hcju/isabelleps huggingface dataset:
l_set_disconnected_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"
symmetric_def: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "symmetric A equiv \<equiv> \<forall>x y. equiv x y \<longrightarrow> x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> equiv y x"
vpa_no_slpa: fixes cs :: "'edge list"   and as :: "'edge list"   and n :: "'node"   and n' :: "'node" assumes "valid_path_aux cs as"   and "n -as\<rightarrow>* n'"   and "valid_call_list cs n"   and "cs \<noteq> []"   and "\<forall>xs ys. as = xs @ ys \<longrightarrow> \<not> same_level_path_aux cs xs \<or> upd_cs cs xs \<noteq> []" shows "\<exists>a Q r fs. valid_edge a \<and> kind a = Q:r\<hookrightarrow>\<^bsub>get_proc n'\<^esub>fs"
Conj_E: fixes A :: "fm"   and B :: "fm"   and H :: "fm set"   and C :: "fm" assumes "insert A (insert B H) \<turnstile> C" shows "insert (A AND B) H \<turnstile> C"
\<theta>_simps_3: shows "trg \<omega>.the_\<theta> = src t"
ffold_fun_left_comm: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'d"   and z :: "'e"   and A :: "'d fset" assumes "comp_fun_commute f" shows "f x (ffold f z A) = ffold f (f x z) A"
finite: fixes i :: "nat" shows "emeasure M (A i) \<noteq> \<infinity>"
continuous_on_chart_inv': fixes c :: "('e, 'c) chart"   and s :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "continuous_on (apply_chart c ` s) (f \<circ> inv_chart c)"   and "s \<subseteq> domain c" shows "continuous_on s f"
tendsto_exp_limit_sequentially: fixes x :: "real" shows "(\<lambda>n. (1 + x / real n) ^ n) \<longlonglongrightarrow> exp x"
filter_a1_map_rel2: shows "filter_a2_map I nt tp a2_map' = (if left I = 0 then rel2 else empty_table)"
free_group_card_infinite: fixes X :: "'c set" assumes "infinite X" shows "|X| =o |carrier \<F>\<^bsub>X\<^esub>|"
iff_allI: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "(\<forall>x. P x) = (\<forall>x. Q x)"
assms: fixes i :: "nat" assumes "i < n" shows "f i = (0::'a)"
zero_hom_0_axioms: shows "zero_hom_0 poly_y_x"
set_mult_closed: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and K :: "'c set" assumes "Group.monoid G"   and "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "H <#>\<^bsub>G\<^esub> K \<subseteq> carrier G"
weakBisimResChainPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<approx> Q"   and "xvec \<sharp>* \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<approx> \<lparr>\<nu>*xvec\<rparr>Q"
of_rat_power: fixes a :: "rat"   and n :: "nat" shows "of_rat (a ^ n) = of_rat a ^ n"
red_setE: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.red F p q" obtains f :: "'a \<Rightarrow>\<^sub>0 'c" and t :: "'a" where "f \<in> F" and "punit.red_single p q f t"
list_rec_Nil: fixes c :: "'b"   and h :: "'c \<Rightarrow> 'c SList.list \<Rightarrow> 'b \<Rightarrow> 'b" shows "list_rec [] c h = c"
1: shows "- (1 ; x) \<cdot> x = 0"
distinct_1: fixes x1 :: "'b"   and x2 :: "int" shows "Var x1 \<noteq> Const x2"
assms: shows "A \<turnstile>\<^sub>R res1 \<approx> res2"
bind_mono_strong: fixes m :: "('b, 'c) state"   and f :: "'c \<Rightarrow> ('b, 'd) state" assumes "mono_state m"   and "\<And>x s s'. run_state m s = (x, s') \<Longrightarrow> mono_state (f x)" shows "mono_state (m \<bind> f)"
