hcju/isabelleps huggingface dataset:
f_def: fixes n :: "nat"   and m :: "nat" shows "f n m = (\<Sum>x\<in>{i. i < n}. p x m)"
shunt1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (inf x y) z = less_eq x (sup (uminus y) z)"
path_CondTrue: fixes prog :: "cmd"   and l :: "nat"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node"   and b :: "expr"   and c\<^sub>2 :: "cmd" assumes "While_CFG.path prog (_ l _) as n'" shows "While_CFG.path (if (b) prog else c\<^sub>2) ((_ l _) \<oplus> 1) (as \<oplus>s 1) (n' \<oplus> 1)"
x: shows "(r \<cdot> s) \<^sup>@ j = x"
bij_def: fixes f :: "'h \<Rightarrow> 'i" shows "bij f = (inj f \<and> surj f)"
cval_add_non_id: fixes b :: "'a \<Rightarrow> real"   and d :: "real" assumes "b \<oplus> d = b"   and "0 < d" shows "False"
gcd_self: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd a a = normalize a"
compile_load: fixes p1 :: "('a, 'l, 'fun) prog"   and p2 :: "('f, 'l, 'fun) prog"   and s2 :: "('f, 'l, ('fun, ('b, 'h, 'i) unboxed) frame) state" assumes "compile p1 = Some p2"   and "Subx.load p2 s2" shows "\<exists>s1. Sinca.load p1 s1 \<and> s1 \<sim> s2"
chain_boundary_chain_map: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and g :: "'c \<Rightarrow> 'd" assumes "singular_chain p X c" shows "chain_boundary p (chain_map p g c) = chain_map (p - Suc 0) g (chain_boundary p c)"
polar_skolemise: fixes P :: "'a \<Rightarrow> bool" assumes "All P = False" shows "P (SOME x. \<not> P x) = False"
simps_3: fixes key :: "'a \<Rightarrow> 'b"   and v :: "'a"   and va :: "'a list" shows "merge key (v # va) [] = v # va"
suffix_conc_length: fixes w :: "'a list"   and w' :: "nat \<Rightarrow> 'a" shows "suffix (length w) (w \<frown> w') = w'"
simps: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) SkewBinomialTree list" shows "tree_to_multiset (Node e a r ts) = {#(e, a)#} + queue_to_multiset ts"
conjugate_dist_mul: fixes conjugate :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (times a b) = times (conjugate a) (conjugate b)"
condB: shows "\<forall>a b c. (a, b) \<in> Rel \<and> (a, c) \<in> Rel \<and> (b, c) \<notin> Rel \<longrightarrow> (b, a) \<in> Rel \<or> (c, a) \<in> Rel"
sel_1: fixes x1 :: "'g"   and x2 :: "'h" shows "fst (x1, x2) = x1"
iUntil_not_iRelease_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<U> t I. Q t) = (\<not> (\<not> P t'. t' \<R> t I. \<not> Q t))"
tree_automaton_def: fixes TA :: "('a, 'b) tree_automaton_rec" shows "tree_automaton TA \<equiv> finite (ta_rules TA) \<and> finite (ta_initial TA)"
homeomorphic_sym: fixes s :: "'a set"   and t :: "'b set" shows "s homeomorphic t = t homeomorphic s"
uf: fixes n :: "nat" shows "(\<lambda>m. u m n)\<up>f n"
OKI_checkI: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and einp :: "'h \<Rightarrow> 'g withERR option" assumes "FixSyn TYPE('a) Delta"   and "liftAll (\<lambda>X. X \<noteq> ERR) einp" shows "OKI (checkI einp) = einp"
9519: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (- x \<sqinter> y) \<sqinter> - - (x \<sqinter> z) = bot"
lhd_LCons_ltl: fixes llist :: "'c llist" assumes "\<not> lnull llist" shows "LCons (lhd llist) (ltl llist) = llist"
length_mtf2: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "length (mtf2 n x xs) = length xs"
sums_minus: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b" assumes "f sums a" shows "(\<lambda>n. - f n) sums - a"
B_trusts_YM4_shrK: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and NB :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key K, Nonce NB\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "\<exists>NA. Says Server A \<lbrace>Nonce NB, Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key K, Nonce NB\<rbrace>\<rbrace> \<in> set evs"
plan_action_path_def: fixes M :: "object atom formula set"   and \<pi>s :: "plan_action list"   and M' :: "object atom formula set" shows "plan_action_path M \<pi>s M' = (Ball (set \<pi>s) wf_plan_action \<and> ground_action_path M (map resolve_instantiate \<pi>s) M')"
ex_consume_NO: fixes vl :: "'value list"   and trn :: "('state, 'act, 'out) trans" assumes "vl \<noteq> []"   and "f trn = hd vl" obtains vl' :: "'value list" where "consume trn vl vl'"
