hcju/isabelleps huggingface dataset:
fg_send_msg_def: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and p :: "'f"   and q :: "'f"   and config :: "('g, 'f, 'h, 'j) config_scheme"   and config' :: "('g, 'f, 'h, 'j) config_scheme" shows "fg_send_msg A p q config config' \<equiv> q \<notin> sent config p \<and> config' = config\<lparr>sent := (sent config)(p := sent config p \<union> {q}), network := network config \<union> {(p, round config p, q, sendMsg A (round config p) p q (state config p))}\<rparr>"
InvariantUniqQAfterApplyLearn: fixes state :: "State" assumes "InvariantUniqQ (getQ state)" shows "let state' = applyLearn state in InvariantUniqQ (getQ state')"
list_of_pdevs_zero_pdevs: shows "list_of_pdevs zero_pdevs = []"
prj_tuple_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.commutative_square C f g h k" shows "C (prj0 f g) (elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k) = k"
empty_domain_fmap_set: shows "{s. fmdom' s = {}} = {fmempty}"
times_discr_def: fixes x :: "'a discr"   and y :: "'a discr" shows "x * y \<equiv> Discr (undiscr x * undiscr y)"
vector_scalar_commute: fixes A :: "(('a, 'c) vec, 'b) vec"   and c :: "'a"   and x :: "('a, 'c) vec" shows "A *v (c *s x) = c *s (A *v x)"
r2f_comp_pres: fixes R :: "('c \<times> 'e) set"   and S :: "('e \<times> 'd) set" shows "\<F> (R ; S) = \<F> R \<circ>\<^sub>K \<F> S"
iwlsFSbSwTR_qSwapAll_igSubstAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "iwlsFSbSwTR MOD" shows "(qGood qX \<longrightarrow> qAFresh ys y1 qX \<longrightarrow> qInit MOD (qX #[[y1 \<and> y]]_ys) = igSubst MOD ys (igVar MOD ys y1) y (qInit MOD qX)) \<and> (qGoodAbs qA \<longrightarrow> qAFreshAbs ys y1 qA \<longrightarrow> qInitAbs MOD (qA $[[y1 \<and> y]]_ys) = igSubstAbs MOD ys (igVar MOD ys y1) y (qInitAbs MOD qA))"
assms: shows "set_integrable (M1 \<Otimes>\<^sub>M M2) (A \<times> B) (\<lambda>(x, y). f x y)"
EF_lem0: fixes x :: "'a"   and f :: "'a set" shows "(x \<in> EF f) = (x \<in> f \<union> EX' (lfp (\<lambda>Z. f \<union> EX' Z)))"
f_def: fixes x :: "int" shows "f x = (if 100 < x then x - 10 else 91)"
cospan_\<mu>\<nu>: shows "C.cospan g.leg0 f.leg1"
Gramian_determinant_1: fixes k :: "nat" assumes "k \<le> length (map of_int_hom.vec_hom fs_init)" shows "fs_init.gs.d k = (\<Prod>j<k. \<parallel>gs.gso j\<parallel>\<^sup>2)"
trancl_trans_induct: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "(x, y) \<in> r\<^sup>+"   and "\<And>x y. (x, y) \<in> r \<Longrightarrow> P x y"   and "\<And>x y z. \<lbrakk>(x, y) \<in> r\<^sup>+; P x y; (y, z) \<in> r\<^sup>+; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x y"
vCons_def: fixes a :: "'a"   and v :: "'a vec" shows "vCons a v \<equiv> vec (Suc (dim_vec v)) (\<lambda>i. case i of 0 \<Rightarrow> a | Suc i \<Rightarrow> v $ i)"
typing_implies_RED: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "lam"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> t : \<tau>" shows "t \<in> RED \<tau>"
denote_while_n_scale: fixes c :: "real"   and x1 :: "nat \<Rightarrow> complex mat"   and x2 :: "com"   and \<rho> :: "complex mat"   and n :: "nat" assumes "0 \<le> c"   and "measurement d 2 x1"   and "well_com x2"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>; trace (complex_of_real c \<cdot>\<^sub>m \<rho>) \<le> 1\<rbrakk> \<Longrightarrow> denote x2 (complex_of_real c \<cdot>\<^sub>m \<rho>) = complex_of_real c \<cdot>\<^sub>m denote x2 \<rho>"   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_while_n (x1 0) (x1 1) (denote x2) n (complex_of_real c \<cdot>\<^sub>m \<rho>) = complex_of_real c \<cdot>\<^sub>m denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>"
lexordp_eq_simps_1: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c list" shows "ord.lexordp_eq less [] ys = True"
pterm_to_sterm_closed: fixes t :: "pterm"   and S :: "name fset" assumes "closed_except t S"   and "pre_strong_term_class.wellformed t" shows "closed_except (pterm_to_sterm t) S"
