hcju/isabelleps huggingface dataset:
elims_3: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list" assumes "\<not> sorted_wrt x xa" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and xb :: "'a" and ys :: "'a list" where "x = P" and "xa = xb # ys" and "\<not> (Ball (set ys) (P xb) \<and> sorted_wrt P ys)"
cor_sqrt_mult_cor_sqrt: fixes A :: "real" shows "cor (sqrt A) * cor (sqrt A) = cor \<bar>A\<bar>"
s2_def: shows "s2 = dfa_steps B 0 bs"
rel_fun_mono: fixes A' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B' :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "A' \<le> A"   and "B \<le> B'" shows "A ===> B \<le> A' ===> B'"
comp_simp: fixes g :: "'a2 \<times> 'a1"   and f :: "'a2 \<times> 'a1" assumes "A2.seq (fst g) (fst f)"   and "A1.seq (snd g) (snd f)" shows "g \<cdot>\<^sub>A\<^sub>2\<^sub>x\<^sub>A\<^sub>1 f = (fst g \<cdot>\<^sub>A\<^sub>2 fst f, snd g \<cdot>\<^sub>A\<^sub>1 snd f)"
mon_loc_def: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and ww :: "('c, 'f) label list el_step list" shows "mon_loc fg ww \<equiv> mon_ww fg (map le_rem_s (loc ww))"
blinding_of_on_F: fixes A :: "'e set"   and rha :: "'e \<Rightarrow> 'f"   and boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'g set"   and rhb :: "'g \<Rightarrow> 'h"   and bob :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "blinding_of_on A rha boa"   and "blinding_of_on B rhb bob" shows "blinding_of_on {x. set1_F\<^sub>m x \<subseteq> A \<and> set3_F\<^sub>m x \<subseteq> B} (root_hash_F rha rhb) (blinding_of_F rha boa rhb bob)"
per_init_def: fixes D :: "'c set" shows "per_init D \<equiv> {(i, i) |i. i \<in> D}"
sp_def: fixes P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "local.sp P c t \<equiv> \<exists>s. P s \<and> \<pi>: (c, s) \<Rightarrow> t"
meval_sched_length: fixes sched :: "nat list"   and gc :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and gc' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" assumes "conc.meval_sched_abv gc sched gc'" shows "length (fst gc') = length (fst gc)"
hom_functor_axioms: shows "hom_functor (\<cdot>\<^sub>D) S.comp (\<lambda>_. inD)"
map_of_eq_None_iff: fixes xys :: "('b \<times> 'a) list"   and x :: "'b" shows "(None = map_of xys x) = (x \<notin> fst ` set xys)"
merge_T_cong: fixes x :: "('e, 'f) T\<^sub>m"   and m :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option"   and m' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option"   and rh :: "'e \<Rightarrow> 'f"   and y :: "('e, 'f) T\<^sub>m" assumes "\<And>a b. a \<in> set1_T\<^sub>m x \<Longrightarrow> m a b = m' a b" shows "merge_T rh m x y = merge_T rh m' x y"
Quotient_circline_mat: shows "Quotient (eq_onp (\<lambda>H. hermitean H \<and> H \<noteq> mat_zero)) Abs_circline_mat Rep_circline_mat cr_circline_mat"
span_sum: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and S :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S" shows "sum f A \<in> span S"
linorder_less_wlog: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.linorder less_eq less"   and "\<And>a b. less a b \<Longrightarrow> P a b"   and "\<And>a. P a a"   and "\<And>a b. P b a \<Longrightarrow> P a b" shows "P a b"
union_multisets_iff: fixes M :: "'b multiset"   and N :: "'b multiset"   and A :: "'b set" shows "(M + N \<in> multisets A) = (M \<in> multisets A \<and> N \<in> multisets A)"
otherassms_6: shows "p \<in> sterms \<Gamma> pp"
assms_3: shows "(f has_vector_derivative f') (at x within {a..b})"
