hcju/isabelleps huggingface dataset:
rk_Ra: fixes Q :: "Points"   and A' :: "Points"   and a :: "Points"   and P :: "Points"   and R :: "Points"   and A :: "Points"   and B :: "Points"   and C :: "Points"   and B' :: "Points"   and C' :: "Points" assumes "rk {Q, A', a} = 2"   and "rk {P, Q, R} = 2"   and "rk {R, Q} = 2"   and "rk {A, A', P} = 2"   and "rk {A', P} = 2"   and "rk {A, B, C, A', B', C', P} = 3"   and "rk {A, B, A'} = 3"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}" shows "rk {R, a} = 2"
simps_2: 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 a :: "'b"   and n :: "nat"   and p :: "'b point" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.point_mult divide times one plus minus uminus a (Suc n) p = ell_field.add divide times one plus minus uminus a p (ell_field.point_mult divide times one plus minus uminus a n p)"
termMOD_igSwapIGOp: shows "igSwapIGOp termMOD"
simps_7: fixes c :: "int"   and e :: "MIR.num" shows "isrlfm (Gt (CN 0 c e)) = (0 < c \<and> numbound0 e)"
coeff_sdiv_poly: fixes f :: "'a poly"   and a :: "'a"   and n :: "nat" shows "coeff (sdiv_poly f a) n = coeff f n div a"
ln_real_def: fixes x :: "real" shows "ln x = (THE u. exp u = x)"
upt_append: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a \<le> b"   and "b \<le> c" shows "[a..<b] @ [b..<c] = [a..<c]"
l2p_aux: fixes list :: "('b, 'c) Combinators list"   and a :: "('b, 'c) Combinators" assumes "list \<noteq> []" shows "list2FWpolicy (a # list) = a \<oplus> list2FWpolicy list"
div_free_RUN: fixes A :: "'b set" shows "D (RUN A) = {}"
assms_2: shows "wf\<^sub>s\<^sub>t\<^sub>s' S (decomp_rm\<^sub>e\<^sub>s\<^sub>t A)"
uw\<theta>\<omega>\<nu>: shows "ide w \<and> \<guillemotleft>\<theta>' : f \<star> w \<Rightarrow> src r\<guillemotright> \<and> \<guillemotleft>\<nu> : local.dom \<r>\<^sup>-\<^sup>1[r] \<Rightarrow> g \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (r \<star> \<theta>') \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w) \<cdot> \<nu> = \<r>\<^sup>-\<^sup>1[r]"
path_def: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" shows "path \<sigma> k = fst ((local.step ^^ k) (local.entry, \<sigma>))"
transfer_cast: fixes s :: "'s net_state"   and mR :: "nat set"   and m :: "'m"   and s' :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state" assumes "(s, mR:*cast(m), s') \<in> automaton.trans (pnet np n)"   and "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n" obtains \<sigma>'' :: "nat \<Rightarrow> 'g" and \<zeta>'' :: "'l net_state" where "((\<sigma>, \<zeta>), mR:*cast(m), \<sigma>'', \<zeta>'') \<in> automaton.trans (opnet onp n)" and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>'' j = \<sigma> j" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>'', \<zeta>'')"
select_convs_3: fixes basepoint :: "'c Point"   and direction :: "'c Vector"   and more :: "'b" shows "Line.more \<lparr>basepoint = basepoint, direction = direction, \<dots> = more\<rparr> = more"
nonstop_true: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "'a" assumes "tm_wf0 tp"   and "\<forall>stp. \<not> TSTD (steps0 (Suc 0, Bk \<up> l, <lm>) tp stp)" shows "\<forall>y. rec_exec rec_nonstop [code tp, bl2wc (<lm>), y] = Suc 0"
