hcju/isabelleps huggingface dataset:
le_cong_l: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> z"
lim_imp_Gromov_converging_at_boundary: fixes u :: "nat \<Rightarrow> 'a"   and x :: "'a Gromov_completion" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x"   and "x \<in> Gromov_boundary" shows "Gromov_converging_at_boundary u"
the_ext_unique: fixes y' :: "'c"   and g :: "'d"   and f :: "'c" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y y' g"   and "\<eta>y.is_ext y' g f" shows "f = \<eta>y.the_ext y' g"
set_shuffles: fixes zs :: "'b list"   and xs :: "'b list"   and ys :: "'b list" assumes "zs \<in> shuffles xs ys" shows "set zs = set xs \<union> set ys"
reachable_Cn: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list"   and k :: "nat" assumes "valid ((Cn n f gs, xs, []) # rest)"   and "\<And>xs rest. valid ((f, xs, []) # rest) \<Longrightarrow> correct ((f, xs, []) # rest, None)"   and "\<And>g xs rest. \<lbrakk>g \<in> set gs; valid ((g, xs, []) # rest)\<rbrakk> \<Longrightarrow> correct ((g, xs, []) # rest, None)"   and "\<forall>i<k. eval (gs ! i) xs \<down>"   and "k \<le> length gs" shows "reachable ((Cn n f gs, xs, []) # rest, None) ((Cn n f gs, xs, take k (map (\<lambda>g. the (eval g xs)) gs)) # rest, None)"
map_filter_def: fixes f :: "'e \<Rightarrow> 'd option"   and xs :: "'e list" shows "List.map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
IntPairVal_def: shows "IntPairVal \<equiv> \<lambda>(x, y). <|IntVal x, IntVal y|>"
basic_trans_rules_31: fixes A :: "'g set"   and B :: "'g set"   and c :: "'g" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
quantale_homset_def: shows "quantale_homset = {f. comp_pres f \<and> Sup_pres f}"
p: shows "p \<in> carrier (K [X]\<^bsub>R\<^esub>)"
RF: fixes z' :: "'v" assumes "z' \<in> set zs" shows "z' \<notin> RF (TER f)"
one_greatest1: fixes x :: "'a" shows "(1::'a) \<oplus> n x = (1::'a)"
antisymD: fixes r :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "antisym r"   and "(a, b) \<in> r"   and "(b, a) \<in> r" shows "a = b"
conv: shows "\<forall>x y m. x \<in> C \<and> y \<in> C \<and> 0 \<le> m \<and> m \<le> 1 \<longrightarrow> f (m *\<^sub>R x + (1 - m) *\<^sub>R y) \<le> ereal m * f x + (1 - ereal m) * f y"
sel_1: fixes x1 :: "'h" shows "projl (Inl x1) = x1"
mult_fract: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "Fraction_Field.Fract a b * Fraction_Field.Fract c d = Fraction_Field.Fract (a * c) (b * d)"
cases_valid_line: fixes l :: "color list" assumes "l = [] \<or> l ! 0 = B \<and> valid_line (tl l) \<or> 3 \<le> length l \<and> (\<forall>i<length l. l ! i = R) \<or> (\<exists>j<length l. 3 \<le> j \<and> (\<forall>i<j. l ! i = R) \<and> l ! j = B \<and> valid_line (drop (j + 1) l))" shows "valid_line l"
simps: fixes b :: "real"   and c :: "real"   and a :: "atomUni" shows "linearSubstitutionUni b c a = (if aEvalUni a (- c / b) then TrueFUni else FalseFUni)"
hmac_trans_lkr: fixes X :: "msg"   and K :: "msg"   and s' :: "l2_state"   and s :: "l2_state"   and A :: "agent" assumes "hmac X K \<in> parts (extr (bad s') (ik s') (chan s'))"   and "K \<notin> synth (analz (extr (bad s) (ik s) (chan s)))"   and "s \<in> l2_inv3"   and "(s, s') \<in> l2_lkr_others A \<union> l2_lkr_after A" shows "hmac X K \<in> parts (extr (bad s) (ik s) (chan s))"
