hcju/isabelleps huggingface dataset:
h: fixes x :: "'var" assumes "x \<in> domA h" shows "\<lbrakk> the (map_of h x) \<rbrakk>\<^bsub>r\<^esub> \<sqsubseteq> r x"
prod_fun_closed: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "\<And>x. f x \<in> A"   and "\<And>x. g x \<in> A" shows "prod_fun f g x \<in> A"
insert_DiffM: fixes x :: "'c"   and M :: "'c multiset" assumes "x \<in># M" shows "add_mset x (remove1_mset x M) = M"
Int_ss_UNIV: shows "\<guillemotleft>\<int>\<guillemotright> \<subseteq> U"
preserves_seq: fixes g :: "'a1"   and f :: "'a1" assumes "A1.seq g f" shows "B1.seq (F1 g) (F1 f)"
quotient_map_compose_eq: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "quotient_map X X' f" shows "quotient_map X X'' (g \<circ> f) = quotient_map X' X'' g"
ide_in_hom: fixes a :: "'a" shows "Right.ide a = Right.in_hom a a a"
collapse_6: fixes call_expr :: "'a call_expr" assumes "is_Operation call_expr" shows "Operation (un_Operation1 call_expr) (call_expr.args call_expr) = call_expr"
addr_loc_type_fun: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and a :: "'b"   and al :: "addr_loc"   and T :: "ty"   and T' :: "ty" assumes "heap_base.addr_loc_type typeof_addr P h a al T"   and "heap_base.addr_loc_type typeof_addr P h a al T'" shows "T = T'"
map_poly_hom_smult: fixes c :: "'a mod_ring poly"   and p :: "'a mod_ring poly poly" shows "map_poly (map_poly rebase) (Polynomial.smult c p) = Polynomial.smult (map_poly rebase c) (map_poly (map_poly rebase) p)"
set_comp_associative: fixes f :: "'b set_arrow"   and U :: "'b set"   and g :: "'b set_arrow"   and h :: "'b set_arrow" assumes "f \<in> Ar\<^bsub>set_cat U\<^esub>"   and "g \<in> Ar\<^bsub>set_cat U\<^esub>"   and "h \<in> Ar\<^bsub>set_cat U\<^esub>"   and "Cod\<^bsub>set_cat U\<^esub> h = Dom\<^bsub>set_cat U\<^esub> g"   and "Cod\<^bsub>set_cat U\<^esub> g = Dom\<^bsub>set_cat U\<^esub> f" shows "f \<bullet>\<^bsub>set_cat U\<^esub> (g \<bullet>\<^bsub>set_cat U\<^esub> h) = f \<bullet>\<^bsub>set_cat U\<^esub> g \<bullet>\<^bsub>set_cat U\<^esub> h"
move_keeps_lanes: fixes v :: "view"   and ts :: "traffic"   and ts' :: "traffic" shows "lan v = lan (move ts ts' v)"
compat_add_opp: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p\<^sub>1 :: "'c point"   and p\<^sub>2 :: "'c point" assumes "ell_field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2"   and "ell_field.add R a p\<^sub>1 p\<^sub>2 = ell_field.add R a p\<^sub>1 (ell_field.opp R p\<^sub>2)"   and "p\<^sub>1 \<noteq> ell_field.opp R p\<^sub>1" shows "p\<^sub>2 = ell_field.opp R p\<^sub>2"
a2: shows "f \<in> {f \<in> space M \<rightarrow> space P. \<forall>y\<in>sets P. f -` y \<inter> space M \<in> sets M}"
nat_pow_Suc2: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier (UP R)" shows "[Suc n] \<cdot>\<^bsub>UP R\<^esub> x = x \<oplus>\<^bsub>UP R\<^esub> [n] \<cdot>\<^bsub>UP R\<^esub> x"
opp_chambers_bij: shows "bij_betw ((`) f) (folding_g.\<C> - f \<turnstile> folding_g.\<C>) (f \<turnstile> folding_g.\<C>)"
W_consts_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n true\<^sub>n W\<^sub>n \<phi>"
f_non_negative: fixes e :: "nat \<times> nat" shows "(0::'capacity) \<le> f' e"
2: fixes i :: "nat" assumes "i \<in> s" shows "least s \<le> i"
simps_1: fixes r :: "'a set"   and f :: "'a \<Rightarrow> 'a" shows "atomics (AnnBasic r f) = {(r, Basic f)}"
lass_val_of_spec: fixes e :: "'c exp"   and a :: "'c \<times> val" assumes "lass_val_of e = \<lfloor>a\<rfloor>" shows "e = fst a:=Val (snd a)"
assms_2: shows "FSwSbImorph h hA MOD MOD'"
