hcju/isabelleps huggingface dataset:
intros_17: fixes t :: "t"   and v :: "v"   and s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and k :: "i32"   and off :: "nat"   and vs :: "v list"   and a :: "nat" assumes "types_agree t v"   and "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "store m (nat_of_int k) off (bits v) (t_length t) = None" shows "\<lparr>s;vs;[$C ConstInt32 k, $C v, $Store t None a off]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Trap]\<rparr>"
H_iy: shows "H $ i $ y \<noteq> (0::'a)"
insert_idem_1: shows "insert t0 (existence_ivl t0 x0 \<inter> {..t0}) = existence_ivl t0 x0 \<inter> {..t0}"
elims: fixes x :: "D"   and y :: "char list option" assumes "fromB x = y" obtains xa :: "char list" where "x = B xa" and "y = Some xa"   | v :: "int" where "x = A v__" and "y = None"   | v :: "bool" where "x = C v" and "y = None"
ph_def: shows "ph = Cn 4 r_phi [recf.Id 4 2, recf.Id 4 0]"
induct: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and packet_tagger :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and state_update :: "'c \<Rightarrow> final_decision \<Rightarrow> 'd \<Rightarrow> 'c"   and \<sigma>\<^sub>0 :: "'c"   and x1 :: "char list \<times> action"   and x2 :: "'d list"   and x3 :: "('d \<times> final_decision) list"   and x4 :: "'c"   and P :: "char list \<times> action \<Rightarrow> 'd list \<Rightarrow> ('d \<times> final_decision) list \<Rightarrow> 'c \<Rightarrow> bool" assumes "semantics_stateful_packet_tagging \<Gamma> \<gamma> packet_tagger state_update \<sigma>\<^sub>0 x1 x2 x3 x4"   and "\<And>built_in_chain default_policy ps. P (built_in_chain, default_policy) ps [] \<sigma>\<^sub>0"   and "\<And>built_in_chain default_policy p ps ps_processed \<sigma>' X. \<lbrakk>semantics_stateful_packet_tagging \<Gamma> \<gamma> packet_tagger state_update \<sigma>\<^sub>0 (built_in_chain, default_policy) (p # ps) ps_processed \<sigma>'; P (built_in_chain, default_policy) (p # ps) ps_processed \<sigma>'; \<Gamma>,\<gamma>,packet_tagger \<sigma>' p\<turnstile> \<langle>[Rule MatchAny (Call built_in_chain), Rule MatchAny default_policy], Undecided\<rangle> \<Rightarrow> Decision X\<rbrakk> \<Longrightarrow> P (built_in_chain, default_policy) ps (ps_processed @ [(p, X)]) (state_update \<sigma>' X p)" shows "P x1 x2 x3 x4"
wf\<^sub>t\<^sub>r\<^sub>m'_def: fixes ar :: "'b \<Rightarrow> nat"   and t :: "('b, 'c) Term.term" shows "wf\<^sub>t\<^sub>r\<^sub>m' ar t \<equiv> \<forall>s\<in>subterms t. is_Fun s \<longrightarrow> ar (the_Fun s) = length (args s)"
fls_shifted_pow: fixes m :: "int"   and f :: "'a fls"   and n :: "nat" shows "fls_shift m f ^ n = fls_shift (int n * m) (f ^ n)"
prv_pls_zer_var: shows "prv (eql (pls (Var xx) zer) (Var xx))"
U_def: shows "U \<equiv> {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
del_left_heap: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "del_left t = (x, t')"   and "t \<noteq> \<langle>\<rangle>"   and "heap t" shows "heap t'"
conj_one_right: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "conj x one = x"
TT_def: shows "TT = Geq (Const 0) (Const 0)"
transversal_section: fixes s :: "'a \<Rightarrow> real"   and Ds :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L real"   and S :: "'a set" assumes "section s Ds S" shows "transversal {x \<in> S. s x = 0}"
prob0_geq_zero: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" shows "0 \<le> prob0 n v i"
