hcju/isabelleps huggingface dataset:
comp_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and g :: "'c \<Rightarrow> 'd" assumes "inj_on f A"   and "inj_on g (f ` A)" shows "inj_on (g \<circ> f) A"
append_fusion_def: fixes g1 :: "('d, 'e) generator"   and g2 :: "('d, 'f) generator"   and s1 :: "'e"   and s2 :: "'f" shows "append_fusion g1 g2 s1 s2 = list.unfoldr g1 s1 @ list.unfoldr g2 s2"
content_ge_0_int: fixes p :: "int poly" shows "0 \<le> content p"
inv_into_f_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a" assumes "inj_on f A"   and "x \<in> A" shows "inv_into A f (f x) = x"
CondT: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "expr"   and e' :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
ttree_restr_both: fixes S :: "'a set"   and t :: "'a ttree"   and t' :: "'a ttree" shows "ttree_restr S (t \<otimes>\<otimes> t') = ttree_restr S t \<otimes>\<otimes> ttree_restr S t'"
zero_in_JointsI: fixes xs :: "'a list"   and XS :: "('a \<times> 'a pdevs) list"   and z :: "'a \<times> 'a pdevs" assumes "xs \<in> Joints XS"   and "z = (0::'a, zero_pdevs)" shows "(0::'a) # xs \<in> Joints (z # XS)"
wf_nabla_aux: fixes P :: "'a set"   and R :: "('a \<times> 'a) set" shows "(P \<subseteq> R\<inverse> `` P \<longrightarrow> P = {}) = (s2r P \<subseteq> relfdia R (s2r P) \<longrightarrow> s2r P = {})"
continuous_on_inverse_closed_map: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "continuous_on S f"   and "f ` S = T"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"   and "\<And>U. closedin (top_of_set S) U \<Longrightarrow> closedin (top_of_set T) (f ` U)" shows "continuous_on T g"
zero: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ring_ops ops R" shows "R (arith_ops_record.zero ops) (0::'c)"
Aw_def: shows "Aw = tensors_from_net (witness_l rs) $ y"
currying_def: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "currying A1 A2 B \<equiv> category A1 \<and> category A2 \<and> category B"
K_meas: fixes N :: "nat" shows "K N \<in> sets M"
in_relationsI: fixes G :: "('a, 'b) monoid_scheme"   and f :: "'a \<Rightarrow> int"   and A :: "'a set" assumes "comm_group G"   and "(\<Otimes>\<^bsub>G\<^esub>a\<in>A. a [^]\<^bsub>G\<^esub> f a) = \<one>\<^bsub>G\<^esub>"   and "f \<in> extensional A" shows "f \<in> comm_group.relations G A"
deleted_ids_last_del: fixes xs :: "('c \<times> ('c, 'd) list_op) list"   and oid :: "'c"   and ref :: "'c" shows "deleted_ids (xs @ [(oid, Delete ref)]) = deleted_ids xs @ [ref]"
hom_in_carrier: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and x :: "'c" assumes "h \<in> hom G H"   and "x \<in> carrier G" shows "h x \<in> carrier H"
Seq: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and Ax :: "('a, 'b) Axioms" assumes "Model I Ax" shows "aS\<^bsub>Ax\<^esub> = iS\<^bsub>I\<^esub>"
assms_2: shows "n < (Suc m)\<^sup>2"
E: shows "y \<in> range p \<or> y \<in> range q"
incseq_sumI2: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
naturality_simplicial_subdivision: fixes p :: "nat"   and q :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and S :: "(nat \<Rightarrow> real) set"   and g :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "simplicial_chain p (standard_simplex q) c"   and "simplicial_simplex q S g" shows "simplicial_subdivision p (chain_map p g c) = chain_map p g (simplicial_subdivision p c)"
l_cancel_one: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and a :: "'b" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> a = x) = (a = \<one>\<^bsub>G\<^esub>)"
dgrad_set_le_subset: fixes S :: "'b set"   and T :: "'b set"   and d :: "'b \<Rightarrow> nat" assumes "S \<subseteq> T" shows "dgrad_set_le d S T"
inv_is_inverse: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "category.inverse_arrows C f (category.inv C f)"
zcount_ne_zero_iff: fixes M :: "'b zmultiset"   and x :: "'b" shows "(zcount M x \<noteq> 0) = (x \<in>#\<^sub>z M)"
below_meet: fixes x :: "'c"   and z :: "'c" assumes "x \<sqsubseteq> z" shows "x \<sqinter> z = x"
