hcju/isabelleps huggingface dataset:
le_ASSERTI: fixes \<Phi> :: "bool"   and M :: "'c nres"   and M' :: "'c nres" assumes "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> ASSERT \<Phi> \<bind> (\<lambda>_. M')"
rtranclp_reflclp_absorb: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "R\<^sup>*\<^sup>*\<^sup>=\<^sup>= = R\<^sup>*\<^sup>*"
language_def: shows "R.language \<equiv> {smap int (p ## S.trace w p) |p w. init p \<and> R.run w p}"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "'state" assumes "a1 \<rightarrow>t a2" obtains atm :: "'atom" and s :: "'state" where "a1 = (Atm atm, s__)" and "a2 = aval atm s__"   | tst :: "'test" and s :: "'state" and c :: "('test, 'atom) com" where "a1 = (while tst do c, s)" and "a2 = s" and "\<not> tval tst s"
swap_msgs_Send_Trans: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isTrans ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
not_ordLeq_iff_ordLess: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "((r', r) \<notin> ordLeq) = (r <o r')"
7: shows "(w \<sqinter> - prim_EP w v e)\<^sup>+ * e \<le> v * top"
removeProcs_length: fixes ps :: "'a pState_scheme list"   and dead :: "bool"   and sd :: "bool"   and ps' :: "'a pState_scheme list"   and dcs :: "integer list" assumes "removeProcs ps = (dead, sd, ps', dcs)" shows "length ps' \<le> length ps"
inject_2: fixes x21 :: "bool"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and y21 :: "bool"   and y22 :: "nat"   and y23 :: "nat"   and y24 :: "nat" shows "(Read\<^sub>s\<^sub>b x21 x22 x23 x24 = Read\<^sub>s\<^sub>b y21 y22 y23 y24) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24)"
igSwapBinp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'b, 'j) Iteration.model_scheme"   and zs :: "'e"   and z1 :: "'h"   and z2 :: "'h"   and binp :: "'a \<Rightarrow> 'b option" shows "igSwapBinp MOD zs z1 z2 binp \<equiv> lift (igSwapAbs MOD zs z1 z2) binp"
eq: shows "(\<lambda>f. f \<circ> uminus) ` (\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if x \<bullet> i \<le> c then (h \<circ> uminus) x else (0::'b)}) = (\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if c \<le> x \<bullet> i then h x else (0::'b)})"
cases: fixes x :: "'h + 'i + 'j" obtains a :: "'h" where "x = Inl a"   | b :: "'i" where "x = Inr (Inl b)"   | c :: "'j" where "x = Inr (Inr c)"
map_of_zip_map: fixes f :: "'b \<Rightarrow> 'c"   and rs :: "'b list"   and k :: "'c"   and v :: "'b" assumes "map_of (zip (map f rs) rs) k = Some v" shows "k = f v"
neutr_eq_iff: fixes a :: "'a"   and b :: "'a" shows "(\<bottom> = a \<squnion> b) = (a = \<bottom> \<and> b = \<bottom>)"
intro: fixes P :: "'b \<Rightarrow> bool" assumes "\<And>t u. P (app t u) = (P t \<and> P u)" shows "simple_syntactic_and P"
gen_kleene_lfp: fixes f :: "'c \<Rightarrow> 'c"   and m :: "'c" assumes "cont f" shows "lfp (\<lambda>x. sup m (f x)) = (SUP i. (f ^^ i) m)"
weight_ge_edge: fixes x :: "'a"   and y :: "'a"   and T :: "'a ugraph"   and w :: "'a set \<Rightarrow> nat" assumes "(x, y) \<in> edges T" shows "w {x, y} \<le> weight w T"
lower_on_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and V :: "'b set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.lower_on trm_ord \<sigma> \<eta> V = (\<forall>x\<in>V. basic_superposition.lower_or_eq trm_ord (Var x \<lhd> \<sigma>) (Var x \<lhd> \<eta>))"
