hcju/isabelleps huggingface dataset:
add_cf_def: fixes S :: "('d, 'e) Ring_scheme"   and c :: "nat \<times> (nat \<Rightarrow> 'd)"   and d :: "nat \<times> (nat \<Rightarrow> 'd)" shows "add_cf S c d = (if fst c < fst d then (fst d, \<lambda>j. if j \<le> fst c then snd c j \<plusminus>\<^bsub>S\<^esub> snd d j else snd d j) else if fst c = fst d then (fst c, \<lambda>j. snd c j \<plusminus>\<^bsub>S\<^esub> snd d j) else (fst c, \<lambda>j. if j \<le> fst d then snd c j \<plusminus>\<^bsub>S\<^esub> snd d j else snd c j))"
tarski_absolute_space_axioms: shows "tarski_absolute_space real_euclid_C real_euclid_B"
Cons_eq_map_D: fixes x :: "'g"   and xs :: "'g list"   and f :: "'h \<Rightarrow> 'g"   and ys :: "'h list" assumes "x # xs = map f ys" shows "\<exists>z zs. ys = z # zs \<and> x = f z \<and> xs = map f zs"
abs_update_map_of''': fixes l :: "('a \<times> 'b) list list"   and x :: "'a"   and k :: "'a"   and v :: "'b" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l"   and "x \<noteq> k" shows "map_of (concat (abs_update k v l)) x = map_of (concat l) x"
every_path_distance: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'" obtains x :: "nat" where "CFG.distance sourcenode targetnode valid_edge n n' x" and "x \<le> length as"
nres_relD: fixes c :: "'c nres"   and a :: "'d nres"   and R :: "('c \<times> 'd) set" assumes "(c, a) \<in> \<langle>R\<rangle>nres_rel" shows "c \<le> \<Down> R a"
conth: shows "continuous_on ({0..1} \<times> T) h"
minus_one_less_multI: fixes e :: "real"   and x :: "real" assumes "- 1 \<le> e"   and "0 < x"   and "x < 1" shows "- 1 < e * x"
assms_6: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "I (\<Gamma>, Lam [y]. e, Arg x # S)" shows "P (\<Gamma>, Lam [y]. e, Arg x # S) (\<Gamma>, e[y::=x], S)"
lseg_append: fixes l :: "'a list"   and p :: "'a node ref option"   and s :: "'a node ref"   and x :: "'a"   and q :: "'a node ref option" shows "lseg l p (Some s) * s \<mapsto>\<^sub>r Node x q \<Longrightarrow>\<^sub>A lseg (l @ [x]) p q"
d: fixes g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex" assumes "valid_path g"   and "valid_path h"   and "\<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d"   and "pathstart h = pathstart g \<and> pathfinish h = pathfinish g" shows "path_image h \<subseteq> S - {z} \<and> (\<forall>f. f holomorphic_on S - {z} \<longrightarrow> contour_integral h f = contour_integral g f)"
deg_nzero_nzero: fixes p :: "nat \<Rightarrow> 'a" assumes "deg R p \<noteq> 0" shows "p \<noteq> \<zero>\<^bsub>UP R\<^esub>"
closedin_Union: fixes S :: "'a set set"   and U :: "'a topology" assumes "finite S"   and "\<And>T. T \<in> S \<Longrightarrow> closedin U T" shows "closedin U (\<Union> S)"
iso_sheaves_of_rings_def: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" shows "iso_sheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> iso_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
accp_downwards: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "Wellfounded.accp r a"   and "r\<^sup>*\<^sup>* b a" shows "Wellfounded.accp r b"
prems_1: fixes i :: "nat" assumes "i < length (x # xs)" shows "length ((x # xs) ! i) = length ((y # ys) ! i)"
least_contains: fixes k :: "'b"   and A :: "'b set" assumes "k \<in> A" shows "least A \<in> A"
not_learn_total_V: shows "\<not> learn_total \<phi> V_constotal s'"
poly_divides_conv0: fixes q :: "'c poly"   and p :: "'c poly" assumes "degree q < degree p"   and "p \<noteq> 0" shows "(p dvd q) = (q = 0)"