poincare_distance_eq_0_iff: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "(poincare_distance u v = 0) = (u = v)"
better_left: fixes M :: "trm"   and a :: "coname"   and x :: "name"   and N :: "trm" assumes "\<not> fic M a" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>c M{a:=x.N}"
suc_subset: fixes X :: "'a set" shows "X \<subseteq> B.suc X"
is_SDG_path_intra_CFG_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" obtains as :: "'a list" where "CFG.intra_path sourcenode targetnode kind valid_edge (parent_node n) as (parent_node n')"
pders_changes_d1: fixes r :: "real poly"   and q :: "real poly" assumes "poly r c \<noteq> 0"   and "q \<noteq> 0"   and "\<forall>x. d1 \<le> x \<and> x < c \<longrightarrow> poly r x \<noteq> 0 \<and> poly q x \<noteq> 0" shows "changes_poly_at (r # pders q) d1 = (if even (order c q) = (sign_r_pos q c = (0 < poly r c)) then changes_poly_at (pders q) d1 else 1 + changes_poly_at (pders q) d1)"
dfs_\<alpha>_def: fixes S :: "'a set \<times> 'a list" shows "dfs_\<alpha> S \<equiv> let (\<Sigma>, W) = S in (\<Sigma>, set W)"
compactin_subtopology: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "compactin (subtopology X S) T = (compactin X T \<and> T \<subseteq> S)"
is_continuation_empty_chars: fixes k :: "nat"   and u :: "nat"   and q :: "(symbol \<times> character list) list"   and ts :: "(symbol \<times> character list) list" assumes "is_continuation k u q ts"   and "charslength (q @ ts) = k" shows "chars ts = []"
rep_eq: fixes x :: "('b, 'a) chart" shows "inv_chart x = (case Rep_chart x of (d, d', f, f') \<Rightarrow> f')"
cases: fixes a1 :: "('i, 'j, 'k, 'l) converter"   and a2 :: "enat" assumes "interaction_any_bounded_converter a1 a2" obtains conv :: "('i, 'j, 'k, 'l) converter" and n :: "enat" where "a1 = conv" and "a2 = n" and "\<And>a. interaction_any_bounded_by (run_converter conv a) n" and "\<And>a b conv'. (b, conv') \<in> results'_gpv (run_converter conv a) \<Longrightarrow> interaction_any_bounded_converter conv' n"
harm_remainder_bound: fixes N :: "nat" assumes "0 < N" shows "\<exists>C. \<forall>n\<ge>1. norm (harm_remainder N n) \<le> C / real n ^ (2 * N + 1)"
continuous_on_closed_gen: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and T :: "'c set" assumes "f ` S \<subseteq> T" shows "continuous_on S f = (\<forall>U. closedin (top_of_set T) U \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` U))"
fbox_whilei_break: fixes p :: "'b ppath set"   and y :: "'b ppath set"   and i :: "'b ppath set"   and t :: "'b ppath set"   and q :: "'b ppath set"   and x :: "'b ppath set" assumes "rdom p \<subseteq> wp y i"   and "rdom i ; pp_a t \<subseteq> rdom q"   and "rdom i ; rdom t \<subseteq> wp x i" shows "rdom p \<subseteq> wp (y ; ppath_aka.whilei t i x) q"
dg_is_arrD_1: fixes \<alpha> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
single_eq_add_mset: fixes a :: "'b"   and b :: "'b"   and M :: "'b multiset" shows "({#a#} = add_mset b M) = (b = a \<and> M = {#})"
natural_isomorphism_axioms: shows "natural_isomorphism A B G F \<tau>'.map"
simps_1: fixes ivl :: "float interval"   and f :: "float interval \<Rightarrow> float interval option" shows "lift_un (Some ivl) f = f ivl"
null_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "null \<equiv> Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<bottom>\<rfloor>"
absorbingr: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seq_comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and neutral :: "'c"   and deactivation :: "'c"   and basic :: "'c set"   and x :: "'c" assumes "class.stimuli leq add seq_comp neutral deactivation basic" shows "seq_comp x deactivation = deactivation"
bv_sadd_returntype: fixes w1 :: "bit list"   and w2 :: "bit list" shows "norm_signed (bv_sadd w1 w2) = bv_sadd w1 w2"
f_p: shows "\<forall>C\<in>CC. enat (f C) < llength Sts \<and> (\<forall>j'. enat (f C) \<le> enat j' \<longrightarrow> enat j' < llength Sts \<longrightarrow> C \<in> Q_of_state (lnth Sts j'))"
