hcju/isabelleps huggingface dataset:
parallel_fixp_induct_2_2: fixes orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and luba :: "'d set \<Rightarrow> 'd"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and lubb :: "'e set \<Rightarrow> 'e"   and F :: "('f \<Rightarrow> 'g \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and G :: "('h \<Rightarrow> 'i \<Rightarrow> 'e) \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'i \<Rightarrow> 'e"   and P :: "('f \<Rightarrow> 'g \<Rightarrow> 'd) \<Rightarrow> ('h \<Rightarrow> 'i \<Rightarrow> 'e) \<Rightarrow> bool" assumes "partial_function_definitions orda luba"   and "partial_function_definitions ordb lubb"   and "\<And>x. monotone (fun_ord orda) orda (\<lambda>f. case x of (x, xa) \<Rightarrow> F (curry f) x xa)"   and "\<And>y. monotone (fun_ord ordb) ordb (\<lambda>g. case y of (x, xa) \<Rightarrow> G (curry g) x xa)"   and "f \<equiv> curry (ccpo.fixp (fun_lub luba) (fun_ord orda) (\<lambda>f (x, y). F (curry f) x y))"   and "g \<equiv> curry (ccpo.fixp (fun_lub lubb) (fun_ord ordb) (\<lambda>g (x, y). G (curry g) x y))"   and "ccpo.admissible (prod_lub (fun_lub luba) (fun_lub lubb)) (rel_prod (fun_ord orda) (fun_ord ordb)) (\<lambda>x. P (curry (fst x)) (curry (snd x)))"   and "P (\<lambda>_ y. luba {}) (\<lambda>_ y. lubb {})"   and "\<And>f g. P f g \<Longrightarrow> P (F f) (G g)" shows "P f g"
cases: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and a :: "('a, 'b, 'c, 'd) converter" assumes "plossless_converter \<I> \<I>' a" obtains conv :: "('a, 'b, 'c, 'd) converter" where "a = conv" and "\<And>a. a \<in> outs_\<I> \<I> \<Longrightarrow> plossless_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> plossless_converter \<I> \<I>' conv'"
s1_def: shows "s1 = dfa_steps A 0 bs"
assms_17: shows "selected_edge h j g \<le> - forest_components f"
rightUniqueFunctionAfterInverse: fixes f :: "('a \<times> 'b) set"   and Y :: "'b set" assumes "runiq f" shows "f `` f\<inverse> `` Y \<subseteq> Y"
rbl_succ: fixes n :: "nat"   and bin :: "int" shows "rbl_succ (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin + 1))"
sw_compose: fixes xs' :: "'b"   and x' :: "'a"   and y' :: "'a"   and zs :: "'b"   and xs :: "'b"   and x :: "'a"   and y :: "'a"   and z :: "'a" shows "z @zs[x \<and> y]_xs @zs[x' \<and> y']_xs' = z @zs[x' \<and> y']_xs' @zs[x @xs[x' \<and> y']_xs' \<and> y @xs[x' \<and> y']_xs']_xs"
isRefV_simps_1: fixes b :: "bool" shows "isRefV (boolV b) = False"
szip_eq: fixes u :: "'c stream"   and v :: "'d stream"   and r :: "'c stream"   and s :: "'d stream" shows "(u ||| v = r ||| s) = (u = r \<and> v = s)"
kruskal_correct_forest: shows "<emp> kruskal (return L) (\<lambda>(u, w, v). return (u, v)) () <\<lambda>r. \<up> (distinct r \<and> set r \<subseteq> E \<and> s.MSF (set (map id r)))>\<^sub>t"
Inf_mono: fixes B :: "'a set"   and A :: "'a set" assumes "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b" shows "\<Sqinter> A \<le> \<Sqinter> B"
fps_integral0_fps_const': fixes c :: "'a" assumes "inverse (1::'a) = (1::'a)" shows "fps_integral0 (fps_const c) = fps_const c * fps_X"
monad_state_prob_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_prob return bind sample" shows "monad_state_prob (return_state return) (bind_state bind) get_state put_state (sample_state sample)"
list_id_3: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "map f [x] = [f x]"
