hcju/isabelleps huggingface dataset:
ordered_map_set_eq: fixes S :: "('a \<times> 'b) set" assumes "finite S"   and "is_map S" shows "set (ordered_map S) = S"
commutative: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "addA x y = addA y x"
case_prodI: fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b"   and b :: "'d" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
is_expansion_aux_coinduct: fixes X :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "X xs f basis"   and "\<And>xs f basis. X xs f basis \<Longrightarrow> xs = MSLNil \<and> (\<forall>\<^sub>F x in at_top. f x = 0) \<and> length basis = Suc (expansion_level TYPE('a)) \<or> (\<exists>xs' b e basis' C. xs = MSLCons (C, e) xs' \<and> basis = b # basis' \<and> X xs' (\<lambda>x. f x - eval C x * b x powr e) (b # basis') \<and> is_expansion C basis' \<and> (\<forall>x>e. f \<in> o(\<lambda>xa. b xa powr x)) \<and> ms_exp_gt e (ms_aux_hd_exp xs'))" shows "is_expansion_aux xs f basis"
abs_le_D1: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less_eq (abs a) b" shows "less_eq a b"
transient_strengthen: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> transient A"   and "B \<subseteq> A" shows "F \<in> transient B"
dX_range': fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and X :: "'d set"   and d :: "'e" assumes "Contracts Xd Xh Pd Ch"   and "x \<in> Contracts.dX Xd X d" shows "x \<in> X \<and> Xd x = d"
starfun_if_eq: fixes w :: "'a star"   and x :: "'a"   and a :: "'b"   and g :: "'a \<Rightarrow> 'b" assumes "w \<noteq> star_of x" shows "(*f* (\<lambda>z. if z = x then a else g z)) w = (*f* g) w"
filterlim_at_top_mult_at_top: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "filterlim f at_top F"   and "filterlim g at_top F" shows "LIM x F. f x * g x :> at_top"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_2: fixes l :: "'e strand_label"   and t :: "('f, 'g) Term.term"   and A :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, receive\<langle>t\<rangle>) # A) = (l, send\<langle>t\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
connected_iff_connected_component: fixes S :: "'b set" shows "connected S = (\<forall>x\<in>S. \<forall>y\<in>S. connected_component S x y)"
LCons_fin: fixes l :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "l \<in> A\<^sup>\<star>"   and "a \<in> A" shows "a ## l \<in> A\<^sup>\<star>"
succ_tran_finite: fixes G :: "'a graph"   and v :: "'a" assumes "wf_graph G" shows "finite (succ_tran G v)"
cod_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VV.cod f = (if VV.arr f then VxV.cod f else VxV.null)"
Un_assoc: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
jkbpCSt_jkbpCS_subset: fixes t :: "('a, 'es, 'as) BEState Trace" shows "spr_jkbpCSt t \<subseteq> SPRdet.jkbpCS"
hyps_2: assumes "yvec \<sharp>* p"   and "xvec \<sharp>* yvec"   and "distinct yvec" shows "xvec \<sharp>* ([xvec yvec] \<bullet>\<^sub>v p)"
rev_drop: fixes i :: "nat"   and xs :: "'g list" shows "rev (drop i xs) = take (length xs - i) (rev xs)"
simps_1: fixes S :: "'fun set"   and a :: "'atom" shows "fresh_pgwt S (Var a) = Fun (SOME c. c \<notin> S \<and> \<Gamma> (Fun c []) = Var a \<and> public c) []"
sub2_preserves_dimensions: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and d :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "sub2 sel_fun mf d l (r, A) = (r', A')" shows "dim_row A' = dim_row A \<and> dim_col A' = dim_col A"
sub_conj_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "sub_conj t \<equiv> \<forall>P Q. sound P \<and> sound Q \<longrightarrow> t P && t Q \<tturnstile> t (P && Q)"
state_is_associated_string: fixes i :: "example_peers \<times> nat" assumes "example.is_valid_state_id i" shows "is_certified_associated_string (set (example.received_messages i)) (example.state i)"
