hcju/isabelleps huggingface dataset:
csmall_def: fixes P :: "'prog"   and \<sigma> :: "'state" shows "csmall P \<sigma> \<equiv> {(\<sigma>', coll). \<sigma>' \<in> small P \<sigma> \<and> collect P \<sigma> \<sigma>' = coll}"
upper_triangular_def: fixes A :: "'a mat" shows "Matrix.upper_triangular A \<equiv> \<forall>i<dim_row A. \<forall>j<i. A $$ (i, j) = (0::'a)"
affine_rv: fixes g :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and b :: "real" assumes "g \<in> rv M" shows "(\<lambda>x. a + g x * b) \<in> rv M"
apsnd_apfst: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'c"   and x :: "'f \<times> 'e" shows "apsnd f (apfst g x) = (g (fst x), f (snd x))"
endpoint_in_S_S_strict_hyp2_distinct: fixes p :: "proj2"   and a :: "proj2" assumes "p \<in> S"   and "a \<in> hyp2" shows "endpoint_in_S p a \<noteq> p"
inteq: shows "to_stream -` {w \<in> space N. w !! n} \<inter> space (Pi\<^sub>M UNIV (\<lambda>i. measure_pmf (bernoulli_pmf p))) = prod_emb UNIV (\<lambda>i. measure_pmf (bernoulli_pmf p)) {n} ({n} \<rightarrow>\<^sub>E {True})"
abs_eq: shows "Containers_Userguide.empty = Abs_trie trie_empty"
separatedin_empty_2: fixes X :: "'c topology"   and S :: "'c set" shows "separatedin X {} S = (S \<subseteq> topspace X)"
k_mkAlgAuto_implements: shows "SPR.implements SPRdet.k_mkAlgAuto"
simps_9: fixes F :: "'f set"   and f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "strip_guards F (Guard f g c) = (if f \<in> F then strip_guards F c else Guard f g (strip_guards F c))"
qSwapped_qSkel: fixes X :: "('f, 'g, 'h, 'i, 'j) qTerm"   and Y :: "('f, 'g, 'h, 'i, 'j) qTerm" assumes "(X, Y) \<in> qSwapped" shows "qSkel Y = qSkel X"
Call: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and s :: "'d" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> (Call p, Normal s) \<rightarrow> (bdy, Normal s)"
assms_2: fixes A :: "'a multiset" assumes "A \<noteq> {#}"   and "\<And>x. x \<in># A \<Longrightarrow> P (A - {#x#})" shows "P A"
altc_parametric: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique R" shows "(rel_cset R ===> (R ===> (=)) ===> (=)) altc altc"
a_Roll_inv_P: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP' :: "Syntax.term"   and eV :: "Syntax.term"   and \<alpha> :: "Syntax.tvar"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> e, Roll eP', eV : Mu \<alpha> \<tau>" obtains e' :: "Syntax.term" and eV' :: "Syntax.term" where "e = Roll e'" and "eV = Roll eV'" and "\<Gamma> \<turnstile> e', eP', eV' : subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>"
assms_3: shows "t \<noteq> \<langle>\<rangle>"
ex_un_null: shows "\<exists>!n. \<forall>f. J' n f = n \<and> J' f n = n"
vec_plus_def: fixes a :: "'b list"   and b :: "'b list" shows "vec_plus a b = map2 (+) a b"
integral_abs_bound_integral: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "\<And>x. x \<in> space M \<Longrightarrow> \<bar>f x\<bar> \<le> g x" shows "\<bar>integral\<^sup>L M f\<bar> \<le> integral\<^sup>L M g"
Sup_empty: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
l_heap_is_wellformed_axioms: shows "l_heap_is_wellformed DocumentClass.type_wf DocumentClass.known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"
finite_enumerate_mono: fixes m :: "nat"   and n :: "nat"   and S :: "'b set" assumes "m < n"   and "finite S"   and "n < card S" shows "wellorder_class.enumerate S m < wellorder_class.enumerate S n"
below_monic_fun_K: fixes f :: "'a \<rightarrow> 'b" assumes "below_monic_cfun f" shows "below_monic_cfun (\<Lambda> c. (\<lambda>_. f\<cdot>c))"
assms: fixes k :: "nat" shows "prime (4 * k + 3) \<longrightarrow> even (multiplicity (4 * k + 3) n)"
