hcju/isabelleps huggingface dataset:
ta_union_rta: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option"   and TA' :: "('a, 'b) tree_automaton_rec" assumes "ranked_tree_automaton TA A"   and "ranked_tree_automaton TA' A" shows "ranked_tree_automaton (ta_union TA TA') A"
M8: fixes F :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and v :: "'d \<Rightarrow> 'e" shows "\<turnstile> \<box>[F \<and> G]_v = (\<box>[F]_v \<and> \<box>[G]_v)"
1: shows "f = incl_of (img f) (cod f) \<cdot> corestr f"
restriction_to_subgroup_is_module: fixes H :: "'g set" assumes "Supgroup.Subgroup H" shows "FGModule H (\<star>) W"
l_remove_shadow_root_get_child_nodes_def: fixes get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_remove_shadow_root_get_child_nodes get_child_nodes_locs remove_shadow_root_locs \<equiv> \<forall>ptr shadow_root_ptr w element_ptr h h' r. ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<longrightarrow> w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_child_nodes_locs ptr \<longrightarrow> r h h'"
fixp_strong_induct_uc: fixes U :: "'e \<Rightarrow> 'c \<Rightarrow> 'd option"   and F :: "'e \<Rightarrow> 'e"   and C :: "('c \<Rightarrow> 'd option) \<Rightarrow> 'e"   and f :: "'e"   and P :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool" assumes "\<And>x. option.mono_body (\<lambda>f. U (F (C f)) x)"   and "f \<equiv> C (option.fixp_fun (\<lambda>f. U (F (C f))))"   and "\<And>f. U (C f) = f"   and "option.admissible P"   and "P Map.empty"   and "\<And>f'. \<lbrakk>P (U f'); option.le_fun (U f') (U f)\<rbrakk> \<Longrightarrow> P (U (F f'))" shows "P (U f)"
is_weak_sim_def: fixes R :: "(('node list \<times> (('var \<Rightarrow> 'val option) \<times> 'ret) list) \<times> 'node list \<times> (('var \<Rightarrow> 'val option) \<times> 'ret) list) set"   and S :: "'node SDG_node set" shows "is_weak_sim R S \<equiv> \<forall>ms\<^sub>1 s\<^sub>1 ms\<^sub>2 s\<^sub>2 ms\<^sub>1' s\<^sub>1' as. ((ms\<^sub>1, s\<^sub>1), ms\<^sub>2, s\<^sub>2) \<in> R \<and> S,kind \<turnstile> (ms\<^sub>1,s\<^sub>1) =as\<Rightarrow> (ms\<^sub>1',s\<^sub>1') \<and> s\<^sub>1' \<noteq> [] \<longrightarrow> (\<exists>ms\<^sub>2' s\<^sub>2' as'. ((ms\<^sub>1', s\<^sub>1'), ms\<^sub>2', s\<^sub>2') \<in> R \<and> S,slice_kind S \<turnstile> (ms\<^sub>2,s\<^sub>2) =as'\<Rightarrow> (ms\<^sub>2',s\<^sub>2'))"
sum2sq_int_def: shows "sum2sq_int = (\<lambda>(a, b). a\<^sup>2 + b\<^sup>2)"
zero_le_power_eq_numeral: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and w :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a (numeral w)) = (even (numeral w) \<or> odd (numeral w) \<and> less_eq zero a)"
SET_INV_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "set_inv\<^bsub>G\<^esub> H = (\<Union>h\<in>H. {inv\<^bsub>G\<^esub> h})"
mono_imp_set_offending_flows_not_empty: fixes G :: "'v graph"   and ff :: "('v \<times> 'v) list"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_mono"   and "wf_graph G"   and "is_offending_flows (set ff) G nP"   and "set ff \<subseteq> edges G"   and "distinct ff" shows "set_offending_flows G nP \<noteq> {}"
elims: fixes x :: "real poly"   and xa :: "real poly"   and y :: "real poly list" assumes "smods x xa = y" obtains p :: "real poly" and q :: "real poly" where "x = p" and "xa = q" and "y = (if p = 0 then [] else p # smods q (- (p mod q)))"
map_le_empty: fixes g :: "'b \<Rightarrow> 'c option" shows "Map.empty \<subseteq>\<^sub>m g"
two_chain_boundary_def: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "two_chain_boundary twoChain \<equiv> \<Union> (boundary ` twoChain)"
emeasure_pmf_single: fixes M :: "'c pmf"   and x :: "'c" shows "emeasure (measure_pmf M) {x} = ennreal (pmf M x)"
