hcju/isabelleps huggingface dataset:
inverse_transformation_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" shows "inverse_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_isomorphism A B F G \<tau>"
wp_under_wlp_Skip: shows "wp_under_wlp Skip"
append_filter: fixes as1 :: "'b list"   and as2 :: "'b list"   and f1 :: "'b \<Rightarrow> bool"   and f2 :: "'b \<Rightarrow> 'b"   and p :: "'b list" assumes "as1 @ as2 = filter f1 (map f2 p)" shows "\<exists>p_1 p_2. p_1 @ p_2 = p \<and> as1 = filter f1 (map f2 p_1) \<and> as2 = filter f1 (map f2 p_2)"
any_moves_remain_winning_Odd: fixes A :: "'c llist set"   and N :: "enat"   and p :: "'c list"   and \<sigma> :: "'c list \<Rightarrow> 'c"   and m :: "'c" assumes "GSgame A N"   and "even (length p)"   and "GSgame.strategy_winning_by_Odd A N \<sigma> p" shows "GSgame.strategy_winning_by_Odd A N \<sigma> (p @ [m])"
assms_6: shows "(A, B) \<in> ns_mul_ext ns s"
conjugation_pair_multr: fixes y :: "'a" shows "conjugation_pair ((\<cdot>) y) ((\<rhd>) y)"
S5Basic_3: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<diamond>\<phi> in v]"
take_add: fixes i :: "nat"   and j :: "nat"   and xs :: "'b list" shows "take (i + j) xs = take i xs @ take j (drop i xs)"
that: fixes q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "q \<in> set (comp_red_basis xs)"   and "lt q adds\<^sub>t lt p" shows "thesis"
IH: fixes es :: "expr list" assumes "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts''" shows "\<exists>Ts'. P,E,h \<turnstile> es [:] Ts' \<and> conformable Ts' Ts''"
comp_ide_arr: fixes b :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "ide b"   and "seq b f" shows "COMP b f = f"
godel_code_get_nth: fixes i :: "nat"   and nl :: "nat list" assumes "i < length nl" shows "Max {u. Pi (Suc i) ^ u dvd godel_code nl} = nl ! i"
ts_okI: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'d" assumes "\<And>t x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> P t x m" shows "ts_ok P ts m"
coeffs_eq_Nil: fixes p :: "'b poly" shows "(coeffs p = []) = (p = 0)"
image_nat_induct: fixes P :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and I :: "'a set"   and a :: "'a" assumes "P (f 0)"   and "\<And>n. P (f n) \<Longrightarrow> P (f (Suc n))"   and "surj_on f UNIV I"   and "a \<in> I" shows "P a"
is_rreq_def: fixes \<xi> :: "'b state_scheme" shows "is_rreq \<xi> \<equiv> case msg \<xi> of Rreq hops' rreqid' dip' dsn' dsk' oip' osn' sip' \<Rightarrow> {\<xi>\<lparr>hops := hops', rreqid := rreqid', dip := dip', dsn := dsn', dsk := dsk', oip := oip', osn := osn', sip := sip'\<rparr>} | _ \<Rightarrow> {}"
simps_5: fixes f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<Pi>\<^sub>1 set \<Rightarrow> 'a"   and x1 :: "\<omega>" shows "(case \<omega>\<nu> x1 of \<omega>\<nu> x \<Rightarrow> f1 x | \<alpha>\<nu> x \<Rightarrow> f2 x) = f1 x1"
IH: fixes v :: "'a" shows "repeat_stabilize n f v = (f ^^ n) v"
pState2HASH_eq: fixes x :: "'b pState_scheme"   and y :: "'b pState_scheme" assumes "pState2HASH x = pState2HASH y" shows "x = y"
bin_cat_eqD2: fixes n :: "nat"   and b :: "int"   and a :: "int"   and d :: "int"   and c :: "int" assumes "concat_bit n b a = concat_bit n d c" shows "take_bit n b = take_bit n d"
assms: shows "character (G\<lparr>carrier := adjoin G H a\<rparr>) \<chi>"
apply_self_in_orbit: fixes f :: "'b perm"   and a :: "'b" shows "apply f a \<in> orbit f a"
foldpairs_sym: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> foldpairs" shows "(g, f) \<in> foldpairs"
in_set_take_conv_nth: fixes x :: "'c"   and n :: "nat"   and xs :: "'c list" shows "(x \<in> set (take n xs)) = (\<exists>m<min n (length xs). xs ! m = x)"
irreducible\<^sub>dD_1: fixes p :: "'a poly" assumes "irreducible\<^sub>d p" shows "0 < degree p"
S_list_endomorphism: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "ChamberComplexEndomorphism X ((\<rightarrow>) (sum_list ss))"