abs_eq: shows "fls_X = Abs_fls (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
neq_le_trans: fixes a :: "'c"   and b :: "'c" assumes "a \<noteq> b"   and "b \<le> a" shows "b < a"
invar_cons_down: fixes t :: "('c, 'd) BinomialTree"   and bq :: "('c, 'd) BinomialTree list" assumes "invar (t # bq)" shows "invar bq"
bet_cong_eq: 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 "Bet A B C"   and "Bet A C D"   and "Cong B C A D" shows "C = D \<and> A = B"
simps_17: fixes I :: "\<I>"   and r :: "Formula.formula Regex.regex" shows "Formula.future_bounded (formula.MatchF I r) = (regex.pred_regex Formula.future_bounded r \<and> right I \<noteq> \<infinity>)"
z_closer_than_new_last: shows "H.distance z v1 < H.distance new_last v1"
digraph_isomorphism_def: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" shows "pre_digraph.digraph_isomorphism G hom \<equiv> wf_digraph G \<and> inj_on (iso_verts hom) (verts G) \<and> inj_on (iso_arcs hom) (arcs G) \<and> (\<forall>a\<in>arcs G. iso_verts hom (tail G a) = iso_tail hom (iso_arcs hom a) \<and> iso_verts hom (head G a) = iso_head hom (iso_arcs hom a))"
unitary_gen_special: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary_gen M"   and "mat_det M = 1" shows "\<exists>a b. M = (a, b, - cnj b, cnj a)"
N_def: fixes xs :: "'b list"   and i :: "nat" shows "??.Newton_Interpolation.N xs i = ??.Newton_Interpolation.Xij xs 0 i"
strategyproofness: shows "(pmf (sds R18) a < pmf (sds R14) a \<or> pmf (sds R18) a + pmf (sds R18) d < pmf (sds R14) a + pmf (sds R14) d \<or> pmf (sds R18) a + (pmf (sds R18) d + pmf (sds R18) c) < pmf (sds R14) a + (pmf (sds R14) d + pmf (sds R14) c)) \<or> pmf (sds R18) a = pmf (sds R14) a \<and> pmf (sds R18) a + pmf (sds R18) d = pmf (sds R14) a + pmf (sds R14) d \<and> pmf (sds R18) a + (pmf (sds R18) d + pmf (sds R18) c) = pmf (sds R14) a + (pmf (sds R14) d + pmf (sds R14) c)"
funPair_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "funPair f g \<equiv> \<lambda>x. (f x, g x)"
fimage_funion: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b fset"   and B :: "'b fset" shows "f |`| (A |\<union>| B) = f |`| A |\<union>| f |`| B"
order_refl: fixes ko :: "'b key_order"   and x :: "'b" shows "ko.le ko x x"
inf: fixes \<sigma> :: "'tp" shows "infinite {a. intT \<sigma> a}"
finite_cartesian_productD2: fixes A :: "'c set"   and B :: "'d set" assumes "finite (A \<times> B)"   and "A \<noteq> {}" shows "finite B"
singleton_sunflower: fixes A :: "'b set" shows "sunflower {A}"
C_def: shows "C = C0 \<inter> Collect (Domainp A)"
vemptyE: fixes A :: "V" assumes "A \<noteq> 0" obtains x :: "V" where "x \<in>\<^sub>\<circ> A"
elims_3: fixes x :: "'b list" assumes "\<not> l_sorted x" obtains x :: "'b" and x' :: "'b" and xs :: "'b list" where "xa__ = x # x' # xs" and "\<not> (x \<le> x' \<and> l_sorted (x' # xs))"
hyps: shows "mset ` set (subseqs xs) = {ps. ps \<subseteq># mset xs}"
IH: fixes gpv :: "('a, 'call, 'ret) gpv"   and s :: "'s" assumes "gpv \<in> sub_gpvs \<I> (GPV p)" shows "lossless_gpv \<I>' (inline callee gpv s)"
