hcju/isabelleps huggingface dataset:
closed_orbit_in_domain: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.closed_orbit f X x" shows "x \<in> X"
strict_coboundedI1: fixes c :: "'c DBMEntry"   and a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "c \<prec> a" shows "c \<prec> linordered_monoid.max a b"
CANDs_imply_SNa_1: fixes a :: "coname"   and M :: "trm"   and B :: "ty" assumes "<a>:M \<in> \<parallel><B>\<parallel>" shows "SNa M"
consequence: shows "P \<subseteq> {s. \<exists>Z. s \<in> I Z \<and> I Z \<inter> - b \<subseteq> Q}"
Fvars_tm_simps_1: shows "Fvars Zero = {}"
less_Suc_eq_0_disj: fixes m :: "nat"   and n :: "nat" shows "(m < Suc n) = (m = 0 \<or> (\<exists>j. m = Suc j \<and> j < n))"
PChain: shows "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"
assms_2: shows "2 * C \<le> dist (c a) (c b)"
is_monomial_monomial: fixes p :: "'d \<Rightarrow>\<^sub>0 'e" assumes "is_monomial p" obtains c :: "'e" and t :: "'d" where "c \<noteq> (0::'e)" and "p = monomial c t"
Abs_dense_inverse: fixes y :: "'c" assumes "y \<in> dense_elements" shows "Rep_dense (Abs_dense y) = y"
dirichlet_prod'_inversion2: fixes f :: "real \<Rightarrow> 'a"   and ainv :: "nat \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and x :: "real"   and a :: "nat \<Rightarrow> 'a" assumes "\<forall>x\<ge>1. f x = dirichlet_prod' ainv g x"   and "1 \<le> x"   and "dirichlet_prod a ainv = (\<lambda>n. if n = 1 then 1::'a else (0::'a))" shows "g x = dirichlet_prod' a f x"
sourcenodes_def: fixes xs :: "'edge list" shows "sourcenodes xs \<equiv> map sourcenode xs"
update_Gamma_log_empty: fixes a :: "action"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and m :: "'c match_expr"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs' :: "'c rule list"   and s :: "state"   and t :: "state" assumes "a = Log \<or> a = Empty" shows "\<Gamma>(chain \<mapsto> Rule m a # rs),\<gamma>,p\<turnstile> \<langle>rs', s\<rangle> \<Rightarrow> t = \<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>rs', s\<rangle> \<Rightarrow> t"
less_1: fixes y :: "nat"   and w :: "'person \<Rightarrow> 'color"   and p :: "'person" assumes "y < n"   and "w p = blue"   and "local.valid w"   and "w guru \<noteq> blue" shows "local.leaves y p w = (card (local.blues_seen w p) \<le> y)"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "Singleton_for_d.Cd d X = greatest (Pd_singletons_for_ds X {Xd x} d) (X \<inter> Field (Pd_singletons_for_ds X {Xd x} d))"
2098: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) \<sqinter> x = x"
distinct_3: shows "ht_NOOP \<noteq> ht_GetWork"
almost_full_on_subset: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<subseteq> B"   and "almost_full_on P B" shows "almost_full_on P A"
span_mem: fixes E :: "'c set"   and u :: "'c" assumes "E \<subseteq> carrier V"   and "u \<in> E" shows "u \<in> span E"
countable_support: fixes p :: "'c spmf" shows "countable {x. Sigma_Algebra.measure (measure_spmf p) {x} \<noteq> 0}"
Sup_real_close: fixes e :: "real"   and S :: "real set" assumes "0 < e"   and "bdd_above S"   and "S \<noteq> {}" shows "\<exists>x\<in>S. \<Squnion> S - e < x"
vimage_small: fixes A :: "V" shows "small {b. \<exists>a\<in>elts (A \<inter>\<^sub>\<circ> \<D>\<^sub>\<circ> r). r\<lparr>a\<rparr> = b}"
n_kat_31: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "!x \<cdot> y \<cdot> t z = (0::'a)" shows "!x \<cdot> y \<cdot> !z = !x \<cdot> y"