sum_eq: fixes b :: "ordinal" assumes "b < a" shows "b + a = a"
floorlog_le_iff: fixes b :: "nat"   and x :: "nat"   and w :: "nat" shows "(floorlog b x \<le> w) = (b \<le> 1 \<or> 1 < b \<and> 0 \<le> w \<and> x < b ^ w)"
simps_1: fixes e :: "name list"   and t :: "tm"   and u :: "tm" shows "trans_fm e (t IN u) = DBMem (trans_tm e t) (trans_tm e u)"
left_tree_initial': fixes s :: "nat"   and t :: "nat" assumes "s \<in> V"   and "t \<in> V"   and "s \<noteq> t" shows "t \<notin> left_tree s t"
sets_T: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "sets (MC_syntax.T K s) = sets (stream_space (count_space UNIV))"
OUTfromVCorrect1_data3: shows "OUTfromVCorrect1 data3"
monoid_mult_hom_axioms: shows "monoid_mult_hom hom"
lens_equiv_trans: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and Z :: "'g \<Longrightarrow> 'e" assumes "X \<approx>\<^sub>L Y"   and "Y \<approx>\<^sub>L Z" shows "X \<approx>\<^sub>L Z"
psubst_Var: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "goodEnv rho" shows "Var xs x #[rho] = (case rho xs x of None \<Rightarrow> Var xs x | Some X \<Rightarrow> X)"
star_left_induct_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> y \<le> y" shows "top \<sqinter> y \<le> y"
h: shows "h \<in> M \<rightarrow>\<^sub>M N"
Image_eq_UN: fixes r :: "('c \<times> 'b) set"   and B :: "'c set" shows "r `` B = (\<Union>y\<in>B. r `` {y})"
free_group_def: fixes gens :: "'c set" shows "\<F>\<^bsub>gens\<^esub> \<equiv> \<lparr>carrier = {l \<in> lists (UNIV \<times> gens). canceled l}, monoid.mult = \<lambda>x y. Cancelation.normalize (x @ y), one = []\<rparr>"
ID_distr_global_conditional_2: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r :: "'b"   and ls :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr" assumes "s r = Some ls" shows "LID\<^sub>G s = LID\<^sub>G (s(r := None)) \<union> LID\<^sub>L ls"
amd1_delete_def: fixes dflt :: "'d"   and k :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "amd1_delete dflt k f \<equiv> f(k := dflt)"
parCasesF: fixes P :: "pi"   and Q :: "pi"   and \<alpha> :: "freeRes"   and PQ' :: "pi"   and F :: "freeRes \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<parallel> Q \<longmapsto> \<alpha> \<prec> PQ'"   and "\<And>P'. P \<longmapsto> \<alpha> \<prec> P' \<Longrightarrow> F \<alpha> (P' \<parallel> Q)"   and "\<And>Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> F \<alpha> (P \<parallel> Q')"   and "\<And>P' Q' a b x. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; Q \<longmapsto> a[b] \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<noteq> b; x \<sharp> Q'; x \<sharp> C; \<alpha> = \<tau>\<rbrakk> \<Longrightarrow> F (\<tau>) (P'[x::=b] \<parallel> Q')"   and "\<And>P' Q' a b x. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; Q \<longmapsto> a<x> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<noteq> b; x \<sharp> P'; x \<sharp> C; \<alpha> = \<tau>\<rbrakk> \<Longrightarrow> F (\<tau>) (P' \<parallel> Q'[x::=b])"   and "\<And>P' Q' a x y. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; Q \<longmapsto> a<\<nu>y> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<noteq> y; x \<sharp> Q'; y \<sharp> P; y \<sharp> Q; y \<noteq> a; y \<sharp> P'; x \<sharp> C; y \<sharp> C; \<alpha> = \<tau>\<rbrakk> \<Longrightarrow> F (\<tau>) (<\<nu>y>(P'[x::=y] \<parallel> Q'))"   and "\<And>P' Q' a x y. \<lbrakk>P \<longmapsto> a<\<nu>y> \<prec> P'; Q \<longmapsto> a<x> \<prec> Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<noteq> y; x \<sharp> P'; y \<sharp> P; y \<sharp> Q; y \<noteq> a; y \<sharp> Q'; x \<sharp> C; y \<sharp> C; \<alpha> = \<tau>\<rbrakk> \<Longrightarrow> F (\<tau>) (<\<nu>y>(P' \<parallel> Q'[x::=y]))" shows "F \<alpha> PQ'"