delete_first_simps_2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "P x" shows "delete_first P (x # xs) = Some xs"
IH: fixes t :: "real" assumes "0 \<le> t" shows "emeasure (Pi\<^sub>M A (\<lambda>_. lborel)) (S A t \<inter> space (Pi\<^sub>M A (\<lambda>_. lborel))) = ennreal (t ^ card A / fact (card A))"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and ternaryvalue :: "ternaryvalue" shows "P (case ternaryvalue of TernaryTrue \<Rightarrow> f1 | TernaryFalse \<Rightarrow> f2 | TernaryUnknown \<Rightarrow> f3) = ((ternaryvalue = TernaryTrue \<longrightarrow> P f1) \<and> (ternaryvalue = TernaryFalse \<longrightarrow> P f2) \<and> (ternaryvalue = TernaryUnknown \<longrightarrow> P f3))"
bounded: fixes a :: "'a" assumes "a \<in> A" shows "c \<le> a"
eq_upto_seq_inconsist_Cons1: fixes P :: "'a prog"   and ob :: "('b, 'c) obs_event action"   and obs :: "('b, 'c) obs_event action list"   and obs' :: "('b, 'c) obs_event action list"   and vs :: "'b \<times> addr_loc \<Rightarrow> ('b val \<times> bool) option" shows "eq_upto_seq_inconsist P (ob # obs) obs' vs = (\<exists>ob' obs''. obs' = ob' # obs'' \<and> (case ob of NormalAction (ReadMem ad al v) \<Rightarrow> if \<exists>b. vs (ad, al) = \<lfloor>(v, b)\<rfloor> then ob' = ob \<and> eq_upto_seq_inconsist P obs obs'' (mrw_value P vs ob) else ob \<approx> ob' | NormalAction _ \<Rightarrow> ob' = ob \<and> eq_upto_seq_inconsist P obs obs'' (mrw_value P vs ob) | _ \<Rightarrow> ob' = ob \<and> eq_upto_seq_inconsist P obs obs'' (mrw_value P vs ob)))"
last_in_set: fixes as :: "'b list" assumes "as \<noteq> []" shows "last as \<in> ran\<^sub>l as"
dual_eq_iff: fixes A :: "'a set"   and A' :: "'a set" shows "(A {.=}\<^bsub>inv_gorder L\<^esub> A') = (A {.=} A')"
summable_sums: fixes f :: "nat \<Rightarrow> 'd" assumes "summable f" shows "f sums suminf f"
Y_car: shows "Q = {vec_first x n |x. x \<in> Y}"
fmdom_of_list: fixes m :: "('e \<times> 'f) list" shows "fmdom (fmap_of_list m) = fst |`| fset_of_list m"
degree_monom_le: fixes a :: "'b"   and n :: "nat" shows "degree (monom a n) \<le> n"
fps_of_poly_add: fixes p :: "'c poly"   and q :: "'c poly" shows "fps_of_poly (p + q) = fps_of_poly p + fps_of_poly q"
dist2: shows "distinct (((adjuster n w us + w) # us) @ ws)"
matrix_vector_mul_lid: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "mat (1::'a) *v x = x"
ker_def: shows "kerT = {v \<in> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}). vec n (\<lambda>i. coeff (v mod m i) 0) = \<zero>\<^bsub>module_vec TYPE('a mod_ring) n\<^esub>}"
len_renamings_apart: fixes Cs :: "('a, nat) Term.term literal multiset list" shows "length (renamings_apart Cs) = length Cs"
exec_move_CondI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (if (e) e1 else e2) h s ta h' s'"
pre_bet': shows "pre_between (verticesFrom f ram1) ram1 ram2"
ProjInd_def: fixes I :: "'e set"   and A :: "'e \<Rightarrow> ('d, 'f) aGroup_scheme"   and f :: "('e \<Rightarrow> 'd) \<Rightarrow> 'c"   and j :: "'e" shows "ProjInd I A f j = compos (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f) (\<pi>\<^bsub>I,A,j\<^esub>) (ainvf\<^bsub>(a\<Pi>\<^bsub>I\<^esub> A),Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f\<^esub> Agii (a\<Pi>\<^bsub>I\<^esub> A) f)"
psubstAbs_vsubstAbs_compose: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "goodEnv rho" shows "A $[y1 // y]_ys $[rho] = A $[rho [y \<leftarrow> Var ys y1 #[rho]]_ys]"
