hcju/isabelleps huggingface dataset:
Y_l: shows "\<forall>(t, n)\<in>set (remdups (map simp_num_pair (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs (uset (rlfm (simpfm p))))))). numbound0 t \<and> 0 < n"
unit_disc_to_complex_inj: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "to_complex u = to_complex v" shows "u = v"
cycle_free: fixes G :: "('a, 'b) graph" assumes "forest G" shows "\<forall>(a, w, b)\<in>edges G. \<not> nodes_connected (delete_edge a w b G) a b"
elims_3: fixes x :: "('a, 'b) fmap"   and xa :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "\<not> sat_precond_as x xa" obtains s :: "('a, 'b) fmap" and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" where "x = s" and "xa = a # as" and "\<not> (fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as)"
index_swap_distinct: fixes xs :: "'c list"   and n :: "nat"   and x :: "'c" assumes "distinct xs"   and "Suc n < length xs" shows "index (swap n xs) x = (if x = xs ! n then Suc n else if x = xs ! Suc n then n else index xs x)"
am_to_adj_nodes_refine: fixes am :: "nat \<Rightarrow> nat list"   and u :: "nat" assumes "is_adj_map am" shows "(am u, adjacent_nodes u) \<in> \<langle>nat_rel\<rangle>list_set_rel"
induces_meta_adjunction: shows "meta_adjunction C D F F.G F.\<phi> F.\<psi>"
startsby_nonzero_power: fixes a :: "'a fps"   and n :: "nat" assumes "fps_nth a 0 \<noteq> (0::'a)" shows "fps_nth (a ^ n) 0 \<noteq> (0::'a)"
finite_Pow_iff: fixes A :: "'a set" shows "finite (Pow A) = finite A"
comm_monoid_def: fixes G :: "('a, 'b) monoid_scheme" shows "Group.comm_monoid G \<equiv> Group.monoid G \<and> Group.comm_monoid_axioms G"
take_Cons: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
iso_img_block_not_in: fixes x :: "'a set" assumes "x \<notin># \<B>"   and "x \<subseteq> \<V>" shows "\<pi> ` x \<notin># \<B>'"
mat_def: fixes nr :: "nat"   and nc :: "nat"   and m :: "'a list list" shows "Matrix_Legacy.mat nr nc m = (length m = nc \<and> Ball (set m) (Matrix_Legacy.vec nr))"
mult_DirProd: fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and g :: "'c"   and h :: "'d"   and g' :: "'c"   and h' :: "'d" shows "(g, h) \<otimes>\<^bsub>G \<times>\<times> H\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"
TC_least: fixes x :: "V"   and a :: "V" assumes "Transset x"   and "a \<le> x" shows "TC a \<le> x"
commute: fixes a :: "'a"   and b :: "'a" shows "a \<^bold>* b = b \<^bold>* a"
plus_bit1_def: fixes x :: "'a bit1"   and y :: "'a bit1" shows "x + y = Abs_bit1' (Rep_bit1 x + Rep_bit1 y)"
assigns_less_Length: fixes x :: "nat"   and m1 :: "nat"   and \<AA> :: "interp"   and k :: "WS1S_Prelim.order" assumes "x |\<in>| m1\<^bsup>\<AA>\<^esup>k" shows "x < Length \<AA>"
STA_one: fixes A :: "'b set"   and n :: "nat"   and branch :: "(('c, 'b) fm list \<times> 'b) list" assumes "A, n \<turnstile> branch" shows "A, 1 \<turnstile> branch"
iso_lunit: fixes f :: "'c" assumes "C.ide f" shows "C.iso \<l>\<^sub>C[f]"
shift_snth: fixes xs :: "'b list"   and s :: "'b stream"   and n :: "nat" shows "(xs @- s) !! n = (if n < length xs then xs ! n else s !! (n - length xs))"
Cd_irc: fixes X :: "'x set"   and d :: "'d" shows "irc_on X (Cd d)"
nvi_12: fixes i :: "nat" assumes "i < length us" shows "normal_vectors ws ! i \<notin> local.span (set (ws @ us) - {us ! i})"
wf_dom_le: fixes ss :: "nat list list"   and w :: "nat list" assumes "wf_dom ss w" shows "ss [\<sqsubseteq>\<^bsub>r\<^esub>] ss"
hyp: shows "map snd (cms'[k := fst z]) \<in> reachable_mode_states (cms', mem')"