Diff_insert_absorb: fixes x :: "'b"   and A :: "'b set" assumes "x \<notin> A" shows "insert x A\<setminus>{x} = A"
WT_plus_oracleD2: fixes \<I>l :: "('a, 'c) \<I>"   and \<I>r :: "('b, 'd) \<I>"   and left :: "'e \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'e) spmf"   and right :: "'e \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) spmf"   and s :: "'e" assumes "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd>" shows "\<I>r \<turnstile>c right s \<surd>"
supp_distinguishing_weak_formula: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c"   and Q :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition"   and "\<not> indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q" shows "supp (indexed_weak_nominal_ts.distinguishing_weak_formula \<tau> satisfies transition P Q) \<subseteq> supp P"
g: shows "(\<Sum>x\<in>UNIV. g x *s X $ x) = w"
prems_1: shows "length (a # xs) - length ys = Suc m"
nonseparation_by_component_eq: fixes S :: "'a set" assumes "open S \<or> closed S" shows "(\<forall>C\<in>components S. connected (- C)) = connected (- S)"
matrix_mul_mat: fixes A :: "(('a, 'b) vec, 'c) vec"   and x :: "'a" shows "A ** mat x = x *k A"
get_attribute_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and k :: "char list" assumes "DocumentClass.type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_attribute element_ptr k"
subfieldI': fixes K :: "'a set" assumes "subring K R"   and "\<And>k. k \<in> K - {\<zero>} \<Longrightarrow> inv k \<in> K" shows "subfield K R"
C_eq_RS3_aux: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "not_MT C p" shows "C (list2FWpolicy p) x = C (list2FWpolicy (rm_MT_rules C p)) x"
sas_plus_equivalent_to_strips_i_a_VIII: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'd option"   and ops' :: "('c \<times> 'd) strips_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "dom s \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom s. the (s v) \<in> \<R>\<^sub>+ \<Psi> v"   and "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "are_all_operators_applicable_in s (map (strips_op_to_sasp \<Psi>) ops') \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent (map (strips_op_to_sasp \<Psi>) ops')" shows "are_all_operators_applicable (\<phi>\<^sub>S \<Psi> s) ops' \<and> STRIPS_Semantics.are_all_operator_effects_consistent ops'"
scaleR_half_double: fixes a :: "'a" shows "(1 / 2) *\<^sub>R (a + a) = a"
C2O_replicate: fixes i :: "nat"   and n :: "C" shows "C2O (C (replicate i n)) = mulO (exp\<omega> (C2O n)) ((S ^^ i) Z)"
genset_order2: fixes s :: "'w" assumes "s \<in> S" shows "add_order s = 2"
wfP_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "wfP r = wf {(x, y). r x y}"
assms_4: shows "num_params (tm_poly t2) \<le> length I"
NAKTr5: fixes A :: "'b set"   and H :: "'a set"   and k :: "nat" assumes "\<not> zeroring R"   and "ideal R A"   and "A \<subseteq> J_rad R"   and "A \<odot>\<^bsub>R\<^esub> M = carrier M"   and "finite_generator R M H"   and "card H = Suc k"   and "0 < k" shows "\<exists>h\<in>{j. j \<le> k} \<rightarrow> carrier M. H = h ` {j. j \<le> k} \<and> h k \<in> linear_span R M A (h ` {j. j \<le> k - Suc 0})"
