hcju/isabelleps huggingface dataset:
rel_F_pos_distr: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co3 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co3' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra1 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2' :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and Contra3 :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra3' :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool"   and uu :: "('af \<times> 'ag \<times> 'ah \<times> 'ai \<times> 'aj \<times> 'ak \<times> 'al \<times> 'am \<times> 'an \<times> 'ao) itself"   and L1 :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and L2 :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and L3 :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and L1' :: "'ag \<Rightarrow> 'ah \<Rightarrow> bool"   and L2' :: "'aj \<Rightarrow> 'ak \<Rightarrow> bool"   and L3' :: "'am \<Rightarrow> 'an \<Rightarrow> bool" assumes "rel_F_pos_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' uu" shows "rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 OO rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3' \<le> rel_F (L1 OO L1') (L2 OO L2') (L3 OO L3') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3')"
cfun_eqI: fixes f :: "'c \<rightarrow> 'd"   and g :: "'c \<rightarrow> 'd" assumes "\<And>x. f\<cdot>x = g\<cdot>x" shows "f = g"
path_conforms_with_strategy_ltl: fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy p P \<sigma>" shows "path_conforms_with_strategy p (ltl P) \<sigma>"
pure_conj_right: fixes Q :: "'a \<Rightarrow> bool"   and P' :: "bool"   and Q' :: "'a \<Rightarrow> bool" shows "(Q \<and>* (\<lambda>s. P' \<and> Q' s)) = (\<lambda>s. P' \<and> (Q \<and>* Q') s)"
All2_Zero: shows "{} \<turnstile> All2 i Zero \<alpha>"
2: shows "singular_chain (p - Suc 0) (subtopology X S) (chain_boundary p e)"
add_divide_distrib: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
eq_integrals: shows "one_chain_line_integral F {i} \<H> = one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
shiftl_mask_is_0: fixes x :: "'a word"   and n :: "nat" shows "(x << n) && mask n = 0"
2: shows "\<Union> (A(x := A x - {Suc j}) ` {1..m}) = {1..j}"
IH: assumes "\<forall>xs\<in>set xss. length xs = d"   and "0 < d" shows "chop d (concat xss) = xss"
assms_3: shows "x \<in> Units (residue_ring (p ^ m))"
test_bit_conj_lt: fixes x :: "'a word"   and m :: "nat" shows "(x !! m \<and> m < LENGTH('a)) = x !! m"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "Preflow c s t f"   and "Finite_Graph c" shows "Finite_Preflow c s t f"
10: shows "\<forall>\<^sub>F k in sequentially. \<bar>\<xi> - (\<Sum>k = 0..k. bb k / aa k)\<bar> < 2 / \<epsilon> * (bb (k + 1) / aa (k + 1)) powr (\<epsilon> / (1 + \<epsilon>))"
rel_gpv''_conversep: fixes A :: "'l \<Rightarrow> 'i \<Rightarrow> bool"   and C :: "'m \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'n \<Rightarrow> 'k \<Rightarrow> bool" shows "rel_gpv'' A\<inverse>\<inverse> C\<inverse>\<inverse> R\<inverse>\<inverse> = (rel_gpv'' A C R)\<inverse>\<inverse>"
eint_ord_code_2: fixes m :: "int"   and n :: "int" shows "(eint m < eint n) = (m < n)"
arith_simps_51: shows "- (0::'e) = (0::'e)"
respection: shows "rel_weakly_respects_barbs (indRelTEQ TRel) (STCalWB SWB TWB)"
subset_eq: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
binopI: fixes A :: "V"   and f :: "V" assumes "nop A (2\<^sub>\<nat>) f" shows "binop A f"
add_less_le_mono: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset"   and d :: "'a multiset" assumes "a \<subset># b"   and "c \<subseteq># d" shows "a + c \<subset># b + d"
neg4: shows "Negligible.negligible (\<lambda>n. local.ddh.advantage n (local.ddh.DDH_\<A>' n (P1_DDH_mal_adv_\<sigma>_false n M z \<A> D)))"
assms: shows "learn_bc \<psi> U (\<phi> i)"
distinct_3: fixes x1 :: "'a"   and x3 :: "'a \<times> nat" shows "CFG_node x1 \<noteq> Formal_out x3"