zippers_rose_tree_same_hash': fixes z :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list \<times> ('a, 'b) rose_tree\<^sub>m"   and e :: "'c \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'b"   and p :: "('c \<times> 'c rose_tree list \<times> 'c rose_tree list) list"   and t :: "'c rose_tree" assumes "z \<in> set (zippers_rose_tree e h (p, t))" shows "hash_tree h (tree_of_zipper\<^sub>m z) = hash_tree h (tree_of_zipper\<^sub>m (embed_path e p, embed_source_tree e t))"
simps: fixes R :: "('b \<times> 'b) set"   and a :: "'b list" shows "Linorder_Relations.sorted_wrt R a = (a = [] \<or> (\<exists>xs x. a = x # xs \<and> Linorder_Relations.sorted_wrt R xs \<and> (\<forall>xa. xa \<in> set xs \<longrightarrow> (x, xa) \<in> R)))"
derivable: fixes x :: "real" assumes "x \<in> {a<..<b}" shows "(f has_real_derivative 0) (at x)"
order: shows "SN_both_mono_ordered_semiring_1 default gt arc_pos"
mat_of_row_index: fixes x :: "nat"   and y :: "'b vec" assumes "x < dim_vec y" shows "mat_of_row y $$ (0, x) = y $ x"
PDG_BS_def: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and control_dependence :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and S :: "'g set" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence" shows "PDG.PDG_BS sourcenode targetnode valid_edge Def Use control_dependence S \<equiv> {n'. \<exists>n. PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n' n \<and> n \<in> S \<and> CFG.valid_node sourcenode targetnode valid_edge n}"
paint_rbt_lookup: fixes c :: "color"   and t :: "('a, 'c) rbt" shows "rbt_lookup (paint c t) = rbt_lookup t"
"2_2": shows "a \<notin> set u\<^sub>1"
IH2: fixes x :: "'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and xa :: "'a"   and y :: "'a \<times> 'a \<times> 'a \<times> 'a"   and xb :: "'a"   and ya :: "'a \<times> 'a \<times> 'a"   and xc :: "'a"   and yb :: "'a \<times> 'a"   and xd :: "'a"   and yc :: "'a"   and xe :: "(('a, 'nc) Finite_Cartesian_Product.vec, 'nr) Finite_Cartesian_Product.vec"   and xf :: "(('a, 'nr) Finite_Cartesian_Product.vec, 'nr) Finite_Cartesian_Product.vec \<times> (('a, 'nc) Finite_Cartesian_Product.vec, 'nc) Finite_Cartesian_Product.vec"   and xg :: "(('a, 'nr) Finite_Cartesian_Product.vec, 'nr) Finite_Cartesian_Product.vec"   and yd :: "(('a, 'nc) Finite_Cartesian_Product.vec, 'nc) Finite_Cartesian_Product.vec"   and P :: "'a mat"   and A :: "'a mat"   and Q :: "'a mat" assumes "\<not> A' $h from_nat i $h from_nat i dvd A' $h from_nat j $h from_nat j"   and "x = bezout (A' $h from_nat i $h from_nat i) (A' $h from_nat j $h from_nat j)"   and "(xa, y) = x"   and "(xb, ya) = y"   and "(xc, yb) = ya"   and "(xd, yc) = yb"   and "xe = diagonal_step A' i j yc xd"   and "xf = diagonal_step_PQ A' i j bezout"   and "(xg, yd) = xf"   and "HMA_M3 (P, A, Q) (xg ** P', xe, Q' ** yd)"   and "i < min CARD('nr) CARD('nc)"   and "\<forall>j\<in>set xs. j < min CARD('nr) CARD('nc)" shows "HMA_M3 (diagonal_to_Smith_i_PQ_JNF xs i bezout (P, A, Q)) (diagonal_to_Smith_i_PQ xs i bezout (xg ** P', xe, Q' ** yd))"
confine_def: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "confine f A = (\<lambda>x. if x \<in> A then f x else (0::'c))"
simps_6: fixes b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "flatten (While b c) = [While b c]"