get_child_nodes_new_document: fixes ptr' :: "(_) object_ptr"   and new_document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "r \<in> get_child_nodes_locs ptr'" shows "r h h'"
asymmetric_master_theorem_ltr: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi>" obtains Y :: "'b ltln set" and i :: "nat" where "Y \<subseteq> subformulas\<^sub>\<nu> \<phi>" and "suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[Y]\<^sub>\<mu>" and "\<forall>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi>\<^sub>1 R\<^sub>n \<psi>\<^sub>2 \<in> Y \<longrightarrow> suffix i w \<Turnstile>\<^sub>n G\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>" and "\<forall>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi>\<^sub>1 W\<^sub>n \<psi>\<^sub>2 \<in> Y \<longrightarrow> suffix i w \<Turnstile>\<^sub>n G\<^sub>n (\<psi>\<^sub>1[Y]\<^sub>\<mu> or\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>)"
path_connected_imp_connected_space: fixes X :: "'c topology" assumes "path_connected_space X" shows "connected_space X"
\<tau>diverge_simulation1: fixes s1 :: "'s1"   and s2 :: "'s2" assumes "s1 -\<tau>1\<rightarrow> \<infinity>"   and "s1 \<approx> s2" shows "s2 -\<tau>2\<rightarrow> \<infinity>"
domain_unique: fixes a :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr"   and a' :: "('obj, 'obj) concrete_category.arr" assumes "a \<in> domains f"   and "a' \<in> domains f" shows "a = a'"
vUniform_def: fixes v :: "bool"   and c :: "('p, 'v, 's) configuration" shows "vUniform v c \<equiv> initReachable c \<and> (\<forall>p. val[p,c] = {v})"
inconsistent: fixes R1 :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_bset R1 OO rel_bset R2 \<le> rel_bset (R1 OO R2)"
abe_ade_bcd_ace: fixes a :: "'a"   and b :: "'a"   and e :: "'a"   and d :: "'a"   and c :: "'a" assumes "[[a b e]]"   and "[[a d e]]"   and "[[b c d]]" shows "[[a c e]]"
reaction_def: fixes \<Delta> :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'state"   and vl :: "'value list"   and s1 :: "'state"   and vl1 :: "'value list" shows "reaction \<Delta> s vl s1 vl1 \<equiv> \<forall>a ou s' vl'. let trn = Trans s a ou s' in validTrans trn \<and> \<not> T trn \<and> consume trn vl vl' \<longrightarrow> match \<Delta> s s1 vl1 a ou s' vl' \<or> ignore \<Delta> s s1 vl1 a ou s' vl'"
frac_frac: fixes x :: "'d" shows "frac (frac x) = frac x"
imageE: fixes b :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "b \<in> f ` A" obtains x :: "'c" where "b = f x" and "x \<in> A"
simps_2: fixes d :: "nat"   and ds :: "nat list"   and a :: "nat" shows "digit_encode (d # ds) a = a mod d # digit_encode ds (a div d)"
is_ground_cls_list_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and CC :: "'b literal multiset list" shows "substitution_ops.is_ground_cls_list subst_atm CC = (\<forall>C\<in>set CC. substitution_ops.is_ground_cls subst_atm C)"
fract_collapse_1: fixes k :: "'b" shows "Fract (0::'b) k = 0"
assms: shows "\<forall>e f g. list_all (eval e f g) G \<longrightarrow> eval e f g p"
output_size_correct_tensors: fixes m :: "real mat convnet" assumes "valid_net' m" shows "output_size' m = dim_vec (tensors_from_net m)"
weakBisimReflexive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<approx> P"
map: fixes C' :: "char list"   and fs :: "fdecl list" assumes "map_of (map (\<lambda>(F, y). ((F, C'), y)) fs) (F, D) = \<lfloor>(b, T)\<rfloor>" shows "D = C'"
closure_complement: fixes S :: "'b set" shows "closure (- S) = - interior S"