simps_5: fixes f1 :: "'d"   and f2 :: "dag \<Rightarrow> ref \<Rightarrow> dag \<Rightarrow> 'd"   and x21 :: "dag"   and x22 :: "ref"   and x23 :: "dag" shows "(case Node x21 x22 x23 of Tip \<Rightarrow> f1 | Node x xa xb \<Rightarrow> f2 x xa xb) = f2 x21 x22 x23"
next_not_nil_left: fixes nil :: "'a"   and "next" :: "('a \<times> 'a) set"   and x :: "'a" assumes "graph nil next" shows "(nil, x) \<notin> next"
lunit_simps_3: fixes f :: "'c" assumes "C.ide f" shows "trg\<^sub>C \<l>\<^sub>C[f] = trg\<^sub>C f"
locally_Times: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and Q :: "'c set \<Rightarrow> bool"   and T :: "'c set"   and R :: "('b \<times> 'c) set \<Rightarrow> bool" assumes "locally P S"   and "locally Q T"   and "\<And>S T. P S \<and> Q T \<Longrightarrow> R (S \<times> T)" shows "locally R (S \<times> T)"
OrdNotEqP_self_contra: fixes x :: "tm" shows "{x NEQ x} \<turnstile> Fls"
arr_dom: fixes f :: "'a" assumes "A.arr f" shows "A.arr (A.dom f)"
distinct_7: fixes x51 :: "expr"   and x52 :: "com" shows "Skip \<noteq> while (x51) x52"
hyps_4: fixes D :: "char list" assumes "D_ \<noteq> Object"   and "P \<turnstile> D_ \<prec>\<^sup>1 D" shows "(D, D_) \<notin> (subcls1 P)\<^sup>*"
stutter_extend_simps_sa: fixes G :: "('b, 'd, 'e) sa_rec_scheme" shows "sa_L (stutter_extend G) = sa_L G"
Fail_eq_assert_gpv: fixes b :: "bool" shows "(Fail = assert_gpv b) = (\<not> b)"
isPath_bwd_cases: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "Graph.isPath c s p t" obtains "p = []" and "t = s"   | p' :: "(nat \<times> nat) list" and u :: "nat" where "p = p' @ [(u, t)]" and "Graph.isPath c s p' u" and "(u, t) \<in> Graph.E c"
DoI_eq_I: shows "I.map = \<F>C.inclusion_of_generators"
arc_imp_inj_on: fixes g :: "real \<Rightarrow> 'a" assumes "arc g" shows "inj_on g {0..1}"
F2in_mono23': fixes A2 :: "'d set"   and A2a :: "'d set"   and A3 :: "'e set"   and A3a :: "'e set"   and c :: "('f, 'd, 'e) F2"   and A1 :: "'f set" assumes "A2 \<subseteq> A2a"   and "A3 \<subseteq> A3a"   and "c \<in> F2in A1 A2 A3" shows "c \<in> F2in A1 A2a A3a"
encode_kind_3: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list" shows "encode_kind (encode (Cn n f gs)) = 3"
le_shiftr: fixes u :: "'a word"   and v :: "'a word"   and n :: "nat" assumes "u \<le> v" shows "u >> n \<le> v >> n"
spoly_swap: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "spoly p q = - spoly q p"
f4: shows "rk {Q, C', R, C} \<le> 3"
scaleR_mtx2: fixes k :: "real"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" shows "k *\<^sub>R mtx [[a, b], [c, d]] = mtx [[k * a, k * b], [k * c, k * d]]"
aprimedivisor_vimage: fixes p :: "'a"   and n :: "'a" assumes "prime p" shows "aprimedivisor -` {p} \<inter> primepow_factors n = {p ^ k |k. 0 < k \<and> p ^ k dvd n}"
in_these_eq: fixes x :: "'d"   and A :: "'d option set" shows "(x \<in> Option.these A) = (Some x \<in> A)"
Key_unique: fixes S :: "agent"   and R :: "agent"   and K :: "nat"   and m :: "nat"   and AO :: "nat"   and cleartext :: "nat"   and q :: "nat"   and hs :: "msg"   and evs :: "event list"   and S' :: "agent"   and R' :: "agent"   and m' :: "nat"   and AO' :: "nat"   and cleartext' :: "nat"   and q' :: "nat"   and hs' :: "msg" assumes "Says S R \<lbrace>Agent S, Agent TTP, Crypt K (Number m), Number AO, Number cleartext, Nonce q, Crypt (pubK TTP) \<lbrace>Agent S, Number AO, Key K, Agent R, hs\<rbrace>\<rbrace> \<in> set evs"   and "Says S' R' \<lbrace>Agent S', Agent TTP, Crypt K (Number m'), Number AO', Number cleartext', Nonce q', Crypt (pubK TTP) \<lbrace>Agent S', Number AO', Key K, Agent R', hs'\<rbrace>\<rbrace> \<in> set evs"   and "Key K \<notin> analz (knows Spy evs)"   and "evs \<in> certified_mail" shows "R' = R \<and> S' = S \<and> AO' = AO \<and> hs' = hs"