FV_Obj: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type" shows "FV (Obj f T) = (\<Union>l\<in>dom f. FVoption (f l))"
log_mult: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "0 < a"   and "a \<noteq> 1"   and "0 < x"   and "0 < y" shows "log a (x * y) = log a x + log a y"
assms_1: shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
drop_perp_same_line_pole_in_S: fixes p :: "proj2"   and l :: "proj2_line" assumes "drop_perp p l = l" shows "pole l \<in> S"
path_conforms_with_strategy_irrelevant': fixes p :: "Player"   and P :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a"   and w :: "'a" assumes "path_conforms_with_strategy p P (\<sigma>(v := w))"   and "v \<notin> lset P" shows "path_conforms_with_strategy p P \<sigma>"
r_cons: fixes e :: "nat"   and es :: "nat" shows "eval r_cons [e, es] \<down>= e_cons e es"
simps_2: fixes y :: "name"   and e :: "lam"   and \<theta> :: "(name \<times> lam) list"   and x :: "name" shows "lookup ((y, e) # \<theta>) x = (if x = y then e else lookup \<theta> x)"
accepting_pair\<^sub>R_restrict: fixes w :: "nat \<Rightarrow> 'd"   and \<Sigma> :: "'d set"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e"   and q\<^sub>0 :: "'e"   and F :: "('e \<times> 'd \<times> 'e) set"   and I :: "('e \<times> 'd \<times> 'e) set" assumes "range w \<subseteq> \<Sigma>" shows "accepting_pair\<^sub>R \<delta> q\<^sub>0 (F, I) w = accepting_pair\<^sub>R \<delta> q\<^sub>0 (F \<inter> reach\<^sub>t \<Sigma> \<delta> q\<^sub>0, I \<inter> reach\<^sub>t \<Sigma> \<delta> q\<^sub>0) w"
assms_2: shows "AE x in M. ennreal \<bar>f x\<bar> \<le> C"
Float_num_3: shows "real_of_float (Float 1 2) = 4"
ord_of_nat_mult: fixes m :: "nat"   and n :: "nat" shows "ord_of_nat (m * n) = ord_of_nat m * ord_of_nat n"
scaleC_add_right: fixes a :: "complex"   and x :: "'a"   and y :: "'a" shows "a *\<^sub>C (x + y) = a *\<^sub>C x + a *\<^sub>C y"
reciprocal_poly_def: fixes p :: "nat"   and P :: "'b poly" shows "reciprocal_poly p P = Poly (rev (coeffs P @ replicate (p - degree P) (0::'b)))"
sub_Suc0_sub_Suc_conv: fixes b :: "nat"   and a :: "nat" shows "b - a - Suc 0 = b - Suc a"
assms_3: shows "\<forall>g\<in>G. \<forall>h\<in>H. \<exists>h'\<in>H. h + g = g + h'"
abs_eq: fixes y :: "'d \<Rightarrow> 'e option" shows "Mapping.lookup (Mapping y) = y"
topological_basisI: fixes "open" :: "'b set \<Rightarrow> bool"   and B :: "'b set set" assumes "class.topological_space open"   and "\<And>B'. B' \<in> B \<Longrightarrow> open B'"   and "\<And>O' x. \<lbrakk>open O'; x \<in> O'\<rbrakk> \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" shows "topological_space.topological_basis open B"
cut_le_inext_nth_card_eq2: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "card I \<le> Suc n" shows "card (I \<down>\<le> (I \<rightarrow> n)) = card I"
elims: fixes x :: "complex \<times> complex \<times> complex \<times> complex"   and xa :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex \<times> complex \<times> complex" assumes "x *\<^sub>m\<^sub>m xa = y" obtains a1 :: "complex" and b1 :: "complex" and c1 :: "complex" and d1 :: "complex" and a2 :: "complex" and b2 :: "complex" and c2 :: "complex" and d2 :: "complex" where "x = (a1, b1, c1, d1)" and "xa = (a2, b2, c2, d2)" and "y = (a1 * a2 + b1 * c2, a1 * b2 + b1 * d2, c1 * a2 + d1 * c2, c1 * b2 + d1 * d2)"
simps_1: shows "FingerTreeStruc.viewLn Empty = None"
simps_1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and xp :: "nat option"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "exec (P, xp, h, []) = None"
once_none_always_none: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and t :: "(String.literal \<times> value list) stream" shows "alw (state_eq None) (make_full_observation e None r p t)"
sgn_div_norm: fixes sgn :: "'a \<Rightarrow> 'a"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.sgn_div_norm sgn norm scaleR" shows "sgn x = scaleR (inverse (norm x)) x"
