hcju/isabelleps huggingface dataset:
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_goon_left_moving_O m rs (l, r) = (\<exists>ml mr rn. l = Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> r = Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc (Suc rs) \<and> 0 < mr)"
m\<mu>t_thr_dom_eq: fixes m :: "'m"   and ts :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and ts' :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option" assumes "m\<mu>t m ts ts'" shows "dom ts = dom ts'"
G_is_full: shows "full_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
Sup_def: fixes D :: "('c, 'd) Order_scheme"   and X :: "'c set" shows "Algebra2.Sup D X = (THE x. minimum_elem D (upper_bounds D X) x)"
intro: fixes \<U> :: "'c set"   and \<A> :: "'c set multiset"   and \<V> :: "'c set"   and \<B> :: "'c set multiset" assumes "sub_incidence_system \<U> \<A> \<V> \<B>"   and "design \<V> \<B>" shows "sub_design \<U> \<A> \<V> \<B>"
rel_dds_eq: shows "rel_dds (=) (=) = (=)"
matched_bracket_call: fixes n\<^sub>0 :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n\<^sub>1 :: "'node SDG_node"   and p :: "'pname"   and n\<^sub>2 :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n\<^sub>3 :: "'node SDG_node"   and n\<^sub>4 :: "'node SDG_node"   and V :: "'var"   and a :: "'edge"   and a' :: "'edge" assumes "matched n\<^sub>0 ns n\<^sub>1"   and "n\<^sub>1 -p\<rightarrow>\<^bsub>call\<^esub> n\<^sub>2"   and "matched n\<^sub>2 ns' n\<^sub>3"   and "n\<^sub>3 -p\<rightarrow>\<^bsub>ret\<^esub> n\<^sub>4 \<or> n\<^sub>3 -p:V\<rightarrow>\<^bsub>out\<^esub> n\<^sub>4"   and "valid_edge a"   and "a' \<in> get_return_edges a"   and "sourcenode a = parent_node n\<^sub>1"   and "targetnode a = parent_node n\<^sub>2"   and "sourcenode a' = parent_node n\<^sub>3"   and "targetnode a' = parent_node n\<^sub>4" shows "matched n\<^sub>0 (ns @ n\<^sub>1 # ns' @ [n\<^sub>3]) n\<^sub>4"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(IMor x2 = IMor y2) = (x2 = y2)"
pt_perm_supp_ineq: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "cp TYPE('a) TYPE('b) TYPE('c)" shows "pi \<bullet> supp x = supp (pi \<bullet> x)"
hensels_lemma_unique_root: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and \<alpha> :: "nat \<Rightarrow> int"   and \<beta> :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp"   and "pderiv f \<bullet> a \<noteq> \<zero>"   and "f \<bullet> a \<noteq> \<zero>"   and "eint 2 * val_Zp (pderiv f \<bullet> a) < val_Zp (f \<bullet> a)"   and "f \<bullet> \<alpha> = \<zero>"   and "\<alpha> \<in> carrier Zp"   and "val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> \<alpha>)"   and "f \<bullet> \<beta> = \<zero>"   and "\<beta> \<in> carrier Zp"   and "val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> \<beta>)"   and "val_Zp (pderiv f \<bullet> \<alpha>) = val_Zp (pderiv f \<bullet> a)" shows "\<alpha> = \<beta>"
l: shows "insert (OrdP k) H \<turnstile> l IN k"
reduce_rules_def: fixes \<delta> :: "('a, 'b) ta_rule set"   and P :: "'a set" shows "reduce_rules \<delta> P \<equiv> \<delta> \<inter> {r. rule_states r \<subseteq> P}"
nextLoc_fresh: fixes h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list" shows "h\<down>nextLoc h = None"
ih_5: shows "InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state')"
distinct_1: fixes x1 :: "'b"   and x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "Var x1 \<noteq> Fun x21 x22"
br_cond_def: shows "br_cond \<equiv> {(Q, W). W \<noteq> {}}"