distinct_13: fixes x21 :: "floatarith"   and x22 :: "floatarith"   and x23 :: "form"   and x31 :: "floatarith"   and x32 :: "floatarith" shows "Assign x21 x22 x23 \<noteq> Less x31 x32"
intro: fixes R :: "('c, 'd) ring_scheme"   and \<phi> :: "'c \<Rightarrow> nat" assumes "\<And>a b. \<lbrakk>a \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}; b \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}\<rbrakk> \<Longrightarrow> \<exists>q r. q \<in> carrier R \<and> r \<in> carrier R \<and> a = b \<otimes>\<^bsub>R\<^esub> q \<oplus>\<^bsub>R\<^esub> r \<and> (r = \<zero>\<^bsub>R\<^esub> \<or> \<phi> r < \<phi> b)" shows "euclidean_domain_axioms R \<phi>"
filterlim_tendsto_add_at_top_iff: fixes f :: "'c \<Rightarrow> real"   and c :: "real"   and F :: "'c filter"   and g :: "'c \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F" shows "(LIM x F. f x + g x :> at_top) = filterlim g at_top F"
Kas_Issues_A: fixes A :: "agent"   and authK :: "nat"   and Peer :: "msg"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Peer, Ta\<rbrace>, authTicket\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "Kas Issues A with Crypt (shrK A) \<lbrace>Key authK, Peer, Ta\<rbrace> on evs"
le_less: fixes y :: "'d"   and x :: "'d" shows "(y \<le> x) = (y < x \<or> x = y)"
7: shows "[a] \<preceq>\<^sub>F\<^sub>I l @- u"
degree_aform_err_add_aform': fixes x :: "(real \<times> real pdevs) \<times> real"   and n :: "nat"   and y :: "(real \<times> real pdevs) \<times> real"   and p :: "nat" assumes "degree_aform_err x \<le> n"   and "degree_aform_err y \<le> n" shows "degree_aform_err (add_aform' p x y) \<le> n"
assms: shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A"
bpE_4: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and S :: "'b set"   and L :: "'b set"   and P :: "'b set set" assumes "BinPacking U w c S L"   and "BinPacking.bp U w c P" shows "\<forall>B\<in>P. sum w B \<le> real c"
coprime_0_right_iff: fixes a :: "'a" shows "coprime a (0::'a) = is_unit a"
WTrts_same_length: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and es :: "char list exp list"   and Ts :: "ty list" assumes "P,E,h,sh \<turnstile> es [:] Ts" shows "length es = length Ts"
top_right_normalize: fixes a :: "'c" shows "gcd a (0::'c) = normalize a"
set: fixes z :: "'a" shows "{g z |g. g \<in> \<G>} \<noteq> {}"
word_msb_alt: shows "msb y = hd (to_bl y)"
plus_absorb1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)" shows "\<Omega>[F](\<lambda>x. f x + g x) = \<Omega>[F](g)"
less_irrefl: fixes x :: "'c word" shows "\<not> x <s x"
ivl_disj_un_one_7: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{l..u} \<union> {u<..} = {l..}"
rep_eq: fixes x :: "'a st_vec" shows "measure_pmf (pmf_of_st_vec x) = measure_of_st_vec x"
ou_witness_open_unresolvable: fixes X :: "ou_witness set" shows "open_unresolvable X"
lfinite_llength_enat: fixes xs :: "'a llist" assumes "lfinite xs" shows "\<exists>n. llength xs = enat n"
irreducible\<^sub>d_mD_1: fixes f :: "int poly" assumes "pl.irreducible\<^sub>d_m f" shows "0 < pl.degree_m f"
fmupd_same: fixes x1 :: "'c"   and x1aa :: "('c, 'd) fmap" assumes "x1 \<in># dom_m x1aa" shows "fmupd x1 (the (fmlookup x1aa x1)) x1aa = x1aa"
logical_instr_sub1_def: fixes instr_name :: "sparc_operation"   and result :: "word32" shows "logical_instr_sub1 instr_name result \<equiv> if instr_name \<in> {logic_type ANDcc, logic_type ANDNcc, logic_type ORcc, logic_type ORNcc, logic_type XORcc, logic_type XNORcc} then do new_psr_val \<leftarrow> gets (logical_new_psr_val result); write_cpu new_psr_val PSR; return () od else return ()"