cong': fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sum' g A = sum' h B"
prems_2: fixes x :: "'a" assumes "x \<in> A" shows "carrier (f x) \<subseteq> A"
assms_2: shows "\<exists>x. x \<notin> A \<and> f x \<noteq> dflt"
sees_fields_fun: fixes Cs :: "char list list"   and T :: "ty"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and F :: "char list"   and T' :: "ty" assumes "(Cs, T) \<in> FieldDecls P C F"   and "(Cs, T') \<in> FieldDecls P C F" shows "T = T'"
canon_ang_unique: fixes \<alpha>\<^sub>1 :: "real"   and \<alpha> :: "real"   and \<alpha>\<^sub>2 :: "real" assumes "canon_ang_P \<alpha> \<alpha>\<^sub>1"   and "canon_ang_P \<alpha> \<alpha>\<^sub>2" shows "\<alpha>\<^sub>1 = \<alpha>\<^sub>2"
diffE: fixes x :: "'a" assumes "x \<in> f.src.carrier" obtains c1 :: "('a, 'b) chart" and c2 :: "('c, 'd) chart" where "c1 \<in> f.src.atlas" and "c2 \<in> f.dest.atlas" and "x \<in> domain c1" and "f ` domain c1 \<subseteq> domain c2" and "k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1)"
IdNatTransMapL: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans f" shows "IdNatTrans (NTDom f) \<bullet> f = f"
iMODb_inext_if: fixes t :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" shows "inext t [ r, mod m, c ] = (if t \<in> [ r, mod m, c ] \<and> t < r + m * c then t + m else t)"
bexi_def: fixes x :: "'var"   and t :: "'trm"   and \<phi> :: "'fmla" shows "bexi x t \<phi> \<equiv> exi x (cnj (LLq (Var x) t) \<phi>)"
iterate_to_list_def: fixes it :: "('b list \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b list \<Rightarrow> 'b list) \<Rightarrow> 'b list \<Rightarrow> 'b list" shows "iterate_to_list it = it (\<lambda>_. True) (#) []"
proj_plus: fixes k :: "'k"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly k (p + q) = proj_poly k p + proj_poly k q"
TS_union: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and i :: "nat" shows "TS M2 M1 \<Omega> V m i = \<Union> (C M2 M1 \<Omega> V m ` set [0..<Suc i])"
ag_eq_diffzero: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier M"   and "b \<in> carrier M" shows "(a = b) = (a \<plusminus> -\<^sub>a b = \<zero>)"
lzip_conv_fixp: shows "lzip \<equiv> curry (llist.fixp_fun (\<lambda>lzip (xs, ys). case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> case ys of LNil \<Rightarrow> LNil | LCons y ys' \<Rightarrow> LCons (x, y) (curry lzip xs' ys')))"
simps_2: fixes list :: "'c list"   and vals :: "'c \<Rightarrow> 'd option"   and oid :: "'c"   and ref :: "'c" shows "interp_op (list, vals) (oid, Delete ref) = (list, vals(ref := None))"
well_Mne: shows "local.well_base {#m \<in># M. m \<noteq> n#}"
trimmed_ms_aux_MSLCons: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist" shows "trimmed_ms_aux (MSLCons x xs) = trimmed (fst x)"
simps: fixes mem :: "var\<^sub>C \<Rightarrow> nat"   and x :: "var\<^sub>C"   and v :: "nat" shows "ev\<^sub>B\<^sub>C mem (bexp\<^sub>C.Eq x v) = (mem x = v)"
minus_equality: fixes y :: "nat \<Rightarrow> int"   and x :: "nat \<Rightarrow> int" assumes "y \<oplus> x = \<zero>"   and "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "\<ominus> x = y"
factorization_m_lead_coeff: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "q.factorization_m f (c, fs)" shows "lead_coeff (q.Mp f) = q.M c"
Um1: shows "um \<in> carrier (residue_ring (p ^ m)) \<and> um * p ^ n = x (n + m)"
silent_move_observable_moves: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms'' :: "'node list"   and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge" assumes "S,f \<turnstile> (ms'',s'') =as\<Rightarrow> (ms',s')"   and "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms'',s'')" shows "S,f \<turnstile> (ms,s) =a # as\<Rightarrow> (ms',s')"