delete_point_strong_block_in_orig: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and bl :: "'b set"   and p :: "'b" assumes "incidence_system point_set block_collection"   and "bl \<in># incidence_system.str_del_point_blocks block_collection p" shows "bl \<in># block_collection"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<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>c s. P (Eq (CNP 0 c s))"   and "\<And>c s. P (NEq (CNP 0 c s))"   and "\<And>c s. P (Lt (CNP 0 c s))"   and "\<And>c s. P (Le (CNP 0 c s))"   and "\<And>p. P (NOT p)"   and "\<And>p q. P (Imp p q)"   and "\<And>p q. P (Iff p q)"   and "P T"   and "P F"   and "\<And>va. P (Le (CP va))"   and "\<And>va. P (Le (tm.Bound va))"   and "\<And>va vb. P (Le (tm.Add va vb))"   and "\<And>va vb. P (Le (tm.Mul va vb))"   and "\<And>va. P (Le (tm.Neg va))"   and "\<And>va vb. P (Le (tm.Sub va vb))"   and "\<And>vd vb vc. P (Le (CNP (Suc vd) vb vc))"   and "\<And>va. P (Lt (CP va))"   and "\<And>va. P (Lt (tm.Bound va))"   and "\<And>va vb. P (Lt (tm.Add va vb))"   and "\<And>va vb. P (Lt (tm.Mul va vb))"   and "\<And>va. P (Lt (tm.Neg va))"   and "\<And>va vb. P (Lt (tm.Sub va vb))"   and "\<And>vd vb vc. P (Lt (CNP (Suc vd) vb vc))"   and "\<And>va. P (Eq (CP va))"   and "\<And>va. P (Eq (tm.Bound va))"   and "\<And>va vb. P (Eq (tm.Add va vb))"   and "\<And>va vb. P (Eq (tm.Mul va vb))"   and "\<And>va. P (Eq (tm.Neg va))"   and "\<And>va vb. P (Eq (tm.Sub va vb))"   and "\<And>vd vb vc. P (Eq (CNP (Suc vd) vb vc))"   and "\<And>va. P (NEq (CP va))"   and "\<And>va. P (NEq (tm.Bound va))"   and "\<And>va vb. P (NEq (tm.Add va vb))"   and "\<And>va vb. P (NEq (tm.Mul va vb))"   and "\<And>va. P (NEq (tm.Neg va))"   and "\<And>va vb. P (NEq (tm.Sub va vb))"   and "\<And>vd vb vc. P (NEq (CNP (Suc vd) vb vc))"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
fresh_vsubst_E1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "fresh zs z (X #[y1 // y]_ys)"   and "zs \<noteq> ys \<or> z \<noteq> y" shows "fresh zs z X"
astI_def: shows "astI \<equiv> case problem of (D, I, G, \<delta>) \<Rightarrow> I"
dbm_entry_dbm_min2: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "dbm_entry_val u None (Some c) (min a b)" shows "dbm_entry_val u None (Some c) b"
set_mset_replicate_mset_subset: fixes n :: "nat"   and x :: "'b" shows "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
bthrs: fixes r :: "int list"   and n :: "nat"   and w :: "nat \<times> int" assumes "r = bin_rsplit n w" shows "r = (if fst w = 0 \<or> n = 0 then [] else let (a, b) = bin_split n (snd w) in bin_rsplit n (fst w - n, a) @ [b])"
simps: fixes f :: "nat \<Rightarrow> 'b"   and j :: "nat" shows "shift f j = (\<lambda>i. f (i + j))"
st: fixes P :: "'s \<Rightarrow> real"   and Q :: "'s \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (t P Q)"
nats_of_boolss_length: fixes n :: "nat"   and bss :: "bool list list" assumes "list_all (is_alph n) bss" shows "length (nats_of_boolss n bss) = n"
remove_child_removes_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h2 :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h2"   and "known_ptrs h"   and "type_wf h" shows "h2 \<turnstile> get_parent child \<rightarrow>\<^sub>r None"
\<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V_def: fixes i :: "nat" shows "\<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i \<equiv> {(aodv_init i, \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V PAodv)}"
lhopital_right_0_at_top: fixes g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and x :: "real" assumes "filterlim g at_top (at_right 0)"   and "\<forall>\<^sub>F x in at_right 0. g' x \<noteq> 0"   and "\<forall>\<^sub>F x in at_right 0. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_right 0. (g has_real_derivative g' x) (at x)"   and "((\<lambda>x. f' x / g' x) \<longlongrightarrow> x) (at_right 0)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> x) (at_right 0)"