handle2: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'c state"   and e :: "exp0"   and s2 :: "'c state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and bv :: "'c state \<times> (v, v) result" assumes "BigStep.evaluate ck env s1 e (s2, Rerr (Rraise v1)) \<and> evaluate_match ck env s2 v1 pes v1 bv" shows "BigStep.evaluate ck env s1 (Handle e pes) bv"
hd_in_awalk_verts_1: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "G.awalk u p v" shows "u \<in> set (pawalk_verts u p)"
sup_left_idem: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup x y) = sup x y"
interior_real_atMost: fixes a :: "real" shows "interior {..a} = {..<a}"
assms: shows "\<forall>\<phi>\<in>\<Phi>. Inf_pres \<phi>"
set_step: fixes s :: "'b list"   and r :: "'b"   and act :: "nat \<times> nat list" shows "set (step s r act) = set s"
s_gt_nil: shows "extf f (>\<^sub>t) [s] []"
kbo_basic_axioms: shows "kbo_basic (>\<^sub>s) wt_sym \<epsilon> extf ground_heads_var"
all_flows_def: fixes \<T> :: "'b stateful_policy" shows "all_flows \<T> \<equiv> flows_fix \<T> \<union> flows_state \<T> \<union> backflows (flows_state \<T>)"
assms_2: shows "(stateA, stateB) \<in> transitionRelation decisionVars"
two_lists_cases_snoc_Cons: fixes as :: "'b list"   and bs :: "'a list" obtains ys :: "'a list" where "as = []" and "bs = ys__"   | xs :: "'b list" where "as = xs__" and "bs = []"   | xs :: "'b list" and x :: "'b" and y :: "'a" and ys :: "'a list" where "as = xs @ [x]" and "bs = y # ys"
ext_inject: fixes mag :: "'a"   and dim :: "(int, 'b) dimvec"   and more :: "'c"   and mag' :: "'a"   and dim' :: "(int, 'b) dimvec"   and more' :: "'c" shows "(\<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = \<lparr>mag = mag', dim = dim', \<dots> = more'\<rparr>) = (mag = mag' \<and> dim = dim' \<and> more = more')"
allUses_def_ass_inv: fixes v' :: "'val"   and n :: "'node" assumes "v' \<in> step.allUses g n"   and "n \<in> set (\<alpha>n g)" shows "step.defAss g n v'"
isMaxim_max_fun_diff: fixes f :: "'b + 'c \<Rightarrow> 'a"   and g :: "'b + 'c \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> rst.FINFUNC"   and "g \<in> rst.FINFUNC" shows "st.isMaxim {a \<in> Field (s +o t). f a \<noteq> g a} (st.max_fun_diff f g)"
A3: shows "(a, b) \<in> R\<^sup>*"
Timeso_undef: fixes \<theta> :: "trm option"   and \<eta> :: "trm option" shows "(Timeso \<theta> \<eta> = undeft) = (\<theta> = undeft \<or> \<eta> = undeft)"
X1_def: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A" shows "perron_frobenius.X1 = perron_frobenius.X \<inter> {x. norm x = 1}"
epo_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool" shows "epo ground_heads_var gt_sym arity_sym arity_var extf \<equiv> Lambda_Free_Term.ground_heads gt_sym arity_sym arity_var ground_heads_var \<and> epo_axioms extf"
run_foldl: fixes \<Delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "run \<Delta> q\<^sub>0 w i = foldl \<Delta> q\<^sub>0 (map w [0..<i])"
Cons_1: fixes xs :: "'a list"   and i :: "nat"   and p :: "nat"   and b :: "'a" assumes "sorted (map fst pxs)"   and "distinct (map fst pxs)"   and "\<forall>a\<in>set pxs. case a of (p, b) \<Rightarrow> p < length xs + length pxs"   and "i < length pxs"   and "pxs ! i = (p, b)" shows "fold (\<lambda>a. case a of (a, b) \<Rightarrow> insert_nth a b) pxs xs ! p = b"
assms_1: fixes i :: "nat" assumes "i < length ss"   and "i < length ts"   and "fst (r (ss ! i) (ts ! i))" shows "fst (r (map f ss ! i) (map f ts ! i))"
