hcju/isabelleps huggingface dataset:
standard_lifting_family: shows "\<forall>q\<in>Q. tiebreaker_lifting Bot_F Inf_F Bot_G (entails_q q) (Inf_G_q q) (Red_I_q q) (Red_F_q q) (\<G>_F_q q) (\<G>_I_q q) (\<lambda>g Cl Cl'. False)"
timpls_transformable_to_pred_finite_aux2: fixes A :: "'d set set"   and a :: "'d set" shows "{s. timpls_transformable_to_pred A (Fun (Abs a) []) s} \<subseteq> (\<lambda>b. Fun (Abs b) []) ` A"
has_bochner_integral_iff: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'c"   and x :: "'c" shows "has_bochner_integral M f x = (integrable M f \<and> integral\<^sup>L M f = x)"
HStartBallot_HInv5: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "HInv5_inner s q" shows "HInv5_inner s' q"
min_algs_mono1: fixes s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e" shows "relChain SucFbd (\<lambda>i. fst (min_algs s1 s2 i))"
complex_of_real_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "complex_of_real x \<le> complex_of_real y"
list_all3_refl: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x x x" shows "list_all3 P xs xs xs"
HMA_SNF: shows "(Mod_Type_Connect.HMA_M ===> (=)) Smith_normal_form_mat Smith_normal_form"
mC_C_sym: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "mC_C theta c d s t P F" shows "mC_C (theta\<inverse>) d c t s (F ` P) (inv_into P F)"
uplus_power_def: fixes P :: "'a hrel" shows "P\<^sup>+ = (\<Sqinter> i \<bullet> power.power II (;;) P (Suc i))"
IdE: fixes p :: "'c \<times> 'c" assumes "p \<in> Id" obtains x :: "'c" where "p = (x, x)"
nth_rotate: fixes n :: "nat"   and xs :: "'a list"   and m :: "nat" assumes "n < length xs" shows "rotate m xs ! n = xs ! ((m + n) mod length xs)"
mono: fixes c :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd"   and d :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" assumes "c \<preceq> a"   and "d \<preceq> b" shows "c \<oplus> d \<preceq> a \<oplus> b"
uniformity_poly_mapping_def: shows "uniformity \<equiv> INF e\<in>{0<..}. principal {(x, y). dist x y < e}"
codomain_compose_chart: fixes p :: "'c \<Rightarrow> 'c"   and p' :: "'c \<Rightarrow> 'c"   and c :: "('d, 'c) chart" assumes "homeomorphism UNIV UNIV p p'" shows "codomain (compose_chart p p' c) = p ` codomain c"
local_time_bound: fixes A :: "'a set"   and n :: "nat"   and x :: "'a" shows "local_time A n x \<le> n"
power_even_eq: fixes a :: "'a"   and n :: "nat" shows "a ^ (2 * n) = (a ^ n)\<^sup>2"
pos_bounded: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "bounded_bilinear prod" shows "\<exists>K>0. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
dbl_dec_simps_3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one one = one"
maximal_element_def: fixes D :: "('c, 'd) Order_scheme"   and m :: "'c" shows "maximal\<^bsub>D\<^esub> m = (m \<in> carrier D \<and> (\<forall>b\<in>carrier D. m \<preceq>\<^bsub>D\<^esub> b \<longrightarrow> m = b))"
sums_cos_x_plus_y: fixes x :: "'a"   and y :: "'a" shows "(\<lambda>p. \<Sum>n\<le>p. if even p then (real_of_int ((- 1) ^ (p div 2) * int (p choose n)) / fact p) *\<^sub>R x ^ n * y ^ (p - n) else (0::'a)) sums cos (x + y)"
new_xcpt_var_def: fixes vn :: "vname" shows "new_xcpt_var vn = (\<lambda>(x, s). Norm (lupd(VName vn\<mapsto>Addr (the_Loc (the_Xcpt (the x)))) s))"
w_def: shows "t.w = perron_frobenius.z (Finite_Cartesian_Product.transpose (Finite_Cartesian_Product.transpose A))"
Pow_bottom: fixes B :: "'a set" shows "{} \<in> Pow B"