l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_to_tree_order_wf_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs to_tree_order \<equiv> l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs \<and> l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_child_nodes"
mapping_of_sum: fixes f :: "'r \<Rightarrow> 'q mpoly"   and A :: "'r set" shows "(\<Sum>x\<in>A. mapping_of (f x)) = mapping_of (sum f A)"
merge_rank_bound: fixes t' :: "'b tree"   and ts\<^sub>1 :: "'b tree list"   and ts\<^sub>2 :: "'b tree list"   and t :: "'c tree" assumes "t' \<in> set (merge ts\<^sub>1 ts\<^sub>2)"   and "\<forall>t\<^sub>1\<in>set ts\<^sub>1. rank t < rank t\<^sub>1"   and "\<forall>t\<^sub>2\<in>set ts\<^sub>2. rank t < rank t\<^sub>2" shows "rank t < rank t'"
not_call_simps_4: fixes l :: "'d"   and fip :: "'a \<Rightarrow> nat"   and fmsg :: "'a \<Rightarrow> 'b"   and p :: "('a, 'b, 'c, 'd) seqp"   and q :: "('a, 'b, 'c, 'd) seqp" shows "not_call ({l}unicast(fip, fmsg) . p \<triangleright> q)"
reduced_GB_univariate_cases: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "F \<subseteq> P[X]" obtains g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "g \<in> P[X]" and "g \<noteq> 0" and "lcf g = (1::'b)" and "punit.reduced_GB F = {g}"   | "punit.reduced_GB F = {}"
arrE: fixes f :: "'c" assumes "C'.arr f" shows "Arr f"
ctor2_nchotomy: fixes y :: "'d IF2" shows "\<exists>x. y = ctor2 x"
in_carrier_2: fixes m :: "'c multiset" shows "P m \<in> carrier R"
InputRes: fixes x :: "name"   and a :: "name"   and y :: "name"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> y" shows "<\<nu>x>a<y>.P \<equiv>\<^sub>s a<y>.(<\<nu>x>P)"
norm_dist_def: fixes x :: "'c"   and y :: "'c" shows "norm_dist x y \<equiv> norm (x - y)"
cond_case_prod_eta: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd"   and g :: "'a \<times> 'b \<Rightarrow> 'd" assumes "\<And>x y. f x y = g (x, y)" shows "(\<lambda>(x, y). f x y) = g"
Max_eqI: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> y \<le>s x"   and "x \<in> A" shows "signed.Max A = x"
comp_lunit_lunit'_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "C (monoidal_category.lunit C T \<alpha> \<iota> a) (category.inv C (monoidal_category.lunit C T \<alpha> \<iota> a)) = a"
preserves_reflects_arr: fixes f :: "'a" shows "arr (UP f) = B.arr f"
of_complex_sub_one: fixes x :: "complex" shows "of_complex x -\<^sub>h 1\<^sub>h = of_complex (x - 1)"
face_of_Int_Int: fixes F :: "'b set"   and T :: "'b set"   and F' :: "'b set"   and t' :: "'b set" assumes "F face_of T"   and "F' face_of t'" shows "F \<inter> F' face_of T \<inter> t'"
lookup_eq_key_not_present: fixes key :: "'a"   and inst :: "('a \<times> 'b) list" assumes "key \<notin> set (map fst inst)" shows "lookup (\<lambda>x. x = key) inst = None"
Unknown_actions_False: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and rs :: "'c rule list"   and t :: "state"   and m :: "'c match_expr"   and a :: "action" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r # rs, Undecided\<rangle> \<Rightarrow> t"   and "r = Rule m a"   and "matches \<gamma> m p"   and "a = Unknown \<or> (\<exists>chain. a = Goto chain)" shows "False"
vertex_subtree_def: fixes v :: "'a" shows "vertex_subtree v \<equiv> {t \<in> V\<^bsub>T\<^esub>. v \<in> bag t}"
set1_var_def: shows "set1_var \<equiv> map_fun rep_var id set1_var0"
insert_Diff: fixes a :: "'b"   and A :: "'b set" assumes "(a::'b) \<in> (A::'b set)" shows "(a::'b) \<triangleright> (A::'b set) - {a} = A"
vnm_utility_is_ordinal_utility: fixes outcomes :: "'b set"   and relation :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real" assumes "vNM_utility outcomes relation u" shows "ordinal_utility (\<P> outcomes) relation (\<lambda>p. measure_pmf.expectation p u)"