power_zero_numeral: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and k :: "num" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero (numeral k) = zero"
1: shows "x\<^sup>\<star> \<sqinter> x\<^sup>T = bot"
convex_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear f"   and "convex S" shows "convex (f ` S)"
crsp: shows "crsp ly (0, lm) (Suc 0, l, r) ires"
Liminf_upto_llist_enat: fixes Xs :: "'a set llist"   and k :: "nat" shows "Liminf_upto_llist Xs (enat k) = (if enat k < llength Xs then lnth Xs k else if lnull Xs then {} else llast Xs)"
mem_tangent_space: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" shows "(X \<in> tangent_space p) = (is_derivation X p \<and> extensional0 diff_fun_space X)"
equiv_transD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "equiv_trans t u"   and "sound P" shows "t P = u P"
elims_2: fixes x :: "'a option" assumes "isSome x" obtains uu_ :: "'a" where "x = Some uu_"
klift_def: fixes f :: "'d \<Rightarrow> 'e set" shows "f\<^sup>\<dagger> = \<mu> \<circ> \<P> f"
si_empty3: fixes x :: "'a event"   and s :: "'a event list"   and X :: "'a event set" shows "setinterleaving (x # s, X, []) = (if x \<in> X then {} else {x # u |u. u setinterleaves ((s, []), X)})"
convex_hull_minus_Collect: fixes A :: "'b set"   and B :: "'b set" shows "{x - y |x y. x \<in> convex hull A \<and> y \<in> convex hull B} = convex hull {x - y |x y. x \<in> A \<and> y \<in> B}"
alw_cong: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream"   and Q1 :: "'c stream \<Rightarrow> bool"   and Q2 :: "'c stream \<Rightarrow> bool" assumes "alw P \<omega>"   and "\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> = Q2 \<omega>" shows "alw Q1 \<omega> = alw Q2 \<omega>"
col_one_side_out: fixes A :: "'p"   and X :: "'p"   and Y :: "'p"   and B :: "'p" assumes "Col A X Y"   and "A B OS X Y" shows "A Out X Y"
R_map: fixes r :: "('f, 'g) aezfun"   and m :: "'v" assumes "r \<in> FG"   and "m \<in> V" shows "T (r \<cdot> m) = r \<star> T m"
neval_loc_reach: fixes lc' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and n :: "nat"   and lc'' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and lc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "abs.neval lc' n lc''"   and "lc' \<in> abs.loc_reach lc" shows "lc'' \<in> abs.loc_reach lc"
assms: shows "(0::'a) \<notin> fst ` set (snd fctrs)"
HFinite_hrabs_iff: fixes x :: "real star" shows "(\<bar>x\<bar> \<in> HFinite) = (x \<in> HFinite)"
hyps_4: fixes bb :: "(name \<times> ty) list"   and b :: "name"   and ba :: "trm" assumes "\<Delta> @ [(x, U)] @ \<Gamma> = bb @ [(b, U)] @ \<Gamma>"   and "\<Gamma> \<turnstile> ba : U" shows "bb @ \<Gamma> \<turnstile> t\<^sub>2[b\<mapsto>ba] : T"
succsr_def: shows "succsr \<equiv> {(x, y). y \<in> set (case x of (uu_, uua_, q) \<Rightarrow> map (\<lambda>\<nu>. (q, \<nu>, \<delta> q \<nu>)) \<Sigma>)}"
MI_pred: fixes intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "MModel intT intF intP" shows "Ex MI_pred"
wffp_State_Idle: fixes sl1 :: "('St, 'U, 'C) state list"   and st1 :: "'St"   and u1 :: "'U"   and c1 :: "'C"   and st2 :: "'St"   and sl2 :: "('St, 'U, 'C) state list" assumes "wffp (sl1 @ [State st1 u1 c1] @ [Idle st2] @ sl2)" shows "st2 = st1"
index_ordered: fixes n :: "nat" shows "index n < index (Suc n)"
span_superset: fixes S :: "'c set" shows "S \<subseteq> ideal S"