typeof_simps_4: fixes "by" :: "int" shows "\<tau> byteV by = ByteT"
l_set_child_nodes_get_disconnected_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_child_nodes_get_disconnected_nodes type_wf set_child_nodes set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_child_nodes type_wf set_child_nodes set_child_nodes_locs \<and> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_child_nodes_get_disconnected_nodes_axioms set_child_nodes_locs get_disconnected_nodes_locs"
sums: shows "(\<lambda>k. x * x\<^sup>2 ^ n * (x ^ 4) ^ k * c k) sums (arctan x - arctan_approx n x)"
lc_not_0: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and cmp_term :: "'b nat_term_order" assumes "p \<noteq> 0" shows "lc_punit cmp_term p \<noteq> (0::'c)"
unl_before: fixes v :: "state \<Rightarrow> 'b"   and s :: "state"   and t :: "state" shows "($v) (s, t) \<equiv> v s"
the_equality: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(THE x. P x) = a"
diff_add_inverse2: fixes m :: "nat"   and n :: "nat" shows "m + n - n = m"
ImagepI: fixes a :: "'c"   and A :: "'c set"   and r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d" assumes "a \<in> A"   and "r a b" shows "b \<in> r ``` A"
induct: fixes P :: "nat list \<Rightarrow> ('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> ('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val) \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" assumes "\<And>c c'. P [] c c'"   and "\<And>n ns c c'. (\<And>x. P ns x c') \<Longrightarrow> P (n # ns) c c'" shows "P a0 a1 a2"
arr_eqI: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b"   and f' :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S f \<and> partial_magma.arr S f' \<and> partial_magma.dom S f = partial_magma.dom S f' \<and> partial_magma.cod S f = partial_magma.cod S f'"   and "set_category_given_img.Fun S img f = set_category_given_img.Fun S img f'" shows "f = f'"
lmeasure_integral: fixes S :: "'c set" assumes "S \<in> lmeasurable" shows "Sigma_Algebra.measure lebesgue S = integral S (\<lambda>x. 1)"
iprev_cut_less_conv: fixes n :: "nat"   and t :: "nat"   and I :: "nat set" assumes "n < t" shows "iprev n (I \<down>< t) = iprev n I"
scnj: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and F :: "'c set" assumes "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls"   and "F \<subseteq> fmla"   and "finite F" shows "Syntax_with_Connectives_False.scnj cnj imp fls F \<in> fmla"
num_unique_vars_def: fixes t :: "('a, 'b) Term.term" shows "num_unique_vars t = card (vars_term t)"
rows_transpose: fixes A :: "(('a, 'c) vec, 'b) vec" shows "rows (Finite_Cartesian_Product.transpose A) = columns A"
ones_inv_def: shows "ones_inv \<equiv> Abs_fps (\<lambda>n. if n = 0 then 1::'a else if n = 1 then - (1::'a) else (0::'a))"
slice_kinds_def: fixes S :: "'node set"   and as :: "'edge list" shows "slice_kinds S as \<equiv> map (slice_kind S) as"
b: shows "\<forall>x y z. T (i (i x (i y z)) (i (i x y) (i x z)))"
indep_vars_compose2: fixes M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> real \<Rightarrow> 'c"   and I :: "'b set"   and Y :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'b \<Rightarrow> 'd measure" assumes "\<mu>.indep_vars M' X I"   and "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> M' i \<rightarrow>\<^sub>M N i" shows "\<mu>.indep_vars N (\<lambda>i x. Y i (X i x)) I"
xivp_2: shows "\<nu> (X t\<^sub>0) \<le> \<mu> (X t\<^sub>0)"
abs_eq: shows "uminus = (\<lambda>x. Abs_atoms (Collect atom - Rep_atoms x))"
conj_cancel_right: fixes x :: "'a" shows "x AND NOT x = (0::'a)"