lookup_union: fixes s :: "('b, 'c) RBT.rbt"   and t :: "('b, 'c) RBT.rbt" shows "RBT.lookup (RBT.union s t) = RBT.lookup s ++ RBT.lookup t"
hyps: fixes xs :: "'a list" shows "ldropn n_ (llist_of xs) = llist_of (drop n_ xs)"
valid_shift_aux_1: shows "valid_mmuaux' args cur nt (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) auxlist"
split_list: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in>. xs" shows "\<exists>ys zs. xs = ys @ x # zs"
coeff_0_reflect_poly: fixes p :: "'a poly" shows "constant_term (reflect_poly p) = lead_coeff p"
residual_network_def: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "res.residual_network f = \<lparr>edge = \<lambda>x y. edge (residual_network f) x y \<or> edge (residual_network f) y x \<and> y \<noteq> source (residual_network f), capacity = \<lambda>(x, y). if edge (residual_network f) x y then capacity (residual_network f) (x, y) - f (x, y) else if y = source (residual_network f) then 0 else f (y, x), source = source (residual_network f), sink = sink (residual_network f), \<dots> = network.more (residual_network f)\<rparr>"
dvdm_def: fixes f :: "int poly"   and g :: "int poly" shows "pl.dvdm f g = (\<exists>h. pl.eq_m g (f * h))"
MERGE2: shows "\<Gamma>1' \<or>\<^sub>A \<Gamma>2' \<Longrightarrow>\<^sub>t \<Gamma>'"
pivot_update_precond_2: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and update :: "nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "PivotUpdate eq_idx_for_lvar pivot update"   and "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "x\<^sub>i \<notin> lvars (\<T> (pivot x\<^sub>i x\<^sub>j s))"
dom_def: fixes h' :: "'b \<Rightarrow> 'b"   and b :: "'b"   and Q :: "'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "recursive h' b" shows "tailrec.dom Q h x = (\<exists>k. Q ((h ^^ k) x))"
sigma_sets_le_sets_iff: fixes x :: "'b measure"   and \<A> :: "'b set set" shows "(sigma_sets (space x) \<A> \<subseteq> sets x) = (\<A> \<subseteq> sets x)"
irrelevant_atom: fixes A :: "'b"   and F :: "'b formula"   and \<A> :: "'b \<Rightarrow> bool"   and V :: "bool" assumes "A \<notin> atoms F" shows "\<A>(A := V) \<Turnstile> F = \<A> \<Turnstile> F"
simps: fixes a :: "('a, 'b) ODE" shows "osafe a = ((\<exists>c. a = OVar c) \<or> (\<exists>\<theta> x. a = OSing x \<theta> \<and> dfree \<theta>) \<or> (\<exists>ODE1 ODE2. a = OProd ODE1 ODE2 \<and> osafe ODE1 \<and> osafe ODE2 \<and> ODE_dom ODE1 \<inter> ODE_dom ODE2 = {}))"
bounded_def: fixes A :: "(nat \<times> 'a) set" shows "bounded A = (\<exists>N. boundedBy N A)"
split_list_first: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
log: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and p :: "'d" assumes "Matching_Ternary.matches \<gamma> m Log p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m Log], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided"
1: fixes g :: "'a \<times> 'a \<Rightarrow> 'b" shows "(k # s) \<langle>star_matrix' s g\<rangle> (k # s) = star_matrix' s g"
vector_N: fixes x :: "'a" assumes "vector x" shows "N x = x \<sqinter> (1::'a)"
openin_continuous_map_preimage: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and U :: "'c set" assumes "continuous_map X Y f"   and "openin Y U" shows "openin X {x \<in> topspace X. f x \<in> U}"
synAssignSC_def: fixes x :: "'c"   and e :: "('c, 'd) Expr"   and \<iota> :: "nat" shows "synAssignSC x e \<iota> \<equiv> \<exists>d. (htchLoc \<iota> \<turnstile>\<^bsub>\<E>\<^esub> e : d \<and> d \<le> DA x) \<and> SubstClosure x e"