a_heap_is_wellformed_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed get_child_nodes get_disconnected_nodes h = (l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_acyclic_heap get_child_nodes h \<and> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_all_ptrs_in_heap get_child_nodes get_disconnected_nodes h \<and> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_distinct_lists get_child_nodes get_disconnected_nodes h \<and> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_owner_document_valid get_child_nodes get_disconnected_nodes h)"
twise_set_pair_index: fixes ps :: "'a set"   and ps2 :: "'a set" assumes "ps \<subseteq> \<V>"   and "ps2 \<subseteq> \<V>"   and "ps \<noteq> ps2"   and "int (card ps) = \<t>"   and "int (card ps2) = \<t>" shows "\<B> index ps = \<B> index ps2"
rr: shows "set rr = {x. poly (real_of_int_poly p) x = 0}"
derivFlip: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/- F \<^esub>P c UNIV,UNIV"
thetaSeqZOT_ZOretrT: shows "thetaSeqZOT \<subseteq> ZOretrT (thetaSeqZOT \<union> ZObisT)"
advantage_advantage1: fixes \<A> :: "(unit, 'a + 'a \<times> 'b, 'b + unit) gpv"   and q :: "nat"   and key_gen :: "'c spmf"   and hash :: "'c \<Rightarrow> 'a \<Rightarrow> 'b" assumes "interaction_bounded_by' (Not \<circ> isl) \<A> q" shows "upf.advantage key_gen hash \<A> \<le> upf.advantage1 key_gen hash (guessing_many_one.reduction q (\<lambda>_. \<A>) ()) * real q"
assms_4: shows "GEN_OP uni (\<union>) (S \<rightarrow> S \<rightarrow> S)"
Lcm_atLeast1AtMost_Suc: fixes n :: "nat" shows "Lcm {1..Suc n} = Lcm {1..n} * (if primepow (Suc n) then aprimedivisor (Suc n) else 1)"
Cidl: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f) ;;\<^bsub>C\<^esub> f = f"
trans_MapOps_update_def: shows "trans_MapOps_update \<equiv> \<lambda>k v m. trie_odlist_update_with (fst (fst k)) m empty_trie (\<lambda>m. trie_odlist_update_with (snd (fst k)) m Mapping.empty (Mapping.update (snd k) v))"
no_loops: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c" assumes "Graph G" shows "(v, v) \<notin> E\<^bsub>G\<^esub>"
assms: shows "x \<in> fv (t \<cdot> \<delta>)"
gval_gNot: fixes x :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (gNot x) s = \<not>? gval x s"
natural_transformation_axioms: shows "natural_transformation CC.comp CC.comp \<Delta>o\<Pi>.map CC.map \<pi>.map"
simps_2: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "stl (siterate f x) = siterate f (f x)"
cmult: fixes c :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "O[F](\<lambda>x. c * f x) = O[F](f)"
dis_introR: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'"   and "\<psi> \<pi>l" shows "Shallow.dis \<phi> \<psi> \<pi>l"
ring_simprules_2: fixes R :: "('d, 'e) ring_scheme" assumes "ring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
rSR1_eq: shows "removeShadowRules1_alternative = removeShadowRules1"
lsteps_comp_map: fixes zs :: "'a list"   and f :: "'b \<Rightarrow> nat"   and g :: "'a \<Rightarrow> 'b"   and l :: "nat"   and u :: "nat"   and v :: "nat" shows "lsteps zs (f \<circ> g) l u v = lsteps (map g zs) f l u v"
simps: fixes xs :: "'a list" shows "IArray.list_of (IArray xs) = xs"
prj_in_hom_4: shows "\<guillemotleft>Prj\<^sub>1 : \<mu>\<nu>\<pi>.chine \<rightarrow>\<^sub>C \<mu>.apex\<guillemotright>"
is_Ref_def2: fixes v :: "'b val" shows "is_Ref v = (v = Null \<or> (\<exists>a. v = Addr a))"
the_inv_into_f_eq: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d"   and y :: "'e" assumes "inj_on f A"   and "f x = y"   and "x \<in> A" shows "the_inv_into A f y = x"
circ_plus_same: fixes x :: "'a" shows "top \<sqinter> x = x \<sqinter> top"
return_def: fixes a :: "'d" shows "return a \<equiv> \<lambda>s. ({(a, s)}, False)"
head: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "comm_monoid_set f z"   and "m \<le> n" shows "comm_monoid_set.F f z g {m..n} = f (g m) (comm_monoid_set.F f z g {m<..n})"