r_univ_exteq_r_normal_form: shows "r_univ \<simeq> r_normal_form"
semiring_norm_89: fixes m :: "num"   and n :: "num" shows "(num.Bit1 m = num.Bit0 n) = False"
productive_imp_produces_Max_atom: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset" assumes "production N C \<noteq> {}" shows "production N C = {Max (atms_of C)}"
alpha_qFresh_qSwap_id: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGood X"   and "qFresh zs z1 X"   and "qFresh zs z2 X" shows "X #[[z1 \<and> z2]]_zs #= X"
simps_1: fixes x :: "real"   and ya :: "real" shows "(Rmax x = Rmax ya) = (x = ya)"
that: shows "Negligible.negligible (\<lambda>\<eta>. local.ddh.advantage \<eta> (elgamal_adversary \<eta> (\<A> \<eta>)))"
aas_ne: shows "\<forall>i<n. AAs ! i \<noteq> {#}"
CollectE: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "a \<in> {x. P x}" obtains "P a"
spr_sim_f: shows "sim_f SPR.MC spr_simMC spr_sim"
assms_3: shows "fst ` set txs \<subseteq> atrm"
lookup_pm_of_idx_pm_distinct: fixes xs :: "'c list"   and i :: "nat"   and f :: "nat \<Rightarrow>\<^sub>0 'd" assumes "distinct xs"   and "i < length xs" shows "lookup (pm_of_idx_pm xs f) (xs ! i) = lookup f i"
rep_eq: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'c fset" shows "fpairwise x xa = Compiler_Utils.pairwise x (fset xa)"
bisimParPresAux: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>R :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi"   and A\<^sub>R :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<sim> Q"   and "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"   and "A\<^sub>R \<sharp>* \<Psi>"   and "A\<^sub>R \<sharp>* P"   and "A\<^sub>R \<sharp>* Q" shows "\<Psi> \<rhd> P \<parallel> R \<sim> Q \<parallel> R"
hyps_5: shows "CFG_node (_Exit_) = CFG_node (targetnode a')"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r # rs, s\<rangle> \<Rightarrow> t"
homeomorphic_punctured_affine_sphere_affine: fixes r :: "real"   and b :: "'c"   and a :: "'c"   and T :: "'c set"   and p :: "'d set" assumes "0 < r"   and "b \<in> sphere a r"   and "affine T"   and "a \<in> T"   and "b \<in> T"   and "affine p"   and "aff_dim T = aff_dim p + 1" shows "sphere a r \<inter> T - {b} homeomorphic p"
trans: fixes b :: "'g multiset"   and a :: "'g multiset"   and c :: "'g multiset" assumes "b \<subseteq># a"   and "c \<subseteq># b" shows "c \<subseteq># a"
simps_3: fixes M :: "('b, 'c) model"   and g :: "'d \<Rightarrow> 'b"   and w :: "'b"   and p :: "('c, 'd) fm" shows "(M, g, w \<Turnstile> \<^bold>\<not> p) = (\<not> M, g, w \<Turnstile> p)"
frac_ge_0: fixes x :: "'b" shows "(0::'b) \<le> frac x"
adds_triv_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus a (plus b a)"
upper_bound_def: fixes A :: "'a set" shows "upper_bound A = {a. \<forall>x\<in>A. x \<le> a}"
pr_simps_1: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "arr \<pp>\<^sub>0[a, b]"
alt_group_carrier': fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p \<in> carrier (alt_group n)" shows "permutation p"
P_lemma: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)"   and "x \<in> S1 p" shows "\<exists>!y. P p a x y"
insort_eq_Cons: fixes xs :: "'a list"   and b :: "'a" assumes "\<forall>a\<in>set xs. b < a"   and "sorted xs" shows "insort b xs = b # xs"
