hcju/isabelleps huggingface dataset:
of_nat_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 0 = zero"
rSym: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "(\<Psi>', R, S) \<in> X" shows "(\<Psi>', S, R) \<in> X \<or> \<Psi>' \<rhd> S \<sim> R"
homotopic_paths_part_circlepath: fixes a :: "real"   and b :: "real"   and c :: "real"   and C :: "complex"   and r :: "real"   and A :: "complex set" assumes "a \<le> b"   and "b \<le> c"   and "path_image (part_circlepath C r a c) \<subseteq> A"   and "0 \<le> r" shows "homotopic_paths A (part_circlepath C r a c) (part_circlepath C r a b +++ part_circlepath C r b c)"
UniversePair: fixes u :: "ZF"   and v :: "ZF" assumes "Elem u U"   and "Elem v U" shows "Elem (Opair u v) U"
map_append: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list" shows "list.map f (xs @ ys) = list.map f xs @ list.map f ys"
inject_6: fixes x61 :: "'a"   and x62 :: "'a exp"   and y61 :: "'a"   and y62 :: "'a exp" shows "(x61:=x62 = y61:=y62) = (x61 = y61 \<and> x62 = y62)"
in_KeysI: fixes s :: "'b"   and f :: "'b \<Rightarrow>\<^sub>0 'c"   and F :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "s \<in> keys f"   and "f \<in> F" shows "s \<in> Keys F"
t_prod_def: fixes X :: "('b \<times> ('c \<times> 'b) list) set"   and Y :: "('b \<times> ('c \<times> 'b) list) set" shows "t_prod X Y = {t_fusion u v |u v. u \<in> X \<and> v \<in> Y \<and> Dioid_Models.last u = first v}"
one_eq_prime_power_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "(one = power.power one times p n) = (n = 0)"
alist_of_inverse: fixes x :: "('c, 'd) pq" shows "Abs_pq (alist_of x) = x"
OO_def: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
f_def: shows "f = Cn 2 r_length [recf.Id 2 1]"
thmD2': fixes X :: "symbol"   and p :: "(symbol \<times> character list) list" assumes "is_terminal X"   and "doc_tokens p"   and "terminals p @ [X] \<in> \<L>\<^sub>P" shows "\<exists>x. pvalid p x \<and> next_symbol x = Some X"
SubstFormP_Ex: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm" shows "{SubstFormP v i x x'} \<turnstile> SubstFormP v i (Q_Ex x) (Q_Ex x')"
suminf_pos: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f"   and "\<And>n. (0::'a) < f n" shows "(0::'a) < suminf f"
subspace_normed_vs: fixes F :: "'c set"   and E :: "'c set"   and norm :: "'c \<Rightarrow> real" assumes "Subspace.subspace F E"   and "normed_vectorspace E norm" shows "normed_vectorspace F norm"
wfrec_def: fixes R :: "('c \<times> 'c) set"   and F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" shows "wfrec R F = (\<lambda>x. THE y. wfrec_rel R (\<lambda>f x. F (cut f R x) x) x y)"
IH_2: assumes "\<not> complete r" shows "size1 r < 2 ^ height r"
dfree_Plus_simps: fixes a :: "('d, 'e) trm"   and b :: "('d, 'e) trm" shows "dfree (Plus a b) = (dfree a \<and> dfree b)"
word_not_le: fixes x :: "'b word"   and y :: "'b word" shows "(\<not> x \<le> y) = (y < x)"
distinct_27: fixes x41 :: "bexp"   and x42 :: "acom"   and x43 :: "acom"   and x61 :: "(((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and x62 :: "bexp"   and x63 :: "acom" shows "IF x41 THEN x42 ELSE x43 \<noteq> {x61} WHILE x62 DO x63"
path_components_of_eq_singleton: fixes X :: "'a topology"   and S :: "'a set" shows "(path_components_of X = {S}) = (path_connected_space X \<and> topspace X \<noteq> {} \<and> S = topspace X)"
assms: shows "\<forall>i<length l. g (l ! i) = f i"
seq_stop_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((Stop ;; c, mds), mem), (c', mds'), mem') \<in> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "c' = c \<and> mds' = mds \<and> mem' = mem"
genlist_def: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "genlist f n = map f [0..<n]"
runit_in_hom_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<r>[f] : f \<star> src f \<Rightarrow> f\<guillemotright>"