IH: shows "\<exists>Di\<subseteq>D. Di \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I> \<subseteq> {d \<cdot>\<^sub>p \<I>} \<and> d \<cdot>\<^sub>p \<I> \<notin> D - Di \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>"
permute_finite: fixes p :: "perm"   and X :: "'c set" shows "finite (p \<bullet> X) = finite X"
has_fields_mono_lem2: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and F :: "char list"   and b :: "staticb"   and T :: "ty" assumes "P \<turnstile> C has_fields FDTs"   and "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "map_of FDTs (F, C) = \<lfloor>(b, T)\<rfloor>" shows "\<exists>FDTs'. FDTs = map (\<lambda>(F, b, T). ((F, C), b, T)) fs @ FDTs' \<and> map_of FDTs' (F, C) = None"
graph_homo_union: fixes A :: "('a, 'b) labeled_graph"   and G :: "('a, 'c) labeled_graph"   and f_a :: "('b \<times> 'c) set"   and B :: "('a, 'b) labeled_graph"   and f_b :: "('b \<times> 'c) set" assumes "graph_homomorphism A G f_a"   and "graph_homomorphism B G f_b"   and "Domain f_a \<inter> Domain f_b = Domain (f_a \<inter> f_b)" shows "graph_homomorphism (graph_union A B) G (f_a \<union> f_b)"
lexord_Nil_left: fixes y :: "'b list"   and r :: "('b \<times> 'b) set" shows "(([], y) \<in> lexord r) = (\<exists>a x. y = a # x)"
card_mono: fixes B :: "'d set"   and A :: "'d set" assumes "finite B"   and "A \<subseteq> B" shows "card A \<le> card B"
vsum_lepoll_self: fixes a :: "V"   and b :: "V" shows "elts a \<lesssim> elts (a \<Uplus> b)"
path_connected_2DIM_I: fixes P :: "real \<Rightarrow> bool"   and a :: "'a" assumes "2 \<le> DIM('a)"   and "path_connected {r. 0 \<le> r \<and> P r}" shows "path_connected {x. P (norm (x - a))}"
prerun_def: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme" shows "prerun sys = (\<lambda>s. (initial_state sys (GST (s 0)) \<and> HST (s 0) = []) \<and> (\<box>system_step_reflclp) s)"
rTest: fixes \<phi> :: "formula"   and \<nu> :: "fin_string \<Rightarrow> real"   and \<omega> :: "fin_string \<Rightarrow> real" assumes "[\<phi>]\<nu> \<downharpoonright> True"   and "\<nu> = \<omega>" shows "[? \<phi>]\<nu> \<downharpoonleft> \<omega>"
spec_def: fixes b :: "bool"   and b' :: "bool"   and P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "spec b b' P \<equiv> \<lambda>(A, A'). P (A\<lparr>wait := b'\<rparr>, A'\<lparr>ok := b\<rparr>)"
cases: fixes a :: "'node SDG_node"   and n :: "'node SDG_node" assumes "a \<in> sum_SDG_slice2 n" obtains "a = n" and "valid_SDG_node n"   | n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a = n''a__" and "n''a__ s\<longrightarrow>\<^bsub>cd\<^esub> n'a__" and "n'a__ \<in> sum_SDG_slice2 n"   | n'' :: "'node SDG_node" and V :: "'var" and n' :: "'node SDG_node" where "a = n''b__" and "n''b__ s-V__\<rightarrow>\<^sub>d\<^sub>d n'b__" and "n'b__ \<in> sum_SDG_slice2 n"   | n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a = n''c__" and "n''c__ s-pa__\<rightarrow>\<^bsub>ret\<^esub> n'c__" and "n'c__ \<in> sum_SDG_slice2 n"   | n'' :: "'node SDG_node" and p :: "'pname" and V :: "'var" and n' :: "'node SDG_node" where "a = n''d__" and "n''d__ s-pb__:V\<rightarrow>\<^bsub>out\<^esub> n'd__" and "n'd__ \<in> sum_SDG_slice2 n"   | n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a = n''" and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'" and "n' \<in> sum_SDG_slice2 n"
valid_W_distinct_simps_3: fixes r :: "'d"   and p :: "'e process_name"   and s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'd) Model.local_state"   and q :: "'e process_name" assumes "r \<in> WL p s"   and "valid_W_inv s" shows "(r \<in> WL q s) = (p = q)"
WhileHigh: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and b :: "BExpr" assumes "G \<rhd> c : HighSec" shows "G \<rhd> While b c : HighSec"
nin_vinsertI: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "a \<noteq> b"   and "a \<notin>\<^sub>\<circ> A" shows "a \<notin>\<^sub>\<circ> vinsert b A"