sets_into_space: fixes x :: "'a set" assumes "x \<in> {D \<in> events. prob (X \<inter> D) = prob X * prob D}" shows "x \<subseteq> space M"
proj_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t: fixes l :: "'a"   and A :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list" shows "proj l (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj l A)"
dominates_mid: fixes g :: "'g"   and n :: "'node"   and x :: "'node"   and m :: "'node"   and ns :: "'node list" assumes "dominates g n x"   and "dominates g x m"   and "g \<turnstile> n-ns\<rightarrow>m"   and "invar g" shows "x \<in> set ns"
step: fixes a :: "'a"   and aa :: "'b"   and b :: "'c"   and ba :: "'d"   and bs :: "'e list"   and ab :: "'a"   and ac :: "'b"   and bb :: "'c"   and bc :: "'d"   and bd :: "'e"   and ad :: "'a"   and ae :: "'b"   and be :: "'c"   and bf :: "'d" assumes "rtrancl3p r (a, (aa, b), ba) bs (ab, (ac, bb), bc)"   and "P a aa b ba bs ab ac bb bc"   and "r (ab, (ac, bb), bc) bd (ad, (ae, be), bf)" shows "P a aa b ba (bs @ [bd]) ad ae be bf"
intro: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat" assumes "wf_nfa A n" shows "aut_nfa A n"
less_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "(x < y) = (fst x < fst y \<or> fst x \<le> fst y \<and> snd x < snd y)"
matches_stateful_matcher_stateful_matcher_tagged: fixes m :: "'m match_expr" shows "matches (stateful_matcher' \<sigma>) m p = matches stateful_matcher_tagged' m (packet_tagger' \<sigma> p)"
Inl_Inr_iff: fixes a :: "V"   and b :: "V" shows "(ZFC_Cardinals.Inl a = ZFC_Cardinals.Inr b) = False"
reachable_def: fixes xs :: "'a set" shows "reachable xs \<equiv> {(x, y). y \<in> set (map (\<delta> x) \<Sigma>)}\<^sup>* `` xs"
in_inv_image: fixes x :: "'b"   and y :: "'b"   and r :: "('c \<times> 'c) set"   and f :: "'b \<Rightarrow> 'c" shows "((x, y) \<in> inv_image r f) = ((f x, f y) \<in> r)"
perp_not_col2: 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.Perp Bet Cong A B C D" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B C \<or> \<not> Tarski_neutral_dimensionless.Col Bet A B D"
that: fixes xs :: "real list"   and ys :: "real list" assumes "zs = xs @ ys"   and "xs \<in> set_of_appr XS"   and "ys \<in> set_of_appr YS" shows "thesis"
inject_2: fixes x31 :: "'e"   and x32 :: "'f"   and y31 :: "'e"   and y32 :: "'f" shows "(Yield x31 x32 = Yield y31 y32) = (x31 = y31 \<and> x32 = y32)"
m_closed: fixes x :: "'a \<Rightarrow> 'a"   and y :: "'a \<Rightarrow> 'a" assumes "x \<in> carrier (function_ring (carrier R) R)"   and "y \<in> carrier (function_ring (carrier R) R)" shows "x \<otimes>\<^bsub>function_ring (carrier R) R\<^esub> y \<in> carrier (function_ring (carrier R) R)"
con_rews: fixes C :: "C" shows "C\<cdot>C \<noteq> \<bottom>"
mindeg_def: fixes A :: "(('b \<Rightarrow>\<^sub>0 'a) \<Rightarrow>\<^sub>0 'c) set" shows "mindeg A = Min (poly_deg ` A)"
subst_var_def: fixes v :: "nat"   and lp' :: "linear_poly"   and lp :: "linear_poly" shows "subst_var v lp' lp \<equiv> lp + coeff lp v *R lp' - coeff lp v *R Abstract_Linear_Poly.Var v"
c_in_aa: shows "c \<in> segment G.face_cycle_succ a a"
pih0: shows "path_image (\<lambda>u. h (u, 0)) \<subseteq> s"
arrI: fixes f :: "'a" assumes "arr f \<and> src f = a \<and> trg f = a" shows "S.arr f"
call_0\<^sub>C_def: fixes M :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option" shows "call_0\<^sub>C M = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some (undefined, \<sigma>) else M \<sigma>)"