positive_mult: fixes x :: "real"   and y :: "real" assumes "Real.positive x"   and "Real.positive y" shows "Real.positive (x * y)"
assms: shows "v \<in> fmdom' (action_needed_asses a s)"
infinite_enumerate: fixes S :: "nat set" assumes "infinite S" shows "\<exists>r. strict_mono r \<and> (\<forall>n. r n \<in> S)"
smult_neg: fixes r :: "('f, 'g) aezfun"   and m :: "'v" assumes "r \<in> FG"   and "m \<in> V" shows "r \<cdot> - m = - (r \<cdot> m)"
coboundedI2: fixes c :: "'g multiset"   and b :: "'g multiset"   and a :: "'g multiset" assumes "c \<subseteq># b" shows "c \<subseteq># a \<union># b"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(CFG_node x1 = CFG_node y1) = (x1 = y1)"
set_transfer: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(llist_all2 R ===> rel_set R) lset lset"
fconf_def: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v\<^sub>m :: "'a \<Rightarrow> val option"   and T\<^sub>m :: "'a \<Rightarrow> ty option" shows "P,h \<turnstile> v\<^sub>m (:\<le>) T\<^sub>m \<equiv> \<forall>FD T. T\<^sub>m FD = \<lfloor>T\<rfloor> \<longrightarrow> (\<exists>v. v\<^sub>m FD = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
prems_5: shows "snd (cs (S t i) cid) = Recording"
equalityD2: fixes A :: "'d set"   and B :: "'d set" assumes "A = B" shows "B \<subseteq> A"
sum_list_nth_eqI: fixes xs :: "'b list"   and ys :: "'b list" assumes "length xs = length ys"   and "\<And>x y. (x, y) \<in> set (zip xs ys) \<Longrightarrow> x = y" shows "sum_list xs = sum_list ys"
order_finite_chain2: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat"   and l :: "nat" assumes "long_ch_by_ord2 f X"   and "finite X"   and "0 \<le> i \<and> i < j \<and> j < l \<and> l < card X" shows "[[f i f j f l]]"
inducts: fixes P :: "'d \<times> 'e \<Rightarrow> bool"   and prod :: "'d \<times> 'e" assumes "\<And>a b. P (a, b)" shows "P prod"
no_\<tau>Move1_\<tau>s_to_no_\<tau>Move2: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and ls2 :: "('d, 'b) locks"   and ts2 :: "'b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option"   and m2 :: "'h"   and ws2 :: "'b \<Rightarrow> 'e wait_set_status option"   and is2 :: "'b set"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWbisimulation_base.mbisim bisim bisim_wait s1 (ls2, (ts2, m2), ws2, is2)" shows "\<exists>ts2'. (\<tau>trsys.silent_move (multithreaded_base.redT final2 r2 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move2))\<^sup>*\<^sup>* (ls2, (ts2, m2), ws2, is2) (ls2, (ts2', m2), ws2, is2) \<and> (\<forall>t. wset s1 t = None \<and> (\<exists>x. thr s1 t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> (\<forall>x' m'. \<not> \<tau>trsys.silent_move (r1 t) \<tau>move1 (x, shr s1) (x', m'))) \<longrightarrow> wset (ls2, (ts2', m2), ws2, is2) t = None \<and> (\<exists>x. thr (ls2, (ts2', m2), ws2, is2) t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> (\<forall>x' m'. \<not> \<tau>trsys.silent_move (r2 t) \<tau>move2 (x, shr (ls2, (ts2', m2), ws2, is2)) (x', m')))) \<and> FWbisimulation_base.mbisim bisim bisim_wait s1 (ls2, (ts2', m2), ws2, is2)"