simp_phi: fixes x :: "('c, 'd) phi" shows "bounded_lattice_homomorphism (Rep_phi x)"
induct: fixes P2 :: "'a tree list \<Rightarrow> bool"   and P1 :: "'a tree \<Rightarrow> bool"   and tree :: "'a tree"   and tree_list :: "'a tree list" assumes "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (NODE x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 tree \<and> P2 tree_list"
ereal_open_greaterThanLessThan: fixes a :: "ereal"   and b :: "ereal" shows "open {a<..<b}"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "comm_semiring_hom hom" shows "map_poly_comm_semiring_hom hom"
AAA: shows "finite {(t, u). s setinterleaves ((t, u), A) \<and> (\<exists>a list. u = a # list \<and> a \<notin> A) \<and> t = []}"
Do: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and f :: "state \<Rightarrow> state set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool" shows "C \<turnstile> {\<lambda>z s. \<forall>t\<in>f s. P z t} Do f {P}"
path_sym: fixes p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "pathfinish p = pathstart q"   and "pathfinish q = pathstart p" shows "path (p +++ q) = path (q +++ p)"
ds_def: fixes r :: "('a \<times> 'a) set"   and S :: "'a set" shows "r \<down>s S = {y. \<exists>x\<in>S. (y, x) \<in> r}"
alist_of_inject: fixes x :: "('c, 'd) pq"   and y :: "('c, 'd) pq" shows "(alist_of x = alist_of y) = (x = y)"
brw_inner_step_proof: fixes \<alpha> :: "'a \<Rightarrow> ('b, 'c) ta_rule set"   and invar :: "'a \<Rightarrow> bool"   and iteratei :: "'a \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> (('b, 'c) ta_rule \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha>s :: "'d \<Rightarrow> ('b \<Rightarrow> 'c tree option) \<times> 'b set \<times> (('b, 'c) ta_rule \<Rightarrow> nat option)"   and \<Sigma> :: "'d"   and \<delta> :: "('b, 'c) ta_rule set"   and cinvar :: "('b, 'c) ta_rule set \<Rightarrow> 'd \<Rightarrow> bool"   and q :: "'b"   and \<Sigma>h :: "'d"   and cstep :: "('b, 'c) ta_rule \<Rightarrow> 'd \<Rightarrow> 'd"   and it_set :: "'a"   and W :: "'b set"   and Q :: "'b \<Rightarrow> 'c tree option"   and rcm :: "('b, 'c) ta_rule \<Rightarrow> nat option" assumes "set_iteratei \<alpha> invar iteratei"   and "\<alpha>s \<Sigma> \<in> brw_invar \<delta>"   and "cinvar {r \<in> \<delta>. q \<in> set (rhsq r)} \<Sigma>h"   and "\<And>it r \<Sigma>. \<lbrakk>r \<in> it; it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}; cinvar it \<Sigma>\<rbrakk> \<Longrightarrow> cinvar (it - {r}) (cstep r \<Sigma>)"   and "\<And>it r \<Sigma>. \<lbrakk>r \<in> it; it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}; cinvar it \<Sigma>\<rbrakk> \<Longrightarrow> (\<alpha>s \<Sigma>, \<alpha>s (cstep r \<Sigma>)) \<in> brw_inner_step r"   and "invar it_set"   and "\<alpha> it_set = {r \<in> \<delta>. q \<in> set (rhsq r)}"   and "q \<in> W"   and "\<alpha>s \<Sigma> = (Q, W, rcm)"   and "\<alpha>s \<Sigma>h = (Q, W - {q}, rcm)" shows "(\<alpha>s \<Sigma>, \<alpha>s (iteratei it_set (\<lambda>_. True) cstep \<Sigma>h)) \<in> brw_step \<delta>"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "S_action.Act x1 \<noteq> S_action.Pred x2"
continuous_on_polymonial_function: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "polynomial_function f" shows "continuous_on S f"
is_unit_div_mult_cancel_right: 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 a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "dvd.dvd times b one" shows "divide a (times b a) = divide one b"
c2a_def: shows "c2a \<equiv> \<Lambda> xs. R\<cdot>xs\<cdot>SetEmpty"