vals_meld_distinct: fixes xs :: "('d, 'c) bintree option list"   and ys :: "('d, 'c) bintree option list" assumes "distinct (vals xs)"   and "distinct (vals ys)"   and "set (vals xs) \<inter> set (vals ys) = {}" shows "distinct (vals (meld xs ys))"
f: shows "(f has_vector_derivative f') (at x within cbox a b)"
get_root_node_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_root_node ptr"
p_prod_def: fixes X :: "'a list set"   and Y :: "'a list set" shows "p_prod X Y = {rs. \<exists>ps\<in>X. \<exists>qs\<in>Y. rs = p_fusion ps qs \<and> p_filter ps qs}"
imply_params_fm: fixes ps :: "('a, 'b) fm list"   and q :: "('a, 'b) fm" shows "params_fm (ps \<^bold>\<leadsto> q) = params_fm q \<union> params (set ps)"
units_of_one: fixes G :: "('e, 'f) monoid_scheme" shows "\<one>\<^bsub>units_of G\<^esub> = \<one>\<^bsub>G\<^esub>"
order_iff: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "semilattice_order f less_eq less" shows "less_eq a b = (a = f a b)"
Suc_1: obtains n :: "nat" where "\<And>l. n \<le> l \<Longrightarrow> card (level A w (2 * k) l) \<le> card (nodes A) - k"
get_child_nodes_reads: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads (get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr) (get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr) h h'"
uw\<theta>\<omega>\<nu>: shows "ide w' \<and> \<guillemotleft>\<theta>' : f \<star> w' \<Rightarrow> f'\<guillemotright> \<and> \<guillemotleft>\<nu>' : local.dom \<rho>' \<Rightarrow> g \<star> w'\<guillemotright> \<and> local.iso \<nu>' \<and> (r \<star> \<theta>') \<cdot> \<a>[r, f, w'] \<cdot> (\<rho> \<star> w') \<cdot> \<nu>' = \<rho>'"
intro: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>h new_character_data_ptr h' r ptr'. \<lbrakk>h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr; h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'; r \<in> get_disconnected_nodes_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_new_character_data_get_disconnected_nodes get_disconnected_nodes_locs"
processedD2: fixes a :: "'e"   and b :: "'e"   and xs :: "'e list"   and ps :: "('e \<times> 'e) list" assumes "processed (a, b) xs ps" shows "b \<in> set xs"
path_append_token: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat"   and t :: "symbol \<times> character list" assumes "p \<in> \<P> k u"   and "t \<in> \<Z> k (Suc u)"   and "admissible (p @ [t])"   and "charslength p = k" shows "p @ [t] \<in> \<P> k (Suc u)"
densU: fixes x :: "'a" assumes "x \<in> G" shows "upper_asymptotic_density (U x) = 0"
IH1': shows "quicksort_post R h (Suc p) hi xs' xs''"
exp_times_self_antimono: fixes y :: "real"   and x :: "real" assumes "y \<le> - 1"   and "x \<le> y" shows "y * exp y \<le> x * exp x"
admit: fixes i :: "'sz" shows "TadmitF \<sigma> (args i)"
restrict_update: fixes y :: "nat"   and A :: "nat set"   and x :: "'a option list"   and z :: "'a option" assumes "y \<notin> A"   and "y < length x" shows "restrict A (x[y := z]) = restrict A x"
hisPrev_nex_less: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and n' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'a" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> Blockchain.his active cmp bc mining t n nid" shows "\<not> (\<exists>x\<in>Blockchain.his active cmp bc mining t n nid. fst x < n' \<and> Blockchain.hisPred active cmp bc mining t n nid n' < fst x)"
Rep_perm_add: fixes p1 :: "'a perm"   and p2 :: "'a perm" shows "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
map_mat_def: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b mat" shows "map_mat f A \<equiv> Matrix.mat (dim_row A) (dim_col A) (\<lambda>ij. f (A $$ ij))"