cases: fixes a :: "'a"   and s :: "'b + 'a" assumes "a \<in> Basic_BNFs.setr s" obtains x :: "'a" where "a = x" and "s = Inr x"
convex_Times: fixes s :: "'a set"   and t :: "'b set" assumes "convex s"   and "convex t" shows "convex (s \<times> t)"
get_parent_locs_def: shows "Shadow_DOM.get_parent_locs \<equiv> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent_locs Shadow_DOM.get_child_nodes_locs"
finite_Linear_order_induct: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "Linear_order r"   and "x \<in> Field r"   and "finite r"   and "\<And>x. \<lbrakk>x \<in> Field r; \<And>y. y \<in> aboveS r x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P x"
these_insert_None: fixes A :: "\<langle>'d\<rangle>\<^sub>\<bottom> set" shows "Option.these (insert None A) = Option.these A"
cases: fixes x :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" obtains A :: "('sf, 'sc, 'sz) formula list" and S :: "('sf, 'sc, 'sz) formula list" where "x = (A, S)"
Gamma_conv_nn_integral_real: fixes s :: "real" assumes "0 < s" shows "ennreal (Gamma s) = \<integral>\<^sup>+ t. ennreal (indicat_real {0..} t * t powr (s - 1) / exp t) \<partial>lborel"
igWlsInp_def: fixes MOD :: "('a, 'b, 'c, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'f option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
elims: fixes x :: "interpolation_algorithm"   and y :: "('a \<times> 'a) list \<Rightarrow> 'a poly" assumes "interpolation_poly x = y" obtains "x = Newton" and "y = newton_interpolation_poly"   | "x = Lagrange" and "y = lagrange_interpolation_poly"   | "x = Neville_Aitken" and "y = neville_aitken_interpolation_poly"
B: shows "dist x (geodesic_segment_param H x t) = t"
f_approx_aux: fixes x :: "real" assumes "x\<^sub>0 \<le> x" shows "1 \<le> 1 + integral (\<lambda>u. g u / u powr (p + 1)) x\<^sub>0 x"
Fvars_\<psi>: shows "Fvars \<psi> = {}"
Q_nonempty: fixes Bot :: "'a set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "consequence_relation_family Bot Q entails_q" shows "Q \<noteq> {}"
\<Omega>: shows "{f -` A \<inter> \<Omega> |A. A \<in> X} \<subseteq> Pow \<Omega>"
Pre_in_segment: fixes a :: "'a" assumes "a \<in> carrier D"   and "ExPre D a" shows "Pre D a \<in> segment D a"
wf_if_measure: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> nat"   and g :: "'c \<Rightarrow> 'c" assumes "\<And>x. P x \<Longrightarrow> f (g x) < f x" shows "wf {(y, x). P x \<and> y = g x}"
locally_finite_in_closure: fixes X :: "'c topology"   and \<A> :: "'c set set" assumes "locally_finite_in X \<A>" shows "locally_finite_in X ((closure_of) X ` \<A>)"
game_A_def: shows "game_A = seed_gen \<bind> (\<lambda>seed. exec_gpv prf.random_oracle \<A> Map.empty \<bind> (\<lambda>(b, \<sigma>). return_spmf (b, \<not> inj_on (h seed) (dom \<sigma>))))"
varsC_def: fixes c :: "('b, 'c) lit list" shows "varsC c = \<Union> (set (map varsL c))"
mem_subring_mem_ring: fixes S :: "('a, 'd) Ring_scheme"   and x :: "'a" assumes "Subring R S"   and "x \<in> carrier S" shows "x \<in> carrier R"
sum_outgoing_pointwise: fixes g :: "nat \<times> nat \<Rightarrow> 'c"   and u :: "nat" shows "sum g (outgoing u) = (\<Sum>v\<in>E `` {u}. g (u, v))"
int_pow_eq: fixes x :: "'c"   and m :: "int"   and n :: "int" assumes "x \<in> carrier H" shows "(x [^]\<^bsub>H\<^esub> m = x [^]\<^bsub>H\<^esub> n) = (int (H.ord x) dvd n - m)"
igSubstIGVar2STR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstIGVar2STR MOD \<equiv> (\<forall>ys y y1. igSubst MOD ys (igVar MOD ys y1) y (igVar MOD ys y) = igVar MOD ys y1) \<and> (\<forall>ys y Y. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igVar MOD ys y) = Y)"
s_subid_closed: fixes x :: "'a" shows "x \<sqinter> nc \<sqinter> 1\<^sub>\<sigma> = x \<sqinter> 1\<^sub>\<sigma>"
assms: shows "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b"