simps_5: fixes \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr" shows "(Ref\<^sub>\<E> \<E>) [e] = Ref (\<E> [e])"
simulation': fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b' :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b' :: "nat \<Rightarrow> bool option"   and ts :: "('a \<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\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "invariant ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b m\<^sub>s\<^sub>b"   and "safe_reach_direct safe_delayed (ts, m, \<S>)" shows "invariant ts\<^sub>s\<^sub>b' \<S>\<^sub>s\<^sub>b' m\<^sub>s\<^sub>b' \<and> (\<exists>ts' \<S>'' m'. (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>'') \<and> (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b') \<sim> (ts', m', \<S>''))"
lunit_eqI: fixes f :: "'a"   and \<mu> :: "'a" assumes "ide f"   and "\<guillemotleft>\<mu> : trg f \<star> f \<Rightarrow> f\<guillemotright>"   and "trg f \<star> \<mu> = (\<i>[trg f] \<star> f) \<cdot> local.inv \<a>[trg f, trg f, f]" shows "\<mu> = \<l>[f]"
Repr_Neg_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and N :: "'c" shows "Repr_Neg var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv enc N \<equiv> Deduct2_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv \<and> Encode var trm fmla Var FvarsT substT Fvars subst num enc \<and> Repr_Neg_axioms var fmla Var FvarsT Fvars subst eql cnj imp all fls bprv enc N"
v_in: shows "v \<in> fv\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"
seqE: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'a \<times> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2"   and "partial_magma.arr (product_category.comp C1 C2) (product_category.comp C1 C2 g f)" obtains "partial_magma.arr C1 (C1 (fst g) (fst f))" and "partial_magma.arr C2 (C2 (snd g) (snd f))"
pfun_upd_comm: fixes x :: "'a"   and y :: "'a"   and f :: "('a, 'b) pfun"   and u :: "'b"   and v :: "'b" assumes "x \<noteq> y" shows "f(y \<mapsto> u, x \<mapsto> v)\<^sub>p = f(x \<mapsto> v, y \<mapsto> u)\<^sub>p"
assms_1: shows "a \<notin> ptrm_fvs X"
counit_in_hom_2: shows "\<guillemotleft>\<xi> : f \<star> g \<Rightarrow> trg f\<guillemotright>"
supports_def: fixes S :: "atom set"   and x :: "'c" shows "S supports x \<equiv> \<forall>a b. a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
a4_def: shows "a4 = Complex (Re a1) (Im a3)"
wp_mod_lhsv_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp_mod \<pi> (lhsv \<pi> c) c Q s = wp \<pi> c Q s"
simps_4: fixes \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "propos (\<phi> or \<psi>) = propos \<phi> \<union> propos \<psi>"
Z: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> R" shows "(b, a\<^sup>\<bullet>) \<in> R\<^sup>* \<and> (a\<^sup>\<bullet>, b\<^sup>\<bullet>) \<in> R\<^sup>*"
61391: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - - y) = - (- x \<sqinter> y)"
simps_3: fixes x2 :: "'c" shows "Some x2 \<noteq> None"
SN: shows "SN (S\<^sup>* O R O S\<^sup>*)"
preserves_iso: fixes f :: "'b" assumes "local.iso f" shows "local.iso (\<rr> f)"
rel_interior_open: fixes S :: "'b set" assumes "open S" shows "rel_interior S = S"
span_substd_basis: fixes d :: "'a set" assumes "d \<subseteq> Basis" shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
imp_recovery1: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<union> {p .\<longrightarrow>. q} \<turnstile> p" shows "A \<turnstile> p"
tranclp_trans: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r\<^sup>+\<^sup>+ x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
UniverseSum: fixes u :: "ZF" assumes "Elem u U" shows "Elem (HOLZF.Sum u) U"