hcju/isabelleps huggingface dataset:
word_mod_div_equality: fixes n :: "'a word"   and b :: "'a word" shows "n div b * b + n mod b = n"
MkIde_Cod': fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide a" shows "J_C.MkIde (J_C.Cod a) = a"
induct: fixes P :: "bool list bdd list \<Rightarrow> bool list \<Rightarrow> bool list bdd \<Rightarrow> bool"   and a0 :: "bool list bdd list"   and a1 :: "bool list"   and a2 :: "bool list bdd" assumes "\<And>bdd. P [] [] bdd"   and "\<And>bdd' bdds b bs bdd. \<lbrakk>b \<Longrightarrow> P bdds bs (bdd_binop bv_or bdd bdd'); \<not> b \<Longrightarrow> P bdds bs bdd\<rbrakk> \<Longrightarrow> P (bdd' # bdds) (b # bs) bdd"   and "\<And>v va c. P (v # va) [] c"   and "\<And>v va c. P [] (v # va) c" shows "P a0 a1 a2"
vec_to_iarray_row': fixes i :: "'f"   and A :: "(('d, 'e) vec, 'f) vec" shows "vec_to_iarray (row i A) = matrix_to_iarray A !! mod_type_class.to_nat i"
vars_partitioned_def: fixes Cs :: "('a, 'b) Term.term literal multiset list" shows "vars_partitioned Cs = (\<forall>i<length Cs. \<forall>j<length Cs. i \<noteq> j \<longrightarrow> vars_clause (Cs ! i) \<inter> vars_clause (Cs ! j) = {})"
abs_def: fixes R :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and Abs :: "'h set \<Rightarrow> 'i"   and Rep :: "'i \<Rightarrow> 'h set"   and x :: "'h" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
Gromov_completion_boundary_limit: fixes x :: "'b Gromov_completion"   and u :: "'c \<Rightarrow> 'b Gromov_completion"   and F :: "'c filter" assumes "x \<in> Gromov_boundary" shows "(u \<longlongrightarrow> x) F = ((\<lambda>n. extended_Gromov_product_at basepoint (u n) x) \<longlongrightarrow> \<infinity>) F"
sound_modes_2: shows "sound_mode_use (cms\<^sub>2, mem\<^sub>2)"
simps_2: fixes f :: "'b \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and a :: "'b option"   and t :: "'b tree"   and u :: "'b tree" shows "update_rightmost f (Node i j a t u) = Node i j (map_option f a) t (update_rightmost f u)"
llist_all2_ltakeI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and n :: "enat" assumes "llist_all2 P xs ys" shows "llist_all2 P (ltake n xs) (ltake n ys)"
lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {..<Suc n} = g 0 * (\<Prod>i<n. g (Suc i))"
inverse_unique: fixes f :: "'a1 \<times> 'a2"   and g :: "'a1 \<times> 'a2" assumes "inverse_arrows f g" shows "local.inv f = g"
integer_winding_number_eq: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "(winding_number \<gamma> z \<in> \<int>) = (pathfinish \<gamma> = pathstart \<gamma>)"
induct: fixes x1 :: "BExpr"   and x2 :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and x3 :: "Var \<Rightarrow> Val"   and x4 :: "Var \<Rightarrow> Val"   and P :: "BExpr \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "(x1, x2, x3, x4) \<in> var"   and "\<And>b t s \<Phi>. \<lbrakk>\<not> evalB b t;  s \<approx> t \<rbrakk> \<Longrightarrow> P b \<Phi> s t"   and "\<And>b t \<Phi> r s. \<lbrakk>evalB b t; \<Phi> (r, t); \<forall>w.  r \<approx> w  \<longrightarrow> (b, \<Phi>, s, w) \<in> var \<and> P b \<Phi> s w\<rbrakk> \<Longrightarrow> P b \<Phi> s t" shows "P x1 x2 x3 x4"
is_sig_red_Un: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and B :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "is_sig_red sing_reg top_tail (A \<union> B) p = (is_sig_red sing_reg top_tail A p \<or> is_sig_red sing_reg top_tail B p)"
fdia_one: fixes x :: "('a \<times> 'a) set" shows "relfdia Id x = rdom x"
inv_Mp_def: fixes m :: "int" shows "poly_mod.inv_Mp m = map_poly (poly_mod.inv_M m)"