le_fin_trivial: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2" shows "traces.le_fin ind w\<^sub>1 w\<^sub>2"
convergent_realpow: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "convergent ((^) x)"
rule_ui: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<beta> :: "'a" assumes "[\<^bold>\<forall>\<alpha>. \<phi> \<alpha> in v]" shows "[\<phi> \<beta> in v]"
Inr: shows "Rep_SkewBinomialHeap q = Inr x"
iso_def: fixes G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" shows "Group.iso G H = {h \<in> hom G H. bij_betw h (carrier G) (carrier H)}"
prems_2: shows "column j (Gram_Schmidt_upt_k A (Suc k)) = column i (Gram_Schmidt_upt_k A (Suc k))"
plossless_converterD: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and a :: "'a" assumes "plossless_converter \<I> \<I>' conv"   and "a \<in> outs_\<I> \<I>" shows "plossless_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). plossless_converter \<I> \<I>' conv')"
f_join_f_expand_iT_Mult: fixes k :: "nat"   and xs :: "'b message_af list"   and I :: "nat set" assumes "0 < k" shows "xs \<odot> k \<Join>\<^sub> (I \<otimes> k) = xs \<Join>\<^sub> I"
nat_inc_prod: fixes m :: "nat"   and n :: "nat" shows "[m] \<cdot> ([n] \<cdot> \<one>) = [(m * n)] \<cdot> \<one>"
the_sym_eq_trivial: fixes x :: "'a" shows "(THE y. x = y) = x"
map_gpv'_id12: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'f"   and gpv :: "('d, 'e, 'f) gpv" shows "map_gpv' f g h gpv = map_gpv' id id h (map_gpv f g gpv)"
lookup_smult: fixes "is" :: "nat list"   and A :: "'b tensor"   and \<alpha> :: "'b" assumes "is \<lhd> dims A" shows "lookup (\<alpha> \<cdot> A) is = \<alpha> * lookup A is"
lex_ext_unbounded_cong: fixes as :: "'b list"   and as' :: "'b list"   and bs :: "'b list"   and bs' :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" assumes "as = as'"   and "bs = bs'"   and "\<And>i. \<lbrakk>i < length as'; i < length bs'\<rbrakk> \<Longrightarrow> f (as' ! i) (bs' ! i) = g (as' ! i) (bs' ! i)" shows "lex_ext_unbounded f as bs = lex_ext_unbounded g as' bs'"
ownership_distinct: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length (t # ts)"   and "j < length (t # ts)"   and "i \<noteq> j"   and "(t # ts) ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "(t # ts) ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>i \<union> all_acquired sb\<^sub>i) \<inter> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) = {}"
dense_r: fixes x :: "real"   and y :: "real" assumes "x < y" shows "\<exists>n. x < r n \<and> r n < y"
group_action_def: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set" shows "group_action G \<phi> M \<equiv> Group.group G \<and> group_action_axioms G \<phi> M"
BINDINGn_elim: fixes x :: "name"   and M :: "trm"   and B :: "ty" assumes "x:M \<in> BINDINGn B (\<parallel><B>\<parallel>)" shows "\<forall>c P. <c>:P \<in> \<parallel><B>\<parallel> \<longrightarrow> SNa (M{x:=<c>.P})"
ord_eq_le_eq_trans: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a = b"   and "b \<le> c"   and "c = d" shows "a \<le> d"
validn_lower: fixes K :: "nat"   and C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and L :: "nat" assumes "\<Turnstile>\<^sub>K \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I "   and "L \<le> K" shows "\<Turnstile>\<^sub>L \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I "
rep_eq: fixes x :: "'c regular"   and xa :: "'c regular" shows "Rep_regular (x \<sqinter> xa) = Rep_regular x \<sqinter> Rep_regular xa"
dominate_def: fixes n1 :: "nat"   and n2 :: "nat" shows "dominate n1 n2 \<equiv> \<forall>pa. path_entry (g_E G) pa n2 \<longrightarrow> n1 \<in> set pa \<or> n1 = n2"
