :: "char list" where "a1 = NT" and "a2 = Class C"