SINK_TER_in'': fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current \<Gamma> g"   and "x \<in> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f \<union> TER\<^bsub>\<Gamma>\<^esub> g)"   and "\<And>x. x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g) \<Longrightarrow> d_OUT g x = 0" shows "x \<in> MFMC_Network.SINK g"
fst_pseudo_divmod_main_as_divide_poly_main: fixes d :: "'b poly"   and q :: "'b poly"   and r :: "'b poly"   and dr :: "nat"   and n :: "nat" assumes "d \<noteq> 0" shows "fst (pseudo_divmod_main (lead_coeff d) q r d dr n) = divide_poly_main (lead_coeff d) (smult (lead_coeff d ^ n) q) (smult (lead_coeff d ^ n) r) d dr n"
assm_6: fixes i :: "'a set" assumes "i \<in> I" shows "\<Union> {K. \<exists>x. (x, K) \<in> pfn i} = i"
4: assumes "P \<turnstile> typeof\<^bsub>h(a \<mapsto> (C, S'))\<^esub> v \<noteq> \<lfloor>NT\<rfloor>"   and "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>NT\<rfloor>" shows "P \<turnstile> typeof\<^bsub>h(a \<mapsto> (C, S'))\<^esub> v = \<lfloor>Class C'\<rfloor>"
fixp_above_Field: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "a \<in> Field leq" shows "bourbaki_witt_fixpoint.fixp_above lub leq f a \<in> Field leq"
valid_call_list_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list"   and n :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.valid_call_list sourcenode kind valid_edge get_proc cs n \<equiv> \<forall>cs' c cs''. cs = cs' @ c # cs'' \<longrightarrow> valid_edge c \<and> (\<exists>Q r p fs. kind c = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<and> p = get_proc (case cs' of [] \<Rightarrow> n | a # list \<Rightarrow> last (CFG.sourcenodes sourcenode cs')))"
tMap_is_tInit: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b Trace"   and s :: "'a" shows "(tMap f t = tInit s) = (\<exists>s'. t = tInit s' \<and> f s' = s)"
closed_map_into_subtopology: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "closed_map X X' f"   and "f ` topspace X \<subseteq> S" shows "closed_map X (subtopology X' S) f"
cx_flow: shows "cx' = flow0 x0' (h' * s1')"
catalan_Suc': fixes n :: "nat" shows "catalan (Suc n) = catalan n * (2 * (2 * n + 1)) div (n + 2)"
permutation_face_cycle_succ: shows "permutation H.face_cycle_succ"
absorb1: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "sup a b = a"
domI: fixes m :: "'c \<Rightarrow> 'b option"   and a :: "'c"   and b :: "'b" assumes "m a = Some b" shows "a \<in> dom m"
state_satisfies_index: fixes v :: "nat \<Rightarrow> 'c"   and s :: "('d, 'c) state"   and I :: "'d set" assumes "v \<Turnstile>\<^sub>s s" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>s s"
into_range_balls: fixes i :: "nat" shows "U i \<in> balls ` carrier"
range_ex1_eq: fixes f :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "inj f" shows "(b \<in> range f) = (\<exists>!x. b = f x)"
vend: shows "\<not>? value_gt (Some (Num 100)) (r $ 2) = trilean.true \<longrightarrow> P (Some 2, [r $ 1], r)"
inv_letter_mono: fixes x :: "accent \<times> 'a"   and y :: "accent \<times> 'a"   and r :: "('a \<times> 'a) set" assumes "(x, y) \<in> letter_less r" shows "(inv_letter x, inv_letter y) \<in> letter_less r"
HPhase0Read_HInv4c_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "HInv4c s q"   and "p \<noteq> q" shows "HInv4c s' q"
widen_Array: fixes P :: "'b prog"   and T :: "ty"   and U :: "ty" shows "P \<turnstile> T \<le> U\<lfloor>\<rceil> = (T = NT \<or> (\<exists>V. T = V\<lfloor>\<rceil> \<and> P \<turnstile> V \<le> U))"
summable_partial_sum_bound: fixes f :: "nat \<Rightarrow> 'a"   and e :: "real" assumes "summable f"   and "0 < e" obtains N :: "nat" where "\<And>m n. N \<le> m \<Longrightarrow> norm (sum f {m..n}) < e"