create_character_data_def: shows "create_character_data \<equiv> l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_character_data set_val get_disconnected_nodes set_disconnected_nodes"
le_cases: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
map\<^sub>0_simps_2: fixes a :: "'d" assumes "D.obj a" shows "src\<^sub>D (\<eta>'.map\<^sub>0 a) = GF.map\<^sub>0 a"
in_set_inline2_lift_spmf1: fixes y :: "'d \<times> 'e \<times> 'f + 'g \<times> ('i \<times> 'f, 'g, 'h) rpv \<times> ('j \<times> 'e, 'k, 'i) rpv \<times> ('d, 'l, 'j) rpv"   and p :: "'f \<Rightarrow> 'k \<Rightarrow> ('i \<times> 'f) spmf"   and callee :: "'e \<Rightarrow> 'l \<Rightarrow> ('j \<times> 'e, 'k, 'i) gpv"   and gpv :: "('d, 'l, 'j) gpv"   and s :: "'e"   and s' :: "'f" assumes "y \<in> set_spmf (inline2 (\<lambda>s x. lift_spmf (p s x)) callee gpv s s')" shows "\<exists>r s s'. y = Inl (r, s, s')"
utp_theory_mono_def: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel" shows "utp_theory_mono hcond \<equiv> utp_theory hcond \<and> utp_theory_mono_axioms hcond"
rbt_lookup_rbt_unionwk: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t1 :: "('c, 'd) rbt"   and t2 :: "('c, 'd) rbt"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and k :: "'c" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t1"   and "ord.rbt_sorted less t2" shows "ord.rbt_lookup less (ord.rbt_union_with_key less f t1 t2) k = (case ord.rbt_lookup less t1 k of None \<Rightarrow> ord.rbt_lookup less t2 k | Some v \<Rightarrow> case ord.rbt_lookup less t2 k of None \<Rightarrow> Some v | Some w \<Rightarrow> Some (f k v w))"
Vagree_def: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<nu>' :: "(real, 'a) vec \<times> (real, 'a) vec"   and V :: "('a + 'a) set" shows "Vagree \<nu> \<nu>' V \<equiv> (\<forall>i. Inl i \<in> V \<longrightarrow> fst \<nu> $ i = fst \<nu>' $ i) \<and> (\<forall>i. Inr i \<in> V \<longrightarrow> snd \<nu> $ i = snd \<nu>' $ i)"
invariant3p_sync_es_ok_lock_ok: fixes P :: "'addr J_prog" assumes "wf_J_prog P" shows "invariant3p (mredT P) {s. sync_es_ok (thr s) (shr s) \<and> lock_ok (locks s) (thr s)}"
tr_par_labeled_rcv_iff: fixes B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and i :: "'lbl strand_label"   and t :: "('fun, 'var) Term.term" assumes "B \<in> set (tr\<^sub>p\<^sub>c A D)" shows "((i, receive\<langle>t\<rangle>\<^sub>s\<^sub>t) \<in> set B) = ((i, receive\<langle>t\<rangle>) \<in> set A)"
card_of_mono2: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r \<le>o r'" shows "|Field r| \<le>o |Field r'|"
simps: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "char list exp list"   and a3 :: "ty list" shows "(P,a1,h,sh \<turnstile> a2 [:] a3) = ((\<exists>E. a1 = E \<and> a2 = [] \<and> a3 = []) \<or> (\<exists>E e T es Ts. a1 = E \<and> a2 = e # es \<and> a3 = T # Ts \<and> P,E,h,sh \<turnstile> e : T \<and> P,E,h,sh \<turnstile> es [:] Ts))"
timpl_closure'_timpls_trancl_subset: fixes c :: "('c set \<times> 'c set) set" shows "timpl_closure' (c\<^sup>+) \<subseteq> timpl_closure' c"
is_sublist_nth2: fixes ls :: "'a list"   and x :: "'a"   and y :: "'a" assumes "\<exists>i j. i < |ls| \<and> j < |ls| \<and> ls ! i = x \<and> ls ! j = y \<and> Suc i = j" shows "is_sublist [x, y] ls"
assms_2: shows "\<R>\<^sub>\<circ> r = B"
lower_one_diff_uppert: fixes i :: "nat"   and k :: "nat"   and B :: "'a mat" assumes "i < n"   and "local.lower_one k B i k" shows "local.diff_ev B i k \<and> local.uppert B i k"