collapse: fixes sem_env :: "'a sem_env" shows "make_sem_env (sem_env.v sem_env) (c sem_env) = sem_env"
yEqAFresh': shows "y = x' \<or> qAFresh xs y Y'"
assms_2: shows "\<forall>n''>n'. \<not> \<parallel>c\<parallel>\<^bsub>t n''\<^esub>"
card_number_partitions_with_only_parts_1_eq_1: fixes n :: "nat"   and x :: "nat" assumes "n \<le> x" shows "card {N. (\<forall>n. n \<in># N \<longrightarrow> n = 1) \<and> number_partition n N \<and> size N \<le> x} = 1"
exp_conj_assoc: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "unitary P"   and "unitary Q"   and "unitary R" shows "P && (Q && R) = P && Q && R"
obj_simps_3: fixes a :: "'a" assumes "obj a" shows "trg a = a"
rSym: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "(\<Psi>, Q, P) \<in> X \<union> weakenBisim"
simps_10: fixes v :: "Core.term"   and vb :: "String.literal"   and vc :: "typ" shows "is_all (v $ Ct vb vc) = False"
loose_bvar_subst_bv1: fixes t :: "Core.term"   and lev :: "nat"   and u :: "Core.term" assumes "\<not> loose_bvar (subst_bv1 t lev u) lev" shows "\<not> loose_bvar t (Suc lev)"
bigtheta_trans2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)"   and "g \<in> L F h" shows "f \<in> L F h"
assms_2: shows "\<forall>i. X i \<le> B"
D_T: fixes s :: "'c event list"   and P :: "'c process" assumes "s \<in> D P" shows "s \<in> T P"
dependent_biggerset: fixes S :: "'a set" assumes "finite S \<Longrightarrow> DIM('a) < card S" shows "dependent S"
AuthReq_msg_in_analz_spies: fixes M :: "agent"   and P_I :: "msg"   and OIData :: "msg"   and HPIData :: "msg"   and evs :: "event list" assumes "Gets M \<lbrace>P_I, OIData, HPIData\<rbrace> \<in> set evs"   and "evs \<in> set_pur" shows "P_I \<in> analz (knows Spy evs)"
lunit_simps_4: fixes f :: "'a" assumes "ide f" shows "local.dom \<l>[f] = trg f \<star> f"
cyclic_on_list_succ: fixes xs :: "'b list" assumes "distinct xs"   and "xs \<noteq> []" shows "cyclic_on (list_succ xs) (set xs)"
subst_bv1_special_case_subst_bvs1: fixes t :: "Core.term"   and lev :: "nat"   and x :: "Core.term" shows "subst_bvs1 t lev [x] = subst_bv1 t lev x"
assms_3: shows "P \<rightarrow> \<langle>\<alpha>,P'\<rangle>"
IF1wit_def: fixes x :: "'b" shows "IF1wit x = ctor1 (wit2_F1 x (ctor2 wit_F2))"
dom_cod: fixes f :: "'c \<times> 'c" assumes "CopxC.arr f" shows "CopxC.dom (CopxC.cod f) = CopxC.cod f"
commutes_with_unit: shows "\<psi> V \<zero>\<^bsub>V\<^esub> = zero"
set_alloc: fixes v :: "'b"   and h :: "heap"   and v' :: "'b" shows "Ref.set (fst (Ref.alloc v h)) v' (snd (Ref.alloc v h)) = snd (Ref.alloc v' h)"
implies_not_eq: fixes A :: "'b set"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "irreflexive A less"   and "less x y"   and "x \<in> A" shows "x \<noteq> y"
w: shows "\<langle>w\<rangle> \<Turnstile>\<^sub>t t'"
ovalidNF_wp_comb2: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool" assumes "ovalidNF P f Q"   and "\<And>s. P' s \<Longrightarrow> P s" shows "ovalidNF P' f Q"
wf_all: fixes \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "TVarB T \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f U" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f (\<forall><:T. U)"
update_def: fixes ks :: "'c list"   and v :: "'d"   and t :: "('c, 'd) trie" shows "update ks v t = Trie2.trie.Trie (update_trie ks v (impl_of t))"
less_vec_def: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
less: fixes y :: "'a" assumes "(y, u) \<in> r" shows "Ord (ordermap A r y)"