dvd_sum: fixes A :: "'b set"   and d :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>a. a \<in> A \<Longrightarrow> d dvd f a" shows "d dvd sum f A"
isometry_on_inverse_4: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "bij_betw f X (f ` X)"
worst_chain_pred: fixes i :: "nat" shows "p (local.worst_chain i) (local.worst_chain (Suc i))"
lm07: fixes N :: "integer set"   and \<Omega> :: "'c list"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "card ((argmax \<circ> sum) (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>)) ((argmax \<circ> sum) b (allAllocations (N \<union> {seller}) (set \<Omega>)))) = 1" shows "the_elem (Outside' {seller} ` (argmax \<circ> sum) (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>)) ((argmax \<circ> sum) b (allAllocations (N \<union> {seller}) (set \<Omega>)))) = Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b)))"
less_trans_Suc: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j < k" shows "Suc i < k"
fold_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('c, 'd) rbt" shows "RBT_Impl.fold f t = List.fold (\<lambda>(x, y). f x y) (RBT_Impl.entries t)"
assms: shows "\<forall>l\<in>dom f. \<forall>x y z a. Px x y z a \<longrightarrow> P x y z a (f l) l"
meval_trm_eval_trm: fixes n :: "nat"   and A :: "nat set"   and x :: "event_data option list"   and t :: "Formula.trm" assumes "wf_tuple n A x"   and "fv_trm t \<subseteq> A"   and "\<forall>i\<in>A. i < n" shows "meval_trm t x = Formula.eval_trm (map the x) t"
cinner_ket_eqI: fixes \<psi> :: "'c ell2"   and \<phi> :: "'c ell2" assumes "\<And>i. ket i \<bullet>\<^sub>C \<psi> = ket i \<bullet>\<^sub>C \<phi>" shows "\<psi> = \<phi>"
wf_partial_state_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and ps :: "(nat \<times> nat) list" shows "ast_problem.wf_partial_state problem ps \<equiv> distinct (map fst ps) \<and> (\<forall>(x, v)\<in>set ps. x < ast_problem.numVars problem \<and> v < ast_problem.numVals problem x)"
extended_Gromov_product_at_diff2: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion"   and z :: "'a Gromov_completion" shows "extended_Gromov_product_at e x y \<le> extended_Gromov_product_at e z y + extended_Gromov_distance x z"
F\<^sub>1_def: shows "F\<^sub>1 = (\<lambda>\<mu>. if hhom\<^sub>C.arr \<mu> then F \<mu> else D.null)"
that: fixes err :: "real" assumes "x = aform_val (e(n := err)) (aform_err_to_aform X n)"   and "- 1 \<le> err"   and "err \<le> 1" shows "thesis"
preserves_reflects_arr: fixes f :: "'c" shows "D.arr (G\<epsilon>'_\<eta>G.map f) = C.arr f"
SystemClasses_def: shows "SystemClasses \<equiv> [NullPointerC, ClassCastC, OutOfMemoryC]"
R_a0iim1_ir_def: shows "R_a0iim1_ir \<equiv> {(s, t). signals s = ir_runs2sigs (runs t) \<and> corrupted s = {}}"
minus_divide_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (uminus a) (uminus b) = divide a b"
edges_of_walkE: fixes v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "(v, w) \<in> H.edges_of_walk xs" shows "\<exists>xs_pre xs_post. xs = xs_pre @ v # w # xs_post"