col_not_col_not_par: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<exists>P. Col A B P \<and> Col C D P"   and "\<exists>Q. Col C D Q \<and> \<not> Col A B Q" shows "\<not> A B Par C D"
7: shows "\<forall>x y. f x \<sqinter> y = bot \<longrightarrow> x \<sqinter> g y = bot"
root_pref_cancel': fixes x :: "'b list"   and y :: "'b list"   and t :: "'b list" assumes "x \<cdot> y \<in> t*"   and "x \<in> t*" shows "y \<in> t*"
IH: fixes S :: "'node SDG_node set" assumes "n'' \<in> HRB_slice {n}"   and "nx \<in> S" shows "n'' \<in> HRB_slice S"
extr_def: shows "extr \<equiv> \<lambda>bad IK H. {x. extrp (\<lambda>x. x \<in> bad) (\<lambda>x. x \<in> IK) (\<lambda>x. x \<in> H) x}"
R2_def: shows "R2 = (\<lambda>x. sum_upto (\<lambda>x. real x powr (- \<alpha> - 1)) x - (\<zeta> - x powr - \<alpha> / \<alpha>))"
atLeastAtMostPlus1_int_conv: fixes m :: "int"   and n :: "int" assumes "m \<le> 1 + n" shows "{m..1 + n} = insert (1 + n) {m..n}"
number_partition_of_1: fixes A :: "'c set"   and B :: "'d set"   and F :: "('c \<Rightarrow> 'd) set" assumes "finite A"   and "finite B"   and "F \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B" shows "number_partition (card A) (number_partition_of A B F)"
subst_idem_def: fixes s :: "'d \<Rightarrow> ('e, 'd) Term.term" shows "subst_idem s \<equiv> s \<circ>\<^sub>s s = s"
nat_pow_Suc: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" shows "x [^] Suc n = x [^] n \<otimes> x"
abs_eq: fixes x :: "((real, 'e) vec, 'f) vec" shows "blinfun_of_vmatrix x = Blinfun ((*v) x)"
openin_topspace: fixes U :: "'d topology" shows "openin U (topspace U)"
mult_isotone: fixes w :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" assumes "w \<preceq> y"   and "x \<preceq> z" shows "w \<odot> x \<preceq> y \<odot> z"
thread_oks_append: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and tas :: "('f, 'i, 'j) new_thread_action list"   and tas' :: "('f, 'i, 'j) new_thread_action list" shows "thread_oks ts (tas @ tas') = (thread_oks ts tas \<and> thread_oks (redT_updTs' ts tas) tas')"
flip_def: fixes a :: "'c"   and b :: "'c" shows "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
f00x: shows "\<forall>x. f (0::'a) (f (0::'a) x) = f (0::'a) x"
assms: shows "h \<turnstile> get_disconnected_document disc_node \<rightarrow>\<^sub>r disc_doc"
inorder_baldR: fixes ba :: "bal"   and l :: "('b \<times> bal) tree"   and a :: "'b"   and r :: "('b \<times> bal) tree alt" assumes "ba = Lh \<longrightarrow> l \<noteq> \<langle>\<rangle>"   and "avl l" shows "Tree2.inorder (tree (baldR l a ba r)) = Tree2.inorder l @ a # Tree2.inorder (tree r)"
thh1: shows "s @ t setinterleaves ((ta, u), A) \<and> (\<forall>t1. t1 \<in> set u1 \<longrightarrow> t1 \<notin> A) \<longrightarrow> s setinterleaves (([], u1), A)"
until_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "until \<phi> \<psi> \<pi>l \<equiv> \<exists>i. \<psi> (map (sdrop i) \<pi>l) \<and> (\<forall>j\<in>{0..<i}. \<phi> (map (sdrop j) \<pi>l))"
x: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and X :: "'d set"   and ds :: "'e set"   and Pd' :: "'e \<Rightarrow> ('d \<times> 'd) set" assumes "TruePrefs Xd Xh Pd Ch x X ds Pd'" shows "x \<in> X"
rt: shows "poly (map_poly complex_of_real fi) x = 0"
is_unifiers_def: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and AAA :: "('a, 'b) Term.term set set" shows "is_unifiers \<sigma> AAA = (\<forall>AA\<in>AAA. is_unifier \<sigma> AA)"