inst_cswitch_consistency: fixes u :: "thread_id_t"   and s :: "rstate_t"   and t :: "rstate_t"   and n :: "nat" assumes "rvpeq u s t" shows "rvpeq u (rcswitch n s) (rcswitch n t)"
ifFalse_deterministic: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and e1 :: "('a, 'b, 'c) expr"   and e2 :: "('a, 'b, 'c) expr"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV F)) e1 e2])" shows "substitution.revision_step subst r s s' = (s' = s(r \<mapsto> (\<sigma>, \<tau>, \<E> [e2])))"
simps_2: fixes c :: "color"   and lt :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and rt :: "('a, 'b) rbt" shows "inv1 (Branch c lt k v rt) = (inv1 lt \<and> inv1 rt \<and> (c = B \<or> color_of lt = B \<and> color_of rt = B))"
s: fixes n :: "nat" shows "\<not> dist (f (s n) x) (F x) < \<epsilon>"
np: shows "\<not> bprv (PPf n \<langle>\<phi>\<rangle>)"
factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "pl.factorization_m f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> pl.eq_m f (Polynomial.smult c (\<Prod>\<^sub># fs)) \<and> (\<forall>f\<in>#fs. pl.irreducible\<^sub>d_m f \<and> monic (pl.Mp f))"
set_map_spmf: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e spmf" shows "set_spmf (map_spmf f p) = f ` set_spmf p"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "F.wtE \<xi>" shows "F.satPB \<xi> \<Phi>"
NF_ND: fixes s :: "'c event list"   and X :: "'c event set"   and P :: "'c process" assumes "(s, X) \<notin> F P" shows "s \<notin> D P"
simps_1: fixes I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset" shows "satisfies I1 I2 T = True"
A: shows "eexp (ereal (epsilonG TYPE('a)) * 1) \<le> esqrt 2"
dickson_grading_lcs: fixes d :: "'b \<Rightarrow> nat"   and s :: "'b"   and t :: "'b" assumes "dickson_grading d" shows "d (lcs s t) \<le> ord_class.max (d s) (d t)"
IH: fixes i1 :: "'index"   and s1 :: "'state1" assumes "match i1 s1 s2" shows "(\<exists>i2 s1'. step1\<^sup>+\<^sup>+ s1 s1' \<and> match i2 s1' s2') \<or> (\<exists>i2. match i2 s1 s2' \<and> (\<sqsubset>)\<^sup>+\<^sup>+ i2 i1)"
fg2: fixes x :: "'a" assumes "x \<in> T" shows "f x = p (g2 x)"
assms_3: shows "\<forall>x\<in>S. \<exists>e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
from_hma\<^sub>v_add: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec"   and w :: "('c, 'd) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v + Mod_Type_Connect.from_hma\<^sub>v w = Mod_Type_Connect.from_hma\<^sub>v (v + w)"
simps_1: fixes uu :: "'a dres" shows "(dSUCCEEDi \<le> uu) = True"
distinct_5: fixes x1 :: "int"   and x4 :: "int" shows "Const' x1 \<noteq> Smaller' x4"
condition2_part5: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f"   and m :: "'f" assumes "is_bezout_ext bezout"   and "echelon_form_upt_k A k"   and "k < ncols A"   and "a < b"   and "\<not> is_zero_row_upt_k a (Suc k) (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k))"   and "\<not> is_zero_row_upt_k b (Suc k) (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k))"   and "mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) < m"   and "A $ m $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "(if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) \<noteq> nrows A" shows "(LEAST n. fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k) $ a $ n \<noteq> (0::'d)) < (LEAST n. fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k) $ b $ n \<noteq> (0::'d))"
exhaustive_natural_def: fixes f :: "natural \<Rightarrow> (bool \<times> term list) option"   and d :: "natural" shows "exhaustive_class.exhaustive f d = Quickcheck_Exhaustive.exhaustive_natural' f d 0"