residue_1_unit_1: fixes m :: "nat" assumes "0 < m" shows "1 \<in> Units (residue_ring (p ^ m))"
homotopic_triviality: fixes S :: "'a set"   and T :: "'b set" shows "(\<forall>f g. continuous_on S f \<and> f ` S \<subseteq> T \<and> continuous_on S g \<and> g ` S \<subseteq> T \<longrightarrow> homotopic_with_canon (\<lambda>x. True) S T f g) = ((S = {} \<or> path_connected T) \<and> (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> T \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c))))"
a_get_child_nodes_locs_def: fixes ptr :: "(_) object_ptr" shows "a_get_child_nodes_locs ptr \<equiv> (if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) child_nodes)} else {}) \<union> (if is_document_ptr_kind ptr then {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) document_element)} else {}) \<union> {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)}"
delta: fixes S :: "'b set"   and a :: "'b"   and b :: "'b \<Rightarrow> 'c" assumes "finite S" shows "(\<Sum>k\<in>S. if k = a then b k else (0::'c)) = (if a \<in> S then b a else (0::'c))"
pred_def: fixes P :: "bool" shows "dynamic_component.pred P \<equiv> \<lambda>t n. P"
simps_4: fixes x :: "char"   and xs :: "char list"   and y :: "char"   and ys :: "char list" shows "lleq_charlist (x # xs) (y # ys) = (if x = y then lleq_charlist xs ys else x < y)"
InvariantNoDecisionsWhenUnit_def: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and level :: "nat" shows "InvariantNoDecisionsWhenUnit F M level = (\<forall>level'<level. \<nexists>clause literal. clause el F \<and> isUnitClause clause literal (elements (prefixToLevel level' M)))"
sibling_swapFourSyms_when_4th_is_sibling: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "consistent t"   and "a \<in> alphabet t"   and "b \<in> alphabet t"   and "c \<in> alphabet t"   and "a \<noteq> b"   and "sibling t c \<noteq> c" shows "sibling (swapFourSyms t a b c (sibling t c)) a = b"
fmupd_lookup: fixes a :: "'f"   and b :: "'e"   and m :: "'f \<rightharpoonup>\<^sub>f 'e"   and a' :: "'f" shows "fmlookup (m(a \<mapsto>\<^sub>f b)) a' = (if a = a' then Some b else fmlookup m a')"
uses_in_\<alpha>n: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and n :: "'node" assumes "v \<in> uses' g n" shows "n \<in> set (\<alpha>n g)"
gtrI: fixes s :: "'f"   and c :: "'f multiset"   and w :: "'g list"   and s' :: "'f"   and c' :: "'f multiset"   and gtrs :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'f \<times> 'f multiset) set" assumes "((s, c), w, s', c') \<in> trcl gtrs" shows "(add_mset s c, w, add_mset s' c') \<in> trcl (gtr gtrs)"
AndR_inv': fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F \<^bold>\<and> G, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n \<and> \<Gamma> \<Rightarrow> G, \<Delta> \<down> n"
carrier_either: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "carrier (t \<oplus>\<oplus> t') = carrier t \<union> carrier t'"
nonempty_Suc: fixes A :: "'b set"   and n :: "nat"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and left :: "(('c, 'b) fm list \<times> 'b) list"   and right :: "(('c, 'b) fm list \<times> 'b) list"   and q :: "('c, 'b) fm" assumes "A, n \<turnstile> (ps, a) # filter nonempty left @ right"   and "q at a in' set ((ps, a) # filter nonempty left @ right)"   and "q \<noteq> Nom a" shows "A, Suc n \<turnstile> filter nonempty ((ps, a) # left) @ right"
right_mult_one_mat': fixes A :: "'a mat"   and n :: "nat" assumes "dim_col A = n" shows "A * 1\<^sub>m n = A"
asym: fixes b :: "'t"   and a :: "'t" assumes "b \<prec>\<^sub>t a"   and "a \<prec>\<^sub>t b" shows "False"
prv_pls_suc_var: shows "prv (eql (pls (Var xx) (suc (Var yy))) (suc (pls (Var xx) (Var yy))))"
el_def: shows "(\<^bold>\<in>) \<equiv> map_fun id (map_fun Rep_nat_int id) (\<in>)"
replicate_add: fixes n :: "nat"   and m :: "nat"   and x :: "'d" shows "x\<^bsup>n + m\<^esup> = x\<^bsup>n\<^esup> @ x\<^bsup>m\<^esup>"
iterate_add_distrib: fixes m :: "nat"   and n :: "nat"   and a :: "'k" shows "iterate_add (m + n) a = iterate_add m a + iterate_add n a"