simps_1: fixes q :: "'a"   and l :: "nat"   and rA :: "'a \<times> 'a mat" shows "local.sub1 q 0 l rA = rA"
H_Inf_pres_fstar: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "Inf_pres f"   and "x \<le> f x" shows "x \<le> fstar f x"
safety_prop_AllowedActs_iff_Allowed: fixes X :: "'c program set"   and F :: "'c program" assumes "safety_prop X" shows "(\<Union> (Acts ` X) \<subseteq> AllowedActs F) = (X \<subseteq> Allowed F)"
exhaust_sel: fixes option :: "'e option" obtains "option = None"   | "option = Some (the option)"
M_bounded: shows "AE x in M. emeasure (N x) (space (N x)) \<le> ennreal B'"
synlr_cal_lr_rep: fixes r :: "(ValD, Prog) synlr dual \<times> (ValD, Prog) synlr" shows "ca_lf_rep r \<in> synlr"
belowD: fixes f :: "'a \<rightarrow> 'a"   and d :: "'a defl"   and x :: "'a" assumes "f \<sqsubseteq> cast\<cdot>d"   and "f\<cdot>x = x" shows "cast\<cdot>d\<cdot>x = x"
insert_monad_def: shows "Set_Impl.insert_monad = insert"
that: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> A"   and "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> b \<le>\<^sub>\<int> x" shows "thesis"
f: shows "f \<in> M \<rightarrow>\<^sub>M count_space UNIV"
inject: fixes x11 :: "'c tree"   and x12 :: "'c"   and x13 :: "bool"   and x14 :: "'c tree"   and y11 :: "'c tree"   and y12 :: "'c"   and y13 :: "bool"   and y14 :: "'c tree" shows "(Node x11 x12 x13 x14 = Node y11 y12 y13 y14) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13 \<and> x14 = y14)"
proj_along_quasiconvex_contraction: fixes C :: "real"   and G :: "'a set"   and px :: "'a"   and x :: "'a"   and py :: "'a"   and y :: "'a" assumes "quasiconvex C G"   and "px \<in> proj_set x G"   and "py \<in> proj_set y G" shows "dist px py \<le> max (5 * deltaG TYPE('a) + 2 * C) (dist x y - dist px x - dist py y + 10 * deltaG TYPE('a) + 4 * C)"
insertion_lowerPoly: fixes p :: "'a mpoly"   and i :: "nat"   and j :: "nat"   and prfx :: "'a list"   and xs :: "'a list"   and L :: "'a list" assumes "vars p \<inter> {i..<i + j} = {}"   and "length prfx = i"   and "length xs = j" shows "insertion (nth_default (0::'a) (prfx @ L)) (lowerPoly i j p) = insertion (nth_default (0::'a) (prfx @ xs @ L)) p"
f2r_kcomp_pres: fixes f :: "'c \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'd set" shows "\<R> (f \<circ>\<^sub>K g) = \<R> f ; \<R> g"
intro: fixes K :: "'a \<Rightarrow> 'a pmf set" assumes "\<And>s. K s \<noteq> {}" shows "Markov_Decision_Process K"
induct: fixes P :: "'a set \<Rightarrow> bool"   and a0 :: "'a set" assumes "\<And>A. \<lbrakk>\<And>x. \<lbrakk>\<not> (infinite A \<or> A = {}); x \<in> set_pmf (pmf_of_set A)\<rbrakk> \<Longrightarrow> P {y \<in> A. y < x}; \<And>x xa. \<lbrakk>\<not> (infinite A \<or> A = {}); x \<in> set_pmf (pmf_of_set A); xa \<in> set_pmf (random_bst {y \<in> A. y < x})\<rbrakk> \<Longrightarrow> P {y \<in> A. x < y}\<rbrakk> \<Longrightarrow> P A" shows "P a0"
Gcd_greatest_nat: fixes A :: "nat set"   and a :: "nat" assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b" shows "a dvd Gcd A"
prems_3: shows "\<forall>vtx. ListMem vtx (a # G) \<longrightarrow> \<not> R vtx vtx"
strong_fusion_in: fixes z :: "i" shows "P (\<sigma>x. z = x) z"
up_top: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c" assumes "class.order_top less_eq less top" shows "{y. less_eq top y} = {top}"
assms_3: shows "regular a \<and> regular b \<and> regular c \<and> regular d \<and> regular H"
basis_col_space: fixes A :: "(('b, 'd) vec, 'c) vec" shows "vec.independent (basis_col_space A) \<and> vec.span (basis_col_space A) = col_space A"