index_append: fixes xs :: "'e list"   and ys :: "'e list"   and x :: "'e" shows "index (xs @ ys) x = (if x \<in> set xs then index xs x else length xs + index ys x)"
constructive_security_obsf_composability: fixes middle :: "('m + 'n, 'o + 'p) resource"   and ideal :: "('q + 'n, 'r + 'p) resource"   and sim_inner :: "('m, 'o, 'q, 'r) converter"   and \<I>_middle :: "('m, 'o) \<I>"   and \<I>_inner :: "('q, 'r) \<I>"   and \<I>_common :: "('n, 'p) \<I>"   and \<A> :: "('s + 'n, 't + 'p) distinguisher_obsf"   and sim_outer :: "('s, 't, 'm, 'o) converter"   and adv1 :: "real"   and real :: "('s + 'n, 't + 'p) resource"   and \<I>_real :: "('s, 't) \<I>"   and adv2 :: "real" assumes "constructive_security_obsf middle ideal sim_inner \<I>_middle \<I>_inner \<I>_common (absorb \<A> (obsf_converter (sim_outer |\<^sub>= 1\<^sub>C))) adv1"   and "constructive_security_obsf real middle sim_outer \<I>_real \<I>_middle \<I>_common \<A> adv2" shows "constructive_security_obsf real ideal (sim_outer \<odot> sim_inner) \<I>_real \<I>_inner \<I>_common \<A> (adv1 + adv2)"
is_regular_zero_sheaf_spec: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U" shows "comm_ring.is_regular R addition multiplication zero unit (comm_ring.zero_sheaf_spec R addition multiplication zero unit U) U"
valid_add_new_table_msaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "'msaux"   and auxlist :: "(nat \<times> event_data option list set) list"   and rel2 :: "event_data option list set" assumes "valid_msaux args cur aux auxlist"   and "table (args_n args) (args_R args) rel2" shows "valid_msaux args cur (add_new_table_msaux args rel2 aux) (case auxlist of [] \<Rightarrow> [(cur, rel2)] | (t, y) # ts \<Rightarrow> if t = cur then (t, y \<union> rel2) # ts else (cur, rel2) # auxlist)"
lin_indpt_subset_cols: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b vec set" assumes "A \<in> carrier_mat n n"   and "invertible_mat A"   and "B \<subseteq> set (cols A)" shows "\<not> module.lin_dep class_ring (module_vec TYPE('b) n) B"
rank_invar_not_empty_hd: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.rank_invar (t # bq)"   and "bq \<noteq> []" shows "BinomialHeapStruc.rank t < BinomialHeapStruc.rank (hd bq)"
takefill_last_def: fixes x :: "'a"   and n :: "nat"   and xs :: "'a list" shows "takefill_last x n xs = takefill (last (x # xs)) n xs"
preserves_hom: fixes f :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr"   and a :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr"   and b :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr" assumes "VV.in_hom f a b" shows "\<guillemotleft>cmp\<^sub>D\<^sub>N f : HoDN_DN.map a \<Rightarrow>\<^sub>B DNoH.map b\<guillemotright>"
up_part: shows "updown' dm lm d (up dm lm d \<alpha>) p = (\<Sum>p''\<in>(\<Union>p'\<in>lgrid (base {0..<d} p) {0..<d} lm. lgrid p' {d} lm - {p'}). \<alpha> p'' * (\<Prod>d' = 0..<Suc d. l2_\<phi> (p'' ! d') (p ! d')))"
homology_dimension_axiom: fixes X :: "'a topology"   and a :: "'a"   and p :: "int" assumes "topspace X = {a}"   and "p \<noteq> 0" shows "trivial_group (homology_group p X)"
union_diff2: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) = sum g (A - B) + sum g (B - A) + sum g (A \<inter> B)"
currentD_weight_out: fixes \<Gamma> :: "('k, 'l) web_scheme"   and h :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k"   and y :: "'k" assumes "current \<Gamma> h" shows "h (x, y) \<le> weight \<Gamma> x"