execute_parallel_operator_effect_2: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list"   and op :: "'b strips_operator"   and v :: "'b" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops" shows "op \<in> set ops \<and> v \<in> set (delete_effects_of op) \<longrightarrow> execute_parallel_operator s ops v = Some False"
inverse_irred_power_poly_def: fixes d :: "'b"   and n :: "nat" shows "inverse_irred_power_poly d n = Poly (map (\<lambda>k. d * of_nat (stirling n (k + 1)) / fact (n - 1)) [0..<n])"
aff_dim_def: fixes V :: "'b set" shows "aff_dim V = (SOME d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = d + 1)"
cast_DEFL_emb: fixes x :: "'a" shows "cast\<cdot>DEFL('a)\<cdot>(EMB('a)\<cdot>x) = EMB('a)\<cdot>x"
assms_1: shows "\<tau>.P \<longmapsto> \<alpha> \<prec> P'"
inv_into_def2: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" shows "inv_into A f x = (SOME y. y \<in> A \<and> f y = x)"
bet_conga__bet: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'" shows "Bet A' B' C'"
swap: fixes a :: "'action"   and b :: "'action"   and u :: "'action list"   and v :: "'action list" assumes "ind a b" shows "u @ [a] @ [b] @ v =\<^sub>S u @ [b] @ [a] @ v"
ub2: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A" shows "y \<sqsubseteq>\<^bsub>r\<^esub> x \<squnion>\<^bsub>f\<^esub> y"
wf_implies_termi_reach_step_case: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "'a" assumes "\<And>c' s'. \<Gamma>\<turnstile> (c, Normal s) \<rightarrow> (c', s') \<Longrightarrow> \<Gamma>\<turnstile>c' \<down> s'" shows "\<Gamma>\<turnstile>c \<down> Normal s"
wt_rOfWax: fixes \<sigma> :: "'tp" assumes "\<not> isRes \<sigma>" shows "TE.wt (rOfWax \<sigma>)"
Pasch: fixes x :: "complex_homo"   and y :: "complex_homo"   and z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo" assumes "x \<in> unit_disc"   and "y \<in> unit_disc"   and "z \<in> unit_disc"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "poincare_between x u z"   and "poincare_between y v z" shows "\<exists>a. a \<in> unit_disc \<and> poincare_between u a y \<and> poincare_between x a v"
simps_1: fixes x :: "'a" shows "insrt\<^sub>0 x ET\<^sub>0 = MKT\<^sub>0 x ET\<^sub>0 ET\<^sub>0"
total_on_restrictPI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "total_onP A r" shows "total_onP (A \<inter> B) (r |` B)"
d2_def: fixes dims :: "nat list"   and vars :: "nat set" shows "partial_state.d2 dims vars = prod_list (partial_state.dims2 dims vars)"
A: shows "[reset' M n cs v d]\<^bsub>v,n\<^esub> = {}"
ODE_sem_OVar: fixes I :: "('b, 'c, 'a) interp"   and x :: "'a" shows "ODE_sem I (OVar x) = ODEs I x"
no_qi: shows "\<not> entails_q qi N1 {B2}"
left_no_excess: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and L_left :: "nat list"   and L_right :: "nat list" assumes "rtf_invar c s t f l n L_left L_right" shows "\<forall>u\<in>set L_left. Network.excess c f u = (0::'a)"
pelims: fixes x :: "complex \<times> complex"   and xa :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex" assumes "x *\<^sub>v\<^sub>m xa = y"   and "mult_vm_dom (x, xa)" obtains xb :: "complex" and ya :: "complex" and a :: "complex" and b :: "complex" and c :: "complex" and d :: "complex" where "x = (xb, ya)" and "xa = (a, b, c, d)" and "y = (xb * a + ya * c, xb * b + ya * d)" and "mult_vm_dom ((xb, ya), a, b, c, d)"
subst_lsst_unlabel: fixes A :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "unlabel (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = unlabel A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>"
iso_is_local_ring_morphism: fixes A :: "'d set"   and addA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroA :: "'d"   and oneA :: "'d"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set"   and addB :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multB :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zeroB :: "'e"   and oneB :: "'e" assumes "local_ring A addA multA zeroA oneA"   and "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "local_ring_morphism f A addA multA zeroA oneA B addB multB zeroB oneB"