jordan_block_bound: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < n"   and "j < n" shows "norm ((jordan_block n a ^\<^sub>m k) $$ (i, j)) \<le> norm a ^ (k + i - j) * max 1 (real k ^ (n - 1))"
after_summary_def: fixes M :: "'t zmultiset"   and S :: "'sum antichain" shows "after_summary M S \<equiv> \<Sum>s\<in>set_antichain S. {#results_in t s. t \<in>#\<^sub>z M#}"
remap_polys_polynomial_bool_def: fixes spec :: "int mpoly" shows "remap_polys_polynomial_bool spec = (\<lambda>\<V> A. SPEC (\<lambda>(st, \<V>', A'). (\<not> is_failed st \<longrightarrow> dom_m A = dom_m A' \<and> (\<forall>i\<in>#dom_m A. the (fmlookup A i) - the (fmlookup A' i) \<in> More_Modules.ideal polynomial_bool) \<and> \<Union> (vars ` set_mset (ran_m A)) \<subseteq> \<V>' \<and> \<Union> (vars ` set_mset (ran_m A')) \<subseteq> \<V>') \<and> (st = FOUND \<longrightarrow> spec \<in># ran_m A')))"
dvd_imp_order_le: fixes q :: "'a poly"   and p :: "'a poly"   and a :: "'a" assumes "q \<noteq> 0"   and "p dvd q" shows "Polynomial.order a p \<le> Polynomial.order a q"
simps: shows "SEND' = (left`?`x \<rightarrow> mid`!`x \<rightarrow> ack \<rightarrow> SEND')"
Q2'_def: shows "Q2' = four_block_mat Q2 (0\<^sub>m 2 (dim_col A - 2)) (0\<^sub>m (dim_col A - 2) 2) (1\<^sub>m (dim_col A - 2))"
sep_disj_addD: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x (plus y z)"   and "sep_disj y z" shows "sep_disj x y \<and> sep_disj x z"
corec_sel_2: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and q22 :: "'c \<Rightarrow> bool"   and g221 :: "'c \<Rightarrow> 'd llist"   and g222 :: "'c \<Rightarrow> 'c" assumes "\<not> p a" shows "ltl (corec_llist p g21 q22 g221 g222 a) = (if q22 a then g221 a else corec_llist p g21 q22 g221 g222 (g222 a))"
prefs_from_table_def: fixes xss :: "('c \<times> 'd set list) list" shows "prefs_from_table xss = (\<lambda>i. case map_of xss i of None \<Rightarrow> \<lambda>_ _. False | Some x \<Rightarrow> of_weak_ranking x)"
Mp_0_smult_sdiv_poly: fixes f :: "int poly" assumes "Mp f = 0" shows "Polynomial.smult p (sdiv_poly f p) = f"
of_nat_dvd_imp_dvd_gauss_cnj: fixes n :: "nat"   and z :: "gauss_int" assumes "of_nat n dvd z" shows "of_nat n dvd gauss_cnj z"
vcard_veqpoll: fixes A :: "V"   and B :: "V" shows "(vcard A = vcard B) = (A \<approx>\<^sub>\<circ> B)"
broken_def: fixes K :: "msg set" shows "broken K \<equiv> {(A, B) |A B. priK A \<in> K \<or> priK B \<in> K \<or> shrK A B \<in> K \<or> shrK B A \<in> K}"
wt_gt_\<delta>_if_superunary: fixes s :: "('s, 'v) tm" assumes "1 < arity_hd (head s)" shows "\<delta> < wt s"
U_LIMCONS_not_in_CONS: shows "U\<^bsub>LIM-CONS\<^esub> \<notin> CONS"
simps_3: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "finite A"   and "A \<noteq> {}" shows "fold_random_permutation f x A = pmf_of_set A \<bind> (\<lambda>a. fold_random_permutation f (f a x) (A - {a}))"
inv_solve_right: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<in> carrier H"   and "b \<in> carrier H"   and "c \<in> carrier H" shows "(a = b \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> c) = (b = a \<otimes>\<^bsub>H\<^esub> c)"
set_disconnected_nodes_get_child_nodes: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" assumes "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
rep_list_vec_of: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "spp_inv sp" shows "rep_list (vec_of sp) = snd sp"
red_lcl_incr: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "dom (lcl s) \<subseteq> dom (lcl s')"
le: shows "\<forall>i\<in>S. i \<le> p i"
simps_16: fixes x1 :: "'b"   and x5 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<l>\<^bold>[x5\<^bold>]"
zero_neq_minf: shows "0 \<noteq> - \<infinity>"