R1_inter_dist: shows "\<bar>spmf (R1 msgs False \<bind> D) True - spmf (local.inter msgs \<bind> D) True\<bar> = local.ddh.advantage (R1_inter_adversary D msgs)"
count_remove1: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat" assumes "p partitions n"   and "0 < p k" shows "sum (p(k := p k - 1)) {..n - k} = sum p {..n} - 1"
has_derivative_transform_within_open: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and t :: "'a set"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within t)"   and "open s"   and "x \<in> s"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "(g has_derivative f') (at x within t)"
circ_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<circ> = (x\<^sup>\<circ> * y)\<^sup>\<circ> * x\<^sup>\<circ>"
sup_idem: fixes x :: "'k multiset" shows "x \<union># x = x"
process_rule_4: fixes xs :: "'c list"   and X :: "'c set"   and P :: "'c process"   and x :: "'c" assumes "(xs, X) \<in> failures P" shows "(xs @ [x], {}) \<in> failures P \<or> (xs, insert x X) \<in> failures P"
silent_moves_call_edge: fixes S :: "'node SDG_node set"   and m :: "'node"   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 rs :: "'edge list"   and cs :: "'edge list" assumes "S,kind \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "valid_node m"   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 "\<forall>i<length rs. rs ! i \<in> get_return_edges (cs ! i)"   and "ms = targetnodes rs"   and "valid_return_list rs m"   and "length rs = length cs" obtains as' :: "'edge list" and a :: "'edge" and as'' :: "'edge list" where "as = as' @ a # as''" and "\<exists>Q r p fs. kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" and "call_of_return_node (hd ms') (sourcenode a)" and "targetnode a -as''\<rightarrow>\<^bsub>sl\<^esub>* m'"   | "ms' = ms"
map_spmf_cong: fixes p :: "'a spmf"   and q :: "'a spmf"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "map_spmf f p = map_spmf g q"
finsertCI: fixes a :: "'c"   and B :: "'c fset"   and b :: "'c" assumes "a |\<notin>| B \<Longrightarrow> a = b" shows "a |\<in>| finsert b B"
homf_preserves_arrows: fixes A :: "'o" shows "Hom(A,_) \<^bsub>\<a>\<^esub> \<in> Ar \<rightarrow> Ar\<^bsub>Set\<^esub>"
pairwise_Suc: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream"   and i :: "nat" assumes "PTA_Reachability.pairwise P xs" shows "P (xs !! i) (xs !! Suc i)"
ex_un_lam_2: fixes b :: "'a"   and c :: "'a" assumes "ide b"   and "ide c" shows "\<guillemotleft>eval b c : local.prod (exp b c) b \<rightarrow> c\<guillemotright>"
S_empty: fixes r :: "'a" shows "Prim.S r {} = {r}"
product_coneI: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<pi> :: "'a \<Rightarrow> 'b" assumes "discrete_diagram J C D"   and "limit_cone J C D a \<pi>" shows "product_cone J C D a \<pi>"
set_map: shows "set (map ((\<otimes>) a) p @ replicate n \<zero>) \<subseteq> carrier R"
Diff_Compl: fixes A :: "'g set"   and B :: "'g set" shows "A - - B = A \<inter> B"
prems_1: shows "f' a = foldr f (x # xs) (f' b)"
shallow_model_def: fixes Y :: "nat"   and Z :: "nat"   and M :: "nat"   and N :: "nat" shows "shallow_model Y Z M N = Conv (Y, Z) (shallow_model' Z M N)"
assms_2: assumes "b"   and "b'" shows "S1 \<le> \<Down> R S1'"