simps: fixes a :: "C"   and b :: "nat" shows "(a \<in> hbase_ext b) = ((\<exists>n. a = n \<and> n \<in> hbase b) \<or> (\<exists>n m. a = C (n # m) \<and> n \<in> hbase_ext b \<and> C m \<in> hbase b \<and> (\<forall>x. x \<in> set m \<longrightarrow> evalC b n \<le> evalC b x)))"
small_Collect: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "small A" shows "small {x \<in> A. P x}"
pick_middlep: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'a"   and c :: "'c" assumes "(P OO Q) a c" shows "P a (BNF_Def.pick_middlep P Q a c) \<and> Q (BNF_Def.pick_middlep P Q a c) c"
mod_mult_self2: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times b c)) b = modulo a b"
path_part_circlepath: fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real" shows "path (part_circlepath z r s t)"
map_of_SomeI: fixes kxs :: "('b \<times> 'c) list"   and k :: "'b"   and x :: "'c" assumes "distinct_fst kxs"   and "(k, x) \<in> set kxs" shows "map_of kxs k = \<lfloor>x\<rfloor>"
Fun_corestr: fixes f :: "'s" assumes "S.arr f" shows "S.Fun (S.corestr f) = S.Fun f"
vector_complement_closed: fixes x :: "'a" assumes "vector x" shows "vector (- x)"
comp_comp_fun_commute: fixes g :: "'c \<Rightarrow> 'a" shows "comp_fun_commute ((\<^bold>*) \<circ> g)"
is_first_power: fixes x :: "'a" shows "is_nth_power 1 x"
qPsubst_preserves_qGood: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGoodEnv rho"   and "qGood X" shows "qGood X #[[rho]]"
set_vzero_eq_ord_of_nat_vone: shows "set {0} = 1\<^sub>\<nat>"
principal_dchar_minus1: shows "\<chi>\<^sub>0 (n - Suc 0) = 1"
RdRequest_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and l :: "Locs" shows "RdRequest ch p l \<equiv> Calling ch p \<and> arg<ch!p> = #(read l)"
hyps: fixes a :: "'a" assumes "\<And>a b. \<lbrakk>a \<in> set xs; b \<in> set xs\<rbrakk> \<Longrightarrow> a \<le> b \<or> b \<le> a"   and "a \<in> set xs" shows "a \<le> Missing_Lemmas.max_list xs"
lin_independent_ConsI: fixes ms :: "'v list"   and m :: "'v" assumes "R_lin_independent ms"   and "\<And>r rs. \<lbrakk>set (r # rs) \<subseteq> UNIV; (r # rs) \<bullet>\<cdot> (m # ms) = (0::'v)\<rbrakk> \<Longrightarrow> r = (0::'f)" shows "R_lin_independent (m # ms)"
lower_abs_interval: fixes x :: "'a interval" shows "lower (abs_interval x) = (if lower x < (0::'a) \<and> (0::'a) < upper x then 0::'a else min \<bar>lower x\<bar> \<bar>upper x\<bar>)"
outstanding_not_volatile_Read\<^sub>s\<^sub>b_refs_conv: fixes sb :: "'b memref list" shows "outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) sb = outstanding_refs is_Write\<^sub>s\<^sub>b sb \<union> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb"
fg_invariant2b: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and n :: "nat"   and q :: "'f"   and p :: "'f"   and m :: "'h" assumes "fg_run A rho HOs SHOs coords"   and "rcvd (rho n) q p = Some m"   and "p \<in> SHOs (round (rho n) q) q" shows "m = sendMsg A (round (rho n) q) p q (state (rho (fg_start_round rho p (round (rho n) q))) p)"
tendsto_exp: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
a_set_shadow_root_def: fixes element_ptr :: "(_) element_ptr" shows "set_shadow_root element_ptr = put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt_update"
Un_left_commute: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
full_normalize_poly_diff_ideal: fixes p :: "(char list list \<times> int) list"   and p' :: "int mpoly" assumes "(p, p') \<in> fully_unsorted_poly_rel O mset_poly_rel" shows "full_normalize_poly p \<le> \<Down> (sorted_poly_rel O mset_poly_rel) (normalize_poly_spec p')"
densg: shows "has_parametrized_subprob_density (state_measure V' \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure dc_True.\<Y> \<rho> \<bind> g) R \<delta>g"