square_free_factorization_int_main_def: fixes f :: "int poly" shows "square_free_factorization_int_main f = (case square_free_heuristic f of None \<Rightarrow> yun_gcd.yun_monic_factorization gcd f | Some p \<Rightarrow> [(f, 0)])"
mersenne_prime_def: fixes p :: "nat" shows "mersenne_prime p \<equiv> 2 < p \<and> prime (2 ^ p - 1)"
less_QF2: fixes k :: "nat"   and j :: "nat"   and i :: "nat" shows "QF k (j, i) \<lless> QF k (j, Suc i)"
one_on_unit_circle: shows "on_circline unit_circle 1\<^sub>h"
multiset_cases: fixes M :: "'b multiset" obtains "M = {#}"   | x :: "'b" and N :: "'b multiset" where "M = add_mset x N"
ceiling_le_zero: fixes x :: "'c" shows "(\<lceil>x\<rceil> \<le> 0) = (x \<le> (0::'c))"
is_singletonE: fixes A :: "'b set" assumes "is_singleton A" obtains x :: "'b" where "A = {x}"
wf3: shows "dsk = unk \<longrightarrow> hops = 1"
vsv_vrange_vsubset: fixes r :: "V"   and A :: "V" assumes "vsv r"   and "\<And>x. x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<Longrightarrow> r\<lparr>x\<rparr> \<in>\<^sub>\<circ> A" shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> A"
\<R>\<^sub>1_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and \<S> :: "'Var set \<times> 'Var set" and P :: "'BExp set" where "c\<^sub>2 = c\<^sub>1" and "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1} \<Gamma>''__,\<S>''__,Pa__" and "tyenv_wellformed mds \<Gamma> \<S> P" and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" and "pred P mem\<^sub>1" and "pred P mem\<^sub>2" and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"
out_next_subseqs: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and base :: "'d"   and elements :: "'c list"   and n :: "nat"   and state :: "'c \<times> 'c iarray \<times> (nat \<times> 'd) list"   and out :: "'d list"   and state' :: "'c \<times> 'c iarray \<times> (nat \<times> 'd) list" assumes "my_subseqs.invariant f base elements n state"   and "my_subseqs.next_subseqs f state = (out, state')" shows "set out = subseqs_foldr_impl.S f base elements (Suc n)"
eq_fold: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "set.F (insert x A) = Finite_Set.fold (\<^bold>*) x A"
gr_st0: shows "grounding_of_clss (wrp.N_of_wstate (wstate_of_dstate St0) \<union> wrp.P_of_wstate (wstate_of_dstate St0) \<union> wrp.Q_of_wstate (wstate_of_dstate St0)) = grounded_N0"
induced_morphism_def: shows "induced_morphism \<equiv> \<lambda>C\<in>stalk.carrier_stalk is_open \<FF> \<rho> (identity S x). let r = SOME r. r \<in> C in direct_lim.class_of \<FF> \<rho> {U. is_open U \<and> x \<in> U} (identity S \<^sup>\<inverse> S (fst r)) (identity (\<FF> (fst r)) (snd r))"
bdd_aboveI2: fixes A :: "'k set"   and f :: "'k \<Rightarrow> 'l multiset"   and M :: "'l multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "subset_mset.bdd_above (f ` A)"
assms: shows "is_subobj P (C, Cs @ [D, D'] @ Cs')"
member_le_sum_list: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "x \<in> set xs" shows "less_eq x (monoid_add.sum_list plus zero xs)"
abs_eq: fixes x :: "'a \<Rightarrow> 'b option" shows "Mapping.keys (Mapping x) = dom x"
measurable_merge: fixes I :: "'d set"   and J :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "merge I J \<in> Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M \<rightarrow>\<^sub>M Pi\<^sub>M (I \<union> J) M"
inv_M2_def: fixes m2 :: "int" shows "inv_M2 m2 = (\<lambda>x. if x \<le> m2 then x else x - m)"
zless_iff_Suc_zadd: fixes w :: "int"   and z :: "int" shows "(w < z) = (\<exists>n. z = w + int (Suc n))"
ncoplanar_perm_4: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet A D B C"
list_induct2: fixes xs :: "'k list"   and ys :: "'l list"   and P :: "'k list \<Rightarrow> 'l list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
exec_strip_to_exec: fixes F :: "'g set"   and \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and c :: "('f, 'e, 'g) com"   and s :: "('f, 'g) xstate"   and t :: "('f, 'g) xstate" assumes "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' \<in> Fault ` (- F) \<longrightarrow> t' = t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