dg_in_Arr_in_Vset: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<DD>\<lparr>Arr\<rparr>" shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
project_set_eq: fixes C :: "'c set" shows "project_set h C = f ` C"
exhaust: fixes y :: "sdim" obtains "y = Length"   | "y = Mass"   | "y = Time"   | "y = Current"   | "y = Temperature"   | "y = Amount"   | "y = Intensity"
circ_loop_fixpoint: fixes y :: "'a"   and z :: "'a" shows "y * (y\<^sup>\<star> * z) \<squnion> z = y\<^sup>\<star> * z"
rep_eq: fixes x :: "'d conjugate_space"   and xa :: "'d conjugate_space" shows "from_conjugate_space (x + xa) = from_conjugate_space x + from_conjugate_space xa"
wset_thread_ok_def: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option" shows "wset_thread_ok ws ts \<equiv> \<forall>t. ts t = None \<longrightarrow> ws t = None"
silent_moves_slpa_path: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and m :: "'node"   and ms'' :: "'node list"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cs :: "'edge list"   and rs :: "'edge list" assumes "S,f \<turnstile> (m # ms'' @ ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_node m"   and "valid_call_list cs m"   and "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "valid_return_list rs m"   and "length rs = length cs"   and "ms'' = targetnodes rs"   and "\<forall>mx\<in>set ms. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "ms'' \<noteq> [] \<longrightarrow> (\<exists>mx'. call_of_return_node (last ms'') mx' \<and> mx' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>)"   and "\<forall>mx\<in>set ms'. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "same_level_path_aux cs as \<and> upd_cs cs as = [] \<and> m -as\<rightarrow>* m' \<and> ms = ms'"
int_a_inv_eq: fixes x :: "int" shows "\<ominus>\<^bsub>\<Z>\<^esub> x = - x"
uint_2p_alt: fixes n :: "nat" assumes "n < LENGTH('b)" shows "uint (2 ^ n) = 2 ^ n"
A: shows "c_fst (c_fst key) mod 7 = 2"
In1_eq: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" shows "(In1 M = In1 N) = (M = N)"
degree_smult_le: fixes a :: "'a"   and p :: "'a poly" shows "degree (smult a p) \<le> degree p"
min_p_min: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list" assumes "is_binqueue l xs"   and "xs \<noteq> []"   and "normalized xs"   and "distinct (vals xs)"   and "distinct (prios xs)" shows "Binomial_Queue.min xs = PQ.priority (pqueue xs) (PQ.min (pqueue xs))"
priK_neq_sessionK: fixes b :: "keymode"   and A :: "agent"   and arg :: "(nat \<times> nat \<times> nat) \<times> role" shows "privateKey b A \<noteq> sessionK arg"
step_def: shows "step \<equiv> err_step (length is) app eff"
linear_sum: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and g :: "'d \<Rightarrow> ('a, 'b) vec"   and S :: "'d set" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
le_add_iff1: fixes a :: "real"   and e :: "'a"   and c :: "'a"   and b :: "real"   and d :: "'a" shows "(a *\<^sub>R e + c \<le> b *\<^sub>R e + d) = ((a - b) *\<^sub>R e + c \<le> d)"
Entry_empty: shows "Def (_Entry_) = {} \<and> Use (_Entry_) = {}"
wk: shows "w \<in> mat_kernel (char_matrix A lam)"
newton_step_def: fixes x :: "nat \<Rightarrow> int" shows "newton_step x = x \<ominus> local.divide (f \<bullet> x) (f' \<bullet> x)"
simps_1: fixes I :: "('b, 'c, 'a) interp"   and t1 :: "('b, 'a) trm"   and t2 :: "('b, 'a) trm" shows "fml_sem I (Geq t1 t2) = {v. dterm_sem I t2 v \<le> dterm_sem I t1 v}"
that: fixes p :: "'a multiset" assumes "p \<in># P"   and "x \<in># p" shows "thesis"