eq_inf_def: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream" shows "w\<^sub>1 =\<^sub>I w\<^sub>2 \<equiv> w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2 \<and> w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1"
face_cycle_succ_def: shows "G.face_cycle_succ \<equiv> edge_succ M \<circ> edge_rev M"
assms_3: shows "(oid, op2) \<in> set op_list"
hyps: shows "below m \<times> below (2 * n) \<in> tiling domino"
non_empty_distribution: fixes nr1 :: "nat"   and n1 :: "nat"   and A1 :: "'a list list"   and nc1 :: "nat"   and A2 :: "'a list list"   and nr2 :: "nat"   and n2 :: "nat"   and B1 :: "'a list list"   and nc2 :: "nat"   and B2 :: "'a list list" assumes "mat nr1 n1 A1"   and "mat n1 nc1 A2"   and "mat nr2 n2 B1"   and "mat n2 nc2 B2"   and "A1 \<noteq> []"   and "B1 \<noteq> []"   and "A2 \<noteq> []"   and "B2 \<noteq> []" shows "A1 \<circ> A2 \<otimes> B1 \<circ> B2 \<noteq> []"
preservation: shows "rel_preserves_binary_pred (indRelLSTPO SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
leq_refl: fixes x :: "'b" shows "x \<le> x"
map_simp: fixes f :: "J.Arr" assumes "J.arr f" shows "E.map f = C.dom p0"
assm: shows "set_mset (r|\<tau>| + r|\<sigma>|) \<subseteq> r \<down>s S"
vpairsE: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> vpairs r" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
MC_action_defs_4: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkReply send rcv cst p = (\<not> $Calling rcv p \<and> $(cst!p) = #clkB \<and> Return send p MClkReplyVal<res<rcv!p>> \<and> (cst!p)$ = #clkA \<and> unchanged (caller rcv!p))"
\<xi>: shows "\<guillemotleft>\<xi> : f \<star> g \<Rightarrow> trg f\<guillemotright> \<and> local.iso \<xi>"
o2: shows "outpt s' (SOME p. outpt s' p \<noteq> NotAnInput) = NotAnInput"
Cantor_nf_exists: fixes \<alpha> :: "V" assumes "Ord \<alpha>" obtains \<alpha>s :: "V list" and ms :: "nat list" where "list.set \<alpha>s \<subseteq> ON" and "list.set ms \<subseteq> {0<..}" and "length \<alpha>s = length ms" and "sorted_wrt (\<lambda>x y. y < x) \<alpha>s" and "\<alpha> = Cantor_sum \<alpha>s ms"
that: fixes x' :: "'a"   and y' :: "'b" assumes "a = f x' y'"   and "P x' y'" shows "thesis"
mult_strict_left_mono: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a < b"   and "(0::'b) < c" shows "c * a < c * b"
wf_mstate_def: fixes \<phi> :: "'b MFOTL.formula"   and \<pi> :: "(char list \<times> 'b list) prefix"   and R :: "'b list set"   and st :: "'b mstate" shows "wf_mstate \<phi> \<pi> R st = (mstate_n st = MFOTL.nfv \<phi> \<and> (\<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> mstate_i st = Monitor.progress \<sigma> \<phi> (plen \<pi>) \<and> wf_mformula \<sigma> (plen \<pi>) (mstate_n st) R (mstate_m st) \<phi>))"
next'_def: fixes c0 :: "('b, 'loc \<times> 't) Exchange.configuration"   and c1 :: "('b, 'loc \<times> 't) Exchange.configuration" shows "cri.next' c0 c1 = (cri.next_performop c0 c1 \<or> cri.next_sendupd c0 c1 \<or> cri.next_recvupd c0 c1 \<or> cri.next_recvcap c0 c1 \<or> c1 = c0)"
List_distinct: fixes h :: "'c \<Rightarrow> 'c ref"   and x :: "'c ref"   and as :: "'c list" assumes "List h x as" shows "distinct as"
mojmir_accept_iff_rabin_accept_rank: fixes i :: "nat" shows "(finite fail \<and> finite (local.merge i) \<and> infinite (succeed i)) = accepting_pair\<^sub>R \<delta>\<^sub>\<R> q\<^sub>\<R> (fail\<^sub>R \<union> merge\<^sub>R i, succeed\<^sub>R i) w"
that: fixes m :: "nat" assumes "m \<le> n"   and "A = replicate_mset m a" shows "thesis"