simps_19: fixes C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "'c exp" shows "ss_exp (INIT C (Cs,b) \<leftarrow> e) = INIT C (Cs,b) \<leftarrow> e"
inj_on_imageI2: fixes f' :: "'d \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd"   and A :: "'b set" assumes "inj_on (f' \<circ> f) A" shows "inj_on f A"
sum_binomial_times_bernoulli: fixes n :: "nat" shows "(\<Sum>k\<le>n. real (Suc n choose k) * bernoulli k) = (if n = 0 then 1 else 0)"
less_eq_option_None_is_None: fixes x :: "'a option" assumes "x \<le> None" shows "x = None"
hyps_1: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "n = sequent_cost \<Gamma> \<Delta>"   and "\<Turnstile> \<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta> \<down> sequent_cost \<Gamma> \<Delta>"
eval_def: fixes envVal :: "('b, 'c, 'd) BEState \<Rightarrow> 'e \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> 'f \<times> 'd option" shows "eval envVal envObs \<equiv> \<lambda>(cec, X). evalS (spr_evalProp envVal) (spr_knowledge envObs cec) (spr_commonKnowledge envObs cec) X"
valid_def: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "(G \<Turnstile> c : A ) = ( \<Turnstile> G  \<longrightarrow>  \<Turnstile> c : A )"
elims_1: fixes x :: "(char list, char list, v) namespace"   and y :: "bool" assumes "is_cupcake_ns x = y" obtains v0 :: "(char list \<times> v) list" where "x = Bind v0 []" and "y = list_all (is_cupcake_value \<circ> snd) v0"   | v :: "(char list \<times> v) list" and vb :: "char list \<times> (char list, char list, v) namespace" and vc :: "(char list \<times> (char list, char list, v) namespace) list" where "x = Bind v (vb # vc)" and "\<not> y"
fBallE: fixes A :: "'j fset"   and P :: "'j \<Rightarrow> bool"   and x :: "'j" assumes "fBall A P" obtains "P x"   | "x |\<notin>| A"
ncols_diagonal_to_Smith: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "ncols (diagonal_to_Smith A bezout) = ncols A"
prems_2: fixes s :: "'a set" assumes "s \<in> insert x F" shows "continuous_on s f"
ltln_of_dnf_prop_equiv: fixes \<phi> :: "'a ltln" shows "ltln_of_dnf (min_dnf \<phi>) \<sim>\<^sub>P \<phi>"
not_zero_jA: shows "\<not> is_zero_row_upt_k j k A"
ex_new_if_finite: fixes A :: "'b set" assumes "infinite UNIV"   and "finite A" shows "\<exists>a. a \<notin> A"
absorb_iff2: fixes b :: "'g"   and a :: "'g" shows "(b \<le> a) = (min a b = b)"
del_UnEdge_non_adj: fixes v :: "'v"   and w :: "'w"   and u :: "'v" assumes "(v, w, u) \<in> E" shows "\<not> valid_unMultigraph.adjacent (del_unEdge v w u G) v u"
bfs_def: fixes src :: "nat"   and dst :: "nat" shows "bfs src dst \<equiv> if src = dst then return (Some []) else pre_bfs src dst \<bind> (\<lambda>br. case br of None \<Rightarrow> return None | Some (d, PRED) \<Rightarrow> extract_rpath src dst PRED \<bind> (\<lambda>p. return (Some p)))"
dg_prod_2_ArrI: fixes g :: "V"   and f :: "V" assumes "g \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>"
supchamberD_2: fixes X :: "'b set set"   and v :: "'b" assumes "ChamberComplex X"   and "v \<in> \<Union> X" shows "v \<in> (SOME C. SimplicialComplex.maxsimp X C \<and> v \<in> C)"
nths_def: fixes xs :: "'g list"   and A :: "nat set" shows "nths xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<length xs]))"
reads_restrict1: fixes \<sigma> :: "'var \<Rightarrow> 'a"   and n :: "'n"   and \<sigma>' :: "'var \<Rightarrow> 'a" assumes "\<sigma> \<restriction> local.reads n = \<sigma>' \<restriction> local.reads n" shows "\<forall>x\<in>local.reads n. \<sigma> x = \<sigma>' x"