cases: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.siso tval aval indis a" obtains c :: "('b, 'c) com" where "a = c" and "\<And>s c' s'. PL.transC tval aval (c, s) (c', s') \<Longrightarrow> PL_Indis.siso tval aval indis c'" and "\<And>s t c' s'. \<lbrakk>(s, t) \<in> indis; PL.transC tval aval (c, s) (c', s')\<rbrakk> \<Longrightarrow> \<exists>t'. (s', t') \<in> indis \<and> PL.transC tval aval (c, t) (c', t')" and "\<And>s t s'. \<lbrakk>(s, t) \<in> indis; PL.transT tval aval (c, s) s'\<rbrakk> \<Longrightarrow> \<exists>t'. (s', t') \<in> indis \<and> PL.transT tval aval (c, t) t'"
refl_trans_closure_of_rel_impl_steps_closure: fixes P :: "'b"   and Q :: "'b"   and Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "(P, Q) \<in> Rel\<^sup>*" shows "P \<R>\<mapsto><Rel,Cal> Q"
ext_irrefl_trans_strong_axioms: shows "ext_irrefl_trans_strong ext"
WT_resourceI: fixes \<I> :: "('h, 'i) \<I>"   and res :: "('h, 'i) resource" assumes "\<And>q r res'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, res') \<in> set_spmf (run_resource res q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> \<I> \<turnstile>res res' \<surd>" shows "\<I> \<turnstile>res res \<surd>"
lfinite_ltakeWhile: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a llist" shows "lfinite (ltakeWhile P xs) = (lfinite xs \<or> (\<exists>x\<in>lset xs. \<not> P x))"
is_weak_ranking_nonempty: fixes xs :: "'b set list" assumes "is_weak_ranking xs" shows "{} \<notin> set xs"
card_equiv_class_restricted_same_size: fixes A :: "'c set"   and R :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool"   and k :: "nat" assumes "equiv A R"   and "P respects R"   and "\<And>F. F \<in> {x \<in> A. P x} // R \<Longrightarrow> card F = k" shows "card {x \<in> A. P x} = k * card ({x \<in> A. P x} // R)"
smult_power: fixes a :: "'a"   and p :: "'a poly"   and n :: "nat" shows "smult a p ^ n = smult (a ^ n) (p ^ n)"
ends_in_int: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "a \<in> MinkowskiBetweenness.interval \<P> betw a b \<and> b \<in> MinkowskiBetweenness.interval \<P> betw a b"
inverse_unique: fixes u :: "('a set \<times> 'b) set"   and v' :: "('a set \<times> 'b) set"   and v :: "('a set \<times> 'b) set" assumes "stx.add_rel u v' = stx.class_of X (zero_str\<^sub>X X)"   and "stx.add_rel v u = stx.class_of X (zero_str\<^sub>X X)"   and "u \<in> stx.carrier_direct_lim"   and "v \<in> stx.carrier_direct_lim"   and "v' \<in> stx.carrier_direct_lim" shows "v = v'"
nsqn\<^sub>r_invalidate: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> kD rt"   and "dip \<in> dom dests" shows "nsqn\<^sub>r (the (invalidate rt dests dip)) = the (dests dip) - 1"
embed_bool_nneg: fixes P :: "'a \<Rightarrow> bool" shows "nneg \<guillemotleft> P \<guillemotright>"
topspace_subtopology_subset: fixes S :: "'d set"   and X :: "'d topology" assumes "S \<subseteq> topspace X" shows "topspace (subtopology X S) = S"
pstn_def: fixes sctn :: "'c sctn" shows "pstn sctn = (case sctn of Sctn x1 x2 \<Rightarrow> x2)"
select_convs_3: fixes Leg0 :: "'c"   and Leg1 :: "'c"   and more :: "'b" shows "span_data.more \<lparr>Leg0 = Leg0, Leg1 = Leg1, \<dots> = more\<rparr> = more"
repSubobj_isSubobj: fixes 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 Xs :: "char list list"   and X :: "char list"   and Y :: "char list" assumes "is_subobj P (C, Xs @ [X])"   and "P \<turnstile> X \<prec>\<^sub>R Y" shows "is_subobj P (C, Xs @ [X, Y])"
borel_set_induct: fixes A :: "real set"   and P :: "real set \<Rightarrow> bool" assumes "A \<in> sets borel"   and "P {}"   and "\<And>a b. a \<le> b \<Longrightarrow> P {a..b}"   and "\<And>A. \<lbrakk>A \<in> sets borel; P A\<rbrakk> \<Longrightarrow> P (- A)"   and "\<And>f. \<lbrakk>disjoint_family f; \<And>i. f i \<in> sets borel; \<And>i. P (f i)\<rbrakk> \<Longrightarrow> P (\<Union> (range f))" shows "P A"