ay_def: shows "ay \<equiv> solution sys' (VAR (Suc 0))"
list_all2_takeI: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list"   and n :: "nat" assumes "list_all2 P xs ys" shows "list_all2 P (take n xs) (take n ys)"
lub_eq: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'b \<Rightarrow> 'a"   and Y :: "'b \<Rightarrow> 'a" assumes "class.cpo below"   and "\<And>i. X i = Y i" shows "po.lub below (range X) = po.lub below (range Y)"
relpowp_0_E: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "(P ^^ 0) x y" obtains "x = y"
ID: shows "case_sum (\<lambda>e. e) (\<lambda>a. undefined) \<circ> Inl = id"
take_lemma: fixes x :: "['b]"   and y :: "['b]" assumes "\<And>n. list_take n\<cdot>x = list_take n\<cdot>y" shows "x = y"
open_blinop_def: fixes U :: "'a blinop set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
range_eq_singletonD: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'b"   and x :: "'c" assumes "range f = {a}" shows "f x = a"
last_seg_impl_def: fixes s :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list" shows "last_seg_impl s \<equiv> GS.last_seg_impl s"
assms_2: shows "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> poly p x \<noteq> 0"
intro: fixes D :: "('h, 'i) Order_scheme" assumes "rel D \<subseteq> carrier D \<times> carrier D"   and "\<And>a. a \<in> carrier D \<Longrightarrow> (a, a) \<in> rel D"   and "\<And>a b. \<lbrakk>a \<in> carrier D; b \<in> carrier D; (a, b) \<in> rel D; (b, a) \<in> rel D\<rbrakk> \<Longrightarrow> a = b"   and "\<And>a b c. \<lbrakk>a \<in> carrier D; b \<in> carrier D; c \<in> carrier D; (a, b) \<in> rel D; (b, c) \<in> rel D\<rbrakk> \<Longrightarrow> (a, c) \<in> rel D" shows "Order D"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (r\<^sub>0\<^sup>* \<star> s\<^sub>1) r\<^sub>0s\<^sub>1.tab r\<^sub>0s\<^sub>1.p\<^sub>0 r\<^sub>0s\<^sub>1.p\<^sub>1 (r\<^sub>0\<^sup>* \<star> s\<^sub>1) r\<^sub>0s\<^sub>1.tab r\<^sub>0s\<^sub>1.p\<^sub>0 r\<^sub>0s\<^sub>1.p\<^sub>1 (r\<^sub>0\<^sup>* \<star> s\<^sub>1)"
injective_codomain: fixes x :: "'a" assumes "injective x" shows "x * (x \<sqinter> (1::'a)) = x \<sqinter> (1::'a)"
CFG_return_edge_fun: 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 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 a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and ins :: "'c list"   and outs :: "'c list"   and vmap :: "'c \<Rightarrow> 'd option"   and vmap' :: "'c \<Rightarrow> 'd option" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"   and "(p, ins, outs) \<in> set procs" shows "f' vmap vmap' = vmap'(ParamDefs (targetnode a) [:=] map vmap outs)"
elims: fixes x :: "nat"   and y :: "char list" assumes "nat_to_string x = y" obtains n :: "nat" where "x = n" and "y = (if is_digit n then [digit_to_char n] else nat_to_string (n div 10) @ [digit_to_char (n mod 10)])"
Bell_eq: fixes n :: "nat" shows "Bell (n + 1) = (\<Sum>k\<le>n. \<Sum>j\<le>1. j ^ (n - k) * Stirling 1 j * (n choose k) * Bell k)"
1: shows "\<F>C.seq (\<F>\<^sub>SC.map (\<F>C.D f)) (\<F>C.\<nu> (\<F>C.dom f))"
rtrancl_onE_ntrancl_onp: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> rtrancl_on F r" obtains n :: "nat" where "ntrancl_onp F r n a b"
assms: shows "Cop.H f \<epsilon> \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> Cop.H \<eta> f = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