derangementsE: fixes d :: "nat \<Rightarrow> nat"   and S :: "nat set" assumes "d \<in> derangements S" obtains "d permutes S" and "\<forall>x\<in>S. d x \<noteq> x"
subst_atm_list_empty: fixes \<sigma> :: "'s" shows "[] \<cdot>al \<sigma> = []"
boundary_def: fixes X :: "'a set" shows "boundary X = closure X - interior X"
CFG_return_edge_fun: fixes a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and ins :: "'c list"   and outs :: "'c list"   and vmap :: "'c \<Rightarrow> 'd option"   and vmap' :: "'c \<Rightarrow> 'd option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"   and "(p, ins, outs) \<in> set procs" shows "f' vmap vmap' = vmap'(ParamDefs (targetnode a) [:=] map vmap outs)"
hyps2: shows "card {M. set_mset M \<subseteq> insert x A \<and> size M = k} = card (insert x A) + k - 1 choose k"
eq_iff: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" shows "(x = y) = (x \<preceq>\<^sub>p y \<and> y \<preceq>\<^sub>p x)"
verit_sko_forall: fixes P :: "'b \<Rightarrow> bool" shows "(\<forall>x. P x) = P (SOME x. \<not> P x)"
component_in_hom: fixes j :: "'j" assumes "J.arr j" shows "\<guillemotleft>\<chi> j : a \<rightarrow> D (J.cod j)\<guillemotright>"
map2_filter: fixes f :: "'d \<rightarrow> 'c"   and P :: "'b \<rightarrow> tr"   and xs :: "[\<langle>'b, 'd\<rangle>]" shows "map\<cdot>(\<Lambda> \<langle>x, y\<rangle>. \<langle>x, f\<cdot>y\<rangle>)\<cdot>(filter\<cdot>(\<Lambda> \<langle>x, y\<rangle>. P\<cdot>x)\<cdot>xs) = filter\<cdot>(\<Lambda> \<langle>x, y\<rangle>. P\<cdot>x)\<cdot>(map\<cdot>(\<Lambda> \<langle>x, y\<rangle>. \<langle>x, f\<cdot>y\<rangle>)\<cdot>xs)"
null_set_Int1: fixes B :: "'c set"   and M :: "'c measure"   and A :: "'c set" assumes "B \<in> null_sets M"   and "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
thin_locs_post_discardE_2: fixes s' :: "('c, 'b, 'a, 'd, 'e) system_state"   and s :: "('f, 'b, 'a, 'g, 'h) system_state"   and p :: "'a"   and lfn :: "'b set"   and l' :: "'b"   and P :: "bool" assumes "AT s' = (AT s)(p := lfn)"   and "l' \<notin> lfn" shows "at p l' s' \<longrightarrow> P"
dim_span: fixes S :: "'d set" shows "cdim (cspan S) = cdim S"
\<epsilon>_leg_simps_1: shows "\<epsilon>.cod.leg0 = f.dtrg"
HOM_is_module: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme" assumes "R module M"   and "R module N" shows "R module HOM\<^bsub>R\<^esub> M N"
induced_morphism_eval: fixes C :: "('c set \<times> 'd) set"   and r :: "'c set \<times> 'd" assumes "C \<in> stfx.carrier_stalk"   and "r \<in> C" shows "induced_morphism C = stx.class_of (f \<^sup>\<inverse> X (fst r)) (\<phi>\<^sub>f (fst r) (snd r))"
norm_le_1_iff_in_hyp2_S: fixes v :: "(real, 2) vec" shows "(norm v \<le> 1) = (proj2_pt v \<in> hyp2 \<union> S)"
distinct_2: fixes x2 :: "'b"   and x1 :: "'a" shows "TargetTerm x2 \<noteq> SourceTerm x1"
b: shows "Vx : B \<turnstile> d'\<equiv>e' : C \<in> Axioms.Theory T"
True_steps_conc: fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "((True # p, q) \<in> NAe.steps (RegExp2NAe.conc L R) w) = ((\<exists>r. (p, r) \<in> NAe.steps L w \<and> q = True # r) \<or> (\<exists>u v. w = u @ v \<and> (\<exists>r. (p, r) \<in> NAe.steps L u \<and> fin L r \<and> (\<exists>s. (start R, s) \<in> NAe.steps R v \<and> q = False # s))))"
regular_def: fixes f :: "'b set \<Rightarrow> 'b set" shows "regular f = (setmonotone f \<and> continuous f)"