jump_ok_IfFalseD: fixes ins :: "'a instr list"   and n :: "nat"   and n' :: "nat"   and pc :: "nat"   and m :: "int" assumes "jump_ok ins n n'"   and "ins ! pc = IfFalse m"   and "pc < length ins" shows "- int (pc + n) \<le> m \<and> m < int (length ins - pc + n')"
simps: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms\<^sub>1 :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and cms\<^sub>2 :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b"   and mems :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.makes_compatible dma \<C>_vars eval (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems = (length cms\<^sub>1 = length cms\<^sub>2 \<and> length cms\<^sub>1 = length mems \<and> (\<forall>i<length cms\<^sub>1. (\<forall>\<sigma>. dom \<sigma> = sifum_security_init.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i \<longrightarrow> ((cms\<^sub>1 ! i, subst \<sigma> (fst (mems ! i))), cms\<^sub>2 ! i, subst \<sigma> (snd (mems ! i))) \<in> sifum_security_init.mm_equiv dma \<C>_vars eval) \<and> (\<forall>x. mem\<^sub>1 x = mem\<^sub>2 x \<or> dma mem\<^sub>1 x = High \<or> x \<in> \<Union> (range \<C>_vars) \<longrightarrow> x \<notin> sifum_security_init.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)) \<and> (length cms\<^sub>1 = 0 \<and> sifum_security_init.low_eq dma mem\<^sub>1 mem\<^sub>2 \<or> (\<forall>x. \<exists>i<length cms\<^sub>1. x \<notin> sifum_security_init.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)))"
scaleR_octo_def: shows "(*\<^sub>R) \<equiv> \<lambda>uu uua. corec_octo (\<lambda>(r, x). r * Ree x) (\<lambda>(r, x). r * Im1 x) (\<lambda>(r, x). r * Im2 x) (\<lambda>(r, x). r * Im3 x) (\<lambda>(r, x). r * Im4 x) (\<lambda>(r, x). r * Im5 x) (\<lambda>(r, x). r * Im6 x) (\<lambda>(r, x). r * Im7 x) (uu, uua)"
atLeast_less_sets: fixes A :: "'a set"   and x :: "'a"   and B :: "'a set" assumes "A \<lless> {x}"   and "B \<subseteq> {x..}" shows "A \<lless> B"
thr_conv: fixes ls :: "'h"   and ts :: "'g"   and m :: "'i"   and ws :: "'j \<times> 'k" shows "thr (ls, (ts, m), ws) = ts"
finfun_apply_def: shows "($) = (\<lambda>f a. finfun_rec (\<lambda>b. b) (\<lambda>a'. If (a = a')) f)"
permute_list_compose: fixes g :: "nat \<Rightarrow> nat"   and xs :: "'c list"   and f :: "nat \<Rightarrow> nat" assumes "g permutes {..<length xs}" shows "permute_list (f \<circ> g) xs = permute_list g (permute_list f xs)"
WCFG_LAss: fixes V :: "char list"   and e :: "expr" shows "V:=e \<turnstile> (_ 0 _) -\<Up>\<lambda>s. s(V := interpret e s)\<rightarrow> (_ 1 _)"
floor_divide_lower: fixes q :: "'a"   and p :: "'a" assumes "(0::'a) < q" shows "of_int \<lfloor>p / q\<rfloor> * q \<le> p"
orderE: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "semilattice_order f less_eq less"   and "less_eq a b" obtains "a = f a b"
bdd_above_finite: fixes A :: "'a multiset set" assumes "finite A" shows "subset_mset.bdd_above A"
A_def: shows "A = A0 \<inter> {N + N0..}"
boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<sqsubseteq> b \<sqinter> c" obtains "a \<sqsubseteq> b" and "a \<sqsubseteq> c"
induct: fixes P :: "'b list \<Rightarrow> bool"   and a0 :: "'b list" assumes "\<And>x. P [x]"   and "\<And>x v va. (\<And>x21 x22. v # va = x21 # x22 \<Longrightarrow> P (v # va)) \<Longrightarrow> P (x # v # va)"   and "P []" shows "P a0"