cJDef: shows "cJ = \<lparr>states = start, msgs = initMsg j\<rparr>"
ACI_Plus_cong: fixes r1 :: "'b rexp"   and s1 :: "'b rexp"   and r2 :: "'b rexp"   and s2 :: "'b rexp" assumes "ACI r1 s1"   and "ACI r2 s2" shows "ACI (Plus r1 r2) (Plus s1 s2)"
card_permutations_of_multiset_insert_aux: fixes A :: "'b multiset"   and x :: "'b" shows "card (permutations_of_multiset (A + {#x#})) * (count A x + 1) = (size A + 1) * card (permutations_of_multiset A)"
mat_gtI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat"   and i :: "nat"   and sd :: "nat"   and j :: "nat"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "ord.mat_ge less_eq A B"   and "i < sd"   and "j < sd"   and "gt (A $$ (i, j)) (B $$ (i, j))" shows "ord.mat_gt less_eq gt sd A B"
nat_numeral_as_int: shows "numeral = (\<lambda>i. nat (numeral i))"
base: fixes a :: "'a" shows "a \<in> iterates_above a"
cases: fixes x :: "nat \<times> nat \<times> nat list \<times> nat" obtains a :: "nat" and xa :: "nat" and xs :: "nat list" and s :: "nat" where "x = (a, xa, xs, s)"
pos: shows "0 < (INF i. emeasure (C 0 i (\<lambda>x. undefined)) (X i))"
fps_inverse_unique: fixes f :: "'b fps"   and g :: "'b fps" assumes "f * g = 1" shows "inverse f = g"
replicate_eq_map: fixes g :: "'a \<Rightarrow> bool"   and n :: "nat"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "list_all g (take n xs)"   and "n \<le> length xs"   and "\<forall>y. g y \<longrightarrow> f y = x" shows "replicate n x = map f (take n xs)"
Subst: fixes xvec :: "name list"   and Tvec :: "'a list" assumes "length xvec = length Tvec" shows "(\<Psi>, P[xvec::=Tvec], Q[xvec::=Tvec]) \<in> Rel"
degree_aform_ivl_err: fixes a :: "real interval" shows "degree_aform_err (ivl_err a) = 0"
arr_char: fixes f :: "'a \<times> 'a" shows "B.VV.arr f = (B.arr (fst f) \<and> B.arr (snd f) \<and> trg (fst f) = src (snd f))"
pq_def: shows "(\<lambda>n. real_of_int (b (n + 1)) / real_of_int (a (n + 1))) sums (real_of_int p / real_of_int q)"
less_kbo_gtotal: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "ground s"   and "ground t" shows "s = t \<or> less_kbo s t \<or> less_kbo t s"
Pd': shows "\<forall>d. d \<noteq> d' \<longrightarrow> Pd' d = Pd d"
tends: fixes i :: "'b"   and x :: "'a" assumes "i \<in> Basis"   and "x \<notin> N i" shows "(\<lambda>n. g i n x) \<longlonglongrightarrow> (f x \<bullet> i) *\<^sub>R i"
simps_1: fixes ss :: "'t list"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b" shows "new_syz_sigs ss bs (Inl (a, b)) = ss"
infinite_UNIV: assumes "class.infinite TYPE('b)" shows "infinite UNIV"
det_upper_triangular: fixes A :: "'b mat"   and n :: "nat" assumes "upper_triangular A"   and "A \<in> carrier_mat n n" shows "det A = prod_list (diag_mat A)"
eval_nat_numeral_2: fixes n :: "num" shows "numeral (num.Bit0 n) = Suc (numeral (Num.BitM n))"
assms_12: shows "bf_between_arcs a b (forest_components h) (d \<squnion> selected_edge h j g)"
Lcm_eqI: fixes a :: "'c"   and A :: "'c set" assumes "normalize a = a"   and "\<And>b. b \<in> A \<Longrightarrow> b dvd a"   and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c" shows "normalization_euclidean_semiring_class.Lcm A = a"
more_temp_simps1_4: fixes F :: "'a \<Rightarrow> bool" shows "\<not> \<diamond>F \<equiv> \<box>\<not> F"
distinct_59: shows "NotEq \<noteq> ShiftRightZeros"
simps: fixes c :: "call"   and \<beta> :: "nat \<Rightarrow> contour option"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and b :: "contour" shows "|(c, \<beta>, ve, b)| = (c, |\<beta>|, |ve|, |b|)"