path2_split_ex: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and x :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "x \<in> set ns" obtains ns\<^sub>1 :: "'node list" and ns\<^sub>2 :: "'node list" where "g \<turnstile> n-ns\<^sub>1\<rightarrow>x" and "g \<turnstile> x-ns\<^sub>2\<rightarrow>m" and "ns = ns\<^sub>1 @ tl ns\<^sub>2" and "ns = butlast ns\<^sub>1 @ ns\<^sub>2"
PChain: shows "P \<parallel> !P \<Longrightarrow>\<^sub>\<tau> P''"
finite_cyclic_subgroup: fixes x :: "'a" assumes "x \<in> carrier G" shows "finite (carrier (subgroup_generated G {x})) = (\<exists>n. n \<noteq> 0 \<and> x [^] n = \<one>)"
distinct_elim_rev: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v1 :: "'a"   and w :: "'b"   and v2 :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.is_trail G v ((v1, w, v2) # ps) v'" shows "((v2, w, v1) \<in> edges (rem_unPath ps G)) = ((v2, w, v1) \<in> edges G)"
assms_3: shows "\<forall>u v. test u \<and> test v \<longrightarrow> ! u \<sqinter> ! v \<sqinter> (u \<squnion> v) = bot"
types_preserved_callcl_native: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and ves :: "e list"   and cl :: "cl"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat"   and t1s :: "t list"   and t2s :: "t list"   and tfs :: "t list"   and es :: "b_e list"   and vs :: "v list"   and n :: "nat"   and k :: "nat"   and m :: "nat"   and zs :: "v list"   and s :: "s" assumes "\<S>\<bullet>\<C> \<turnstile> ves @ [Callcl cl] : ts _> ts'"   and "cl = Func_native i (t1s _> t2s) tfs es"   and "ves = $$* vs"   and "length vs = n"   and "length tfs = k"   and "length t1s = n"   and "length t2s = m"   and "n_zeros tfs = zs"   and "store_typing s \<S>" shows "\<S>\<bullet>\<C> \<turnstile> [Local m i (vs @ zs) [$Block ([] _> t2s) es]] : ts _> ts'"
Un: fixes a :: "'a set"   and A :: "'a set set"   and t :: "'a set" assumes "a \<in> A"   and "t \<in> tiling A"   and "a \<inter> t = {}" shows "a \<union> t \<in> tiling A"
has_limits_if_has_products: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category_with_equalizers C"   and "category J"   and "category.has_products C (Collect (partial_magma.ide J))"   and "category.has_products C (Collect (partial_magma.arr J))" shows "category.has_limits_of_shape C J"
apply_cltn2_line_id: fixes l :: "proj2_line" shows "apply_cltn2_line l cltn2_id = l"
RFc: shows "RF\<^sup>\<circ> (TER h) = RF\<^sup>\<circ> (TER f)"
assms_1: shows "A o\<^sub>C\<^sub>L B = id_cblinfun"
mopup_even: fixes x :: "nat"   and n :: "nat" shows "is_even (2 * x + length (mopup n))"
rtranclp_unfold: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" shows "r\<^sup>*\<^sup>* a b \<equiv> a = b \<or> r\<^sup>+\<^sup>+ a b"
x_def: shows "x = fast_select (((length xs + 4) div 5 - 1) div 2) (map median_le_5 (chop 5 xs))"
suba12_open: shows "{a<--<flow0 x t1} \<subseteq> {a<--<flow0 x t2}"
transp_trans: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "transp r = trans {(x, y). r x y}"
rOutput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and C :: "'d"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "Prop C \<Psi> (M\<langle>N\<rangle>.P) (K\<langle>N\<rangle>) P"
par_strict_not_col_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "\<not> Col A B C"
comp1: fixes g :: "'f \<Rightarrow> 'a" assumes "bounded_clinear g" shows "bounded_sesquilinear (\<lambda>x. (**) (g x))"
from_nat_into_surj: fixes A :: "'c set"   and a :: "'c" assumes "countable A"   and "a \<in> A" shows "\<exists>n. from_nat_into A n = a"
imp_simps_3: fixes Q :: "fm" shows "imp T Q = Q"
EQ_def: shows "EQ \<equiv> Ordering_abs\<cdot>(sinr\<cdot>(sinl\<cdot>ONE))"
less_le_not_le: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(x \<prec> y) = strict (\<preceq>) x y"
irreflp_on_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "irreflp_on P A = (\<forall>a\<in>A. \<not> P a a)"