compl_sup: fixes x :: "'a upred"   and y :: "'a upred" shows "(\<not> (x \<or> y)) = (\<not> x \<and> \<not> y)"
in_arith_progI: fixes x :: "int"   and a :: "int"   and b :: "nat" assumes "[x = a] (mod int b)" shows "x \<in> arith_prog a b"
elims_3: fixes x :: "'at set"   and xa :: "'at Literal set set" assumes "\<not> x \<Turnstile> xa" obtains I :: "'at set" and S :: "'at Literal set set" where "x = I" and "xa = S" and "\<not> (\<forall>C. C \<in> S \<longrightarrow> I \<Turnstile> C)"
assms_2: shows "inverse (f $$ fls_subdegree f) \<noteq> (0::'a)"
simps_2: fixes err :: "'b error_result"   and rs :: "('a, 'b) result list" shows "sequence_result (Rerr err # rs) = Rerr err"
not_R: fixes c :: "color" assumes "c \<noteq> R" shows "c = B"
apply_operations_def: fixes es :: "'a list" shows "apply_operations es \<equiv> foldl (\<rhd>) Some (map interp es)"
PSQ: shows "diagonal_to_Smith_PQ_JNF A bezout = (P, S, Q)"
fdomain_vsubset_VUnion2: fixes r :: "V" shows "\<D>\<^sub>\<bullet> r \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> r))"
simps_1: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "longest_common_prefix (x # xs) (y # ys) = (if x = y then x # longest_common_prefix xs ys else [])"
eval_monom_expr: fixes a :: "int" assumes "a \<in> carrier R" shows "eval R R id a (monom (UP R) \<one>\<^bsub>R\<^esub> 1 \<ominus>\<^bsub>UP R\<^esub> monom (UP R) a 0) = \<zero>\<^bsub>R\<^esub>"
Inf_action: fixes S :: "('c, 'd) action set" shows "Inf S \<equiv> if S = {} then top else action_of (\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> action.relation_of ` S})"
iso: shows "is_pr_iso \<phi> Fs\<^sub>1 Fs\<^sub>2"
hlog_starfun_ln: fixes x :: "real star" shows "(*f* real_ln) x = hlog ((*f* exp) 1) x"
set_sort_oalist: fixes comp1 :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and comp2 :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator2 comp1 comp2"   and "comparator.oalist_inv_raw comp2 xs" shows "set (comparator.sort_oalist comp1 xs) = set xs"
Simple_WPC_def: fixes t :: "'e itself"   and Rs :: "(('f, 'g) labeled_graph \<times> ('f, 'g) labeled_graph) set"   and S :: "nat \<Rightarrow> ('f, 'h) labeled_graph" shows "Simple_WPC t Rs S \<equiv> set_of_graph_rules Rs \<and> (\<forall>i. graph (S i) \<and> S i = S (Suc i) \<or> (\<exists>R\<in>Rs. pushout_step t R (S i) (S (Suc i))))"
normalize_manual_orderPrQ_def: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" shows "normalize_manual_orderPrQ p l = removeAllDuplicates (insertDenies (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l)))"
fs1: fixes x :: "'a" assumes "fs TYPE('a) TYPE('b)" shows "finite (supp x)"
efficient_power_correct: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a"   and n :: "nat" assumes "\<And>x z. f x (f x z) = f (f x x) z" shows "efficient_power f y x n = (f x ^^ n) y"
cong_bigtheta: fixes f :: "real \<Rightarrow> real"   and F :: "real filter"   and g :: "real \<Rightarrow> real" assumes "f \<in> \<Theta>[F](g)" shows "L F f = L F g"
extend_LeadsTo: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "Extend h" shows "(extend h F \<in> extend_set h A \<longmapsto>w extend_set h B) = (F \<in> A \<longmapsto>w B)"
prime_ideal_ideal: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "prime_ideal R I" shows "ideal R I"
simps_30: fixes \<C> :: "t_context"   and t :: "t"   and tp :: "tp option"   and a :: "nat"   and off :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Store t tp a off) ts = (if memory \<C> \<noteq> None \<and> load_store_t_bounds a tp t then type_update ts [TSome T_i32, TSome t] (Type []) else Bot)"
elims_3: fixes x :: "('c \<times> nat) option"   and xa :: "'c"   and xb :: "lock_action list" assumes "\<not> lock_actions_ok x xa xb" obtains l :: "('c \<times> nat) option" and t :: "'c" and L :: "lock_action" and Ls :: "lock_action list" where "x = l" and "xa = t" and "xb = L # Ls" and "\<not> (lock_action_ok l t L \<and> lock_actions_ok (upd_lock l t L) t Ls)"