times_OrdSum_def: fixes a :: "('a, 'b) OrdSum"   and b :: "('a, 'b) OrdSum" shows "a * b \<equiv> if second a = (1::'b) \<and> second b = (1::'b) then Abs_OrdSum (first a * first b, 1::'b) else if first a = (1::'a) \<and> first b = (1::'a) then Abs_OrdSum (1::'a, second a * second b) else if first a = (1::'a) \<and> second b = (1::'b) then b else a"
ereal_inv_inv: fixes x :: "ereal" shows "inverse (inverse x) = (if x \<noteq> - \<infinity> then x else \<infinity>)"
abs_eq_iff: fixes x :: "'b ltln"   and y :: "'b ltln" shows "(abs_ltln\<^sub>Q x = abs_ltln\<^sub>Q y) = x \<sim>\<^sub>Q y"
rel_conversep: fixes R1 :: "'j \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'k \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_prod R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_prod R1 R2)\<inverse>\<inverse>"
bin_to_bl_def: fixes n :: "nat"   and w :: "int" shows "bin_to_bl n w = bin_to_bl_aux n w []"
bilinear_lzero: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'b" assumes "bilinear h" shows "h (0::'a) x = (0::'c)"
emptyE: fixes a :: "'e"   and P :: "bool" assumes "a \<in> {}" shows "P"
9: shows "prim_E w v e\<^sup>T \<le> - v"
simps_1: fixes l :: "'b tree"   and v :: "'b"   and r :: "'b tree" shows "right \<langle>l, v, r\<rangle> = r"
sub_rel_of: shows "SubRel.vertices subs \<subseteq> Graph.vertices g"
zone_delay_def: fixes Z :: "('d \<Rightarrow> 'e) set" shows "Z\<^sup>\<up> = {u \<oplus> d |u d. u \<in> Z \<and> (0::'e) \<le> d}"
f_shrink_last_last: fixes xs :: "'b list"   and k :: "nat" assumes "length xs mod k = 0" shows "last (xs \<div>\<^sub>l k) = last xs"
1: shows "termFSbMorph (rec MOD) (recAbs MOD) MOD"
meet_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
option_rel_Some2: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b option"   and y :: "'c" shows "rel_option A x (Some y) = (\<exists>x'. x = Some x' \<and> A x' y)"
vbrelation_vdiffI: fixes s :: "V" shows "vbrelation (r -\<^sub>\<circ> s)"
word_add_le_iff2: fixes i :: "'b word"   and k :: "'b word"   and j :: "'b word" assumes "i \<le> i + k"   and "j \<le> j + k" shows "(i + k \<le> j + k) = (i \<le> j)"
type_global_elim: fixes cs :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) list" assumes "\<turnstile> cs" obtains "list_all (\<lambda>(c, m). (\<exists>\<Gamma>' a b. Ex (has_type (\<Gamma>_of_mds m) (\<S>_of_mds m) {} c \<Gamma>' (a, b))) \<and> mds_yields_stable_types m) cs" and "\<forall>mem. list_all (\<lambda>cm. locally_sound_mode_use (cm, mem)) cs \<and> globally_sound_mode_use (cs, mem)"
span_eq: fixes S :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "(local.punit.pmdl S = local.punit.pmdl T) = (S \<subseteq> local.punit.pmdl T \<and> T \<subseteq> local.punit.pmdl S)"
CallParams: fixes E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 v :: "val"   and Copt :: "char list option"   and M :: "char list" assumes "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>" shows "P,E \<turnstile> \<langle>Call (Val v) Copt M es,s\<rangle> \<rightarrow> \<langle>Call (Val v) Copt M es',s'\<rangle>"
prime_imp_euler_liar: fixes p :: "nat"   and x :: "int" assumes "prime p"   and "2 < p"   and "0 < x"   and "x < int p" shows "euler_liar x p"
axioms_2: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "ring_hom_ring R S h" shows "ring S"
set_def: fixes X :: "V set" shows "set X \<equiv> if small X \<Rightarrow> inv elts X | otherwise \<Rightarrow> inv elts {}"