virt_to_phys_def: fixes va :: "word32"   and mmu :: "MMU_state"   and m :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option" shows "virt_to_phys va mmu m \<equiv> let ctp_val = mmu_reg_val mmu 256; cnr_val = mmu_reg_val mmu 512; mmu_cr_val = registers mmu CR in if mmu_cr_val AND 1 \<noteq> 0 then case (ctp_val, cnr_val) of (None, b) \<Rightarrow> None | (Some v1, None) \<Rightarrow> None | (Some v1, Some v2) \<Rightarrow> let context_table_entry = (v1 >> 11 << 11) OR (v2 AND 511 << 2); context_table_data = mem_context_val_w32 (word_of_int 9) (ucast context_table_entry) m in case context_table_data of None \<Rightarrow> None | Some lvl1_page_table \<Rightarrow> ptd_lookup va lvl1_page_table m 1 else Some (ucast va, 239)"
not_SN_onE: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "\<not> SN_on r A" obtains f :: "nat \<Rightarrow> 'a" where "f 0 \<in> A" and "\<forall>i. (f i, f (Suc i)) \<in> r"
simps_5: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and x1 :: "'c" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
chine_assoc_def: shows "\<mu>\<nu>\<pi>.chine_assoc \<equiv> \<langle>\<mu>\<nu>\<pi>.Prj\<^sub>1\<^sub>1 \<lbrakk>\<mu>.leg0, \<nu>.leg1 \<cdot> \<nu>\<pi>.prj\<^sub>1\<rbrakk> \<langle>\<mu>\<nu>\<pi>.Prj\<^sub>0\<^sub>1 \<lbrakk>\<nu>.leg0, \<pi>.leg1\<rbrakk> \<mu>\<nu>\<pi>.Prj\<^sub>0\<rangle>\<rangle>"
simps_6: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and M :: "'b ltln set" shows "\<phi> or\<^sub>n \<psi>[M]\<^sub>\<Sigma>\<^sub>2 = \<phi>[M]\<^sub>\<Sigma>\<^sub>2 or\<^sub>n \<psi>[M]\<^sub>\<Sigma>\<^sub>2"
\<alpha>sdec: shows "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"
fun_upd_mono: fixes \<rho>1 :: "'a \<Rightarrow> 'b"   and \<rho>2 :: "'a \<Rightarrow> 'b"   and v1 :: "'b"   and v2 :: "'b"   and x :: "'a" assumes "\<rho>1 \<sqsubseteq> \<rho>2"   and "v1 \<sqsubseteq> v2" shows "\<rho>1(x := v1) \<sqsubseteq> \<rho>2(x := v2)"
lexordp_simps_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "ord.lexordp less \<epsilon> ys = (ys \<noteq> \<epsilon>)"
A: shows "((f, l), p, f', l') \<in> trcl pr_algo_lts'"
typings_index: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and es :: "exp list"   and Cs :: "nat list"   and i :: "nat" assumes "CT;\<Gamma> \<turnstile>+ es : Cs"   and "i < length es" shows "CT;\<Gamma> \<turnstile> es ! i : Cs ! i"
lF_char_1: fixes f :: "'c" assumes "C.ide f" shows "\<guillemotleft>lF f : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright>"
hyp_port_outPort: fixes f :: "'form"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list" assumes "f \<in> hyps_along t is" shows "hyp_port_h_for t is f |\<in>| outPorts (iNodeOf (tree_at t (hyp_port_path_for t is f)))"
finite_vimageI: fixes F :: "'a set"   and h :: "'b \<Rightarrow> 'a" assumes "finite F"   and "inj h" shows "finite (h -` F)"
rank_invar_insert: fixes bq :: "('c, 'd) BinomialTree list"   and e :: "'c"   and a :: "'d" assumes "rank_invar bq" shows "rank_invar (local.insert e a bq)"
dghm_id_ObjMap_app: fixes x :: "V"   and \<CC> :: "V" assumes "x \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "dghm_id \<CC>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr> = x"
E: shows "xs @ ipurge_tr I D (D y) (ys @ [y']) \<in> traces P"
simps_1: fixes a :: "('a, 'b) Node" shows "FingerTreeStruc.is_measured_digit (One a) = FingerTreeStruc.is_measured_node a"
sup_def: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder_lattice inf sup less_eq less" shows "sup x y = (if less_eq y x then x else y)"