add_def: fixes e :: "char list \<Rightarrow> 'b option"   and x :: "char list"   and a :: "'b" shows "Env e\<lparr>x:a\<rparr> = (if x \<notin> dom e then Env (e(x \<mapsto> a)) else Malformed)"
SFAccInitDoneReds: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and sfs :: "sfields"   and v :: "val"   and b :: "bool" assumes "P \<turnstile> C has F,Static:t in D"   and "shp s D = \<lfloor>(sfs, Done)\<rfloor>"   and "sfs F = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},s,b\<rangle> \<rightarrow>* \<langle>Val v,s,False\<rangle>"
rotater_def: fixes n :: "nat" shows "rotater n = rotater1 ^^ n"
lookup_default_empty: fixes d :: "'b"   and k :: "'c" shows "Mapping.lookup_default d Mapping.empty k = d"
lemma_W2: fixes a :: "'b G"   and b :: "'b G" shows "(a r\<rightarrow> b) l\<rightarrow> b = (b r\<rightarrow> a) l\<rightarrow> a"
Approximation1_5Tr3: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "'a"   and j :: "nat"   and m :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "x \<in> carrier K"   and "Ostrowski_elem K (Suc n) vv x"   and "j \<in> nset (Suc 0) (Suc n)" shows "vv j ((1\<^sub>r \<plusminus> -\<^sub>a x)^\<^bsup>K m\<^esup>) = 0"
zero_le_one: shows "(0::'b) \<le> (1::'b)"
inf_not_le_Lt: fixes x :: "'d" shows "(\<infinity> \<le> Lt x) = False"
rep_eq: fixes x :: "'a sq_mtx"   and xa :: "'a sq_mtx" shows "to_vec (x + xa) = to_vec x + to_vec xa"
Pairs_def: fixes AAA :: "'c set set" shows "Pairs AAA = concat (compare_linorder.sorted_list_of_set ((pairs \<circ> compare_linorder.sorted_list_of_set) ` AAA))"
intro: fixes Bot :: "'a set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Inf :: "'a inference set"   and Red_I_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a inference set"   and Red_F_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set" assumes "consequence_relation_family Bot Q entails_q"   and "intersection_calculus_axioms Bot Inf Q entails_q Red_I_q Red_F_q" shows "intersection_calculus Bot Inf Q entails_q Red_I_q Red_F_q"
deny_dom: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set" assumes "twoNetsDistinct a b c d" shows "dom (Cp (DenyAllFromTo a b)) \<inter> dom (Cp (DenyAllFromTo c d)) = {}"
true_PL: fixes A :: "'a set" shows "A \<turnstile> \<top>"
intro: fixes g :: "'a \<Rightarrow> 'b" assumes "\<And>x. \<lbrakk>x \<noteq> \<bottom>; x \<noteq> null\<rbrakk> \<Longrightarrow> g x \<noteq> \<bottom>" shows "profile_mono\<^sub>d_axioms g"
WTrtThrow: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T' :: "ty"   and T :: "ty" assumes "P,E,h \<turnstile> e : T'"   and "is_refT T'" shows "P,E,h \<turnstile> throw e : T"
finite_Inf_prod: fixes A :: "('a \<times> 'b) set" shows "Inf A = Finite_Set.fold inf top A"
bet_neq12__neq: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C"   and "A \<noteq> B" shows "A \<noteq> C"
cycle_free_diag: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and i :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "linordered_ab_monoid_add.cycle_free plus less_eq neutral m n"   and "i \<le> n" shows "less_eq neutral (m i i)"
ih: fixes s :: "('s, 'v) tm" assumes "s \<in> set ss"   and "wary s" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt (subst \<rho> s))) = eval_tpoly (subst_zpassign \<rho> A) (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
map_add_comm: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option" assumes "dom m1 \<inter> dom m2 = {}" shows "m1 ++ m2 = m2 ++ m1"