test_sup_neg_1: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "x \<squnion> y \<squnion> ! x \<sqinter> ! y = top"
3: shows "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m (store c)"
remove1_mset_eqE: fixes x1 :: "'c multiset"   and L :: "'c"   and M :: "'c multiset" assumes "remove1_mset L x1 = M" obtains "L \<in># x1" and "x1 = M + {#L#}"   | "L \<notin># x1" and "x1 = M"
divide_add_eq_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "plus (divide x z) y = divide (plus x (times y z)) z"
Suc: shows "echelon_form (foldl (Hermite_of_row_i ass res) A (map mod_type_class.from_nat [0..<k]))"
infm: shows "\<forall>i. \<exists>j>i. P j"
subst_eqG: shows "Fsubst (foldr (&&) \<Gamma> TT) \<sigma> = foldr (&&) (map (\<lambda>\<phi>. Fsubst \<phi> \<sigma>) \<Gamma>) TT"
Well_Scoped_Graph_axioms: shows "Well_Scoped_Graph task.nodes task.inPorts task.outPorts vertices nodeOf task_edges task.hyps"
comp_cod_arr: fixes f :: "'b"   and b :: "'b" assumes "Adj.D.arr f"   and "Adj.D.cod f = b" shows "b \<cdot>\<^sub>D f = f"
free_lift: fixes t :: "Core.term"   and k :: "nat"   and i :: "nat" shows "loose_bvar1 (lift t k) i = (i < k \<and> loose_bvar1 t i \<or> k < i \<and> loose_bvar1 t (i - 1))"
is_projection_on_rank1: fixes t :: "'b" shows "is_projection_on (\<lambda>x. (\<langle>t, x\<rangle> / \<langle>t, t\<rangle>) *\<^sub>C t) (cspan {t})"
normal_invE_2: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set"   and x :: "'c"   and h :: "'c" assumes "Group.group G"   and "N \<lhd> G"   and "x \<in> carrier G"   and "h \<in> N" shows "x \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x \<in> N"
0: fixes v :: "nat \<times> 'b"   and u :: "nat \<times> 'b" assumes "v \<in> gunodes A w"   and "u \<in> gusuccessors A w v" shows "f u \<le> f v"
distinct_7: fixes x21 :: "fmla"   and x22 :: "fmla"   and x3 :: "fmla" shows "x21 OR x22 \<noteq> Robinson_Arithmetic.neg x3"
raw_term_cases: fixes y :: "('d, 'e) trm" assumes "y \<in> {\<theta>. dfree \<theta>}" obtains x :: "('d, 'e) strm" where "y = raw_term x"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [$Loop (t1s _> t2s) es] : ts _> ts'"
simps_1: fixes l :: "'a extended"   and uu :: "'a"   and u :: "'a extended"   and uv :: "'b" shows "deps (Insert (InsertMessage l uu u uv)) = extended_to_set l \<union> extended_to_set u"
SF1_general: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b"   and F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and w :: "'a \<Rightarrow> 'c" assumes "|~ P \<and> N \<longrightarrow> \<circle>P \<or> \<circle>Q"   and "|~ P \<and> N \<and> \<langle>A\<rangle>_v \<longrightarrow> \<circle>Q"   and "\<turnstile> \<box>P \<and> \<box>N \<and> \<box>F \<longrightarrow> \<diamond>Enabled \<langle>A\<rangle>_v"   and "|~ P \<and> Unchanged w \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>N \<and> SF(A)_v \<and> \<box>F \<longrightarrow> P \<leadsto> Q"
rel_setD2: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d set"   and B :: "'e set"   and y :: "'e" assumes "rel_set R A B"   and "y \<in> B" shows "\<exists>x\<in>A. R x y"