deriv_mult: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable at z"   and "g field_differentiable at z" shows "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
cooper_def: fixes p :: "fm" shows "cooper p \<equiv> let (q, B, d) = unit p; js = [1..d]; mq = simpfm (minusinf q); md = evaldjf (\<lambda>j. simpfm (subst0 (C j) mq)) js in if md = T then T else let qd = evaldjf (\<lambda>t. simpfm (subst0 t q)) (remdups (map (\<lambda>(b, j). simpnum (Add b (C j))) (concat (map (\<lambda>b. map (Pair b) js) B)))) in decr (MIR.disj md qd)"
msg_sender_def: fixes m :: "msg" shows "msg_sender m \<equiv> case m of Rreq x xa xb xc xd xe ipc xf \<Rightarrow> ipc | Rrep x xa xb xc ipc \<Rightarrow> ipc | Rerr x ipc \<Rightarrow> ipc | Pkt x xa ipc \<Rightarrow> ipc"
sub_addI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P Q s. \<lbrakk>sound P; sound Q\<rbrakk> \<Longrightarrow> t P s + t Q s \<le> t (\<lambda>s'. P s' + Q s') s" shows "sub_add t"
assms_3: shows "ops \<in> set (\<Phi>\<inverse> \<Pi> \<A> t)"
cSup_le_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'a set"   and a :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "S \<noteq> {}"   and "preorder.bdd_above less_eq S" shows "less_eq (Sup S) a = (\<forall>x\<in>S. less_eq x a)"
lex_ord_raw_valE: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ys :: "('a \<times> 'f) list \<times> 'o"   and ko :: "'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> order option"   and aux :: "order option" assumes "oalist_inv xs"   and "oalist_inv ys"   and "lex_ord_raw ko f xs ys = aux"   and "aux \<noteq> Some Eq" obtains k :: "'a" where "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)" and "aux = f k (lookup_raw xs k) (lookup_raw ys k)" and "\<And>k'. \<lbrakk>k' \<in> fst ` set (fst xs) \<union> fst ` set (fst ys); lt ko k' k\<rbrakk> \<Longrightarrow> f k' (lookup_raw xs k') (lookup_raw ys k') = Some Eq"
cases: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and a :: "('h, 'i, 'j, 'k) converter" assumes "pfinite_converter \<I> \<I>' a" obtains conv :: "('h, 'i, 'j, 'k) converter" where "a = conv" and "\<And>a. a \<in> outs_\<I> \<I> \<Longrightarrow> pfinite_gpv \<I>' (run_converter conv a)" and "\<And>a b conv'. \<lbrakk>a \<in> outs_\<I> \<I>; (b, conv') \<in> results_gpv \<I>' (run_converter conv a)\<rbrakk> \<Longrightarrow> pfinite_converter \<I> \<I>' conv'"
assn_times_comm: fixes P :: "assn"   and Q :: "assn" shows "P * Q = Q * P"
sd_exists: fixes A :: "'a set"   and Is :: "'b set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set" assumes "has 3 A"   and "finite Is"   and "has 2 Is"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain" shows "\<exists>j u v. hasw [u, v] A \<and> semidecisive swf A Is {j} u v"
finite_imp_card_less: fixes A :: "'a set" assumes "finite A" shows "|A| <o natLeq"
refl_onD1: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "refl_on A r"   and "(x, y) \<in> r" shows "x \<in> A"
simps: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and t :: "taylor_model" shows "tm_norm' prec ord I a t = tm_round_floats_of_normed prec I a (tm_lower_order_of_normed prec ord I a t)"
sg2: shows "x t = [] \<or> x t = [sc_ack]"
continuous_disconnected_range_constant: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and t :: "'b set" assumes "connected S"   and "continuous_on S f"   and "f ` S \<subseteq> t"   and "\<And>y. y \<in> t \<Longrightarrow> connected_component_set t y = {y}" shows "f constant_on S"
R1_of_list: fixes x :: "'c list" shows "list_R1.list_R1_to_list (list_R1.list_to_list_R1 x) = x"