homeomorphism_moving_points_exists: fixes S :: "'b set"   and T :: "'b set"   and K :: "'c set"   and x :: "'c \<Rightarrow> 'b"   and y :: "'c \<Rightarrow> 'b" assumes "2 \<le> DIM('b)"   and "open S"   and "connected S"   and "S \<subseteq> T"   and "finite K"   and "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"   and "pairwise (\<lambda>i j. x i \<noteq> x j \<and> y i \<noteq> y j) K"   and "S \<subseteq> T"   and "T \<subseteq> affine hull S"   and "connected S" obtains f :: "'b \<Rightarrow> 'b" and g :: "'b \<Rightarrow> 'b" where "homeomorphism T T f g" and "\<And>i. i \<in> K \<Longrightarrow> f (x i) = y i" and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" and "bounded {x. \<not> (f x = x \<and> g x = x)}"
pred_uprod_code: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "pred_uprod P (Upair x y) = (P x \<and> P y)"
restrict_relp_apply: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" shows "(R \<upharpoonleft> P \<otimes> Q) x y = (R x y \<and> P x \<and> Q y)"
leadsETo_Basis: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set"   and CC :: "'d set set" assumes "F \<in> A ensures B"   and "A - B \<in> insert {} CC" shows "F \<in> A leadsTo[CC] B"
ofilter_linord: fixes A :: "'a set"   and B :: "'a set" assumes "local.ofilter A"   and "local.ofilter B" shows "A \<subseteq> B \<or> B \<subseteq> A"
RF_def: shows "RF \<equiv> \<lambda>j i. if j = k then QF k (j, m - Suc i) else QF k (j, i)"
finfun_curry_update: fixes f :: "('a \<times> 'b, 'c) finfun"   and a :: "'a"   and b :: "'b"   and c :: "'c" shows "finfun_curry (finfun_update f (a, b) c) = finfun_update (finfun_curry f) a (finfun_update (finfun_apply (finfun_curry f) a) b c)"
new_psubst_image: fixes c :: "'a"   and p :: "('a, 'b) form"   and d :: "'c"   and f :: "'a \<Rightarrow> 'c" assumes "new c p"   and "d \<notin> f ` params p" shows "new d (psubst (f(c := d)) p)"
confTs_Cons1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and x :: "val"   and xs :: "val list"   and ys :: "ty err list" shows "(P,h \<turnstile> x # xs [:\<le>\<^sub>\<top>] ys) = (\<exists>z zs. ys = z # zs \<and> P,h \<turnstile> x :\<le>\<^sub>\<top> z \<and> P,h \<turnstile> xs [:\<le>\<^sub>\<top>] zs)"
neutral_left: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> Pi\<^sub>E A B - (\<lambda>f. restrict f A) ` S" shows "(\<Prod>xa\<in>A. pmf (p xa) (f xa)) = 0"
A_def: fixes x :: "real"   and y :: "real" shows "A x y = blinfun_of_matrix ((\<lambda>_ _. 0)((1, 0) := (\<lambda>_. 0)((1, 0) := A11 x y, (0, 1) := A12 x y), (0, 1) := (\<lambda>_. 0)((1, 0) := A21 x y, (0, 1) := A22 x y)))"
lsided_eq_var1: fixes x :: "'b \<times> 'c" shows "less_eq_prod x (prod.bres top_prod x) = (x = prod.bres top_prod x)"
nexts_code_def: fixes prog :: "program"   and g :: "gState" shows "nexts_code prog g = the_res (dSUCCEED_abort STR ''The Universe is broken!'' (dRETURN (ls.sng g)) (nexts_code_aux prog g))"
y_fresh: shows "qAFresh xs y X \<and> qAFresh xs y X'"
borel_measurable_integrable: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" shows "f \<in> borel_measurable M"
prime_elem_not_zeroI: fixes p :: "'a" assumes "prime_elem p" shows "p \<noteq> (0::'a)"
bisimSubstParNil: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> \<zero> \<sim>\<^sub>s P"
derivative_norm_bounded_subset: fixes X' :: "'a set" assumes "X' \<noteq> {}"   and "X' \<subseteq> X"   and "convex X'" shows "derivative_norm_bounded T X' f f' B B'"
exec_instr_IAdd: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr IAdd P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (None, h, (Intg (the_Intg (hd (tl stk)) + the_Intg (hd stk)) # tl (tl stk), loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
finite_Collect: fixes S :: "'b set"   and f :: "'c \<Rightarrow> 'b" assumes "finite S"   and "inj f" shows "finite {a. f a \<in> S}"