iTILL_eq_conv: fixes n :: "nat"   and n' :: "nat" shows "([\<dots>n] = [\<dots>n']) = (n = n')"
fds_nth_inverse: fixes f :: "'b fds" shows "($) (inverse f) = dirichlet_inverse (($) f) (inverse (f $ 1))"
natural_transformation_axioms: shows "natural_transformation A B F H \<rho>\<sigma>.map"
odd_less_0_iff: fixes z :: "int" shows "(1 + z + z < 0) = (z < 0)"
inv_closed: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "\<ominus>\<^bsub>F\<^esub> x \<in> carrier F"
atLeastatMost_empty_iff2: fixes a :: "'b word"   and b :: "'b word" shows "({} = signed.atLeastAtMost a b) = (\<not> a \<le>s b)"
sample_cong: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and sample :: "'f pmf \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and p :: "'f pmf"   and q :: "'f pmf"   and f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e" assumes "monad_prob return bind sample"   and "p = q"   and "\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x" shows "sample p f = sample q g"
assms_1: shows "\<forall>ii<i. \<not> sublist_at s t ii"
rank_transpose: shows "vec_space.rank (length signs) (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) = vec_space.rank (length (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T"
run_ask_env: fixes f :: "'c \<Rightarrow> ('c, 'm) envT"   and r :: "'c" shows "run_env (local.ask_env f) r = run_env (f r) r"
finite_trancl_ntranl: fixes R :: "('g \<times> 'g) set" assumes "finite R" shows "R\<^sup>+ = ntrancl (card R - 1) R"
state_absc: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" shows "state (absc cfg) = abss (state cfg)"
imp: assumes "\<exists>q\<in>{x. a * x\<^sup>2 + b * x + c = 0}. r < q" shows "\<exists>q\<in>{x. a * x\<^sup>2 + b * x + c = 0}. r < q \<and> (\<forall>x\<in>{x. a * x\<^sup>2 + b * x + c = 0}. r < x \<longrightarrow> q \<le> x)"
\<rr>_ide_simp: fixes f :: "'a" assumes "S.ide f" shows "S.\<rr> f = S.runit f"
pos_divideR_le_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "real"   and b :: "'b"   and a :: "'b" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less_eq (scaleR (inverse c) b) a = less_eq b (scaleR c a)"
sum_length_filter_compl: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "|filter P xs| + |filter (\<lambda>x. \<not> P x) xs| = |xs|"
assms_1: shows "mcont lub ord lSup (\<sqsubseteq>) f"
sqrt_def: shows "sqrt = sqrt_approx_main (\<bar>x\<bar> + (1::'a))"
lookup_tabulate: fixes xs :: "mregex list"   and f :: "mregex \<Rightarrow> 'a option list set"   and x :: "mregex" assumes "distinct xs" shows "lookup (mrtabulate xs f) x = (if x \<in> set xs then f x else empty_table)"
jjn: shows "[jj..<i] = jj # [j..<i]"
attach_def: shows "attach \<equiv> \<lambda>uu uua. corec_resource (\<lambda>(conv, res) a. map_spmf (\<lambda>uu. (id (fst (fst uu)), Inr (snd (fst uu), snd uu))) (exec_gpv run_resource (run_converter conv a) res)) (uu, uua)"
Mp_inv_Mp_id: fixes f :: "int poly" shows "inv_Mp f =m f"
poly_gt_def: fixes p :: "('d monom \<times> 'a) list"   and q :: "('d monom \<times> 'a) list" shows "p >p q = (\<forall>\<alpha>. pos_assign \<alpha> \<longrightarrow> eval_poly \<alpha> p \<succ> eval_poly \<alpha> q)"
map_eq_upd_eq: fixes tab :: "'b \<Rightarrow> 'a option"   and vn :: "'b"   and tab' :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a" assumes "tab vn = tab' vn" shows "(tab(x \<mapsto> y)) vn = (tab'(x \<mapsto> y)) vn"
eqv: shows "lst ! (db + i) = vec_to_lpoly (0\<^sub>v (dim_col A) @\<^sub>v row A\<^sup>T i)"