vector_derivative_linepath_within: fixes x :: "real"   and a :: "'a"   and b :: "'a" assumes "x \<in> {0..1}" shows "vector_derivative (linepath a b) (at x within {0..1}) = b - a"
fset_inverse: fixes x :: "'a fset" shows "Abs_fset (fset x) = x"
before_id: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "xs ! i < xs ! j in xs = (i < j)"
lookup_times_monomial_right: fixes p :: "'b \<Rightarrow>\<^sub>0 'a"   and t :: "'b"   and c :: "'a"   and s :: "'b" shows "lookup (p * monomial c t) s = (lookup p (s - t) * c when t adds s)"
SUP_L_P_def: fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> 'f"   and u :: "'c"   and i :: "'b" shows "SUP_L_P X u i = (SUP v\<in>{v. pair v i \<sqsubset> u}. X v i)"
obj_self_adjunction: fixes a :: "'a" assumes "obj a" shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg a a \<l>\<^sup>-\<^sup>1[a] \<r>[a]"
subgraph_preserves_hom: fixes A :: "('d, 'e) labeled_graph"   and B :: "('d, 'e) labeled_graph"   and X :: "('d, 'f) labeled_graph"   and h :: "('f \<times> 'e) set" assumes "subgraph A B"   and "graph_homomorphism X A h" shows "graph_homomorphism X B h"
simps_2: fixes n :: "nat"   and d :: "nat \<Rightarrow> nat" shows "from_digits (Suc n) d = d 0 + base * from_digits n (d \<circ> Suc)"
borel_measurable_vimage_halfspace_component_le: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i \<le> a} \<in> sets (lebesgue_on S))"
Lazy_tllist_def: fixes xs :: "unit \<Rightarrow> 'a \<times> ('a, 'b) tllist + 'b" shows "Lazy_tllist xs = (case xs () of Inl (x, ys) \<Rightarrow> TCons x ys | Inr b \<Rightarrow> TNil b)"
IH: fixes \<rho> :: "complex mat" assumes "well_com x2a"   and "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "trace (complex_of_real c \<cdot>\<^sub>m \<rho>) \<le> 1" shows "denote x2a (complex_of_real c \<cdot>\<^sub>m \<rho>) = complex_of_real c \<cdot>\<^sub>m denote x2a \<rho>"
cont: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and x :: "real" assumes "right_continuous_mono f a b" shows "continuous (at_right x) f"
index_rev: fixes j :: "nat" shows "lj \<le> j \<and> j_to_jb_index n_as (j - lj) = (bj, lj - lj) \<and> (j_to_jb_index n_as j = (bj, lj - lj) \<longrightarrow> j < n \<longrightarrow> j = j - lj)"
prefix_contains_msg: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and es :: "'c event list"   and i :: "nat"   and m :: "'c" assumes "network history msg_id"   and "node_histories.prefix_of_node_history history es i"   and "m \<in> set (trivial_network.node_deliver_messages es)" shows "Deliver m \<in> set es"
cong: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" shows "ab_semigroup_add_0.sum_0 plus zero \<equiv> ab_semigroup_add_0.sum_0 plus zero"
singular_subdivision_diff: fixes p :: "nat"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_subdivision p (a - b) = singular_subdivision p a - singular_subdivision p b"
phi_def: fixes k :: "nat" shows "phi k = (Max (card ` {S. S \<subseteq> {0..k} \<and> gen_mono_on f (\<le>) S \<and> k \<in> S}), Max (card ` {S. S \<subseteq> {0..k} \<and> gen_mono_on f (\<lambda>x y. y \<le> x) S \<and> k \<in> S}))"
op_inter_ivl_coll_def: shows "op_inter_ivl_coll = (\<inter>)"