actionsE: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> actions E" obtains "enat a < llength E"
induct: fixes P2 :: "s list \<Rightarrow> bool"   and P1 :: "s \<Rightarrow> bool"   and s :: "s" assumes "\<And>x. P2 x \<Longrightarrow> P1 (s_block x)"   and "\<And>x1 x2. P1 (s_ass x1 x2)"   and "\<And>x1 x2 x3. P1 (s_read x1 x2 x3)"   and "\<And>x1 x2 x3. P1 (s_write x1 x2 x3)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P1 x3; P1 x4\<rbrakk> \<Longrightarrow> P1 (s_if x1 x2 x3 x4)"   and "\<And>x1 x2 x3. P1 (s_new x1 x2 x3)"   and "\<And>x1 x2 x3 x4. P1 (s_call x1 x2 x3 x4)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 s"
a_2: shows "0 < PZ.prob {x. a < \<bar>x\<bar>}"
fresh_2: fixes a :: "atom"   and fmla1 :: "fmla"   and fmla2 :: "fmla" shows "a \<sharp> fmla1 OR fmla2 = (a \<sharp> fmla1 \<and> a \<sharp> fmla2)"
that: fixes k :: "'a\<cdot>'b error\<cdot>'c" assumes "y = ErrorT\<cdot>k" shows "thesis"
that: fixes x :: "nat" assumes "distance n n' x"   and "x \<le> length as" shows "thesis"
r_consist_upto_recfn: fixes r_psi :: "recf" assumes "recfn 2 r_psi" shows "recfn 3 (r_consist_upto r_psi)"
simps: fixes a :: "'b set"   and A :: "'b set set" shows "(a \<in> sigma A) = ((\<exists>a. a = a \<and> a \<in> A) \<or> a = {} \<or> (\<exists>a. a = - a \<and> a \<in> sigma A) \<or> (\<exists>a. a = \<Union> (range a) \<and> (\<forall>x. a x \<in> sigma A)))"
simps: fixes gpv :: "('b, 'c2, 'r2) gpv"   and s2 :: "'s2"   and s1 :: "'s1" shows "local.inline2 gpv s2 s1 = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s2, s1))) (\<lambda>out rpv. inline1 callee1 (callee2 s2 out) s1 \<bind> case_sum (\<lambda>((r2, s2), s1). local.inline2 (rpv r2) s2 s1) (\<lambda>(x, rpv'', rpv'). return_spmf (Inr (x, rpv'', rpv', rpv))))"
assms_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t A)"
Y: shows "Y = Y - {#y \<in># Y. \<exists>x. x \<in># M \<and> P y x#} + {#y \<in># Y. \<exists>x. x \<in># M \<and> P y x#}"
while_unfold: fixes b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "while b c s = (if b s then while b c (c s) else s)"
ret: shows "(\<pi> @\<^bsup>n\<^esup> \<pi>') (n + r) = local.return"
invalidate_changes: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "invalidate rt dests dip = Some (dsn, dsk, flag, hops, nhip)" shows "dsn = (case dests dip of None \<Rightarrow> \<pi>\<^sub>2 (the (rt dip)) | Some rsn \<Rightarrow> rsn) \<and> dsk = \<pi>\<^sub>3 (the (rt dip)) \<and> flag = (if dests dip = None then \<pi>\<^sub>4 (the (rt dip)) else Aodv_Basic.inv) \<and> hops = \<pi>\<^sub>5 (the (rt dip)) \<and> nhip = \<pi>\<^sub>6 (the (rt dip))"
SINK_Sup: fixes Y :: "('k \<times> 'k \<Rightarrow> ennreal) set" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "countable (support_flow (\<Squnion> Y))" shows "MFMC_Network.SINK (\<Squnion> Y) = \<Inter> (MFMC_Network.SINK ` Y)"
disjnt_insert: fixes x :: "'c"   and N :: "'c set"   and M :: "'c set" assumes "x \<notin> N"   and "disjnt M N" shows "disjnt (insert x M) N"
nsum_mem: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j \<in> carrier A" shows "\<Sigma>\<^sub>e A f n \<in> carrier A"
cfg_onD_cont: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)"   and t :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> MDP.cfg_on s"   and "t \<in> set_pmf (action cfg)" shows "cont cfg t \<in> MDP.cfg_on t"
discount_factor_init: fixes r :: "real" shows "discount_factor r 0 = (\<lambda>w. 1)"