pmat_extension_id: shows "st1.pmat_extension (1\<^sub>m st2.d1) = 1\<^sub>m st1.d0"
RPDi_Star: fixes j :: "nat"   and i :: "nat"   and r :: "mregex" assumes "j \<le> i" shows "RPDi j (MStar r) \<subseteq> {MStar r} \<union> MTimesL (MStar r) (\<Union>j\<le>i. RPDi j r)"
Di_def: fixes i :: "nat"   and y :: "nat list" shows "Di i y = {local.dij i j - 1 |j. j < length y \<and> local.eij i j \<le> y ! j}"
add_nonpos_nonpos: fixes a :: "'a"   and b :: "'a" assumes "a \<le> (0::'a)"   and "b \<le> (0::'a)" shows "a + b \<le> (0::'a)"
elims: fixes x :: "('form, 'rule) graph_node"   and y :: "('form, 'var) out_port fset" assumes "outPorts x = y" obtains r :: "'rule" where "x = Rule r__" and "y = outPortsRule r__"   | r :: "'form" where "x = Assumption ra__" and "y = {|Reg ra__|}"   | r :: "'form" where "x = Conclusion r" and "y = {||}"   | "x = Helper" and "y = {|Reg anyP|}"
mult_zero_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "class.mult_zero times zero \<equiv> (\<forall>a. times zero a = zero) \<and> (\<forall>a. times a zero = zero)"
corec_1: 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 "p a" shows "corec_llist p g21 q22 g221 g222 a = LNil"
mayDiverge_Par_L: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and c2 :: "('test, 'atom) com" assumes "mayDiverge c1 s" shows "mayDiverge (c1 | c2) s"
filter_base: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" obtains Vs :: "'a list" where "set Vs \<subseteq> carrier R" and "local.independent K Vs" and "Span K Vs = Span K Us"
upset_set_def: fixes X :: "'b set" shows "\<Up> X = {y. \<exists>x\<in>X. x \<le> y}"
simps_8: fixes C :: "char list"   and F :: "char list"   and D :: "char list" shows "max_vars (C\<bullet>\<^sub>sF{D}) = 0"
fls_lr_inverse_X_inv_2: fixes y :: "'a" shows "fls_right_inverse fls_X_inv y = fls_shift (- 1) (fls_const y)"
unlabel_Cons_1: fixes l :: "'b"   and a :: "'a"   and A :: "('b \<times> 'a) list" shows "unlabel ((l, a) # A) = a # unlabel A"
fmap_distinct_2: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) fmap" shows "fmupd k v m \<noteq> fmempty"
fine_min: fixes g1 :: "real \<Rightarrow> real"   and g2 :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list" assumes "fine (\<lambda>x. min (g1 x) (g2 x)) (a, b) D" shows "fine g1 (a, b) D \<and> fine g2 (a, b) D"
wsorted'_sorted_wrt_edges_less_eq: fixes s :: "(nat \<times> int \<times> nat) list" assumes "\<forall>(a, wv, b)\<in>set s. weight (\<alpha> (a, wv, b)) = wv"   and "sorted_wrt edges_less_eq s" shows "wsorted' s"
Utrans: fixes a :: "ZF"   and b :: "ZF" assumes "Elem a U"   and "Elem b a" shows "Elem b U"
dims_tensor_deep_model: fixes m :: "real mat convnet" assumes "remove_weights m = deep_model_l rs" shows "dims (tensors_from_net m $ y) = replicate (2 * N_half) (last rs)"
assoc_coherence: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "local.map \<a>\<^sub>C[f, g, h] \<cdot>\<^sub>D cmp (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (cmp (f, g) \<star>\<^sub>D local.map h) = cmp (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (local.map f \<star>\<^sub>D cmp (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[local.map f, local.map g, local.map h]"
assms: shows "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b"
llength_mono: fixes xs :: "'c llist" shows "lfp.mono_body (\<lambda>llength. case xs of LNil \<Rightarrow> 0 | LCons x xs \<Rightarrow> eSuc (llength xs))"