vsv_vimage_vsubsetI: fixes A :: "V"   and B :: "V" assumes "\<And>a. \<lbrakk>a \<in>\<^sub>\<circ> A; a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r\<rbrakk> \<Longrightarrow> r\<lparr>a\<rparr> \<in>\<^sub>\<circ> B" shows "r `\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> B"
reindex: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and h :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'d \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "inj_on h A" shows "abel_semigroup_set.F f z g (h ` A) = abel_semigroup_set.F f z (g \<circ> h) A"
nn_integral_count_space_indicator: fixes X :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "NO_MATCH UNIV X" shows "integral\<^sup>N (count_space X) f = \<integral>\<^sup>+ x. f x * indicator X x \<partial>count_space UNIV"
valid_W_invE_5: fixes r :: "'f"   and fl :: "bool"   and s :: "'g process_name \<Rightarrow> ('d, 'g, 'e, 'f) Model.local_state"   and p :: "'g process_name" assumes "mw_Mark r fl \<in> set (sys_mem_store_buffers p s)"   and "valid_W_inv s" shows "r \<in> ghost_honorary_grey (s p)"
assms_2: shows "v \<notin> set (delete_effects_of op)"
distinct_249: shows "BinAnd \<noteq> ShiftRightZeros"
mset_ce_eq: shows "mset (C @ E) = mset (C' @ E)"
assms_3: shows "DBM_reset (FW M n) n (v c) d M'"
openin_inverse: fixes x :: "'c topology" shows "topology (openin x) = x"
wnf_lemma_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- x \<squnion> - y) \<sqinter> (- z \<squnion> - - y) \<squnion> - y = - x \<squnion> - y"
most_recent_write_for_fun: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ra :: "nat"   and wa :: "nat"   and wa' :: "nat" assumes "P,E \<turnstile> ra \<leadsto>mrw wa"   and "P,E \<turnstile> ra \<leadsto>mrw wa'" shows "wa = wa'"
length_eq_card_rows: fixes A :: "(('c, 'd) vec, 'e) vec" shows "IArray.length (matrix_to_iarray A) = CARD('e)"
function_ring_car_memI: fixes R :: "('a, 'b) ring_scheme"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "ring_functions R"   and "\<And>a. a \<in> S \<Longrightarrow> f a \<in> carrier R"   and "\<And>a. a \<notin> S \<Longrightarrow> f a = undefined" shows "f \<in> carrier (function_ring S R)"
has_bochner_integral_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and i :: "'d" assumes "\<Omega> \<inter> space M \<in> sets M" shows "has_bochner_integral (restrict_space M \<Omega>) f i = has_bochner_integral M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) i"
chain_boundary_simplicial_cone_of: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and v :: "nat \<Rightarrow> real" assumes "simplicial_simplex p S f" shows "chain_boundary (Suc p) (simplicial_cone p v (frag_of f)) = frag_of f - (if p = 0 then frag_of (\<lambda>u\<in>standard_simplex p. v) else simplicial_cone (p - 1) v (chain_boundary p (frag_of f)))"
distributed_integral: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> real"   and g :: "'d \<Rightarrow> real" assumes "distributed M N X (\<lambda>x. ennreal (f x))"   and "g \<in> borel_measurable N"   and "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x" shows "LINT x|N. f x * g x = LINT x|M. g (X x)"
Interleaves_interleaves: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "xs \<cong> {ys, zs, P}" shows "xs \<simeq> {ys, zs, P}"
kth_state_def: fixes prob :: "('c, 'a, 'b, 'd) ifc_problem"   and \<sigma> :: "'a \<Rightarrow> 'b"   and k :: "nat" shows "IFC_def.kth_state prob \<sigma> k = snd ((IFC_def.step prob ^^ k) (IFC_def.entry prob, \<sigma>))"
assms: shows "\<forall>l<k. v \<notin> local.writes (path \<sigma> l)"
pick_UNIV: fixes j :: "nat" shows "pick UNIV j = j"