equivalence_of_bicategories_and_pseudonatural_equivalence_left_def: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'s \<Rightarrow> 'r"   and \<Phi>\<^sub>F :: "'s \<times> 's \<Rightarrow> 'r"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and \<eta>\<^sub>0 :: "'s \<Rightarrow> 's"   and \<eta>\<^sub>1 :: "'s \<Rightarrow> 's"   and \<epsilon>\<^sub>0 :: "'r \<Rightarrow> 'r"   and \<epsilon>\<^sub>1 :: "'r \<Rightarrow> 'r"   and F' :: "'s \<Rightarrow> 'r"   and \<Phi>\<^sub>F' :: "'s \<times> 's \<Rightarrow> 'r"   and \<tau>\<^sub>0 :: "'s \<Rightarrow> 'r"   and \<tau>\<^sub>1 :: "'s \<Rightarrow> 'r" shows "equivalence_of_bicategories_and_pseudonatural_equivalence_left 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 \<eta>\<^sub>0 \<eta>\<^sub>1 \<epsilon>\<^sub>0 \<epsilon>\<^sub>1 F' \<Phi>\<^sub>F' \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D) \<and> equivalence_of_bicategories 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 \<eta>\<^sub>0 \<eta>\<^sub>1 \<epsilon>\<^sub>0 \<epsilon>\<^sub>1 \<and> pseudonatural_equivalence V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F F' \<Phi>\<^sub>F' \<tau>\<^sub>0 \<tau>\<^sub>1"
cr_spmf_option_def: shows "cr_option \<equiv> \<lambda>x y. x = return_pmf y"
power_add: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "a ^ (m + n) = a ^ m \<cdot> a ^ n"
Times_Plus_Plus: fixes r :: "('a \<times> bool) language"   and s :: "('a \<times> bool) language" shows "Times_Plus (Plus r s) = Plus (Times_Plus r) (Times_Plus s)"
total_add_var: fixes x :: "'a"   and y :: "'a" assumes "1' \<le> x\<^sup>\<smile> ; x" shows "1' \<le> (x + y)\<^sup>\<smile> ; (x + y)"
wtPB: fixes \<Phi> :: "('a, 'b) lit list set" assumes "F.wtPB \<Phi>" shows "local.wtPB \<Phi>"
subspace_neg: fixes S :: "'a set"   and x :: "'a" assumes "csubspace S"   and "x \<in> S" shows "- x \<in> S"
f: fixes x :: "real" assumes "x \<in> h ` K" shows "f x \<in> h ` U"
det_bound_def: fixes n :: "nat"   and x :: "'b" shows "det_bound n x = fact n * x ^ n"
intro: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and Pair :: "'a"   and label_witness1 :: "'d"   and label_witness2 :: "'d" assumes "stateful_typed_model arity public Ana \<Gamma> Pair"   and "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2" shows "labeled_stateful_typed_model arity public Ana \<Gamma> Pair label_witness1 label_witness2"
zeros_rone: fixes n :: "nat"   and v :: "nat" shows "zeros (Suc n) v = zeros n v @ zeros 1 v"
fds_abs_converges_norm_mult: fixes f :: "'a fds"   and s :: "real"   and g :: "'a fds" assumes "fds_abs_converges (fds_norm f) s"   and "fds_abs_converges (fds_norm g) s" shows "fds_abs_converges (fds_norm (f * g)) s"
circuit_in_subI: fixes \<E>' :: "'a set"   and \<E> :: "'a set"   and C :: "'a set" assumes "\<E>' \<subseteq> \<E>"   and "\<E> \<subseteq> \<E>"   and "\<E>.circuit_in \<E>' C" shows "\<E>.circuit_in \<E> C"
puf: shows "(\<Prod>i\<leftarrow>[0..<n]. [:- cis (real i * 2 * pi / real n), 1:]) = root_unity n * f"
li_le_dim_1: fixes A :: "'c set" assumes "fin_dim"   and "A \<subseteq> carrier V"   and "lin_indpt A" shows "finite A"