IH_1: assumes "M \<subseteq> M'" shows "intruder_deduct M' t"
minus_diff_minus: fixes a :: "'b"   and b :: "'b" shows "- a - - b = - (a - b)"
hyps_2: shows "N = mset_set ((\<lambda>D. (D, n)) ` concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) (set_mset (image_mset fst Q)) C'))"
admissibleI: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and lub :: "'c set \<Rightarrow> 'c" assumes "\<And>A. \<lbrakk>Complete_Partial_Order.chain ord A; A \<noteq> {}; \<forall>x\<in>A. P x\<rbrakk> \<Longrightarrow> P (lub A)" shows "ccpo.admissible lub ord P"
Since: fixes valid_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> event_data option list set) list \<Rightarrow> bool"   and init_msaux :: "Monitor.args \<Rightarrow> 'a"   and add_new_ts_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and join_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and add_new_table_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and result_msaux :: "Monitor.args \<Rightarrow> 'a \<Rightarrow> event_data option list set"   and valid_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and init_muaux :: "Monitor.args \<Rightarrow> 'b"   and add_new_muaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and length_muaux :: "Monitor.args \<Rightarrow> 'b \<Rightarrow> nat"   and eval_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> event_data option list set list \<times> 'b"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('a, 'b) mformula"   and \<phi>' :: "Formula.formula"   and \<psi> :: "('a, 'b) mformula"   and \<psi>' :: "Formula.formula"   and args :: "Monitor.args"   and \<phi>'' :: "Formula.formula"   and I :: "\<I>"   and buf :: "event_data option list set list \<times> event_data option list set list"   and nts :: "nat list"   and aux :: "'a" assumes "maux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux"   and "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R \<phi> \<phi>'"   and "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R \<psi> \<psi>'"   and "if args_pos args then \<phi>'' = \<phi>' else \<phi>'' = formula.Neg \<phi>'"   and "safe_formula \<phi>'' = args_pos args"   and "args_ivl args = I"   and "args_n args = n"   and "args_L args = fv \<phi>'"   and "args_R args = fv \<psi>'"   and "fv \<phi>' \<subseteq> fv \<psi>'"   and "wf_mbuf2' \<sigma> P V j n R \<phi>' \<psi>' buf"   and "wf_ts \<sigma> P j \<phi>' \<psi>' nts"   and "msaux.wf_since_aux valid_msaux \<sigma> V R args \<phi>' \<psi>' aux (Monitor.progress \<sigma> P (formula.Since \<phi>'' I \<psi>') j)" shows "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R (MSince args \<phi> \<psi> buf nts aux) (formula.Since \<phi>'' I \<psi>')"
Node_3: shows "avl \<langle>l, (a, h), r\<rangle>"
INT_I: fixes A :: "'d set"   and b :: "'e"   and B :: "'d \<Rightarrow> 'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> b \<in> B x" shows "b \<in> \<Inter> (B ` A)"
hull_gen_free_hull: fixes G :: "'b list set" shows "\<langle>\<langle>G\<rangle>\<rangle>\<^sub>F = \<langle>G\<rangle>\<^sub>F"
autoref_WHILEI: fixes I :: "'d \<Rightarrow> bool"   and R :: "('c \<times> 'd) set"   and c :: "'c \<Rightarrow> bool"   and c' :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and f' :: "'d \<Rightarrow> 'd nres"   and s' :: "'d"   and s :: "'c" assumes "\<And>x x'. \<lbrakk>I x'; (x, x') \<in> R\<rbrakk> \<Longrightarrow> (c x, c' $ x') \<in> bool_rel"   and "\<And>x x'. \<lbrakk>REMOVE_INTERNAL c' x'; I x'; (x, x') \<in> R\<rbrakk> \<Longrightarrow> (f x, f' $ x') \<in> \<langle>R\<rangle>nres_rel"   and "I s' \<Longrightarrow> (s, s') \<in> R" shows "(WHILE c f s, (OP WHILE\<^bsup>I\<^esup> ::: (R \<rightarrow> bool_rel) \<rightarrow> (R \<rightarrow> \<langle>R\<rangle>nres_rel) \<rightarrow> R \<rightarrow> \<langle>R\<rangle>nres_rel) $ c' $ f' $ s') \<in> \<langle>R\<rangle>nres_rel"