EX: fixes v :: "'a"   and u' :: "'a"   and p :: "'b" assumes "s' = v # u' # r"   and "(([u], c), ee, [v, u'], c') \<in> ntrs fg"   and "(u, Call p, u') \<in> edges fg" shows "P"
strict_trans1: fixes a :: "nat list"   and b :: "nat list"   and c :: "nat list" assumes "a \<le>\<^sub>v b"   and "b <\<^sub>v c" shows "a <\<^sub>v c"
inv_imps_work_sum_zcount_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration" assumes "dataflow_topology summary results_in" shows "dataflow_topology.inv_imps_work_sum_zcount summary results_in c \<equiv> \<forall>loc t. zcount (c_imp c loc + c_work c loc) t = zcount (zmset_of (mset_set (set_antichain (frontier (c_pts c loc)))) + (\<Sum>loc'\<in>UNIV. dataflow_topology.after_summary results_in (zmset_of (mset_set (set_antichain (frontier (c_imp c loc'))))) (summary loc' loc))) t"
assms_1: shows "w1 \<equiv>\<^sub>E r1"
left_unit: fixes a :: "'a set" assumes "a \<in> Partition" shows "Class \<one> [\<cdot>] a = a"
has_suc_notempty: fixes n :: "nat"   and S :: "'b set" assumes "has (Suc n) S" shows "{} \<noteq> S"
not_in_own_radicals: fixes r :: "expr" shows "r \<notin> radicals r"
sset_def: fixes xs :: "[:'b:]" shows "sset xs = {x. slistmem x xs}"
oalist_eqI: fixes xs :: "('a, 'b) oalist"   and ys :: "('a, 'b) oalist" assumes "list_of_oalist xs = list_of_oalist ys" shows "xs = ys"
assms_2: assumes "vassn_tag (hn_ctxt A x y)"   and "vassn_tag (hn_ctxt A' x y)"   and "P x" shows "P' x"
deadlockD1: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x" assumes "deadlock s"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "\<not> final x"   and "wset s t = None" obtains "t \<turnstile> \<langle>x, shr s\<rangle> \<wrong>" and "\<forall>LT. t \<turnstile> \<langle>x, shr s\<rangle> LT \<wrong> \<longrightarrow> (\<exists>lt\<in>LT. must_wait s t lt (dom (thr s)))"
BtoW_def: shows "BtoW \<equiv> \<lambda>(e, M). M ! fst e = Black \<and> M ! snd e \<noteq> Black"
eucl_down_prod_def: fixes p :: "int"   and a :: "'a \<times> 'b" shows "eucl_down p a = (eucl_down p (fst a), eucl_down p (snd a))"
truncate_up_nonneg_mono: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "0 \<le> x"   and "x \<le> y" shows "truncate_up prec x \<le> truncate_up prec y"
union_diff_assoc: fixes C :: "'a multiset"   and B :: "'a multiset"   and A :: "'a multiset" assumes "C - B = {#}" shows "A + B - C = A + (B - C)"
simps_1: fixes f :: "'i \<Rightarrow> 'g"   and t :: "'f"   and x :: "'i"   and m :: "'h" shows "convert_new_thread_action f (NewThread t x m) = NewThread t (f x) m"
ghost_honorary_grey_mo_co_mark_1: fixes s :: "'c process_name \<Rightarrow> ('b, 'c, 'd, 'a) Model.local_state"   and p :: "'c process_name"   and b :: "'a"   and r :: "'a" assumes "ghost_honorary_grey (s p) = {}" shows "black b (s(p := s p\<lparr>ghost_honorary_grey := {r}\<rparr>)) = (black b s \<and> b \<noteq> r)"
final_state_correct: fixes n :: "nat"   and x :: "'a \<Rightarrow> int"   and ts :: "(('a \<times> 'b) \<times> int) list"   and start :: "'b \<Rightarrow> int"   and ts' :: "(('a \<times> 'b) \<times> int) list"   and ms :: "nat \<Rightarrow> (('a \<times> 'b) \<times> int) option"   and r :: "'b \<Rightarrow> int" assumes "0 < n"   and "reachable x n ts start (ts', ms, r)"   and "final n (ts', ms, r)" shows "r = fold (\<lambda>((r, c), v) y. y(c := y c + x r * v)) ts start"
match_list_to_match_expr_disjunction: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and ms :: "'b match_expr list"   and a :: "action"   and p :: "'c" shows "match_list \<gamma> ms a p = matches \<gamma> (match_list_to_match_expr ms) a p"
real_greaterThanLessThan_minus_infinity_eq: fixes N :: "ereal" shows "real_of_ereal ` {- \<infinity><..<N} = (if N = \<infinity> then UNIV else if N = - \<infinity> then {} else {..<real_of_ereal N})"
oid_preserve: fixes x :: "'\<AA>" assumes "x \<in> ran (heap (fst \<tau>)) \<or> x \<in> ran (heap (snd \<tau>))" shows "oid_of (H x) = oid_of x"