simps: fixes as :: "nat"   and lm' :: "nat list"   and ss :: "nat"   and x :: "nat"   and rsx :: "nat"   and suf_lm :: "nat list"   and lm :: "nat list" shows "mn_ind_inv (as, lm') ss x rsx suf_lm lm = (if as = ss then lm' = lm @ x # rsx # suf_lm else if as = ss + 1 then \<exists>y. lm' = lm @ x # y # suf_lm \<and> y \<le> rsx else if as = ss + 2 then \<exists>y. lm' = lm @ x # y # suf_lm \<and> y \<le> rsx else if as = ss + 3 then lm' = lm @ x # 0 # suf_lm else if as = ss + 4 then lm' = lm @ Suc x # 0 # suf_lm else if as = 0 then lm' = lm @ Suc x # 0 # suf_lm else False)"
simps_1: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and signalsInit2 :: "signal \<Rightarrow> nat"   and signalsResp2 :: "signal \<Rightarrow> nat"   and more :: "'a" shows "signalsInit2 \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, signalsInit2 = signalsInit2, signalsResp2 = signalsResp2, \<dots> = more\<rparr> = signalsInit2"
act_emp_var: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c" assumes "unital_quantale_module act" shows "bot_pres (act x)"
avl_split_max: fixes t :: "('b \<times> bal) tree"   and t' :: "('b \<times> bal) tree alt"   and a :: "'b" assumes "split_max t = (t', a)"   and "avl t"   and "t \<noteq> \<langle>\<rangle>" shows "case t' of Same t' \<Rightarrow> avl t' \<and> height t = height t' | Diff t' \<Rightarrow> avl t' \<and> height t = height t' + 1"
nz: shows "p \<noteq> 0 \<or> q \<noteq> 0"
Berlekamp_subspace_dim: shows "Berlekamp_subspace.dim = card P"
IK_card: shows "card {f. IK f} \<le> 7"
XY: shows "(x, y) \<in> \<langle>A\<rangle>list_rel"
abs_venv_def: fixes ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" shows "|ve| = (\<lambda>(v, b_a). \<Union> {case ve (v, b) of None \<Rightarrow> {} | Some d \<Rightarrow> |d| |b. |b| = b_a})"
filter_Un: fixes f :: "'a \<Rightarrow> bool"   and x :: "'a set"   and y :: "'a set" shows "Set.filter f (x \<union> y) = Set.filter f x \<union> Set.filter f y"
cinner_sum_left: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (comm_monoid_add.sum plus zero f A) y = (\<Sum>x\<in>A. cinner (f x) y)"
are_endpoints_in_S: fixes a :: "proj2"   and b :: "proj2"   and p :: "proj2"   and q :: "proj2" assumes "a \<noteq> b"   and "are_endpoints_in_S p q a b" shows "p = endpoint_in_S a b \<and> q = endpoint_in_S b a \<or> q = endpoint_in_S a b \<and> p = endpoint_in_S b a"
meetD: fixes b :: "'b"   and c :: "'b"   and A :: "('b \<times> 'b) set" assumes "(b, c) \<in> A\<^sup>\<up>" shows "\<exists>a. (a, b) \<in> A\<^sup>* \<and> (a, c) \<in> A\<^sup>*"
trancl_trans: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set"   and z :: "'b" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
hyps_4: fixes Vs :: "char list list"   and ls :: "val list" assumes "fvs es \<subseteq> set Vs"   and "l\<^sub>1 \<subseteq>\<^sub>m [Vs [\<mapsto>] ls]"   and "length Vs + max_varss es \<le> length ls" shows "\<exists>ls'. compP\<^sub>1 P \<turnstile>\<^sub>1 \<langle>compEs\<^sub>1 Vs es,(h\<^sub>1, ls)\<rangle> [\<Rightarrow>] \<langle>compEs\<^sub>1 Vs (map Val vs),(h\<^sub>2, ls')\<rangle> \<and> l\<^sub>2 \<subseteq>\<^sub>m [Vs [\<mapsto>] ls']"
conga_refl: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B" shows "A B C CongA A B C"
writes_singleton2: fixes f :: "('l, 'm, 'n) prog"   and h :: "'l"   and h' :: "'l" shows "writes {f} f h h'"