res_lim_Zp_cauchy_0_2: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and f :: "int"   and k :: "nat" assumes "is_Zp_cauchy s"   and "f = res_lim s k" shows "f \<in> carrier (residue_ring (p ^ k))"
any_node_exits_path: fixes v :: "nat" assumes "v \<in> set (g_V G)" shows "\<exists>p. path_entry (g_E G) p v"
vrrestriction_vid_on: fixes A :: "V"   and B :: "V" shows "vid_on A \<restriction>\<^sup>r\<^sub>\<circ> B = vid_on (A \<inter>\<^sub>\<circ> B)"
MRList1: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and b :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C a)" shows "applied_rule_rev C x (b @ [a]) = \<lfloor>a\<rfloor>"
canonical_tabulation: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>*) (tab\<^sub>0.trnr\<^sub>\<eta> (tab\<^sub>1 r) (tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>*)) (tab\<^sub>0 r) (tab\<^sub>1 r)"
\<section>: assumes "u \<noteq> 0" shows "g u \<in> closure S"
no_msg_change_if_no_event: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "isSend ev \<longrightarrow> getId ev \<noteq> i"   and "isRecv ev \<longrightarrow> getId ev \<noteq> i"   and "regular_event ev" shows "msgs c i = msgs c' i"
right_in_interval: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "interval r a b \<noteq> {}" shows "b \<in> interval r a b"
nn_integral_count_space_reindex: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'d \<Rightarrow> ennreal" assumes "inj_on f A" shows "integral\<^sup>N (count_space (f ` A)) g = (\<Sum>\<^sup>+ x\<in>A. g (f x))"
Seq_NoFaultStuckD1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` F" shows "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` F"
Q: fixes i :: "'a" assumes "P i" shows "pred M (Q i)"
piecewise_C1_differentiable_neg: fixes f :: "real \<Rightarrow> 'b"   and S :: "real set" assumes "f piecewise_C1_differentiable_on S" shows "(\<lambda>x. - f x) piecewise_C1_differentiable_on S"
isint_Floor: fixes n :: "MIR.num"   and bs :: "real list" shows "isint (Floor n) bs"
N_sums: fixes n :: "int" shows "(\<lambda>k. if k = 0 \<or> int k dvd n then 0 else 1 / q ^ k) sums N n"
subst_term_empty_no_change: fixes t :: "Core.term" shows "subst_term [] t = t"
in_rel_def: fixes R :: "('c \<times> 'd) set"   and x :: "'c"   and y :: "'d" shows "in_rel R x y \<equiv> (x, y) \<in> R"
gtrace_alt_def: fixes r :: "(nat \<times> 'a) stream"   and v :: "nat \<times> 'a" shows "gtrace r v = r"
trivial_empty_or_singleton: fixes X :: "'b set" shows "trivial X = (X = {} \<or> X = {the_elem X})"
prime_factors_gt_0_nat: fixes p :: "nat"   and x :: "nat" assumes "p \<in># prime_factorization x" shows "0 < p"
saturated_upto_def: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and N :: "'b literal multiset set" assumes "redundancy_criterion \<Gamma> Rf Ri" shows "redundancy_criterion.saturated_upto \<Gamma> Rf Ri N = (inference_system.inferences_from \<Gamma> (N - Rf N) \<subseteq> Ri N)"
ivp_unique_solution: fixes s :: "'a"   and U :: "'a \<Rightarrow> real set"   and t :: "real"   and Y\<^sub>1 :: "real \<Rightarrow> 'a"   and Y\<^sub>2 :: "real \<Rightarrow> 'a" assumes "s \<in> S"   and "is_interval (U s)"   and "U s \<subseteq> T"   and "t \<in> U s"   and "Y\<^sub>1 \<in> Sols (\<lambda>t. f) U S 0 s"   and "Y\<^sub>2 \<in> Sols (\<lambda>t. f) U S 0 s" shows "Y\<^sub>1 t = Y\<^sub>2 t"
transcendental_tan: fixes z :: "complex" assumes "algebraic z"   and "z \<noteq> 0" shows "\<not> algebraic (tan z)"
mem_simps_11: fixes A :: "'c"   and C :: "'c set set" shows "s2p (\<Inter> C) A = (\<forall>X\<in>C. s2p X A)"
minus_ivl_def: fixes iv1 :: "ivl"   and iv2 :: "ivl" shows "iv1 - iv2 = iv1 + - iv2"
face_of_refl_eq: fixes S :: "'a set" shows "(S face_of S) = convex S"
rGamma_0: shows "rGamma (0::'a) = (0::'a)"
zpcarr: shows "zp \<in> carrier (relative_homology_group (int n) (lsphere n) (equator n))"