less_imp_not_eq2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "less x y" shows "(y = x) = False"
lt_max_keys: fixes v :: "'a option \<Rightarrow>\<^sub>0 nat"   and p :: "('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "extended_ord v (extended_ord.lpp p)"
IH: fixes as1 :: "'a list"   and as2 :: "'a list"   and f1 :: "'a \<Rightarrow> bool"   and f2 :: "'a \<Rightarrow> 'a" assumes "as1 @ as2 = filter f1 (map f2 p)" shows "\<exists>p_1 p_2. p_1 @ p_2 = p \<and> as1 = filter f1 (map f2 p_1) \<and> as2 = filter f1 (map f2 p_2)"
env_restr_eqD: fixes S :: "'d set"   and m\<^sub>1 :: "'d \<Rightarrow> 'e"   and m\<^sub>2 :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "m\<^sub>1 f|` S = m\<^sub>2 f|` S"   and "x \<in> S" shows "m\<^sub>1 x = m\<^sub>2 x"
real_of_ereal: fixes x :: "ereal" shows "real_of_ereal (- x) = - real_of_ereal x"
epiE: fixes f :: "'d"   and g :: "'d"   and g' :: "'d" assumes "D.epi f"   and "D.seq g f"   and "D.seq g' f"   and "g \<cdot>\<^sub>D f = g' \<cdot>\<^sub>D f" shows "g = g'"
neg_fun_distr2: fixes R' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S' :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'i \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_unique R'"   and "left_total R'"   and "left_unique S'"   and "right_total S'" shows "rel_fun (R OO R') (S OO S') \<le> rel_fun R S OO rel_fun R' S'"
rv_ge_iff: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and f :: "'b \<Rightarrow> real" assumes "measure_space M" shows "(f \<in> rv M) = (\<forall>a. {w. a \<le> f w} \<in> measurable_sets M)"
distinct_indices_stateD: fixes s :: "('a, 'b) state"   and x :: "nat"   and i :: "'a"   and b :: "'b"   and x' :: "nat"   and b' :: "'b" assumes "distinct_indices_state s"   and "Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b)"   and "Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x' = Some (i, b') \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x' = Some (i, b')" shows "x = x' \<and> b = b'"
simps: fixes ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and a :: "'a atom formula list"   and d :: "'a atom formula list" shows "wf_effect ty_ent (Effect a d) = (Ball (set a) (wf_fmla_atom ty_ent) \<and> Ball (set d) (wf_fmla_atom ty_ent))"
compile_z_correct: fixes ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and n :: "nat"   and r :: "nat"   and anything :: "nat list" assumes "rec_ci z = (ap, arity, fp)"   and "rec_exec z [n] = r" shows "{\<lambda>nl. nl = n # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = n # r # 0 \<up> (fp - Suc arity) @ anything}"
bot_eq_sup_iff: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and bot :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(bot = sup x y) = (x = bot \<and> y = bot)"
lunit_naturality: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "V \<mu> (prebicategory_with_homs_and_units.lunit V H \<a> \<i> trg (partial_magma.dom V \<mu>)) = V (prebicategory_with_homs_and_units.lunit V H \<a> \<i> trg (partial_magma.cod V \<mu>)) (H (trg \<mu>) \<mu>)"
Q_hit'_v0_v1_H_x: shows "v0 \<leadsto>Q_hit'\<leadsto>\<^bsub>H_x\<^esub> v1"
x_def: shows "x \<in> set_of_k_words_length_m k 0"
transdb_inv_open: fixes e :: "db"   and c :: "nat"   and k :: "nat"   and \<Gamma> :: "nat \<Rightarrow> nat"   and \<Gamma>' :: "nat list" assumes "\<forall>v. freedb e v \<longrightarrow> v < c + k"   and "\<forall>v. freedb e v \<longrightarrow> \<Gamma> v = (if k \<le> v then v - k else c + k - v - 1)"   and "\<forall>v. freedb e v \<longrightarrow> (if k \<le> v then index \<Gamma>' (v - k) 0 = v else index \<Gamma>' (c + k - v - 1) 0 = v)" shows "transdb (transdb_inv e \<Gamma> c k) \<Gamma>' = e"