rho_le_x_Ax_imp_rho_le_rx: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and x :: "(real, 'a) Finite_Cartesian_Product.vec"   and \<rho> :: "real" assumes "perron_frobenius A"   and "x \<in> perron_frobenius.X"   and "\<forall>i. (\<rho> *s x) $h i \<le> (A *v x) $h i" shows "\<rho> \<le> perron_frobenius.r A x"
A'_def: shows "A' = the ` (A \<inter> range Some)"
ln_inverse: fixes x :: "real" assumes "0 < x" shows "ln (inverse x) = - ln x"
eqvt_def: fixes X :: "'a set" shows "eqvt X \<equiv> \<forall>x\<in>X. \<forall>p. p \<bullet> x \<in> X"
span_mono: fixes A :: "('b \<Rightarrow>\<^sub>0 'c) set"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "A \<subseteq> B" shows "phull A \<subseteq> phull B"
filter_fset: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b fset" shows "fset (filter_fset P xs) = Collect P \<inter> fset xs"
RInitFailFinal: fixes P :: "(char list list \<times> char list exp) prog"   and e' :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and a :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh'' :: "sheap"   and e :: "char list exp" assumes "P \<turnstile> \<langle>e',s\<rangle> \<Rightarrow> \<langle>throw a,(h', l', sh')\<rangle>"   and "sh' C = \<lfloor>(sfs, i)\<rfloor>"   and "sh'' = sh'(C \<mapsto> (sfs, Error))" shows "P \<turnstile> \<langle>RI (C,e') ; [] \<leftarrow> e,s\<rangle> \<Rightarrow> \<langle>throw a,(h', l', sh'')\<rangle>"
split_max_height: fixes k :: "nat"   and t :: "'a btree"   and sub :: "'a btree"   and sep :: "'a" assumes "split_max k t = (sub, sep)"   and "nonempty_lasttreebal t"   and "t \<noteq> Leaf" shows "height sub = height t"
in_prime_factors_iff: fixes p :: "'a"   and x :: "'a" shows "(p \<in> prime_factors x) = (x \<noteq> (0::'a) \<and> p dvd x \<and> prime p)"
atLeastLessThan_shift_bounds: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "sum g {m + k..<n + k} = sum (g \<circ> (+) k) {m..<n}"
not_final_thread_iff: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "r2.not_final_thread s t = (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (\<not> final2 x \<or> ln \<noteq> no_wait_locks \<or> (\<exists>w. wset s t = \<lfloor>w\<rfloor>)))"
cases_scheme: fixes r :: "'a State_scheme" obtains getM :: "(Literal \<times> bool) list" and more :: "'a" where "r = \<lparr>getM = getM, \<dots> = more\<rparr>"
focus_in_Polys: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "focus X p \<in> P[X]"
word_rev_gal': fixes u :: "'b word"   and w :: "'b word" assumes "u = word_reverse w" shows "w = word_reverse u"
is_on_stack_impl_def: fixes v :: "'c"   and s :: "'c list \<times> nat list \<times> ('c \<Rightarrow> node_state option) \<times> (nat \<times> 'c set) list" shows "is_on_stack_impl v s \<equiv> GS.is_on_stack_impl s v"
add_edge_wf: fixes G :: "'c list_graph"   and v :: "'c"   and v' :: "'c" assumes "wf_list_graph G" shows "wf_list_graph (FiniteListGraph.add_edge v v' G)"
Arr_in_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "t \<in> mkarr t"
assms: shows "GEN_OP un op_union_coll (clw_rel A \<rightarrow> clw_rel A \<rightarrow> clw_rel A)"