map_filter_snd_map_filter: fixes P :: "'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "('b \<times> 'c) list" shows "List.map_filter (\<lambda>(a, b). if P b then None else Some (f a b)) xs = map (\<lambda>(a, b). f a b) (filter (\<lambda>x. \<not> P (snd x)) xs)"
DmdPrime: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> \<diamond>P$ \<longrightarrow> \<diamond>P"
hta_is_empty_witness_def: fixes H :: "('b, 'a, 'c) hashedTa_scheme" shows "hta_is_empty_witness H \<equiv> let rqrm = build_rqrm (hta_\<delta> H); (Q, uu_, uu_, qwit) = while brec_cond (brec_step rqrm (hta_Qi H)) (brec_initial (hta_Qi H) (hta_\<delta> H)) in case qwit of None \<Rightarrow> None | Some q \<Rightarrow> hm.lookup q Q"
subspace_direct_decomp: fixes A :: "'b set"   and ss :: "'b set list" assumes "direct_decomp A ss"   and "\<And>s. s \<in> set ss \<Longrightarrow> local.subspace s" shows "local.subspace A"
c2: shows "dim_row (u \<Otimes> v) = 2 ^ (m + n)"
min_set_min_product_1: fixes A :: "'b fset set"   and B :: "'b fset set" shows "min_set A \<otimes>\<^sub>m B = A \<otimes>\<^sub>m B"
Cons_4: fixes i :: "nat" assumes "i < length (x # xs)" shows "length ((y # ys) ! i) = length ((x # xs) ! i)"
the_Some_image: fixes A :: "'b set" shows "the ` Some ` A = A"
trms\<^sub>s\<^sub>s\<^sub>t_subst: fixes S :: "('e, 'd) stateful_strand_step list"   and \<theta> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "bvars\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}" shows "trms\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>) = trms\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
right_adjoint_is_ide: fixes f :: "'b" assumes "D.is_right_adjoint f" shows "D.ide f"
left_lang_def: fixes q :: "hf" shows "Accessible.left_lang q \<equiv> {u. Accessible.nextl (init Accessible_dfa) u = q}"
assms_5: shows "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r"
incseq_sumI: fixes f :: "nat \<Rightarrow> 'b" assumes "\<And>i. (0::'b) \<le> f i" shows "incseq (\<lambda>i. sum f {..<i})"
scaleC: fixes r :: "complex"   and b :: "'a" shows "f (r *\<^sub>C b) = r *\<^sub>C f b"
l2_inv8_trans: shows "{l2_inv8 \<inter> l2_inv1 \<inter> l2_inv3} TS.trans l2 {> l2_inv8}"
is_valid_problem_sas_plus_then_6: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>dom (\<Psi>\<^sub>G\<^sub>+). the ((\<Psi>\<^sub>G\<^sub>+) v) \<in> \<R>\<^sub>+ \<Psi> v"
domain: shows "Domainp pcr_real_int = (\<lambda>x. \<exists>y. rel_prod (=) (=) x y \<and> fst y \<le> snd y)"
belong_ar_coset2: fixes I :: "'a set"   and a :: "'a"   and x :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "x \<in> a \<uplus>\<^bsub>R\<^esub> I" shows "x \<plusminus> -\<^sub>a a \<in> I"
distinct_105: fixes x91 :: "rat"   and x92 :: "rat"   and x101 :: "nat"   and x102 :: "msg" shows "Auth_Data x91 x92 \<noteq> Crypt x101 x102"
initial: fixes p :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and V :: "(nat \<times> 'c) set"   and w :: "'d stream" assumes "p \<in> ginitial A \<inter> V" shows "p \<in> gnodes A w V"
simps_4: fixes a :: "tm" shows "bound0 (Le a) = tmbound0 a"
assms_4: shows "ps (S t (i + 1)) p \<noteq> None"
offending_notevalD: fixes F :: "('a \<times> 'a) set"   and G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" assumes "F \<in> set_offending_flows G nP" shows "\<not> sinvar G nP"
cexp_def: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('b \<times> 'b) set" shows "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
le_less_imp_div: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k * m \<le> n"   and "n < Suc k * m" shows "n div m = k"