assms: shows "vinsert a A \<subseteq>\<^sub>\<circ> B"
r_into_trancl: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a, b) \<in> r" shows "(a, b) \<in> r\<^sup>+"
altc_cong: fixes C :: "'c cset"   and f :: "'c \<Rightarrow> 'm"   and g :: "'c \<Rightarrow> 'm" assumes "cBall C (\<lambda>x. f x = g x)" shows "altc C f = altc C g"
fork_transitions_def: fixes i :: "nat"   and s :: "nat" shows "Tr\<^sub>f i s = (if s = 0 then {picks i i} \<union> {picks ((i + 1) mod N) i} else if s = 1 then {putsdown i i} else if s = 2 then {putsdown ((i + 1) mod N) i} else {})"
clock_repMC_aec: fixes t :: "'s Trace"   and aec :: "'s odlist"   and a :: "'a"   and cec :: "'s odlist"   and x :: "'s"   and y :: "'s" assumes "t \<in> Clock.jkbpC"   and "toSet aec = agent_abs a t"   and "toSet cec = common_abs t"   and "x \<in> toSet aec"   and "(x, y) \<in> relations (clock_repMC (toSet cec)) a" shows "y \<in> toSet aec"
swap_same: fixes l :: "'c list"   and i :: "nat" shows "swap l i i = l"
vfsequence_at_last: fixes xs :: "V"   and i :: "V"   and x :: "V" assumes "vfsequence xs"   and "i = vcard xs" shows "(xs #\<^sub>\<circ> x)\<lparr>i\<rparr> = x"
writerT_eqI: fixes a :: "'e\<cdot>('g, 'f) writerT"   and b :: "'e\<cdot>('g, 'f) writerT" assumes "runWriterT\<cdot>a = runWriterT\<cdot>b" shows "a = b"
consist: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> U" shows "eval r_consist_phi [the (eval t [f \<triangleright> n]), f \<triangleright> n] \<down>= (if \<forall>k\<le>n. \<phi> (the (eval t [f \<triangleright> n])) k = f k then 0 else 1)"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f :: "tcp_flag set \<Rightarrow> tcp_flag set \<Rightarrow> 'a"   and ipt_tcp_flags :: "ipt_tcp_flags" shows "P (case ipt_tcp_flags of TCP_Flags x xa \<Rightarrow> f x xa) = (\<nexists>x1 x2. ipt_tcp_flags = TCP_Flags x1 x2 \<and> \<not> P (f x1 x2))"
max_def: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
heads_face_cycle_set: fixes a :: "'b" assumes "a \<in> arcs G" shows "head G ` face_cycle_set a = tail G ` face_cycle_set a"
vanishes_mult_bounded: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"   and "Real.vanishes Y" shows "Real.vanishes (\<lambda>n. X n * Y n)"
SeqRedsThrow: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>throw e',s'\<rangle>" shows "P \<turnstile> \<langle>e;; e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>throw e',s'\<rangle>"
hyps_13: shows "A\<^sub>P \<sharp>* K"
assms_1: shows "\<forall>x. f x (0::'a) = x"
finite: fixes y :: "real" shows "finite {p. prime p \<and> real p \<le> y}"
axioms_1: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>F :: "'d \<times> 'd \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'d \<Rightarrow> 'e"   and \<tau>\<^sub>1 :: "'d \<Rightarrow> 'e" assumes "pseudonatural_transformation_whisker_right V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1" shows "bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"
pg_4: shows "path_image pg \<subseteq> s - insert p pts"
StdSet_no_invar_axioms: shows "StdSet_no_invar hs_ops"
IH: shows "setops\<^sub>s\<^sub>s\<^sub>t (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)) = setops\<^sub>s\<^sub>s\<^sub>t (unlabel A)"
induct: fixes x1 :: "'b literal set set"   and x2 :: "'b literal set"   and P :: "'b literal set set \<Rightarrow> 'b literal set \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>C S. C \<in> S \<Longrightarrow> P S C"   and "\<And>S C D k. \<lbrakk>S \<turnstile> C; P S C; S \<turnstile> D; P S D; k\<^sup>+ \<in> C; k\<inverse> \<in> D\<rbrakk> \<Longrightarrow> P S (C - {k\<^sup>+} \<union> (D - {k\<inverse>}))" shows "P x1 x2"
last_rev: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "last (rev xs) = hd xs"
positive_proj_fst_k: fixes k :: "nat" shows "positive (proj_fst_k k)"
carrier_submanifold: shows "a.dest.carrier = codomain c \<inter> manifold_eucl.dest.carrier"