rep_eq: fixes f :: "'c \<Rightarrow> real" assumes "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1" shows "measure_pmf (embed_pmf f) = density (count_space UNIV) (ennreal \<circ> f)"
mem_Times_iff: fixes x :: "'a \<times> 'b"   and A :: "'a set"   and B :: "'b set" shows "(x \<in> A \<times> B) = (fst x \<in> A \<and> snd x \<in> B)"
Scheduler_L1: fixes C :: "Config"   and activation :: "nat \<Rightarrow> nat list"   and t :: "nat" assumes "Scheduler C activation"   and "activation t \<noteq> []" shows "t mod cycleLength C mem schedule C"
le_funI: fixes f :: "'k \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'l" assumes "\<And>x. f x \<le> g x" shows "f \<le> g"
rel_sel: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and a :: "('g, 'i) tllist"   and b :: "('h, 'j) tllist" shows "tllist_all2 R1 R2 a b = (is_TNil a = is_TNil b \<and> (is_TNil a \<longrightarrow> is_TNil b \<longrightarrow> R2 (terminal a) (terminal b)) \<and> (\<not> is_TNil a \<longrightarrow> \<not> is_TNil b \<longrightarrow> R1 (thd a) (thd b) \<and> tllist_all2 R1 R2 (ttl a) (ttl b)))"
asym: fixes a :: "'c list"   and b :: "'c list" assumes "strict_prefix a b"   and "strict_prefix b a" shows "False"
fds_deriv_fds_integral: fixes f :: "'a fds"   and c :: "'a" assumes "fds_nth f (Suc 0) = (0::'a)" shows "fds_deriv (fds_integral c f) = f"
EF_step_star_rev: fixes x :: "'a"   and s :: "'a set" assumes "x \<in> EF s" shows "\<exists>y\<in>s. x \<rightarrow>\<^sub>i* y"
assms: fixes x :: "'c" shows "mcont lSup (\<sqsubseteq>) Sup (\<le>) (\<lambda>xs. f x xs (LCons x xs))"
openin_topology_base_unique: fixes X :: "'c topology"   and P :: "'c set \<Rightarrow> bool" shows "(openin X = arbitrary union_of P) = ((\<forall>V. P V \<longrightarrow> openin X V) \<and> (\<forall>U x. openin X U \<and> x \<in> U \<longrightarrow> (\<exists>V. P V \<and> x \<in> V \<and> V \<subseteq> U)))"
count_pre_img: fixes x :: "'c"   and B :: "'d set"   and Y :: "'c \<Rightarrow> 'd"   and n :: "nat" assumes "x \<in> countable_preimages B Y n" shows "Y x = from_nat_into B n"
bdd_above_bot: fixes A :: "'c list set" shows "prefix_order.bdd_below A"
int_and_ge0: fixes x :: "int"   and y :: "int" shows "(0 \<le> x AND y) = (0 \<le> x \<or> 0 \<le> y)"
no_zero_divisors: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_no_zero_divisors plus zero times"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "times a b \<noteq> zero"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "S.par f f'"   and "R f = R f'" shows "f = f'"
diff_mult_distrib2: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> V" shows "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
"4_3": shows "Ind (set s\<^sub>2) (set s\<^sub>3)"
assms: shows "x \<le> 636 / 10\<^sup>2"
delta: fixes S :: "'c set"   and a :: "'c"   and b :: "'c \<Rightarrow> 'd" assumes "finite S" shows "sup_monoid.sum (\<lambda>k. if k = a then b k else bot) S = (if a \<in> S then b a else bot)"
oghoare_Spec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and r :: "('e \<times> 'e) set"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Spec r) Q, A" shows "\<exists>P'. P = AnnExpr P' \<and> P' \<subseteq> {s. (\<forall>t. (s, t) \<in> r \<longrightarrow> t \<in> Q) \<and> (\<exists>t. (s, t) \<in> r)}"
Seq_sound: fixes P1 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and C1 :: "com"   and P2 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and C2 :: "com"   and P3 :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<Turnstile>\<^sub>2 {P1} C1 {P2}"   and "\<Turnstile>\<^sub>2 {P2} C2 {P3}" shows "\<Turnstile>\<^sub>2 {P1} C1;; C2 {P3}"