E'_def: fixes x :: "'a"   and y :: "'a" shows "x \<rightarrow> y \<equiv> E x y \<and> reachable x \<and> reachable y"
assms: shows "cs\<^bsup>\<pi>\<^esup> i = [x]"
preserves_reflects_arr: fixes f :: "'c" shows "C.arr (src\<^sub>C f) = C.arr f"
partial_span: fixes i :: "nat" assumes "i \<le> length fs" shows "local.span (gso ` {0..<i}) = local.span (set (take i fs))"
wset_find_path'_refine: fixes E :: "('b \<times> 'b) set"   and U0 :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "wset_find_path' E U0 P \<le> \<Down> Id (wset_find_path E U0 P)"
simps_10: fixes bbs :: "bool list"   and bs :: "int list"   and i :: "int"   and b :: "Cooper.num" shows "Ifm bbs bs (NDvd i b) = (\<not> i dvd Inum bs b)"
rsd_winners_def: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and alts :: "'a set"   and agents :: "'b list" shows "rsd_winners R alts agents = foldr (\<lambda>i. Max_wrt_among (R i)) agents alts"
xt1_4: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "order_ow U le ls"   and "le b a"   and "b = c" shows "le c a"
INF_top_conv_1: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and B :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Inf (B ` A) = top) = (\<forall>x\<in>A. B x = top)"
simps_8: fixes v :: "nat \<Rightarrow> 'a"   and l1 :: "linear_poly"   and l2 :: "linear_poly" shows "v \<Turnstile>\<^sub>c LEQPP l1 l2 = (l1 \<lbrace> v \<rbrace> \<le> l2 \<lbrace> v \<rbrace>)"
simps_4: fixes A :: "hfm" shows "closed_hfm (HNeg A) = closed_hfm A"
mset_tree_heap_of_A: fixes xs :: "'b list" shows "mset_tree (heap_of_A xs) = mset xs"
in_multiset_in_set: fixes x :: "'d"   and xs :: "'d list" shows "(x \<in># mset xs) = (x \<in> set xs)"
partitions_eq_implies: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and x' :: "'c" assumes "f \<in> A \<rightarrow>\<^sub>E B"   and "f' \<in> A \<rightarrow>\<^sub>E B"   and "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = (\<lambda>b. {x \<in> A. f' x = b}) ` B - {{}}"   and "x \<in> A"   and "x' \<in> A"   and "f x = f x'" shows "f' x = f' x'"
le_fininfI: fixes v\<^sub>2 :: "'action list"   and w\<^sub>2 :: "'action stream"   and w\<^sub>1 :: "'action list" assumes "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
fds_deriv_uminus: fixes f :: "'a fds" shows "fds_deriv (- f) = - fds_deriv f"
WTrts_Val: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and vs :: "val list"   and Ts :: "ty list" shows "(P,E,h,sh \<turnstile> map Val vs [:] Ts) = (map typeof\<^bsub>h\<^esub> vs = map Some Ts)"
iterator_to_list_def: fixes cnt :: "'a \<Rightarrow> bool"   and nxt :: "'a \<Rightarrow> 'a" shows "iterator_defs.iterator_to_list cnt nxt \<equiv> wfrec {(nxt x, x) |x. cnt x} (\<lambda>rec x. if cnt x then x # rec (nxt x) else [])"
g: shows "\<guillemotleft>g : L a' \<Rightarrow> L a\<guillemotright>"
tri_caseE: fixes P :: "bool"   and Q :: "bool" obtains "\<not> P" and "\<not> Q"   | "P"   | "\<not> P" and "Q"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "Right.in_hom f a b" obtains "Right.arr f" and "Right.dom f = a" and "Right.cod f = b"
product_in_first_summand: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "CEM"   and "\<not> O y z" shows "P ((x \<oplus> y) \<otimes> (x \<oplus> z)) x"
z2'_def: shows "z2' = of_nat (fst z2) + \<i>\<^sub>\<int> * of_nat (snd z2)"
subst_atms_comp_subst: fixes AA :: "'a set"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "AA \<cdot>as (\<tau> \<odot> \<sigma>) = AA \<cdot>as \<tau> \<cdot>as \<sigma>"
division_wellfounded: shows "wf {(x, y). x \<in> carrier G \<and> y \<in> carrier G \<and> properfactor G x y}"