not_isAction_isChoice: fixes p :: "'a process" shows "\<not> (isAction p \<and> isChoice p)"
perm_fresh_fresh_2: fixes a :: "vrs"   and x :: "'a"   and b :: "vrs" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
drop_Cons: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
fg_def: shows "fg \<equiv> \<lambda>z. f z + g z"
compl_structD2: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "compl_struct compl"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps" shows "component_of_term ` Keys (fst ` set (fst (compl gs bs (ps -- sps) sps data))) \<subseteq> component_of_term ` Keys (args_to_set (gs, bs, ps))"
set_sorted_list_of_set: fixes A :: "'b set" assumes "finite A" shows "set (compare_linorder.sorted_list_of_set A) = A"
independent_imp_trivial_combine: fixes K :: "'a set"   and Us :: "'a list"   and Ks :: "'a list" assumes "subfield K R"   and "local.independent K Us"   and "set Ks \<subseteq> K"   and "combine Ks Us = \<zero>" shows "set (take (length Us) Ks) \<subseteq> {\<zero>}"
submodule_sum_def: fixes N1 :: "('c \<times> 'c) set"   and N2 :: "('c \<times> 'c) set" shows "l.submodule_sum N1 N2 = (\<lambda>(x, y). x \<oplus>\<^bsub>direct_sum (md N1) (md N2)\<^esub> y) ` {(x, y). x \<in> N1 \<and> y \<in> N2}"
SbisL_Sbis: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat" assumes "SbisL cl dl"   and "n < length cl \<or> n < length dl" shows "cl ! n \<approx>s dl ! n"
map_iterator_genord_rule_P: assumes "map_iterator_genord it m (\<lambda>_ _. True)"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>True; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> it - {k} \<and> m k' = Some v' \<longrightarrow> True; \<forall>k' v'. k' \<notin> it \<and> m k' = Some v' \<longrightarrow> True\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> True; I it \<sigma>; \<forall>k v k' v'. k \<notin> it \<and> m k = Some v \<and> k' \<in> it \<and> m k' = Some v' \<longrightarrow> True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
inv_atD: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and m :: "nat"   and i :: "nat" assumes "inv_at n p A m"   and "i < n" shows "p A i m"
get_scdom_component_ptrs_same_owner_document: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc"   and "h \<turnstile> Shadow_DOM.get_owner_document ptr \<rightarrow>\<^sub>r owner_document" shows "h \<turnstile> Shadow_DOM.get_owner_document ptr' \<rightarrow>\<^sub>r owner_document"
normalize_consistent: fixes K :: "'a set" assumes "subring K R" shows "ring.normalize (R\<lparr>carrier := K\<rparr>) = local.normalize"
bidirected_digraph: shows "bidirected_digraph (with_proj H) (swap_in (parcs H))"
R_lin_independent_Cons: fixes m :: "'m"   and ms :: "'m list" shows "R_lin_independent (m # ms) = (R_lin_independent ms \<and> (\<forall>r rs. set (r # rs) \<subseteq> R \<and> (r # rs) \<bullet>\<cdot> (m # ms) = (0::'m) \<longrightarrow> r = (0::'r)))"
simps: fixes outPorts :: "'a \<Rightarrow> 'b fset"   and n :: "'a" shows "Vertex_Graph.terminal_node outPorts n = (outPorts n = {||})"
has_derivative_continuous: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" assumes "(f has_derivative f') (at x within s)" shows "continuous (at x within s) f"
simps_1: fixes x :: "vrs"   and y :: "vrs"   and t' :: "trm" shows "Var x[y \<mapsto> t'] = (if x = y then t' else Var x)"
dens: shows "has_density M (stock_measure (PRODUCT REAL REAL)) \<delta>"
OO_def: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
strict_trans2: fixes b :: "'b zmultiset"   and a :: "'b zmultiset"   and c :: "'b zmultiset" assumes "b \<subset>#\<^sub>z a"   and "c \<subseteq>#\<^sub>z b" shows "c \<subset>#\<^sub>z a"
refines_def: fixes e :: "'e \<times> ('f \<times> 'e) list"   and s :: "'g"   and a :: "'f"   and t :: "'g"   and A :: "('e, 'f) ioa"   and f :: "'g \<Rightarrow> 'e" shows "refines e s a t A f \<equiv> fst e = f s \<and> last_state e = f t \<and> is_exec_frag_of A e \<and> (let tr = trace (ioa.asig A) e in if a \<in> externals (ioa.asig A) then tr = [a] else tr = [])"