l4_16R1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" assumes "A B C D FSC A' B' C' D'"   and "A \<noteq> B"   and "Bet A B C" shows "Cong C D C' D'"
F_def: fixes K :: "'b set" shows "F K = {a. u - Rep_G a \<in> K}"
always_induct: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "P \<sigma>"   and "(\<box>(\<lambda>s. P s \<longrightarrow> (\<circle>P) s)) \<sigma>" shows "(\<box>P) \<sigma>"
evnodd_Diff: fixes A :: "(nat \<times> nat) set"   and B :: "(nat \<times> nat) set"   and b :: "nat" shows "evnodd (A - B) b = evnodd A b - evnodd B b"
add_arc_simps_1: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" assumes "tail G a \<in> verts G"   and "head G a \<in> verts G" shows "verts (pre_digraph.add_arc G a) = verts G"
induct: fixes x1 :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and x2 :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and P :: "('d, 'e, 'varSort, 'var, 'f) qTerm set \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set \<Rightarrow> bool" assumes "(x1, x2) \<in> swapped"   and "\<And>X. P X X"   and "\<And>X Y Z. \<lbrakk>(X, Y) \<in> swapped; P X Y; (Y, Z) \<in> swapped; P Y Z\<rbrakk> \<Longrightarrow> P X Z"   and "\<And>X Y x y zs. \<lbrakk>(X, Y) \<in> swapped; P X Y\<rbrakk> \<Longrightarrow> P X (Y #[x \<and> y]_zs)" shows "P x1 x2"
flow_initial_time: fixes x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> X" shows "flow t0 x0 t0 = x0"
Sup2_I: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "('c \<Rightarrow> 'd \<Rightarrow> bool) set"   and a :: "'c"   and b :: "'d" assumes "r \<in> A"   and "r a b" shows "Sup A a b"
io_target_target: fixes M :: "('e, 'f, 'd) FSM"   and q1 :: "'d"   and io :: "('e \<times> 'f) list"   and q2 :: "'d"   and tr :: "'d list" assumes "io_targets M q1 io = {q2}"   and "path M (io || tr) q1"   and "length io = length tr" shows "target (io || tr) q1 = q2"
isFaultE: fixes s :: "('a, 'b) xstate" assumes "isFault s" obtains f :: "'b" where "s = Fault f"
compress_normalize_primitive_nnf: fixes disc_sel :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'd)"   and C :: "'d \<Rightarrow> 'c"   and m :: "'c match_expr"   and f :: "'d negation_type list \<Rightarrow> ('d list \<times> 'd list) option"   and m' :: "'c match_expr" assumes "wf_disc_sel disc_sel C"   and "normalized_nnf_match m"   and "compress_normalize_primitive disc_sel C f m = Some m'" shows "normalized_nnf_match m'"
current_of_bipartite_def: fixes f :: "('v + 'v) \<times> ('v + 'v) \<Rightarrow> ennreal" shows "current_of_bipartite f = (\<lambda>(x, y). f (Inl x, Inr y) * indicator \<^bold>E (x, y))"
lookup_trie_update_with: fixes ks :: "'c list"   and v :: "'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "('c, 'b) trie"   and ks' :: "'c list" shows "lookup_trie (trie_update_with' ks t v f) ks' = (if ks = ks' then Some (f (case lookup_trie t ks of None \<Rightarrow> v | Some v' \<Rightarrow> v')) else lookup_trie t ks')"
mono: shows "mono (\<lambda>p x1 x2. (\<exists>y ys. x1 = [] \<and> x2 = y # ys) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> x < y) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> \<not> x < y \<and> \<not> y < x \<and> p xs ys))"
eqvt_at_apply'': fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and p :: "perm" assumes "eqvt_at f x" shows "(p \<bullet> f) (p \<bullet> x) = f (p \<bullet> x)"
children_eq: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "ptr \<noteq> ptr'" shows "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