splitAt_elements: fixes ram1 :: "'c"   and vs :: "'c list"   and ram2 :: "'c" assumes "ram1 \<in> set vs"   and "ram2 \<in> set vs" shows "ram2 \<in> set (fst (splitAt ram1 vs)) \<or> ram2 \<in> set [ram1] \<or> ram2 \<in> set (snd (splitAt ram1 vs))"
showsp_float10_append: fixes p :: "nat"   and r :: "float10"   and x :: "char list"   and y :: "char list" shows "showsp_float10 p r (x @ y) = showsp_float10 p r x @ y"
nonz: fixes i :: "nat" shows "0 < (enum N \<circ> Suc) i"
comp_maps: fixes \<eta> :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and \<theta> :: "'c \<Rightarrow> 'd"   and C :: "'d set" assumes "map \<eta> A B"   and "map \<theta> B C" shows "map (\<theta> \<circ> \<eta> \<down> A) A C"
inline1_fixp_induct_strong2: fixes P :: "(('d, 'e, 'f) gpv \<Rightarrow> 'g \<Rightarrow> ('d \<times> 'g + 'h \<times> ('f \<times> 'g, 'h, 'i) rpv \<times> ('d, 'e, 'f) rpv) spmf) \<Rightarrow> bool"   and callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g, 'h, 'i) gpv" assumes "spmf.admissible (\<lambda>inline1'. P (\<lambda>gpv s. inline1' (gpv, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>inline1'. \<lbrakk>\<And>gpv s. ord_spmf (=) (inline1' gpv s) (inline1 callee gpv s); \<And>gpv s. ord_spmf (=) (inline1' gpv s) (the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1' (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))); P inline1'\<rbrakk> \<Longrightarrow> P (\<lambda>gpv s. the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1' (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv)))))" shows "P (inline1 callee)"
elims_3: fixes x :: "atom"   and xa :: "real list" assumes "\<not> I\<^sub>R x xa" obtains r :: "real" and cs :: "real list" and xs :: "real list" where "x = Less r__ cs__" and "xa = xsa__" and "\<not> r__ < \<langle>cs__,xsa__\<rangle>"   | r :: "real" and cs :: "real list" and xs :: "real list" where "x = Eq r cs" and "xa = xs" and "r \<noteq> \<langle>cs,xs\<rangle>"
iarray_to_vec_nth: fixes A :: "'a iarray"   and i :: "'b" shows "iarray_to_vec A $ i = A !! mod_type_class.to_nat i"
F2_add_f: shows "Fubx_add F2 f (rewrite_fundef_body fd2 pc instr) = F2'"
A_7: shows "\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y"
simps_5: fixes m :: "nat"   and r :: "mregex"   and s :: "mregex" shows "ok m (MTimes r s) = (ok m r \<and> ok m s)"
group_relcycle_group: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "Group.group (relcycle_group p X S)"
gSwapAbsPresGWlsAbs_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gSwapAbsPresGWlsAbs MOD \<equiv> \<forall>zs z1 z2 us s A' A. isInBar (us, s) \<and> wlsAbs (us, s) A' \<and> gWlsAbs MOD (us, s) A \<longrightarrow> gWlsAbs MOD (us, s) (gSwapAbs MOD zs z1 z2 A' A)"
snoc_1: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat"   and n' :: "nat" assumes "xs = cs\<^bsup>\<pi>\<^esup> n"   and "is_path \<pi>"   and "\<pi> n \<noteq> local.return"   and "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>\<^esup> n'" shows "n = n'"
remove_comp_portfolio: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'c" assumes "portfolio pf1" shows "portfolio (qty_rem_comp pf1 x)"
prv_imp_eqvER: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "prv (eqv \<phi>1 \<phi>2)" shows "prv (imp \<phi>2 \<phi>1)"
insert_3: assumes "swapidseq_ext A n p" shows "swapidseq_ext (A \<union> B) n p"