Ch_domain: 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 h :: "'f"   and A :: "'d set" assumes "ContractsWithIRC Xd Xh Pd Ch" shows "Ch h (A \<inter> {x. Xh x = h}) = Ch h A"
intros_11: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "F \<triangleright> \<Gamma> \<turnstile> G" shows "\<Gamma> \<turnstile> F \<^bold>\<rightarrow> G"
right_adjoint_preserved_by_iso': fixes g :: "('a S.E.term, 'a) S.arr"   and g' :: "('a S.E.term, 'a) S.arr" assumes "S.is_right_adjoint g"   and "S.isomorphic g g'" shows "S.is_right_adjoint g'"
static_def: fixes s :: "'c list"   and rs :: "'c list" shows "static s rs = (set rs \<subseteq> set s)"
fcard_funion_fsubset: fixes B :: "'b fset"   and A :: "'b fset" assumes "B |\<subseteq>| A" shows "fcard (A |-| B) = fcard A - fcard B"
wordinterval_list_to_set_def: fixes ws :: "'a wordinterval list" shows "wordinterval_list_to_set ws = \<Union> (set (map wordinterval_to_set ws))"
intersect_ts_def: fixes A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'b set" shows "A \<Rightarrow>\<inter>  B = (\<lambda>k. A k \<inter> B k)"
topological_basis_iff: fixes "open" :: "'d set \<Rightarrow> bool"   and B :: "'d set set" assumes "class.topological_space open"   and "\<And>B'. B' \<in> B \<Longrightarrow> open B'" shows "topological_space.topological_basis open B = (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
abs_minus_commute: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (minus a b) = abs (minus b a)"
order_iff: fixes b :: "'d multiset"   and a :: "'d multiset" shows "(b \<subseteq># a) = (a = a \<union># b)"
Swap: fixes B :: "fmla"   and H :: "fmla set"   and A :: "fmla" assumes "insert (neg B) H \<turnstile> A" shows "insert (neg A) H \<turnstile> B"
alw_InvGlobalRecordCount: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw (holds InvGlobalRecordCount) s"
that: fixes k :: "nat" assumes "fixp_above a = (f ^^ k) a" shows "thesis"
token_succeeds_def: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and w :: "nat \<Rightarrow> 'c"   and F :: "'b set"   and x :: "nat" shows "mojmir_def.token_succeeds \<delta> q\<^sub>0 w F x = (\<exists>n. semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n \<in> F)"
power2_eq_iff_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero x"   and "less_eq zero y" shows "(power.power one times x 2 = power.power one times y 2) = (x = y)"
x: shows "x \<cdot> 1\<^sub>\<pi> \<le> 1\<^sub>\<pi>"
polyfun_def: fixes N :: "nat set"   and f :: "(nat \<Rightarrow> 'b) \<Rightarrow> 'b" shows "polyfun N f = (\<exists>p. vars p \<subseteq> N \<and> (\<forall>x. insertion x p = f x))"
bicategory_axioms: shows "bicategory (\<cdot>) (\<star>) \<a> \<i> src trg"
mkPoint_in_hom: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b"   and t :: "'b" assumes "set_category S \<AA>"   and "partial_magma.ide S a"   and "t \<in> set_category_data.set S (set_category.some_img S \<AA>) a" shows "partial_magma.in_hom S (set_category.mkPoint S \<AA> a t) (set_category_data.unity S) a"
WT_elim_cases_3: 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 C :: "char list"   and e :: "expr"   and T :: "ty" assumes "P,E \<turnstile> \<lparr>C\<rparr>e :: T" obtains D :: "char list" where "T = Class C" and "P,E \<turnstile> e :: Class D" and "is_class P C" and "P \<turnstile> Path D to C unique \<or> P \<turnstile> C \<preceq>\<^sup>* D \<and> (\<forall>Cs. P \<turnstile> Path C to D via Cs  \<longrightarrow> Subobjs\<^sub>R P C Cs)"