assms_2: fixes \<sigma> :: "'\<sigma>"   and x :: "'x"   and S :: "'x set" assumes "c \<sigma>"   and "x \<in> S0 - S"   and "I S \<sigma>"   and "S \<subseteq> S0"   and "\<forall>y\<in>S0 - S - {x}. R x y"   and "\<forall>y\<in>S. R y x" shows "I (insert x S) (f x \<sigma>)"
opp_chambers_distinct_map: fixes Cs :: "'a set list" assumes "order.greater_eq (folding_g.\<C> - g \<turnstile> folding_g.\<C>) (set Cs)"   and "distinct Cs" shows "distinct (g \<Turnstile> Cs)"
B: shows "(xps @ yps, Y) \<in> c_failures step out s\<^sub>0"
aff_dim_cball: fixes e :: "real"   and a :: "'b" assumes "0 < e" shows "aff_dim (cball a e) = int DIM('b)"
strict_mono_imp_inj_on: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "inj_on f A"
powr_neg_one: fixes x :: "real" assumes "0 < x" shows "x powr - 1 = 1 / x"
sig_inv_set_zero: fixes j :: "nat" shows "0 \<in> sig_inv_set' j"
complete_eq_closed: fixes S :: "'b set" shows "complete S = closed S"
mult_Bit0_left: fixes m :: "num"   and n :: "num" shows "num.Bit0 m * n = num.Bit0 (m * n)"
wf_ci_check_instr: fixes typeof_addr :: "'b \<Rightarrow> 'a \<Rightarrow> htype option" shows "wf_ci (JVM_heap_base.check_instr typeof_addr)"
par_strict_not_col_2: 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 C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ParStrict Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Col Bet B C D"
assms_5: shows "\<V> (insert (1 - punit.monom_mult (1::'a) (monomial 1 x) f) F) = {}"
r_inv: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.group G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
row_add_code_nth: fixes A :: "(('c, 'd) vec, 'b) vec"   and a :: "'b"   and b :: "'b"   and q :: "'c" shows "($) (row_add A a b q) = row_add_row A a b q"
free_tv_bound_typ_inst1: fixes v :: "nat"   and sch :: "type_scheme"   and S :: "nat \<Rightarrow> nat" assumes "v \<notin> free_tv sch"   and "v \<in> free_tv (bound_typ_inst (TVar \<circ> S) sch)" shows "\<exists>x. v = S x"
authKeys_used: fixes K :: "nat"   and evs :: "event list" assumes "K \<in> authKeys evs" shows "Key K \<in> used evs"
finfun_fst_comp_conv: fixes f :: "'d \<Rightarrow> 'b \<times> 'c"   and g :: "('a, 'd) finfun" shows "finfun_fst (finfun_comp f g) = finfun_comp (fst \<circ> f) g"
choose_rising_sum_1: fixes n :: "nat"   and m :: "nat" shows "(\<Sum>j\<le>m. n + j choose n) = n + m + 1 choose (n + 1)"
ide_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "C.ide (trg\<^sub>C \<mu>)"
wf_subtype: fixes \<Gamma> :: "binding list"   and P :: "POPLmark.type"   and Q :: "POPLmark.type" assumes "\<Gamma> \<turnstile> P <: Q" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f P \<and> \<Gamma> \<turnstile>\<^sub>w\<^sub>f Q"
conservation: fixes x :: "nat" assumes "x \<noteq> s"   and "x \<noteq> t" shows "sum max_flow (incoming x) = sum max_flow (outgoing x)"
Arr_rep_ARR: fixes f :: "'c local.term set" assumes "\<F>C.ARR f" shows "Arr (\<F>C.rep f)"
fundamental_theorem_2: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and \<Gamma>' :: "(name \<times> ty) list"   and \<theta> :: "(name \<times> trm) list"   and \<theta>' :: "(name \<times> trm) list" assumes "\<Gamma> \<turnstile> s \<equiv> t : T"   and "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"   and "valid \<Gamma>'" shows "\<Gamma>' \<turnstile> \<theta>
 is \<theta>'<t> : T"