fp_wf_induction: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'b \<Rightarrow> 'a" assumes "f x = x"   and "mono f"   and "\<forall>w. y w \<sqsubseteq> f (Sup_less y w)" shows "\<Squnion>range y \<sqsubseteq> x"
invertible_Hermite_reduce_above: fixes n :: "nat"   and i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and j :: "'c"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "n \<le> mod_type_class.to_nat i" shows "\<exists>P. invertible P \<and> Hermite_reduce_above A n i j res = P ** A"
strip_while_idem_iff: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "(strip_while P xs = xs) = no_trailing P xs"
well_founded_axioms: shows "well_founded A (\<sqsubset>)"
absorb1: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "min a b = a"
filterlim_tendsto_add_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 "filterlim g at_top F" shows "LIM x F. f x + g x :> at_top"
free_tv_S': fixes A :: "type_scheme list"   and t :: "typ"   and S :: "nat \<Rightarrow> typ" shows "free_tv (\<lambda>n. if n \<in> free_tv A \<union> free_tv t then S n else TVar n) \<subseteq> free_tv A \<union> free_tv ($ S A) \<union> free_tv t \<union> free_tv ($ S t)"
Darii: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A Q B"   and "B I C" shows "A I C"
cases: fixes x :: "'b list \<times> 'b list" obtains ys :: "'b list" where "x = ([], ys__)"   | xs :: "'b list" where "x = (xs__, [])"   | xa :: "'b" and xs :: "'b list" and y :: "'b" and ys :: "'b list" where "x = (xa # xs, y # ys)"
acc_bacc_eq: fixes A :: "('b \<times> 'b) set" assumes "finite A" shows "Wellfounded.acc A = bacc A CARD('b)"
i_set_defs_4: shows "iMOD_set \<equiv> {[ r, mod m ] |r m. True}"
cycle_free_altD: fixes g :: "'b ugraph"   and u :: "'b"   and v :: "'b" assumes "cycle_free g"   and "(u, v) \<in> edges g" shows "(u, v) \<notin> (edges g - {(u, v), (v, u)})\<^sup>*"
fps_of_poly_def: fixes p :: "'a poly" shows "fps_of_poly p = Abs_fps (coeff p)"
choice_iff': fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) = (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))"
conjugI1: fixes u :: "'b list"   and r :: "'b list"   and v :: "'b list" assumes "u \<cdot> r = r \<cdot> v" shows "u \<sim> v"
observable_move_slice_edge: 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 a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and rs :: "'edge list"   and cs :: "'edge list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow> (ms',s')"   and "tl ms = targetnodes rs"   and "length rs = length cs"   and "\<forall>i<length cs. call_of_return_node (tl ms ! i) (sourcenode (cs ! i))" shows "slice_edge S cs a"
hyperb_ineq': fixes e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" shows "min (Gromov_product_at e x y) (Gromov_product_at e y z) \<le> Gromov_product_at e x z + deltaG TYPE('a)"
length_pos_if_in_set: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "0 < length xs"
intro: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" assumes "bounded_lattice_homomorphism phi" shows "triple phi"
red_def: shows "red \<equiv> \<lambda>P. {(x, (xb, xa), xd, xc). redp P x xb xa xd xc}"
ins_trees: fixes t :: "('b \<times> 'c) tree"   and A :: "('b \<times> 'c) set"   and x :: "'b"   and y :: "'c" assumes "t \<in> trees A"   and "(x, y) \<in> A" shows "ins x y t \<in> trees A"
sublist_trans: fixes l1 :: "'a list"   and l2 :: "'a list"   and l3 :: "'a list" assumes "subseq l1 l2"   and "subseq l2 l3" shows "subseq l1 l3"