tau_mpj_bilin2: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "('g, 'a, 'f) Module_scheme"   and P :: "('e \<times> 'g, 'a, 'f) Module_scheme"   and m :: "'e"   and n1 :: "'g"   and n2 :: "'g" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "m \<in> carrier M"   and "n1 \<in> carrier N"   and "n2 \<in> carrier N" shows "mpj P (TR\<^bsub>R\<^esub> M N P) (m, n1 \<plusminus>\<^bsub>N\<^esub> n2) = mpj P (TR\<^bsub>R\<^esub> M N P) (m, n1) \<plusminus>\<^bsub>M \<^bsub>P\<^esub>\<Otimes>\<^bsub>R\<^esub> N\<^esub> mpj P (TR\<^bsub>R\<^esub> M N P) (m, n2)"
sin_arg: fixes z :: "complex" assumes "z \<noteq> 0" shows "sin (arg z) = Im z / cmod z"
dependent_explicit: fixes s :: "('c \<Rightarrow>\<^sub>0 'b) set" shows "phull.dependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v \<cdot> v) = 0 \<and> (\<exists>v\<in>t. u v \<noteq> (0::'b)))"
TS_b': fixes x :: "nat"   and y :: "nat"   and h :: "nat list"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs \<le> 2 * T\<^sub>p [x, y] qs (OPT2 qs [x, y]) \<and> TS_inv' (Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs) (last qs) [x, y]"
tail_def: fixes S :: "'a list" shows "tail S \<equiv> if S = [] then [] else tl S"
iptables_bigstep_process_ret_cases3: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_ret rs, Undecided\<rangle> \<Rightarrow> Undecided" obtains "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided"   | rs\<^sub>1 :: "'c rule list" and rs\<^sub>2 :: "'c rule list" and m :: "'c match_expr" where "rs = rs\<^sub>1 @ [Rule m Return] @ rs\<^sub>2" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided" and "matches \<gamma> m p"
bind_state_def: fixes bind :: "'h \<Rightarrow> ('i \<times> 'g \<Rightarrow> 'h) \<Rightarrow> 'h" shows "bind_state bind \<equiv> rec_stateT (\<lambda>x f. StateT (\<lambda>s. bind (x s) (\<lambda>(a, s'). run_state (f a) s')))"
contour_integral_neg: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" shows "contour_integral g (\<lambda>x. - f x) = - contour_integral g f"
sequentially_imp_eventually_at_right: fixes a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "a < b"   and "\<And>f. \<lbrakk>\<And>n. a < f n; \<And>n. f n < b; decseq f; f \<longlonglongrightarrow> a\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F n in sequentially. P (f n)" shows "eventually P (at_right a)"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> R" shows "(P x, P y) \<in> R"
last_drop: fixes n :: "nat"   and xs :: "'d list" assumes "n < length xs" shows "last (drop n xs) = last xs"
cardSuc_UNION_Cinfinite: fixes r :: "('d \<times> 'd) set"   and As :: "'d set \<Rightarrow> 'e set"   and B :: "'e set" assumes "Cinfinite r"   and "relChain (cardSuc r) As"   and "B \<subseteq> \<Union> (As ` Field (cardSuc r))"   and "|B| \<le>o r" shows "\<exists>i\<in>Field (cardSuc r). B \<subseteq> As i"
image_ident: fixes Y :: "'b set" shows "(\<lambda>x. x) ` Y = Y"
is_gcd_gcd: fixes a :: "'a"   and b :: "'a" shows "is_gcd (gcd a b) a b"
eq_iff: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly" shows "(to_int_poly x = to_int_poly y) = (x = y)"
annir: fixes x :: "'b \<times> 'c" shows "times_prod' x bot_prod = bot_prod"
update_Mapping: fixes k :: "'a"   and v :: "'b"   and xs :: "('a \<times> 'b) list" shows "Mapping.update k v (AList_Mapping.Mapping xs) = AList_Mapping.Mapping (AList.update k v xs)"
fresh_fact1: fixes a :: "name"   and P :: "pi"   and c :: "name"   and b :: "name" assumes "a \<sharp> P"   and "a \<noteq> c" shows "a \<sharp> P[b::=c]"