subset_mset_imp_subset_add_mset: fixes A :: "'b multiset"   and B :: "'b multiset"   and x :: "'b" assumes "A \<subseteq># B" shows "A \<subseteq># add_mset x B"
map_pmf_of_set_bij_betw: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B"   and "A \<noteq> {}"   and "finite A" shows "map_pmf f (pmf_of_set A) = pmf_of_set B"
hyp: shows "\<forall>i<k. (\<exists>c' s'. f (i + 1) = (Catch c' c\<^sub>2, s')) \<and> \<Gamma>\<turnstile> head (f i) \<rightarrow> head (f (i + 1))"
prime_factorization_cong: 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 x :: "'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalize x = normalize y" shows "factorial_semiring.prime_factorization zero times one normalize x = factorial_semiring.prime_factorization zero times one normalize y"
can_simps_1: 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 t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u" shows "partial_magma.arr V (bicategory.can V H \<a> \<i> src trg u t)"
set_filter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "set (List.filter P xs) = rel_ext (\<lambda>x. x \<in> set xs \<and> P x)"
order_iff_strict: fixes b :: "'a"   and a :: "'a" shows "(b \<preceq> a) = (b \<prec> a \<or> a = b)"
funion_mono: fixes A :: "'b fset"   and C :: "'b fset"   and B :: "'b fset"   and D :: "'b fset" assumes "A |\<subseteq>| C"   and "B |\<subseteq>| D" shows "A |\<union>| B |\<subseteq>| C |\<union>| D"
depth_pair_2: fixes B :: "'b trm"   and A :: "'b trm" shows "depth B < depth (SimplyTyped.Pair A B)"
sorted_wrt_generate: fixes A :: "nat"   and B :: "nat"   and m :: "nat"   and n :: "nat" shows "sorted_wrt (<\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2) (generate A B m n)"
bin_log_esimps_12: fixes x :: "int" shows "- 1 XOR x = NOT x"
g: shows "steps0 (s, l, r) A stp = (s1, l1, r1)"
map_of_constant_assignments_dom: fixes m :: "'a \<Rightarrow> 'b option"   and d :: "'b"   and vs :: "'a list" assumes "m = map_of (map (\<lambda>v. (v, d)) vs)" shows "dom m = set vs"
Bossy'_stable: fixes X :: "'a set" shows "Bossy'.stable_on ds X = (X = {d1h3, d3h1} \<or> X = {d1h1, d3h3})"
set_product: fixes xs :: "'g list"   and ys :: "'h list" shows "set (List.product xs ys) = set xs \<times> set ys"
Set_set_keys: fixes x :: "('b, unit) RBT.rbt" shows "Set x = dom (RBT.lookup x)"
bij_betw_number_partition_of: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "bij_betw (number_partition_of A B) ({f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // domain_and_range_permutation A B) {N. (\<forall>n. n \<in># N \<longrightarrow> n = 1) \<and> number_partition (card A) N \<and> size N \<le> card B}"
dehomogenize_zero: fixes x :: "'b" shows "dehomogenize x 0 = 0"
simps_4: fixes master :: "clock"   and slave :: "clock" shows "\<lbrakk> master implies not slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {\<rho>. \<forall>n. hamlet (Rep_run \<rho> n master) \<longrightarrow> \<not> hamlet (Rep_run \<rho> n slave)}"
c12_2: shows "fst (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3) = fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1''')))"
delay_gr0_converg: fixes i :: "nat"   and x :: "nat" assumes "0 < the (delay i x)" shows "\<phi> i (e_take (the (delay i x)) x) \<down>"
nonequiv_segment: fixes a :: "'a" assumes "a \<in> carrier D" shows "\<not> ord_equiv D (Iod D (segment D a))"
poly_mod_prime_type_def: fixes m :: "int" shows "poly_mod_prime_type TYPE('b) m \<equiv> poly_mod_type TYPE('b) m"