term_ok_Types_imp_typ_ok_pre: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and \<tau> :: "typ" assumes "is_std_sig \<Sigma>"   and "term_ok' \<Sigma> t"   and "\<tau> \<in> Types t" shows "typ_ok_sig \<Sigma> \<tau>"
distinct_69: fixes x71 :: "int"   and x72 :: "MIR.num"   and x91 :: "int"   and x92 :: "MIR.num"   and x93 :: "MIR.num" shows "Mul x71 x72 \<noteq> CF x91 x92 x93"
Acute_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "Acute A B C \<equiv> \<exists>A' B' C'. Per A' B' C' \<and> A B C LtA A' B' C'"
CCCD: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans f" shows "CatCod (NTCod f) = CatCod (NTDom f)"
iteratei_correct: fixes S :: "'s" assumes "invar S" shows "set_iterator (iteratei S) (\<alpha> S)"
hf_of_prod_def: shows "hf_of \<equiv> \<lambda>(x, y). \<langle>hf_of x, hf_of y\<rangle>"
diff_intersect_left_idem: fixes M :: "'a multiset"   and N :: "'a multiset" shows "M - M \<inter># N = M - N"
sub_def: fixes z :: "complex_homo"   and w :: "complex_homo" shows "z -\<^sub>h w = (if z = \<infinity>\<^sub>h \<and> w = \<infinity>\<^sub>h then 0\<^sub>h else z +\<^sub>h ~\<^sub>h w)"
fp_cop_F_theorem_5_inv: fixes ds :: "'b set" shows "theorem_5_inv ds (fp_cop_F ds)"
hyperplane_eq_Ex: fixes a :: "'b"   and b :: "real" assumes "a \<noteq> (0::'b)" obtains x :: "'b" where "a \<bullet> x = b"
pow_mat_dim_square_2: fixes A :: "'a mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n" shows "dim_col (A ^\<^sub>m k) = n"
filterlim_tendsto_pos_mult_at_top: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "0 < c"   and "filterlim g at_top F" shows "LIM x F. f x * g x :> at_top"
box_midpoint: fixes a :: "'a"   and b :: "'a" assumes "box a b \<noteq> {}" shows "(1 / 2) *\<^sub>R (a + b) \<in> box a b"
unitary_is_length_preserving: fixes U :: "complex mat"   and v :: "complex vec" assumes "unitary U"   and "dim_vec v = dim_col U" shows "\<parallel>col_fst (U * |v\<rangle>)\<parallel> = \<parallel>v\<parallel>"
vcons_neq_vempty: fixes ys :: "V"   and y :: "V" shows "ys #\<^sub>\<circ> y \<noteq> []\<^sub>\<circ>"
imp_ab_rational: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "\<forall>n. 0 < a n"   and "\<forall>\<^sub>F n in sequentially. 1 < a n"   and "(\<lambda>n. real_of_int \<bar>b n\<bar> / real_of_int (a (n - 1) * a n)) \<longlonglongrightarrow> 0"   and "\<exists>B>0. \<exists>c. \<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) \<and> real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2" shows "(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) \<in> \<rat>"
unsat_state_core_def: fixes s :: "('c, 'd) state" shows "unsat_state_core s = (set (the (\<U>\<^sub>c s)) \<subseteq> indices_state s \<and> (\<nexists>v. (set (the (\<U>\<^sub>c s)), v) \<Turnstile>\<^sub>i\<^sub>s s))"
subset_insertI: fixes B :: "'d set"   and a :: "'d" shows "B \<subseteq> insert a B"
primes_coprime: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "p \<noteq> q" shows "algebraic_semidom.coprime times one p q"
part_7: fixes i :: "nat" assumes "hi < i"   and "i < length xs'" shows "xs' ! i = xs ! i"
r_normal_form_def: shows "r_normal_form \<equiv> Cn 2 r_nf_u [Mn 2 r_nf_t]"
assms_3: shows "i' = is ! nat_of_int c \<and> nat_of_int c < length is \<or> i' = i"
all_union_tsI2: fixes c :: "'a"   and B :: "'a set"   and A :: "'b \<Rightarrow> 'a set"   and k :: "'b" assumes "c \<in> B" shows "c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs" shows "l_set_disconnected_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
nth_rows: fixes i :: "nat"   and A :: "'b mat" assumes "i < dim_row A" shows "Matrix.rows A ! i = Matrix.row A i"
ang_distinct: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "QCongA a"   and "a A B C" shows "A \<noteq> B \<and> C \<noteq> B"