supp_Pair: fixes x :: "'b"   and y :: "'c" shows "supp (x, y) = supp x \<union> supp y"
Carmichael_prod_coprime: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite A"   and "\<And>i j. \<lbrakk>i \<in> A; j \<in> A; i \<noteq> j\<rbrakk> \<Longrightarrow> coprime (f i) (f j)" shows "Carmichael (prod f A) = (LCM i\<in>A. Carmichael (f i))"
negligible_on_intervals: fixes s :: "'c set" shows "negligible s = (\<forall>a b. negligible (s \<inter> cbox a b))"
NIL_I: fixes A :: "('b, unit) Old_Datatype.node set set" shows "NIL \<in> list A"
induct: fixes P :: "nat \<times> nat \<Rightarrow> (nat \<times> nat) list \<Rightarrow> bool"   and a0 :: "nat \<times> nat"   and a1 :: "(nat \<times> nat) list" assumes "\<And>block. P block []"   and "\<And>m_start m_end i_start i_end blocks. \<lbrakk>m_end - m_start \<le> i_end - i_start \<Longrightarrow> P (i_start, i_end) blocks; \<not> m_end - m_start \<le> i_end - i_start \<Longrightarrow> P (m_start, m_end) blocks\<rbrakk> \<Longrightarrow> P (m_start, m_end) ((i_start, i_end) # blocks)" shows "P a0 a1"
tObsC_abs_conv: fixes u :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState"   and v :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" shows "((u, v) \<in> tObsC_abs t) = (\<exists>t'. t' \<in> SPR.jkbpC \<and> tObsC t' = tObsC t \<and> u = tFirst t' \<and> v = tLast t')"
assms: shows "\<forall>i. f i < f (Suc i)"
supp_of_finite_union: fixes S :: "'a set"   and T :: "'a set" assumes "finite S"   and "finite T" shows "supp (S \<union> T) = supp S \<union> supp T"
0: fixes f :: "'fsym" assumes "f \<in> fsymsA at" shows "intF1 f = intF2 f"
NEQ_DEP_IMP_IN_DOM: fixes v :: "'b"   and v' :: "'b"   and PROB :: "(('b, 'c) fmap \<times> ('b, 'c) fmap) set" assumes "v \<noteq> v'"   and "dep PROB v v'" shows "v \<in> prob_dom PROB \<and> v' \<in> prob_dom PROB"
weak_unit_self_composable_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "weak_composition V H"   and "weak_composition.weak_unit V H a" shows "partial_magma.ide V a"
UN: shows "\<Union> (range (binary (main_part M S) (main_part M T))) = (\<Union>i. main_part M (binary S T i))"
div_mult_self4: 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 b :: "'e"   and c :: "'e"   and a :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus (times b c) a) b = plus c (divide a b)"
IH: assumes "rga_ops xs" shows "set (interp_rga xs) = set (map fst xs)"
aeX1: shows "AE x in T \<Otimes>\<^sub>M P. Pz (snd x) = 0 \<longrightarrow> Pyz x = 0"
out_two_sides_two_sides: fixes A :: "'p"   and B :: "'p"   and PX :: "'p"   and X :: "'p"   and P :: "'p"   and Y :: "'p" assumes "Col A B PX"   and "PX Out X P"   and "A B TS P Y" shows "A B TS X Y"
card_le_inj: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "card A \<le> card B" shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
induct: fixes P :: "'a bdd \<Rightarrow> bool list \<Rightarrow> bool"   and a0 :: "'a bdd"   and a1 :: "bool list" assumes "\<And>x bs. P (Leaf x) bs"   and "\<And>l r b bs. P (if b then r else l) bs \<Longrightarrow> P (Branch l r) (b # bs)"   and "\<And>v va. P (Branch v va) []" shows "P a0 a1"