simps_2: fixes v :: "'c" shows "dbm_add (Le v) \<infinity> = \<infinity>"
BuildSeq_def: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and y :: "hf" shows "BuildSeq B C s k y \<equiv> LstSeq s k y \<and> HBall (succ k) (Builds B C s)"
drop_empty_tokens: fixes p :: "(symbol \<times> character list) list"   and r :: "nat" assumes "p \<in> \<PP>"   and "r \<le> length p"   and "charslength (take r p) = 0"   and "admissible (drop r p)" shows "drop r p \<in> \<PP>"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) r"
rbt_greater_simps_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b" shows "ord.rbt_greater less k rbt.Empty = True"
Rep_real_inverse: fixes x :: "real" shows "Abs_real (Rep_real x) = x"
mem_ring_n_pd_mem_K: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and x :: "'a" assumes "distinct_pds K n P"   and "x \<in> carrier (O\<^bsub>K P n\<^esub>)" shows "x \<in> carrier K"
atLeastatMost_empty_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b = {}) = (\<not> less_eq a b)"
nat_in_between_eq_1: fixes a :: "nat"   and b :: "nat" shows "(a < b \<and> b \<le> Suc a) = (b = Suc a)"
C: shows "pairwise (\<lambda>x y. disjnt (C x) (C y)) A"
max_hi_is_max: fixes t :: "('b ivl \<times> 'b) tree"   and a :: "'b ivl" assumes "inv_max_hi t"   and "a \<in> Tree2.set_tree t" shows "high a \<le> max_hi t"
lfinite_lfilterI: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "lfinite xs" shows "lfinite (lfilter P xs)"
sp_same_arities: fixes F1 :: "'a \<Rightarrow> 'b fundef option"   and F2 :: "'a \<Rightarrow> 'c fundef option" assumes "all_same_arities F1 F2" shows "Subx.sp F1 = Subx.sp F2"
prems_2: shows "\<forall>s\<in>insert x f. \<exists>p. p division_of s"
listset_ConsI: fixes x :: "'a"   and X :: "'a set"   and xs :: "'a list"   and Xs :: "'a set list" assumes "x \<in> X"   and "xs \<in> listset Xs" shows "x # xs \<in> listset (X # Xs)"
M1_mult_ket_k: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat"   and k :: "nat" assumes "grover_state_sig n f dims R K"   and "k < R" shows "grover_state_sig.M1 R K *\<^sub>v grover_state_sig.ket_k K k = grover_state_sig.ket_k K k"
CD_on_f_range_on: fixes A :: "'x set"   and ds :: "'d set" shows "f_range_on A (CD_on ds)"
j_def: shows "j = (LEAST j. u <p concat (take (Suc j) ws))"
assms: shows "\<forall>l\<in>set L1. set L2 = f2 (set l) N"
height_balR: fixes l :: "('c \<times> nat) tree"   and r :: "('c \<times> nat) tree"   and a :: "'c" assumes "avl l"   and "avl r"   and "height r = height l + 2" shows "height (balR l a r) \<in> {height l + 2, height l + 3}"
det_upperdiagonal: fixes A :: "(('b, 'a) vec, 'a) vec" assumes "\<And>i j. j < i \<Longrightarrow> A $ i $ j = (0::'b)" shows "det A = (\<Prod>i\<in>UNIV. A $ i $ i)"
finite_sysI: assumes "finite (bl\<^sup>c)" shows "finite_incidence_system (bl\<^sup>c) \<B>\<^sup>R"
lookup_update: fixes ks :: "'b list"   and v :: "'a"   and t :: "('b, 'a) Trie.trie"   and ks' :: "'b list" shows "lookup_trie (update_trie ks v t) ks' = (if ks = ks' then Some v else lookup_trie t ks')"
exp_base_closed: fixes x :: "'a list"   and n :: "nat" assumes "x \<in> carrier (K [X])" shows "set (UP.exp_base x n) \<subseteq> carrier (K [X])"
un_rec: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1'"   and "g \<in> PrimRec1'" shows "UnaryRecOp f g \<in> PrimRec1'"
Ints_mat_def: shows "\<int>\<^sub>m = {A. \<forall>i<dim_row A. \<forall>j<dim_col A. A $$ (i, j) \<in> \<int>}"