disjoint_insert_2: fixes A :: "'d set"   and b :: "'d"   and B :: "'d set" shows "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
isCont_ego_other2: fixes x :: "real" shows "isCont ego_other2 x"
lm061: fixes G :: "'b set"   and N :: "'a set" shows "allAllocations N G \<subseteq> allocationsUniverse \<and> allAllocations N G \<subseteq> {a. Domain a \<subseteq> N \<and> \<Union> (Range a) = G}"
l_comb_scTr: fixes A :: "'b set"   and H :: "'a set"   and r :: "'b"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "r \<in> carrier R"   and "H \<noteq> {}" shows "s \<in> {j. j \<le> n} \<rightarrow> A \<and> g \<in> {j. j \<le> n} \<rightarrow> H \<longrightarrow> r \<cdot>\<^sub>s \<Sigma>\<^sub>e M (\<lambda>k. s k \<cdot>\<^sub>s g k) n = \<Sigma>\<^sub>e M (\<lambda>k. r \<cdot>\<^sub>s (s k \<cdot>\<^sub>s g k)) n"
sel_2: fixes x11 :: "'a"   and x12 :: "'a process" shows "contOf (Action x11 x12) = x12"
has_integral_substitution: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and c :: "real"   and d :: "real"   and f :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> real" assumes "a \<le> b"   and "g a \<le> g b"   and "g ` {a..b} \<subseteq> {c..d}"   and "continuous_on {c..d} f"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x within {a..b})" shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral integral {g a..g b} f) {a..b}"
nataction_add_flip: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "(a + b) +^ Suc n = a + (b + a) +^ n + b"
scale_cancel_left: fixes a :: "complex"   and x :: "'a"   and y :: "'a" shows "(a *\<^sub>C x = a *\<^sub>C y) = (x = y \<or> a = 0)"
items_pri_agrk_out: fixes x :: "rat"   and H :: "msg set" assumes "Pri_AgrK x \<notin> parts H" shows "items (insert (Pri_AgrK x) H) = insert (Pri_AgrK x) (items H)"
rreq_nsqn_is_fresh: fixes \<sigma> :: "nat \<Rightarrow> state"   and sip :: "nat"   and hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and handled :: "bool" assumes "rreq_rrep_fresh (rt (\<sigma> sip)) (Rreq hops dip dsn dsk oip osn sip handled)"   and "rreq_rrep_sn (Rreq hops dip dsn dsk oip osn sip handled)" shows "msg_fresh \<sigma> (Rreq hops dip dsn dsk oip osn sip handled)"
simple_function_def: fixes M :: "'e measure"   and g :: "'e \<Rightarrow> 'f" shows "simple_function M g = (finite (g ` space M) \<and> (\<forall>x\<in>g ` space M. g -` {x} \<inter> space M \<in> sets M))"
basis: fixes a :: "'a"   and b :: "'b" assumes "a \<in> basisA"   and "b \<in> basisB" shows "\<exists>F\<in>basis. \<forall>a'\<in>basisA. F *\<^sub>V a' = (if a' = a then b else (0::'b))"
f_i': fixes i :: "nat"   and e :: "'v \<times> 'v" shows "f_i i e = ennreal (f_i' i e)"
supr_def: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "wo_rel r" shows "supr r A \<equiv> minim r (Above r A)"
conv_order: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<preceq> y) = (x\<^sup>t \<preceq> y\<^sup>t)"
bind_commute: fixes m :: "'m"   and m' :: "'m"   and f :: "'a option \<Rightarrow> 'a option \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m' (f x)) = bind m' (\<lambda>y. bind m (\<lambda>x. f x y))"
assms: shows "h \<turnstile> get_host shadow_root_ptr \<rightarrow>\<^sub>r host"
unique_class: fixes A :: "'d set"   and B :: "'d set set"   and a :: "'d" assumes "Congruence.partition A B"   and "a \<in> A" shows "\<exists>!b. b \<in> B \<and> a \<in> b"
\<S>: fixes S' :: "('fun, 'var) strand_step list" assumes "S' \<in> update\<^sub>s\<^sub>t \<S> (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)" shows "S' = S \<or> S' \<in> \<S>"
rep_eq: fixes x :: "32 word" shows "Rep_uint32 (Abs_uint32' x) = x"
spectral_radius_ev_def: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "HMA_Connect.spectral_radius A = Max (cmod ` Collect (eigen_value A))"