pred_inject_2: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and aa :: "'d list" shows "list_all P (a # aa) = (P a \<and> list_all P aa)"
eqd: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v"   and "\<^bold>|x\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "\<exists>t. x \<cdot> t = u \<and> t \<cdot> v = y"
eventually_at_bot_dense: fixes P :: "'c \<Rightarrow> bool" shows "eventually P at_bot = (\<exists>N. \<forall>n<N. P n)"
delete_elt_on_set: fixes t :: "interval_tree"   and lt :: "interval_tree"   and x :: "nat idx_interval"   and m :: "nat"   and rt :: "interval_tree" assumes "t \<noteq> Tip" shows "tree_set (delete_elt_tree (Node lt x m rt)) = tree_set lt \<union> tree_set rt"
from_to_view\<^sub>m: fixes x :: "((view_metadata, view_metadata) blindable\<^sub>m \<times> (view_data, view_data) blindable\<^sub>m, view_metadata blindable\<^sub>h \<times> view_data blindable\<^sub>h) rose_tree\<^sub>m" shows "view_merkle.from_view\<^sub>m (view_merkle.to_view\<^sub>m x) = x"
sorted1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq [x] = True"
steps_ConsD: fixes x :: "'a"   and xs :: "'a list" assumes "local.steps (x # xs)"   and "xs \<noteq> []" shows "local.steps xs"
hv_minus: fixes x :: "(real, 3) vec" shows "Hv (- x) = - Hv x"
singleton_abelian_group: fixes a :: "'e" shows "comm_group (singleton_group a)"
interchange: fixes B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<nu> :: "'d \<Rightarrow> 'e"   and D :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and K :: "'e \<Rightarrow> 'f"   and L :: "'e \<Rightarrow> 'f"   and \<sigma> :: "'e \<Rightarrow> 'f"   and M :: "'e \<Rightarrow> 'f"   and \<mu> :: "'e \<Rightarrow> 'f" assumes "natural_transformation B C F G \<tau>"   and "natural_transformation B C G H \<nu>"   and "natural_transformation C D K L \<sigma>"   and "natural_transformation C D L M \<mu>" shows "vertical_composite.map C D \<sigma> \<mu> \<circ> vertical_composite.map B C \<tau> \<nu> = vertical_composite.map B D (\<sigma> \<circ> \<tau>) (\<mu> \<circ> \<nu>)"
total_def: fixes f :: "recf" shows "Partial_Recursive.total f \<equiv> \<forall>xs. length xs = arity f \<longrightarrow> eval f xs \<down>"
bisims1_ThrowD: fixes P :: "'a prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and vs :: "'addr val list"   and a :: "'addr"   and es' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,es,h \<turnstile> (map Val vs @ Throw a # es', xs) [\<leftrightarrow>] (stk, loc, pc, xcp)" shows "pc < length (compEs2 es) \<and> (xcp = \<lfloor>a\<rfloor> \<or> xcp = None) \<and> xs = loc"
L: shows "(\<lambda>n. integral (cbox a b) (f n)) \<longlonglongrightarrow> L"
assms_2: shows "Q \<simeq>\<^sup>s R"
carr_Ssegment_adjunct_ord: fixes D :: "'c Order"   and a :: "'c" assumes "Order D"   and "a \<notin> carrier D" shows "carrier D = Ssegment (adjunct_ord D a) a"
simps_2: fixes k :: "'c"   and v :: "'d"   and p :: "'c \<times> 'd"   and ps :: "('c \<times> 'd) list" shows "update k v (p # ps) = (if fst p = k then (k, v) # ps else p # update k v ps)"
multpw_map: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and ns' :: "('b \<times> 'b) set" assumes "\<And>x y. \<lbrakk>x \<in># X; y \<in># Y; (x, y) \<in> ns\<rbrakk> \<Longrightarrow> (f x, g y) \<in> ns'"   and "(X, Y) \<in> multpw ns" shows "(image_mset f X, image_mset g Y) \<in> multpw ns'"
fvi_Suc: fixes x :: "nat"   and b :: "nat"   and \<phi> :: "'a MFOTL.formula" shows "(x \<in> MFOTL.fvi (Suc b) \<phi>) = (Suc x \<in> MFOTL.fvi b \<phi>)"