topcontinuous_at_def: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" shows "topcontinuous_at X Y f x = (x \<in> topspace X \<and> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>V. openin Y V \<and> f x \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> (\<forall>y\<in>U. f y \<in> V))))"
int_simps_14: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<longrightarrow> #False) = \<not> P"
monotone: fixes X :: "(variable \<Rightarrow> real) set"   and Y :: "(variable \<Rightarrow> real) set"   and I :: "interp"   and \<alpha> :: "game" assumes "X \<subseteq> Y" shows "game_sem I \<alpha> X \<subseteq> game_sem I \<alpha> Y"
strongly_accepts_subset: fixes \<F> :: "nat set set"   and S :: "nat set"   and M :: "nat set"   and N :: "nat set" assumes "strongly_accepts \<F> S M"   and "N \<subseteq> M" shows "strongly_accepts \<F> S N"
diff_nonempty: fixes A :: "'c set"   and B :: "'c set" assumes "A \<noteq> B"   and "B \<subseteq> A" shows "A - B \<noteq> {}"
int_and_comm: fixes x :: "int"   and y :: "int" shows "x AND y = y AND x"
zero_le_scaleC_iff: fixes a :: "complex"   and b :: "'b" assumes "a \<in> \<real>" shows "((0::'b) \<le> a *\<^sub>C b) = (0 < a \<and> (0::'b) \<le> b \<or> a < 0 \<and> b \<le> (0::'b) \<or> a = 0)"
Abs_freeletter_prod_conv_Abs_freeword: fixes x :: "'b \<times> bool" assumes "snd x" shows "Abs_freeletter (fst x) = Abs_freeword [x]"
helper3: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and ss :: "'b vec list"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "distinct ss"   and "set ss \<subseteq> set (cols A)"   and "v \<in> carrier_vec (length ss)" obtains c :: "'b vec" where "mat_of_cols nr ss *\<^sub>v v = A *\<^sub>v c" and "dim_vec c = nc"
intros_1: fixes e :: "'e"   and D :: "'e set"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" assumes "e \<in> D" shows "({}, e) \<in> foldSetD D f e"
AsSpecSkip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and p :: "'e set"   and r :: "('e \<times> 'e) set" shows "assertionsR \<Gamma> \<Theta> Q A (AnnExpr p) (Spec r) Q"
fmlookup_drop: fixes a :: "'d"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmlookup (fmdrop a m) x = (if x \<noteq> a then fmlookup m x else None)"
assms_4: shows "F\<epsilon> = (\<lambda>r. \<forall>rel\<in>R. \<forall>(a, b, c)\<in>Atoms rel. \<exists>y>r. \<forall>x\<in>{r<..y}. rel (a * x\<^sup>2 + b * x + c) 0)"
sum_list_lists_in_genby_sym: fixes S :: "'b set"   and ss :: "'b list" assumes "order.greater_eq S (uminus ` S)"   and "ss \<in> lists S" shows "sum_list ss \<in> \<langle>S\<rangle>"
c_def: shows "c = Cn 1 r_inconsist [m, recf.Id 1 0]"
inject_1: fixes x2 :: "nat"   and y2 :: "nat" shows "(WAIT x2 = WAIT y2) = (x2 = y2)"
distinct_add_tvars': fixes acc :: "(variable \<times> String.literal set) list"   and t :: "Core.term" assumes "distinct acc" shows "distinct (add_tvars' t acc)"
get_element_by_is_l_get_element_by: shows "l_get_element_by get_element_by_id get_elements_by_tag_name to_tree_order"
subst_compose2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Z :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var" assumes "good X"   and "good Y"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "X #[Y / y]_ys #[Z / z]_zs = X #[Z / z]_zs #[Y #[Z / z]_zs / y]_ys"
cases: fixes x :: "'a \<times> nat" obtains i :: "'a" and j :: "nat" where "x = (i, j)"
morphism_locally_ringed_spaces_def: fixes X :: "'d set"   and is_open\<^sub>X :: "'d set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e" shows "morphism_locally_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> morphism_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<and> morphism_locally_ringed_spaces_axioms X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f"
subjectEqvt: fixes p :: "(name \<times> name) list"   and \<alpha> :: "'d action" shows "p \<bullet> subject \<alpha> = subject (p \<bullet> \<alpha>)"