stable_no_blocking_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "ManiPref.stable_no_blocking_on ds X = (\<forall>h X'. \<not> ManiPref.blocking_on ds X h X')"
lepoll_trans: fixes A :: "'a set"   and B :: "'b set"   and C :: "'c set" assumes "A \<lesssim> B"   and "B \<lesssim> C" shows "A \<lesssim> C"
Ident: fixes A :: "fm" shows "A IMP A \<in> boolean_axioms"
invar_MST_def: fixes \<pi>i :: "'m" shows "invar_MST \<pi>i \<equiv> M_invar \<pi>i"
all_AA_def: shows "all_AA = \<Union> (atms_of ` (CC \<union> {C}))"
inject_2: fixes x2 :: "('e, 'f) Term.term"   and y2 :: "('e, 'f) Term.term" shows "(receive\<langle>x2\<rangle> = receive\<langle>y2\<rangle>) = (x2 = y2)"
wf_tuple_restrict_simple: fixes n :: "nat"   and B :: "nat set"   and v :: "'b option list"   and A :: "nat set" assumes "wf_tuple n B v"   and "A \<subseteq> B" shows "wf_tuple n A (restrict A v)"
iMODb_iTILL_subset_conv: fixes r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and n :: "nat" shows "([ r', mod m', c' ] \<subseteq> [\<dots>n]) = (r' + m' * c' \<le> n)"
comp_ide_arr: fixes b :: "'b"   and f :: "'b" assumes "\<phi>.B.ide b"   and "\<phi>.B.seq b f" shows "b \<cdot>\<^sub>B f = f"
ideD_3: fixes a :: "'a1" assumes "C1.ide a" shows "C1.cod a = a"
domain_restrict_chart_if: fixes S :: "'c set"   and c :: "('c, 'd) chart" shows "domain (restrict_chart S c) = (if open S then domain c \<inter> S else {})"
gfp_star_least: fixes f :: "'c \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b \<Rightarrow> 'c" assumes "\<And>u. u \<le> Omega_fun f g u \<Longrightarrow> u \<le> A" shows "star_fun f \<circ> g \<le> A"
exec_elim_cases_3: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> \<Rightarrow> t" obtains "t = Abrupt s"
LEAST_eq_floor: fixes h :: "nat" shows "int (LEAST n. int h \<le> \<lceil>c * log 2 (real n + 1)\<rceil>) = \<lfloor>2 powr ((real h - 1) / c)\<rfloor>"
mult_nonpos_nonpos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq a zero"   and "less_eq b zero" shows "less_eq zero (times a b)"
dsumag_sub_prodag: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "carr_dsumag I A \<subseteq> carr_prodag I A"
select_weight_def: fixes xs :: "(natural \<times> 'b) list" shows "select_weight xs = local.range (sum_list (map fst xs)) \<circ>\<rightarrow> (\<lambda>k. Pair (pick xs k))"
append_same_eq: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" shows "(ys @ xs = zs @ xs) = (ys = zs)"
Mg: shows "g \<in> N \<rightarrow>\<^sub>M subprob_algebra N'"
i0_less: fixes n :: "enat" shows "(0 < n) = (n \<noteq> 0)"
pm_prod_E_def: shows "pm_prod_E = {x. fst x \<in> E \<and> snd x \<in> E}"
unrest_bop: fixes x :: "'a \<Longrightarrow> 'b"   and u :: "('c, 'b) uexpr"   and v :: "('d, 'b) uexpr"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "x \<sharp> u"   and "x \<sharp> v" shows "x \<sharp> bop f u v"
ord_term_canc_left: fixes t :: "'a"   and v :: "'t"   and s :: "'a" assumes "t \<oplus> v \<preceq>\<^sub>t s \<oplus> v" shows "t \<preceq> s"
orderI: fixes a :: "'d"   and b :: "'d" assumes "a = inf a b" shows "a \<le> b"
lessThan_subset_Collect: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool" shows "({..<x} \<subseteq> Collect P) = (\<forall>y<x. P y)"
map_pmf_of_set: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "A \<noteq> {}" shows "map_pmf f (pmf_of_set A) = pmf_of_multiset (image_mset f (mset_set A))"
fin: fixes x :: "real" shows "emeasure M {x<..} < \<infinity>"
integral_singleton: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "integral {a} f = (0::'a)"