Suc_5: shows "\<S>\<bullet>\<C> \<turnstile> LI : ts _> ts'"
inv_in_hhom: fixes \<mu> :: "'b"   and a :: "'b"   and b :: "'b" assumes "D.iso \<mu>"   and "src\<^sub>D \<mu> = a"   and "trg\<^sub>D \<mu> = b" shows "\<guillemotleft>D.inv \<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>"
sfilter_bottom_iff: fixes p :: "'a \<rightarrow> tr"   and xs :: "[:'a:]" shows "(sfilter\<cdot>p\<cdot>xs = \<bottom>) = (xs = \<bottom> \<or> (\<exists>x\<in>sset xs. p\<cdot>x = \<bottom>))"
Sigma_Un_distrib1: fixes I :: "'e set"   and J :: "'e set"   and C :: "'e \<Rightarrow> 'f set" shows "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
arctan_upper_13_def: shows "arctan_upper_13 \<equiv> \<lambda>x. x"
sqrt_differentiable: fixes f :: "'c \<Rightarrow> real"   and x :: "'c"   and S :: "'c set" assumes "f differentiable at x within S"   and "f x \<noteq> 0" shows "(\<lambda>x. sqrt (f x)) differentiable at x within S"
d1_def: fixes dims :: "nat list"   and vars :: "nat set" shows "partial_state.d1 dims vars = prod_list (partial_state.dims1 dims vars)"
list_update_same_conv: fixes i :: "nat"   and xs :: "'d list"   and x :: "'d" assumes "i < length xs" shows "(xs[i := x] = xs) = (xs ! i = x)"
child_length: fixes p :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat" shows "length (child p dir d) = length p"
a: shows "\<forall>i<length cms\<^sub>1. \<forall>x\<in>differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i. doesnt_read_or_modify (fst (cms\<^sub>1 ! i)) x \<and> doesnt_read_or_modify (fst (cms\<^sub>2 ! i)) x"
Liminf_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" shows "Liminf F f = (\<Squnion>P\<in>{P. eventually P F}. \<Sqinter> (f ` {x. P x}))"
assms_1: shows "run (case x of None \<Rightarrow> n | Some y \<Rightarrow> s y) \<sigma> \<sigma>' r"
integrableI_bounded: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "f \<in> borel_measurable M"   and "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>" shows "integrable M f"
cr_prod2_def: fixes x :: "'i"   and A :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "cr_prod2 x A = (\<lambda>b (a, c). A b c \<and> x = a)"
finite_cartesian_product: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "finite B" shows "finite (A \<times> B)"
intAg: shows "integrable M (\<lambda>x. indicat_real A x * g x)"
subtopology_eq_discrete_topology_gen: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<inter> X derived_set_of S = {}" shows "subtopology X S = discrete_topology (topspace X \<inter> S)"
zero_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<equiv> mat nr nc (\<lambda>ij. 0::'a)"
assigns_backward_hoare_r: fixes \<sigma> :: "'a \<Rightarrow> 'a"   and p :: "'a upred" shows "\<lbrace>\<sigma> \<dagger> p\<rbrace> \<langle>\<sigma>\<rangle>\<^sub>a \<lbrace>p\<rbrace>\<^sub>u"
smult_assoc1: fixes a :: "'a"   and b :: "'a"   and x :: "'e" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier N" shows "a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>N\<^esub> x = a \<odot>\<^bsub>N\<^esub> (b \<odot>\<^bsub>N\<^esub> x)"
int_decomp_ord: fixes n :: "int"   and l :: "int"   and k :: "nat" assumes "n = l * p ^ k"   and "l mod p \<noteq> 0" shows "ord_Zp ([n] \<cdot> \<one>) = int k"
fMax_fold: fixes a :: "'a"   and as :: "'a list" shows "fMax (fset_of_list (a # as)) = fold max as a"
simps_1: fixes n :: "int"   and s :: "String.literal" shows "(Num n < Str s) = True"