assms_2: shows "sKS m \<notin> specKeysSecrets Q"
vrestr_same_emeasure_f: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "f \<in> measure_preserving M M"   and "A \<in> sets M" shows "emeasure M (f --` A) = emeasure M A"
prime_n_pd_def: fixes K :: "('c, 'd) Ring_scheme"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set"   and n :: "nat"   and j :: "nat" shows "P\<^bsub>K P n\<^esub> j = {x \<in> carrier (O\<^bsub>K P n\<^esub>). 0 < (\<nu>\<^bsub>K P j\<^esub>) x}"
sset_induct: fixes y :: "'a"   and s :: "'a stream"   and P :: "'a \<Rightarrow> 'a stream \<Rightarrow> bool" assumes "y \<in> sset s"   and "\<And>s. P (shd s) s"   and "\<And>s y. \<lbrakk>y \<in> sset (stl s); P y (stl s)\<rbrakk> \<Longrightarrow> P y s" shows "P y s"
zero_cap_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> cf.E" shows "cf (u, v) = (0::'a)"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "src\<^sub>B (g \<cdot>\<^sub>B f) = src\<^sub>B g \<cdot>\<^sub>B src\<^sub>B f"
unit_simps_2: fixes a :: "'a" assumes "B.obj a" shows "src\<^sub>B \<i>[a] = a"
inv_inv: fixes f :: "'d" assumes "D.iso f" shows "D.inv (D.inv f) = f"
assert_in_carrier: shows "h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>"
setops\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('a, 'b) stateful_strand_step list" shows "setops\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
induces_limit_situation: shows "limit_situation (\<cdot>\<^sub>J) (\<cdot>) D (\<cdot>\<^sub>S) \<phi> \<iota> a \<Phi> \<chi>"
ordertype_interval_eq: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "ordertype ({\<alpha>..<\<alpha> + \<beta>} \<inter> ON) VWF = \<beta>"
ReprF: fixes f :: "'trm \<Rightarrow> 'trm"   and n :: "'trm" assumes "f \<in> F"   and "n \<in> num" shows "prv (eql (encF f n) (f n))"
P'_def: fixes x :: "'a \<times> 'b"   and y :: "'c \<times> 'd" shows "P' x y = P (fst x) (snd x) (fst y) (snd y)"
mgu_Some: fixes S :: "nat \<Rightarrow> typ"   and t1 :: "typ"   and t2 :: "typ" assumes "$ S t1 = $ S t2" shows "\<exists>U. mgu t1 t2 = Some U"
subspace_scale: fixes S :: "'c set"   and x :: "'c"   and c :: "'a" assumes "m2.subspace S"   and "x \<in> S" shows "c *b x \<in> S"
bsub: fixes i :: "'a set" assumes "i \<in> F"   and "a i \<bullet> z = b i" shows "ball z e \<inter> affine hull \<Inter> {S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F \<and> a h \<bullet> z = b h} \<subseteq> affine hull S \<inter> \<Inter> F \<inter> \<Inter> {{x. a h \<bullet> x = b h} |h. h \<in> F \<and> a h \<bullet> z = b h}"
leadsTo_empty: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> A \<longmapsto> {}"   and "all_total F" shows "A = {}"
m2_1: shows "Polynomial.monom (1::'a) 2 = [:0::'a, 0::'a, 1::'a:]"
subprob_not_empty: fixes M :: "'a pmf" shows "space (measure_pmf M) \<noteq> {}"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "partial_order L"   and "total_order_axioms L" shows "total_order L"
cupcake_opapp_preserve_2: fixes vs :: "v list"   and env :: "v sem_env"   and e :: "exp0" assumes "do_opapp vs = Some (env, e)"   and "list_all is_cupcake_value vs" shows "is_cupcake_exp e"
simps: fixes succs :: "'d \<Rightarrow> 'd list"   and ins :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and memb :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c"   and wl :: "'d list" shows "gen_dfs succs ins memb S wl = (case wl of [] \<Rightarrow> S | x # xs \<Rightarrow> if memb x S then gen_dfs succs ins memb S xs else gen_dfs succs ins memb (ins x S) (succs x @ xs))"