sets_PiM_finite: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" shows "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {Pi\<^sub>E I X |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}"
copyp_distrib: fixes k :: "nat"   and x :: "'a"   and y :: "'a" shows "k \<otimes> (x \<squnion> y) = k \<otimes> x \<squnion> k \<otimes> y"
zero_index: fixes i :: "'e" shows "0 $ i = (0::'d)"
lNAct_def: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<equiv> LEAST n'. n = n' \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. k < n \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)"
bound_intersect_2d_le: fixes prec :: "nat"   and Xs :: "((real \<times> real) \<times> real \<times> real) list"   and x :: "real"   and m :: "real"   and M :: "real" assumes "bound_intersect_2d prec Xs x = Some (m, M)" shows "m \<le> M"
simstep_disj1: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "s \<approx> t" shows "\<exists>m. s |\<^sub>s n \<approx> t |\<^sub>s m"
assms_3: shows "mapM (concurrent w l u) (substr w l u) = Inr d"
linear_subspace_image: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and S :: "('a, 'b) vec set" assumes "Vector_Spaces.linear (*s) (*s) f"   and "vec.subspace S" shows "vec.subspace (f ` S)"
that: fixes A :: "'a fset"   and B :: "'a fset" shows "f (A |\<union>| B) = bind (f A) (\<lambda>x. bind (f B) (\<lambda>y. return (x |\<union>| y)))"
small_vproduct: fixes I :: "V"   and A :: "V \<Rightarrow> V" shows "small {f. vsv f \<and> \<D>\<^sub>\<circ> f = I \<and> (\<forall>i\<in>elts I. f\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i)}"
lprod_3_1: fixes x' :: "'a"   and x :: "'a"   and R :: "('a \<times> 'a) set"   and y :: "'a"   and z :: "'a" assumes "(x', x) \<in> R" shows "([y, z, x'], [x, y, z]) \<in> lprod R"
card_eq: shows "card (([^]) a ` {0..G.ord a - 1}) = card {0..G.ord a - 1}"
degree_msum_pdevs_le: fixes n :: "nat"   and f :: "'a pdevs"   and g :: "'a pdevs" shows "degree (msum_pdevs n f g) \<le> n + degree g"
finite_blocks: fixes b :: "'a set" assumes "b \<in># add_block bl" shows "finite b"
alphaAbs_preserves_qGoodAbs2: fixes A :: "('f, 'g, 'e, 'd, 'h) qAbs"   and A' :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)"   and "A $= A'"   and "FixVars.qGoodAbs A'" shows "FixVars.qGoodAbs A"
openin_path_component_of_locally_path_connected_space: fixes X :: "'c topology"   and x :: "'c" assumes "locally_path_connected_space X" shows "openin X (path_component_of_set X x)"
lex_ord_pair_valE: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option"   and aux :: "order option" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "lex_ord_pair_tc f xs ys = aux"   and "aux \<noteq> Some Eq" obtains k :: "'a" where "k \<in> fst ` set xs \<union> fst ` set ys" and "aux = f k (lookup_pair_tc xs k) (lookup_pair_tc ys k)" and "\<And>k'. \<lbrakk>k' \<in> fst ` set xs \<union> fst ` set ys; tc.lt k' k\<rbrakk> \<Longrightarrow> f k' (lookup_pair_tc xs k') (lookup_pair_tc ys k') = Some Eq"
WTCond: fixes 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 :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty"   and T :: "ty" assumes "P,E \<turnstile> e :: Boolean"   and "P,E \<turnstile> e\<^sub>1 :: T\<^sub>1"   and "P,E \<turnstile> e\<^sub>2 :: T\<^sub>2"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2 \<or> P \<turnstile> T\<^sub>2 \<le> T\<^sub>1"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2 \<longrightarrow> T = T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T\<^sub>1 \<longrightarrow> T = T\<^sub>1" shows "P,E \<turnstile> if (e) e\<^sub>1 else e\<^sub>2 :: T"