norm_eff_def: fixes i :: "instr"   and P :: "'b prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list" shows "norm_eff i P pc \<tau> = map (\<lambda>pc'. (pc', \<lfloor>eff\<^sub>i (i, P, \<tau>)\<rfloor>)) (succs i \<tau> pc)"
compactin_subset_topspace: fixes X :: "'b topology"   and S :: "'b set" assumes "compactin X S" shows "S \<subseteq> topspace X"
integrable_const_bound: fixes f :: "'a \<Rightarrow> 'b"   and B :: "real" assumes "AE x in M. norm (f x) \<le> B"   and "f \<in> borel_measurable M" shows "integrable M f"
integrable': shows "set_integrable lborel D (\<lambda>(x, y). P x * (x * y * z) ^ n * (1 - y) ^ n / (1 - (1 - x * y) * z) ^ (n + 1))"
ptensor_vec_dim: fixes v1 :: "'a vec"   and v2 :: "'a vec" shows "dim_vec (st.ptensor_vec v1 v2) = st.d0"
map_upds_apply_nontin: fixes x :: "'c"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'd option"   and ys :: "'d list" assumes "x \<notin> set xs" shows "(f(xs [\<mapsto>] ys)) x = f x"
e2: fixes \<G> :: "'N set set" assumes "\<G> \<subseteq> fbx ` \<D>"   and "finite \<G>" shows "sum (Sigma_Algebra.measure lebesgue) \<G> \<le> e / 2"
n_ne: shows "(deriv ^^ n) f \<xi> \<noteq> 0"
map_fun_parametric: fixes A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and D :: "'i \<Rightarrow> 'k \<Rightarrow> bool" shows "rel_fun (rel_fun A B) (rel_fun (rel_fun C D) (rel_fun (rel_fun B C) (rel_fun A D))) map_fun map_fun"
infinite_set_mset_mset_set: fixes A :: "'b set" assumes "infinite A" shows "set_mset (mset_set A) = {}"
mat_mult_assoc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and nr :: "nat"   and n1 :: "nat"   and m1 :: "'c list list"   and n2 :: "nat"   and m2 :: "'c list list"   and nc :: "nat"   and m3 :: "'c list list" assumes "class.semiring_0 plus zero times"   and "mat nr n1 m1"   and "mat n1 n2 m2"   and "mat n2 nc m3" shows "mat_multI zero plus times nr (mat_multI zero plus times nr m1 m2) m3 = mat_multI zero plus times nr m1 (mat_multI zero plus times n1 m2 m3)"
dvdnumcoeff_aux2: fixes t :: "Ferrack.num" assumes "1 < numgcd t" shows "dvdnumcoeff t (numgcd t) \<and> 0 < numgcd t"
l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs"
assms: shows "\<bar>Re z\<bar> < pi / 2 \<or> \<bar>Re z\<bar> = pi / 2 \<and> Im z = 0"
enum_strict_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(s.enum i < s.enum j) = (i < j)"
inner_axis: fixes x :: "('a, 'b) vec"   and i :: "'b"   and y :: "'a" shows "inner x (axis i y) = inner (x $ i) y"
applyConflict_def: fixes state :: "State" shows "applyConflict state = (let conflictClause = getF state ! getConflictClause state in setConflictAnalysisClause conflictClause state)"
bij_f: shows "bij_betw f (carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})) (carrier_vec (card P))"
uniqueness_algorithm_division_of_int_poly: fixes g :: "int poly"   and F :: "'a mod_ring poly"   and f :: "int poly"   and G :: "'a mod_ring poly"   and Q :: "'a mod_ring poly"   and R :: "'a mod_ring poly"   and q :: "int poly"   and r :: "int poly" assumes "Mp g \<noteq> 0"   and "F = of_int_poly f"   and "G = of_int_poly g"   and "F = G * Q + R"   and "R = 0 \<or> degree R < degree G"   and "Mp f = Mp g * q + r"   and "r = 0 \<or> degree r < degree_m g" shows "Q = of_int_poly q \<and> R = of_int_poly r"
AllowedActs_JN: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program" shows "AllowedActs (JOIN I F) = (\<Inter>i\<in>I. AllowedActs (F i))"
norm_divide: fixes a :: "'b"   and b :: "'b" shows "norm (a / b) = norm a / norm b"