kbo_code: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" shows "kbo'.kbo s t = (let wt = weight t; ws = weight s in if vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> wt \<le> ws then if wt < ws then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> True | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> let ff = (f, length ss); gg = (g, length ts) in if pr_strict' ff gg then (True, True) else if pr_weak' ff gg then lex_ext_unbounded kbo'.kbo ss ts else (False, False) else (False, False))"
filter_M_holds_for_result: fixes h :: "'x"   and P :: "'z \<Rightarrow> ('x, 'y, bool) prog"   and xs :: "'z list"   and ys :: "'z list"   and x :: "'z" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"   and "x \<in> set ys"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "h \<turnstile> P x \<rightarrow>\<^sub>r True"
acc_eq_limit: fixes r :: "nat \<Rightarrow> 'Q" assumes "finite (range r)" shows "is_acc r = (\<forall>A\<in>F. limit r \<inter> A \<noteq> {})"
rec_lo_def: shows "rec_lo = (let rR = Cn 3 rec_eq [Cn 3 rec_mod [recf.id 3 0, Cn 3 rec_power [recf.id 3 1, recf.id 3 2]], Cn 3 (constn 0) [recf.id 3 1]]; rb = Cn 2 (rec_maxr rR) [recf.id 2 0, recf.id 2 1, recf.id 2 0]; rcond = Cn 2 rec_conj [Cn 2 rec_less [Cn 2 (constn 1) [recf.id 2 0], recf.id 2 0], Cn 2 rec_less [Cn 2 (constn 1) [recf.id 2 0], recf.id 2 1]]; rcond2 = Cn 2 rec_minus [Cn 2 (constn 1) [recf.id 2 0], rcond] in Cn 2 rec_add [Cn 2 rec_mult [rb, rcond], Cn 2 rec_mult [Cn 2 (constn 0) [recf.id 2 0], rcond2]])"
seqsim_suffix_seqsim: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "s \<approx> t" shows "nextsuffix s \<approx> nextsuffix t"
vrestr_eq: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a"   and A :: "'a set" shows "(a \<in> f --` A) = (a \<in> space M \<and> f a \<in> A \<inter> space M)"
mk_doubleton_def: fixes X :: "V"   and a :: "V" shows "mk_doubleton X a = set {a, X}"
fresh_heap_expr': fixes a :: "atom"   and \<Gamma> :: "('d \<times> 'e) list"   and e :: "'e" assumes "a \<sharp> \<Gamma>"   and "e \<in> snd ` set \<Gamma>" shows "a \<sharp> e"
change_eval: fixes x :: "real"   and y :: "real"   and At :: "atomUni"   and a :: "real"   and b :: "real"   and c :: "real" assumes "x < y"   and "aEvalUni At x \<noteq> aEvalUni At y"   and "getPoly At = (a, b, c)" shows "\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0"
zero_in_Field: assumes "Field s \<noteq> {}" shows "s.zero \<in> Field s"
atd_lem: fixes n :: "nat"   and xs :: "'a list"   and t :: "'a list"   and d :: "'a list" assumes "take n xs = t"   and "drop n xs = d" shows "xs = t @ d"
span_prj: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "span \<p>\<^sub>0[f, g] \<p>\<^sub>1[f, g]"
ipresIGSwapAbs_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'd, 'j, 'k, 'a, 'l, 'g, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGSwapAbs hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresSwapAbs Delta hA MOD"
map_zip_map: fixes f :: "'c \<times> 'd \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and xs :: "'e list"   and ys :: "'d list" shows "map f (zip (map g xs) ys) = Map2.map2 (\<lambda>x y. f (g x, y)) xs ys"
igSubstInpIPresIGWlsInpSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstInpIPresIGWlsInpSTR MOD \<equiv> \<forall>ys y Y delta inp. igWls MOD (asSort ys) Y \<longrightarrow> igWlsInp MOD delta (igSubstInp MOD ys Y y inp) = igWlsInp MOD delta inp"
Symm: fixes Ax :: "('a, 'b) Axioms"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and e2 :: "('a, 'b) Expression"   and B :: "'a" assumes "Axioms Ax"   and "Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Axioms.Theory Ax" shows "Vx : A \<turnstile> e2\<equiv>e1 : B \<in> Axioms.Theory Ax"
changing_atoms_works': fixes M :: "'b \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'b"   and phi :: "'a formula" shows "M \<circ> g \<Turnstile> phi = M \<Turnstile> map_formula g phi"
comp_apply: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and x :: "'h" shows "(f \<circ> g) x = f (g x)"
rel_simps_52: fixes n :: "num" shows "\<not> numeral n < (0::'a)"