hyps: assumes "p \<noteq> 0" shows "if even (order c p) = sign_r_pos p c then \<forall>\<^sub>F x in at_left c. 0 < poly p x else \<forall>\<^sub>F x in at_left c. poly p x < 0"
borel_measurable_uminus': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "- f \<in> borel_measurable M"
inv_inv: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.inv (CC.inv f) = f"
assms_2: fixes j :: "'e" assumes "p = Inr j" shows "thesis"
assms_7: shows "Roots \<equiv> \<lambda>p. {\<alpha>. ipoly p \<alpha> = 0}"
Vset: fixes i :: "V" shows "Vset i = \<Union>\<^sub>\<circ> ((\<lambda>j\<in>\<^sub>\<circ>i. VPow (Vset j)) `\<^sub>\<circ> i)"
lexordp_antisym: fixes xs :: "'a list"   and ys :: "'a list" assumes "ord_class.lexordp xs ys"   and "ord_class.lexordp ys xs" shows "False"
cardE_nz_aux_2: shows "Suc 0 \<le> card E"
simps_6: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre ({(I, S, E)} WHILE b DO C) Q = I"
ID_combination_subset_union_2: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" shows "LID\<^sub>S (\<sigma>;;\<tau>) \<subseteq> LID\<^sub>S \<sigma> \<union> LID\<^sub>S \<tau>"
sconjunctive_INF_simp: fixes x :: "'a"   and X :: "'a set"   and S :: "'b \<Rightarrow> 'c"   and Q :: "'a \<Rightarrow> 'b" assumes "x \<in> X"   and "sconjunctive S" shows "S (Inf (Q ` X)) = Inf (S ` Q ` X)"
unit_simps_1: fixes a :: "'b" assumes "D.obj a" shows "D.arr \<i>\<^sub>D[a]"
must_syncE: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2" assumes "t \<turnstile> \<langle>x, m\<rangle> \<wrong>2" obtains taa :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action" and x' :: "'x2" and m' :: "'m2" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "r2.r_syntax t x m taa x' m'" and "r2.actions_ok s t taa" and "m = shr s"
trans: fixes P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and R :: "('a, 'b) Proc" assumes "(P, Q) \<in> encoding.indRelLPO Enc"   and "(Q, R) \<in> encoding.indRelLPO Enc" shows "(P, R) \<in> encoding.indRelLPO Enc"
carrier_nxt_subset: fixes t :: "'b ttree"   and x :: "'b" shows "carrier (nxt t x) \<subseteq> carrier t"
eval_rec_models: fixes X :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and Y :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and s :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState"   and \<phi> :: "('a, 'p) Kform" assumes "toSet X \<subseteq> toSet Y"   and "s \<in> toSet X" shows "(s \<in> toSet (eval_rec (spr_evalProp envVal) (spr_knowledge envObs Y) (spr_commonKnowledge envObs Y) X \<phi>)) = spr_repMC (toSet Y), s \<Turnstile> \<phi>"
emeasure_restrict_spmf: fixes p :: "'c spmf"   and A :: "'c set"   and X :: "'c set" shows "emeasure (measure_spmf (p \<upharpoonleft> A)) X = emeasure (measure_spmf p) (X \<inter> A)"
pickT_def: fixes \<sigma> :: "'tp" shows "pickT \<sigma> \<equiv> SOME a. intT \<sigma> a"
full_abstraction_and_enc_respects_pred_impl_SRel_iff_TRel_reflects: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "enc_respects_pred Pred" shows "rel_reflects_pred (_Collect (P, Q) (\<exists>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<and> (SP, SQ) \<in> SRel)) Pred = rel_reflects_pred (_Collect (P, Q) (\<exists>SP SQ. \<lbrakk>SP\<rbrakk> \<in>T P \<and> \<lbrakk>SQ\<rbrakk> \<in>T Q \<and> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel)) Pred"