P_eq: fixes t :: "real" shows "(0 < t \<and> t \<in> existence_ivl0 x \<and> flow0 x t \<in> A \<and> (\<forall>s\<in>{0<..<t}. flow0 x s \<notin> A)) = (0 < t \<and> t \<in> Y.existence_ivl0 x \<and> Y.flow0 x t \<in> A \<and> (\<forall>s\<in>{0<..<t}. Y.flow0 x s \<notin> A))"
ccField_CCexp: fixes e :: "exp"   and a :: "Arity" shows "ccField (ccExp_syn a e) \<subseteq> fv e"
synthetic_div_unique: fixes p :: "'b poly"   and c :: "'b"   and q :: "'b poly"   and r :: "'b" assumes "p + smult c q = pCons r q" shows "r = poly p c \<and> q = synthetic_div p c"
cindex_pathE_subpath_combine: fixes g :: "real \<Rightarrow> complex"   and z :: "complex"   and a :: "real"   and b :: "real"   and c :: "real" assumes "finite_ReZ_segments g z"   and "path g"   and "0 \<le> a"   and "a \<le> b"   and "b \<le> c"   and "c \<le> 1" shows "cindex_pathE (subpath a b g) z + cindex_pathE (subpath b c g) z = cindex_pathE (subpath a c g) z"
remove1_append: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "remove1 x (xs @ ys) = (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
setsetmapim_cong_subset: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set set"   and B :: "'b set set" assumes "fun_eq_on g f (\<Union> A)"   and "order.greater_eq A B" shows "order.greater_eq (f \<turnstile> B) (g \<turnstile> B)"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (\<lambda>l s. if bval b s then pre C\<^sub>1 Q l s else pre C\<^sub>2 Q l s)"
path_valid_node_2: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.path n as n'" shows "CFGExit.valid_node n'"
exhaust: fixes y :: "'a Trace" obtains x1 :: "'a" where "y = tInit x1"   | x21 :: "'a Trace" and x22 :: "'a" where "y = x21 \<leadsto> x22"
strict_implies_order: fixes a :: "'b list"   and b :: "'b list" assumes "a <s b" shows "a \<le>s b"
th: shows "\<forall>j. isrlfm (MIR.conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s)))))"
module_hom_axioms: shows "module_hom (*\<^sub>R) (*\<^sub>R) f"
assms: shows "\<forall>i\<in>I. p i \<le>o r i"
defNode_eq: fixes n :: "'node"   and g :: "'g"   and v :: "'var \<times> 'node \<times> Def" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> braun_ssa.allDefs g n" shows "braun_ssa.defNode g v = n"
sum_of_2_squares_nat_altdef: fixes n :: "nat" shows "sum_of_2_squares_nat n = (n \<in> range gauss_int_norm)"
ipassmt_iface_replace_srcip_mexpr_def: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce :: "iface" shows "ipassmt_iface_replace_srcip_mexpr ipassmt ifce \<equiv> case ipassmt ifce of None \<Rightarrow> Match (IIface ifce) | Some ips \<Rightarrow> match_list_to_match_expr (map (Match \<circ> Src) (map (uncurry IpAddrNetmask) ips))"
ODNmap_self: fixes D :: "'c Order"   and c :: "'c"   and a :: "'c Order set" assumes "Worder D"   and "c \<in> carrier D"   and "a = ordinal_number (Iod D (segment D c))" shows "ODNmap D a = c"
rep_eq: fixes x :: "mini_alg_unique" shows "Rep_mini_alg_unique (mau_uminus x) = ma_uminus (Rep_mini_alg_unique x)"
mdc_eq_last: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C Cs"   and "last Cs = C"   and "wf_prog wf_md P" shows "Cs = [C]"
facet_unique_other_chamber: fixes X :: "'b set set"   and B :: "'b set"   and z :: "'b set"   and C :: "'b set"   and D :: "'b set" assumes "ThinishChamberComplex X"   and "SimplicialComplex.maxsimp X B"   and "z \<lhd> B"   and "SimplicialComplex.maxsimp X C"   and "z \<lhd> C"   and "SimplicialComplex.maxsimp X D"   and "z \<lhd> D"   and "C \<noteq> B"   and "D \<noteq> B" shows "C = D"
U_def: fixes p :: "'a" shows "U p = inv_chart (\<psi> p) ` ball (0::'b) 1"
fresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm set" shows "fresh xs x X \<equiv> qFresh xs x (pick X)"
continuous_at_sequentially: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "isCont f a = (\<forall>x. x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
insert_iff: fixes a :: "'d"   and b :: "'d"   and A :: "'d set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"