vsubstAbs_trans: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "freshAbs ys y1 A" shows "A $[y1 // y]_ys $[y2 // y1]_ys = A $[y2 // y]_ys"
DiffConstSubst_def: shows "DiffConstSubst = \<lparr>SFunctions = \<lambda>f. if f = fid1 then Some (Const 0) else None, SPredicates = Map.empty, SContexts = Map.empty, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
gf_cong1: shows "fun_eq_on gf (g \<circ> f) (\<Union> A)"
one: shows "bv_to_int xs = bv_to_int ys"
wfP_app: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "wfP p" shows "wfP (\<lambda>x y. p (f x) (f y))"
shared_switch_release_False: fixes sb :: "'b memref list"   and a :: "nat"   and \<S> :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "a \<notin> dom \<S>"   and "a \<in> dom (share sb \<S>)" shows "release sb (dom \<S>) \<R> a = Some False"
linear_inj_imp_surj: fixes f :: "'a \<Rightarrow> 'a" assumes "clinear f"   and "inj f" shows "surj f"
hyp3: assumes "k < ncols A" shows "snd (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k]) \<le> nrows A"
elims_2: fixes x :: "'b tree" assumes "braun x" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and xa :: "'b" and r :: "'b tree" where "x = \<langle>l, xa, r\<rangle>" and "(size l = size r \<or> size l = size r + 1) \<and> braun l \<and> braun r"
l3bis: shows "(abscissa X - abscissa D)\<^sup>2 + (ordinate X - ordinate D)\<^sup>2 = (abscissa E - abscissa F)\<^sup>2 + (ordinate E - ordinate F)\<^sup>2"
Terminal_not_Linear: fixes a :: "'f"   and b :: "'g"   and r :: "('f, 'g, 'h, 'i) resumption" shows "Terminal a \<noteq> Linear b r"
rPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' Q')"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* C"   and "xvec \<sharp>* P"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* M"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' (P \<parallel> Q'))"
weakCongSum2: fixes Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs"   and P :: "ccs" assumes "Q \<Longrightarrow>\<alpha> \<prec> Q'" shows "P \<oplus> Q \<Longrightarrow>\<alpha> \<prec> Q'"
perzeta_altdef2: fixes q' :: "real"   and s :: "complex" assumes "q' \<notin> \<int>"   and "0 < Re s" shows "perzeta q' s = eval_fds (fds_perzeta q') s"
af_simps_1: fixes w :: "'a set list" shows "af true w = true"
y_2: shows "\<forall>y'\<in>snd ` A. (y', y) \<notin> r'"
assms: shows "poly p x\<^sub>0 \<noteq> 0"
brw_inner_invar_step: fixes Q :: "'d \<Rightarrow> 'e tree option"   and W :: "'d set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set"   and q :: "'d"   and r :: "('d, 'e) ta_rule"   and it :: "('d, 'e) ta_rule set"   and \<Sigma>h :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and \<Sigma>' :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" assumes "(Q, W, rcm) \<in> brw_invar \<delta>"   and "q \<in> W"   and "r \<in> it"   and "it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}"   and "brw_inner_invar {r \<in> \<delta>. q \<in> set (rhsq r)} q (Q, W - {q}, rcm) it \<Sigma>h"   and "(\<Sigma>h, \<Sigma>') \<in> brw_inner_step r" shows "brw_inner_invar {r \<in> \<delta>. q \<in> set (rhsq r)} q (Q, W - {q}, rcm) (it - {r}) \<Sigma>'"
augment_edge_same: fixes e :: "nat \<times> nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and \<Delta> :: "'a" assumes "e \<in> E" shows "augment_edge f e \<Delta> e = f e + \<Delta>"
axioms_1: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop A n f" shows "vsv f"
th: shows "\<forall>x. Ifm (x # bs) (plusinf p)"
boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b \<sqinter> c" obtains "a \<le> b" and "a \<le> c"
insert_messages_def: fixes M :: "('a, 'b) message set" shows "insert_messages M = {x. Insert x \<in> M}"
JK_disj: shows "emb (J \<union> K) J X \<inter> emb (J \<union> K) K Y = {}"