gt_sym_def: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "gt_sym gt_sym \<equiv> ((\<forall>f. \<not> gt_sym f f) \<and> (\<forall>h g f. gt_sym h g \<longrightarrow> gt_sym g f \<longrightarrow> gt_sym h f)) \<and> (\<forall>f g. gt_sym f g \<or> gt_sym g f \<or> g = f) \<and> wfP (\<lambda>f g. gt_sym g f)"
sheaf_of_rings_axioms: shows "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
arr_char: fixes f :: "'a" shows "hom.arr f = \<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"
exhaust_sel: fixes le_constraint :: "'a le_constraint" obtains "le_constraint = Le_Constraint (lec_rel le_constraint) (lec_poly le_constraint) (lec_const le_constraint)"
bet_col1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B D"   and "Bet A C D" shows "Tarski_neutral_dimensionless.Col Bet A B C"
contractible_Times: fixes S :: "'a set"   and T :: "'b set" assumes "contractible S"   and "contractible T" shows "contractible (S \<times> T)"
uminus_eq_inj: fixes a :: "'b literal"   and b :: "'b literal" shows "(- a = - b) = (a = b)"
simps: fixes S :: "'c set set"   and a :: "'c set" shows "generate_topology_on S a = (a = {} \<or> (\<exists>aa b. a = aa \<inter> b \<and> generate_topology_on S aa \<and> generate_topology_on S b) \<or> (\<exists>K. a = \<Union> K \<and> (\<forall>x. x \<in> K \<longrightarrow> generate_topology_on S x)) \<or> (\<exists>s. a = s \<and> s \<in> S))"
the_cm_eq_apply_cm: fixes c :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "\<exists>c'. next_change_multiplicity' c c' loc t n" shows "the_cm c loc t n = apply_cm c loc t n"
gsC: shows "mat_of_rows n gs \<in> carrier_mat (length gs) n"
simps_6: fixes x2 :: "Vars.var"   and x11 :: "exp"   and x12 :: "exp" shows "Arg x2 \<noteq> Alts x11 x12"
card_eq_UNIV_imp_eq_UNIV: fixes A :: "'a set" assumes "finite UNIV"   and "card A = CARD('a)" shows "A = UNIV"
adj_reachable_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<rightarrow>\<^bsub>H\<^esub> b"   and "b \<rightarrow>\<^sup>*\<^bsub>H\<^esub> c" shows "a \<rightarrow>\<^sup>*\<^bsub>H\<^esub> c"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order L"   and "weak_complete_lattice_axioms L" shows "weak_complete_lattice L"
assms_1: shows "v \<in> braun_ssa.allUses g m"
cases: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" assumes "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>L a2" obtains S :: "'procS" where "\<lbrakk>S\<rbrakk> \<in>T a1" and "S \<in>S a2"
fv_def: fixes e :: "'e" shows "fv e = {v. atom v \<in> supp e}"
not_knows_tail_not_knows_conc: fixes P :: "specID"   and e :: "Expression list"   and a :: "Expression" assumes "\<not> knows P e" shows "\<not> knows P (a # e)"
bounded_matrix_for_marginals_nat: fixes f :: "nat \<Rightarrow> real"   and s :: "real"   and g :: "nat \<Rightarrow> real"   and R :: "(nat \<times> nat) set" assumes "f sums s"   and "g sums s"   and "\<And>x. 0 \<le> f x"   and "\<And>y. 0 \<le> g y"   and "\<And>X. suminf (zero_on f (- X)) \<le> suminf (zero_on g (- R `` X))" obtains h :: "nat \<Rightarrow> nat \<Rightarrow> real" where "\<And>x y. 0 \<le> h x y" and "\<And>x y. 0 < h x y \<Longrightarrow> (x, y) \<in> R" and "\<And>x. h x sums f x" and "\<And>y. (\<lambda>x. h x y) sums g y"
interpret_floatariths_eq_take_max_VarI: fixes ea :: "floatarith list"   and ys :: "real list"   and zs :: "real list" assumes "take (max_Var_floatariths ea) ys = take (max_Var_floatariths ea) zs" shows "interpret_floatariths ea ys = interpret_floatariths ea zs"
tso_store_refs_def: fixes m :: "'f" shows "mut_m.tso_store_refs m = (\<lambda>s. \<Union> (store_refs ` set (sys_mem_store_buffers (mutator m) s)))"
fin_cut_same_idem: fixes x :: "'b"   and xs :: "'b list" shows "fin_cut_same x (fin_cut_same x xs) = fin_cut_same x xs"