sum_mset_ge0: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c multiset" assumes "\<And>x. (0::'a) \<le> f x" shows "(0::'a) \<le> \<Sum>\<^sub># (image_mset f A)"
currentD_outside_OUT: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f"   and "x \<notin> \<^bold>V\<^bsub>\<Gamma>\<^esub>" shows "d_OUT f x = 0"
assms: shows "\<exists>D\<in>MajoritySet. \<forall>d\<in>D. P d"
b_bounds: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and b :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "b \<in> set bs" shows "b \<in> {0<..<1}"
i_set_Div_not_closed: fixes k :: "nat" assumes "Suc 0 < k" shows "\<exists>I\<in>i_set. I \<oslash> k \<notin> i_set"
run_return_option: fixes return :: "'e option \<Rightarrow> 'd"   and x :: "'e" shows "run_option (return_option return x) = return (Some x)"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "B.cod (G f) = G (A.cod f)"
unit_factor: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" assumes "Group.comm_monoid G"   and "a \<otimes>\<^bsub>G\<^esub> b \<in> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<in> Units G"
word_length_le: fixes as :: "'a list"   and A :: "'a set"   and a :: "'a" assumes "as \<in> lists A"   and "sum_list as = a" shows "order.greater_eq (length as) (word_length A a)"
satC_iff_set: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list" shows "GE.satC \<xi> c = (\<exists>l\<in>set c. GE.satL \<xi> l)"
Id_on_def: fixes A :: "'a set" shows "Id_on A = \<mu> (\<P> (\<lambda>x. \<eta> (x, x)) A)"
mapCollectI: fixes m :: "'f \<Rightarrow> 'e option"   and k :: "'f"   and v :: "'e"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'g" assumes "m k = Some v" shows "f k v \<in> mapCollect f m"
Lindelof_space_subtopology_subset: fixes S :: "'a set"   and X :: "'a topology" assumes "S \<subseteq> topspace X" shows "Lindelof_space (subtopology X S) = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> S \<subseteq> \<Union> \<U> \<longrightarrow> (\<exists>V. countable V \<and> V \<subseteq> \<U> \<and> S \<subseteq> \<Union> V))"
arrow_in_bicategory_of_spans_axioms: shows "arrow_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg s s s"
oreachable_local': fixes \<sigma> :: "'c"   and p :: "'d"   and A :: "('c \<times> 'd, 'e) automaton"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and U :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'e"   and \<sigma>' :: "'c"   and p' :: "'d" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "(\<sigma>', p') \<in> oreachable A S U"
less_top_Top_is_Value: fixes x :: "'a top" assumes "x < Top" shows "\<exists>z. x = top.Value z"
fmdom_fmap_of_list: fixes xs :: "('c \<times> 'd) list" shows "fmdom (fmap_of_list xs) = fset_of_list (map fst xs)"
acyclic_def: fixes r :: "('c \<times> 'c) set" shows "acyclic r = (\<forall>x. (x, x) \<notin> r\<^sup>+)"
is_GB_cofactor_bound_def: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and b :: "nat" shows "is_GB_cofactor_bound F b = (\<exists>G. punit.is_Groebner_basis G \<and> ideal G = ideal F \<and> \<Union> (indets ` G) \<subseteq> \<Union> (indets ` F) \<and> (\<forall>g\<in>G. \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f\<in>F'. poly_deg (q f * f) \<le> b)))"
assms: shows "raw_match' (Ty a Ts) (Ty b Us) subs = Some subs'"