E9: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> G) \<longrightarrow> \<diamond>F \<longrightarrow> \<diamond>G"
connected_component_disjoint: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" shows "(connected_component_set S a \<inter> connected_component_set S b = {}) = (a \<notin> connected_component_set S b)"
h_def: shows "h \<equiv> \<lambda>y. \<langle>f (\<phi> y), g (\<phi> y) y\<rangle>"
le_Liminf_iff: fixes C :: "'b"   and F :: "'c filter"   and X :: "'c \<Rightarrow> 'b" shows "(C \<le> Liminf F X) = (\<forall>y<C. \<forall>\<^sub>F x in F. y < X x)"
a0: shows "\<forall>b\<ge>a0. integrable (\<lambda>u. g' u / u powr (p + 1)) a0 b"
right_unit: fixes a :: "'a" assumes "a \<in> A" shows "multA a oneA = a"
HC_intros_10: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F \<^bold>\<and> G"
af\<^sub>G_keeps_F_and_S: fixes ys :: "'b set list"   and S :: "'b ltl set"   and \<phi> :: "'b ltl"   and xs :: "'b set list" assumes "ys \<noteq> []"   and "S \<Turnstile>\<^sub>P af\<^sub>G \<phi> ys" shows "S \<Turnstile>\<^sub>P af\<^sub>G (F \<phi>) (xs @ ys)"
ord_embed_def: fixes f :: "'a \<Rightarrow> 'b" shows "ord_embed f = (\<forall>x y. (f x \<le> f y) = (x \<le> y))"
vifunion_vsingleton_is: fixes i :: "V"   and f :: "V \<Rightarrow> V" shows "\<Union>\<^sub>\<circ> (VLambda (set {i}) f `\<^sub>\<circ> set {i}) = f i"
Integral_def2: shows "Integral = (\<lambda>(a, b) f k. \<forall>e>0. \<exists>\<delta>. gauge {a..b} \<delta> \<and> (\<forall>D. fine \<delta> (a, b) D \<longrightarrow> \<bar>rsum D f - k\<bar> \<le> e))"
padd_in_carrierp: fixes a :: "'a"   and p\<^sub>1 :: "'a \<times> 'a \<times> 'a"   and p\<^sub>2 :: "'a \<times> 'a \<times> 'a" assumes "a \<in> carrier R"   and "in_carrierp p\<^sub>1"   and "in_carrierp p\<^sub>2" shows "in_carrierp (padd a p\<^sub>1 p\<^sub>2)"
less_vec_def: fixes x :: "('c, 'd) vec"   and y :: "('c, 'd) vec" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
Cut: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and P :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "G \<rhd> Call : P"   and "insert P G \<rhd> c : A" shows "G \<rhd> c : A"
F_def: shows "F \<equiv> \<lambda>llength xs. case xs of LNil \<Rightarrow> 0 | LCons x xs \<Rightarrow> eSuc (llength xs)"
embedding_imp_closed_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "embedding_map X Y f"   and "closedin Y (f ` topspace X)" shows "closed_map X Y f"
list_seq_nth: fixes seq :: "nat \<Rightarrow> 'b list"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "\<And>i. \<exists>x. seq (Suc i) = x # seq i"   and "i < length (seq j)"   and "j \<le> k" shows "rev (seq k) ! i = rev (seq j) ! i"
comp_ide_arr: fixes b :: "'a"   and f :: "'a" assumes "V.ide b"   and "V.seq b f" shows "V b f = f"
Weaks_def: shows "Weaks = {z. \<exists>A Gamma. z = (A # Gamma, {Gamma})}"
doubleton_eq_iff: fixes a :: "hf"   and b :: "hf"   and c :: "hf"   and d :: "hf" shows "(\<lbrace>a, b\<rbrace> = \<lbrace>c, d\<rbrace>) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
fls_X_times_comm: fixes f :: "'a fls" shows "fls_X * f = f * fls_X"
numeral_le_one_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (numeral.numeral one plus n) one = (n \<le> num.One)"