swap_cs_Send_Snapshot: 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 "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
option_Inf_Sup: fixes A :: "'b option set set" shows "\<Sqinter> (Sup ` A) \<le> \<Squnion> (Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_plus" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> vreal_plus). vreal_plus\<lparr>a\<rparr> = b"
small_image_nat_V: fixes g :: "nat \<Rightarrow> V"   and N :: "nat set" shows "small (g ` N)"
sources_aux_append_nil: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and U :: "'c set"   and ys :: "'d list"   and xs :: "'d list" assumes "sources_aux I D U ys = U" shows "sources_aux I D U (xs @ ys) = sources_aux I D U xs"
has_integral_AE: fixes \<Omega> :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "AE x\<in>\<Omega> in lborel. f x = g x" shows "(f has_integral x) \<Omega> = (g has_integral x) \<Omega>"
is_span: shows "C.span \<nu>'.cod.leg0 \<nu>'.cod.leg1"
positions_of_a_non_atomic_term: fixes t1 :: "'b trm"   and t2 :: "'b trm" shows "pos_of (t1 \<cdot> t2) = left_app ` pos_of t1 \<union> right_app ` pos_of t2 \<union> {[]}"
A_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "A i \<le> A j"
simps_2: fixes p :: "real"   and x :: "bool"   and xs :: "bool list" shows "lprob_comp p (x # xs) = (if x then p else 1 - p) * lprob_comp p xs"
mem_to_nonempty: fixes \<omega> :: "'a"   and S :: "'a set" assumes "\<omega> \<in> S" shows "S \<noteq> {}"
in_dgrad_p_set_iff: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and d :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> nat"   and m :: "nat" shows "(p \<in> punit.dgrad_p_set d m) = (\<forall>v\<in>keys p. d (local.punit.pp_of_term v) \<le> m)"
rep_eq: fixes p :: "real" assumes "0 < p"   and "p \<le> 1" shows "pmf (geometric_pmf p) = (\<lambda>n. (1 - p) ^ n * p)"
ub: fixes x :: "'a" shows "spmf p x \<le> f x"
fmadd_fmupd: fixes m :: "('c, 'd) fmap"   and a :: "'c"   and b :: "'d"   and n :: "('c, 'd) fmap" shows "m ++\<^sub>f fmupd a b n = fmupd a b (m ++\<^sub>f n)"
hyps: shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
e''step: shows "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>throw a,(h', l', sh')\<rangle>"
solve_start: fixes \<Psi> :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk> [], 0 \<turnstile> \<Psi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
complies_formal_def: shows "SecurityInvariant_complies_formal_def m_impl m_spec"
measurableI: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<in> space N"   and "\<And>A. A \<in> sets N \<Longrightarrow> f -` A \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M N"
agree_empty_fresh: fixes e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and a :: "Syntax.var" assumes "{$$} \<turnstile> e, eP, eV : \<tau>" shows "{atom a} \<sharp>* {e, eP, eV}"
NT: shows "NT \<eta> : YFtor C @@ X \<Longrightarrow> F"
less_eq_sqmatrix_def: fixes A :: "('a, 'b) sqmatrix"   and B :: "('a, 'b) sqmatrix" shows "A \<le> B \<equiv> A + B = B"
ign_res_def2: fixes P :: "'b assn"   and Y :: "vals"   and s :: "state"   and Z :: "'b" shows "P\<down> Y s Z = (\<exists>Y. P Y s Z)"
is_serial_solution_for_problem_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list" shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi> \<equiv> let I = \<Psi>\<^sub>I\<^sub>+; G = \<Psi>\<^sub>G\<^sub>+; ops = \<Psi>\<^sub>\<O>\<^sub>+ in G \<subseteq>\<^sub>m execute_serial_plan_sas_plus I \<psi> \<and> list_all (\<lambda>op. ListMem op ops) \<psi>"
S21: shows "g_pair (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair n x) = c_cons (c_pair (c_pair n x) (c_pair (c_assoc_value (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair (c_fst (c_snd n)) x)) (c_assoc_value (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair (c_snd (c_snd n)) x)))) (pr_gr (c_pair (c_pair n x) (upb1 + upb2)))"