generate_idem_fUn: fixes f :: "'d \<Rightarrow> 'a set"   and A :: "'d" assumes "f A \<subseteq> carrier G" shows "generate G (\<Union> {generate G {x} |x. x \<in> f A}) = generate G (f A)"
outs_exception_\<I>: fixes \<I> :: "('y, 'z) \<I>" shows "outs_\<I> (exception_\<I> \<I>) = outs_\<I> \<I>"
pdec2_encode_Id: fixes m :: "nat"   and n :: "nat" shows "pdec2 (encode (recf.Id m n)) = prod_encode (m, n)"
dom_rhs: shows "\<D>\<^sub>\<circ> (dg_SemiCAT \<alpha>) = 4\<^sub>\<nat>"
l11_50_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 B' :: "'a"   and C' :: "'a"   and A' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.CongA Bet Cong B C A B' C' A'"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Cong A B A' B'" shows "Cong A C A' C' \<and> Cong B C B' C' \<and> Tarski_neutral_dimensionless.CongA Bet Cong C A B C' A' B'"
K: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<in> \<F>"   and "Y \<in> \<F>"   and "\<not> X \<subseteq> Y"   and "\<not> Y \<subseteq> X" shows "X \<inter> Y \<subseteq> K"
irreducible_mult_unit_left: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and p :: "'a" assumes "class.idom times one plus zero minus uminus"   and "dvd.dvd times a one" shows "comm_semiring_1.irreducible times one zero (times a p) = comm_semiring_1.irreducible times one zero p"
subprob_space_null_measure_iff: fixes M :: "'a measure" shows "subprob_space (null_measure M) = (space M \<noteq> {})"
deduct_zero_in_ik: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "\<langle>M; 0\<rangle> \<turnstile>\<^sub>n t" shows "t \<in> M"
finite_char_closed: fixes C :: "'a set set" assumes "finite_char C" shows "subset_closed C"
assms_2: shows "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"
bisim_can_sync_preserve2: fixes t :: "'t"   and x1 :: "'x1"   and m1 :: "'m1"   and x2 :: "'x2"   and m2 :: "'m2"   and LT :: "('l + 't + 't) set" assumes "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "t \<turnstile> \<langle>x2, m2\<rangle> LT \<wrong>2" shows "t \<turnstile> \<langle>x1, m1\<rangle> LT \<wrong>1"
table_subset: fixes n :: "nat"   and V :: "nat set"   and T :: "'a option list set"   and S :: "'a option list set" assumes "table n V T"   and "S \<subseteq> T" shows "table n V S"
vbrelation_vinsert: fixes a :: "V"   and b :: "V"   and r :: "V" shows "vbrelation (vinsert \<langle>a, b\<rangle> r) = vbrelation r"
simps_2: fixes nm_name :: "char list"   and nm_receiver_violation :: "bool"   and nm_default :: "'b"   and nm_sinvar :: "'a list_graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"   and nm_offending_flows :: "'a list_graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) list list"   and nm_node_props :: "('a, 'b) TopoS_Params \<Rightarrow> 'a \<Rightarrow> 'b"   and nm_eval :: "'a list_graph \<Rightarrow> ('a, 'b) TopoS_Params \<Rightarrow> bool"   and more :: "'c" shows "nm_receiver_violation \<lparr>nm_name = nm_name, nm_receiver_violation = nm_receiver_violation, nm_default = nm_default, nm_sinvar = nm_sinvar, nm_offending_flows = nm_offending_flows, nm_node_props = nm_node_props, nm_eval = nm_eval, \<dots> = more\<rparr> = nm_receiver_violation"
fmember_possible_steps: fixes s' :: "nat"   and t :: "transition"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" shows "((s', t) |\<in>| possible_steps e s r l i) = (((s, s'), t) \<in> {((origin, dest), t). ((origin, dest), t) \<in> fset e \<and> origin = s \<and> Label t = l \<and> length i = Arity t \<and> apply_guards (Guards t) (join_ir i r)})"