filter_simp: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "filter (\<lambda>x. P x \<and> Q x) xs = filter Q xs"
kuhn_labelling_lemma: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a"   and Q :: "'a \<Rightarrow> bool" assumes "\<forall>x. P x \<longrightarrow> P (f x)"   and "\<forall>x. P x \<longrightarrow> (\<forall>i\<in>Basis. Q i \<longrightarrow> 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)" shows "\<exists>l. (\<forall>x. \<forall>i\<in>Basis. l x i \<le> 1) \<and> (\<forall>x. \<forall>i\<in>Basis. P x \<and> Q i \<and> x \<bullet> i = 0 \<longrightarrow> l x i = 0) \<and> (\<forall>x. \<forall>i\<in>Basis. P x \<and> Q i \<and> x \<bullet> i = 1 \<longrightarrow> l x i = 1) \<and> (\<forall>x. \<forall>i\<in>Basis. P x \<and> Q i \<and> l x i = 0 \<longrightarrow> x \<bullet> i \<le> f x \<bullet> i) \<and> (\<forall>x. \<forall>i\<in>Basis. P x \<and> Q i \<and> l x i = 1 \<longrightarrow> f x \<bullet> i \<le> x \<bullet> i)"
Field_r_def: shows "Field_r = {f. current \<Gamma> f \<and> wave \<Gamma> f}"
choose_minimal: fixes v :: "'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "v \<in> S"   and "(\<sigma>', choose v) \<in> r - Id" shows "\<sigma>' \<notin> good v"
abs_eq: fixes x :: "complex \<times> complex \<times> complex \<times> complex" assumes "eq_onp (\<lambda>M. mat_det M \<noteq> 0) x x" shows "unit_circle_fix_mmat (Abs_moebius_mat x) = unitary11_gen x"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>nat. P (TVar nat)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<rightarrow> ty2)" shows "P ty"
convex_lincomb_list_def: fixes n :: "nat"   and c :: "nat \<Rightarrow> 'b"   and Vs :: "'b vec list"   and b :: "'b vec" shows "gram_schmidt.convex_lincomb_list n c Vs b = (gram_schmidt.nonneg_lincomb_list n c Vs b \<and> sum c {0..<length Vs} = (1::'b))"
bits_mod_div_trivial: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide (modulo a b) b = zero"
free: shows "dfree (Times \<theta>\<^sub>1 \<theta>\<^sub>2)"
u_def: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "u x y n = T.prob x {\<omega> \<in> space (T x). ev_at (HLD {y}) n \<omega>}"
eSuc_Max: fixes A :: "enat set" assumes "finite A"   and "A \<noteq> {}" shows "eSuc (Max A) = Max (eSuc ` A)"
A_eq_B: shows "adjugate (\<^bold>X - \<^bold>C A) = (\<Sum>i\<le>n. X ^ i *\<^sub>S \<^bold>C (B i))"
bisimWeakCoinduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto>[X] Q"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<sim> Q"
poly_infinity: fixes p :: "'a poly"   and d :: "real"   and a :: "'a" assumes "p \<noteq> 0" shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
reflexive_def: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" shows "reflexive x = (\<forall>y. x y y)"
monom_diff: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and b :: "'b" shows "monom m (a - b) = monom m a - monom m b"
prems_5: shows "InvariantWatchListsUniq (getWatchList state')"
in_set_dropD: fixes x :: "'b"   and n :: "nat"   and xs :: "'b list" assumes "x \<in> set (drop n xs)" shows "x \<in> set xs"
abs_summable_on_cmult_left: fixes c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "c \<noteq> (0::'b) \<Longrightarrow> f abs_summable_on A" shows "(\<lambda>x. f x * c) abs_summable_on A"
card_a_I: fixes I :: "nat set" assumes "I \<subseteq> {0..<m}" shows "card (a ` I) = card I"
adapted_is_dsp: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "adapt_stoch_proc F X N" shows "X \<in> stoch_procs M N"
mod_subset_def: fixes p :: "nat"   and X :: "'b topology" shows "mod_subset p X \<equiv> {(a, b). singular_chain p X (a - b)}"