empty_forall_equiv: fixes m :: "'c \<Rightarrow> 'd option" shows "(m = Map.empty) = (\<forall>x. m x = None)"
rtranclp_PAC_Format_subset_ideal: fixes \<V> :: "nat set"   and A :: "int mpoly multiset"   and \<V>' :: "nat set"   and B :: "int mpoly multiset" assumes "PAC_Format\<^sup>*\<^sup>* (\<V>, A) (\<V>', B)"   and "\<Union> (vars ` set_mset A) \<subseteq> \<V>" shows "restricted_ideal_to\<^sub>I \<V> B \<subseteq> restricted_ideal_to\<^sub>I \<V> A \<and> \<V> \<subseteq> \<V>' \<and> \<Union> (vars ` set_mset B) \<subseteq> \<V>'"
image_affinity_atLeastAtMost_div: 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and m :: "'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(\<lambda>x. plus (divide x m) c) ` ord.atLeastAtMost less_eq a b = (if ord.atLeastAtMost less_eq a b = {} then {} else if less_eq zero m then ord.atLeastAtMost less_eq (plus (divide a m) c) (plus (divide b m) c) else ord.atLeastAtMost less_eq (plus (divide b m) c) (plus (divide a m) c))"
compact_space_alt: fixes X :: "'d topology" shows "compact_space X = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> topspace X \<subseteq> \<Union> \<U> \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> topspace X \<subseteq> \<Union> \<F>))"
l1_obs_iagreement: shows "oreach l1 \<subseteq> l2_iagreement"
restrict_on_source: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "Set_Theory.map f S T" shows "restrict f S = f"
stop_def: shows "stop = expectation_gpv fail (stop_\<I> \<I>) (\<lambda>_. c)"
chart_basis_last_defined: fixes b :: "'b"   and xa :: "'b proj_space" assumes "b \<in> Basis"   and "chart_last_domainP xa"   and "chart_basis_domainP b xa" shows "chart_basis b xa \<bullet> b \<noteq> 0"
good_item_simps_50: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and ys :: "'varSort"   and xa :: "'var"   and y :: "'var" assumes "good Y"   and "good X"   and "xsa \<noteq> ys \<or> xa \<noteq> y"   and "fresh xsa xa Y" shows "Abs xsa xa X $[Y / y]_ys = Abs xsa xa (X #[Y / y]_ys)"
l_get_scdom_component_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_scdom_component_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_scdom_component is_strongly_scdom_component_safe is_weakly_scdom_component_safe get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_disconnected_nodes to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
ub3: shows "1 / 3 \<le> real_of_float (rapprox_rat (max prec 1) 1 3)"
assms_1: fixes \<sigma> :: "'s" assumes "is_ground_subst \<sigma>" shows "I \<Turnstile> C \<cdot> \<sigma>"
Ints_of_nat: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "semiring_1.of_nat one plus zero n \<in> ring_1.Ints one plus zero minus"
run_detI: fixes c :: "'a Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'a"   and \<sigma> :: "heap option"   and s :: "'a" assumes "run c (Some h) (Some h') r"   and "run c (Some h) \<sigma> s" shows "\<sigma> = Some h' \<and> r = s"
regular_trapG: shows "0 \<notin> (\<lambda>(x, y). (gx x y, gy x y)) ` trapG"
XE2: fixes u :: "'a" assumes "u \<in> N2" shows "(y, u) \<in> (edges T2)\<^sup>*"
vconverse_small: fixes r :: "V" shows "small {\<langle>b, a\<rangle> |a b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
image_mset_single: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "image_mset f {#x#} = {#f x#}"
finsert_absorb2: fixes x :: "'b"   and A :: "'b fset" shows "finsert x (finsert x A) = finsert x A"
simps_2: fixes x :: "brick"   and y :: "brick list" shows "block_length (x # y) = 1 + block_length y"