set_integral_Un: fixes A :: "'a set"   and B :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "A \<inter> B = {}"   and "set_integrable M A f"   and "set_integrable M B f" shows "set_lebesgue_integral M (A \<union> B) f = set_lebesgue_integral M A f + set_lebesgue_integral M B f"
rbl_mult_gt1: fixes bl :: "bool list"   and m :: "nat"   and binb :: "int" assumes "length bl \<le> m" shows "rbl_mult bl (rev (bin_to_bl m binb)) = rbl_mult bl (rev (bin_to_bl (length bl) binb))"
IH: fixes x2 :: "'a rule list"   and rs' :: "'a rule list"   and s :: "state" assumes "\<Gamma> chain = Some x2"   and "\<not> \<not> terminal_chain x2"   and "rewrite_Goto_chain_safe \<Gamma> rs = Some rs'" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs', s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"
is_path_split: fixes v :: "'V"   and p1 :: "('V \<times> 'W \<times> 'V) list"   and p2 :: "('V \<times> 'W \<times> 'V) list"   and v' :: "'V" shows "is_path v (p1 @ p2) v' = (\<exists>u. is_path v p1 u \<and> is_path u p2 v')"
assms: shows "sorted_wrt (\<succ>\<^sub>t) ts"
InvariantConflictFlagCharacterizationAfterApplyLearn: fixes state :: "State" assumes "getConflictFlag state"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)" shows "let state' = applyLearn state in InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
add_zerol: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and x :: "'c" assumes "class.ab_near_semiring_one_zerol plus times one zero" shows "plus zero x = x"
pivotal_ind: fixes D :: "indi set"   and b :: "alt"   and P :: "indi \<Rightarrow> alt \<Rightarrow> real" assumes "finite D"   and "D = {i. b \<cdot>< P i}"   and "Extreme P b"   and "b \<cdot>< F P" shows "\<exists>i. pivotal i b"
nn_integral_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N M f = Sup (integral\<^sup>S M ` {g. simple_function M g \<and> g \<le> f})"
real_root_mult: fixes n :: "nat"   and x :: "real"   and y :: "real" shows "root n (x * y) = root n x * root n y"
axioms_2: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "valid_ownership_and_sharing \<S> ts" shows "outstanding_non_volatile_writes_unshared \<S> ts"
section_def: fixes f :: "'a" shows "section f \<equiv> \<exists>g. ide (g \<cdot> f)"
cexpr_typing_code: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type" shows "(\<Gamma> \<turnstile>\<^sub>c e : t) = (cexpr_type \<Gamma> e = Some t)"
less_imp_add_positive: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "\<exists>k>0. i + k = j"
IH: fixes a :: "nat" assumes "a < prod_list ds" shows "digit_encode ds a \<lhd> ds"
graph_empty: fixes A :: "('d, 'c) nba"   and w :: "'d stream" assumes "finite (nodes A)"   and "w \<notin> language A" shows "graph A w (Suc (2 * card (nodes A))) = {}"
assms_3: shows "M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a)"
in_affected: fixes a :: "'b"   and f :: "'b perm" shows "(a \<in> affected f) = (apply f a \<noteq> a)"
runit'_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<r>\<^sup>-\<^sup>1[a] : a \<rightarrow> a \<otimes> \<one>\<guillemotright>"
eq: shows "segments_of_aform X = x # y # z # zs"
part: shows "partition bound0 (conjuncts p) = (fst (partition bound0 (conjuncts p)), snd (partition bound0 (conjuncts p)))"
assms: shows "(Q, P) \<in> R \<rightarrow> bool_rel"
set_plus_mono3_b: fixes a :: "'c"   and C :: "'c set"   and x :: "'c"   and D :: "'c set" assumes "a \<in> C"   and "x \<in> a +o D" shows "x \<in> C + D"
no_outgoing_t: shows "\<forall>u. (t, u) \<notin> E"
may_lock_upd_locks_conv: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and Ls :: "lock_action list" assumes "lock_actions_ok l t Ls" shows "may_lock (upd_locks l t Ls) t = may_lock l t"
doctor_optimal_match_unique: fixes ds :: "'d set"   and X :: "'x set"   and Y :: "'x set" assumes "doctor_optimal_match ds X"   and "doctor_optimal_match ds Y" shows "X = Y"
simps_6: fixes r :: "'a rexp" shows "lang (Star r) = star (lang r)"
le_reflexivity: fixes A :: "'p"   and B :: "'p" shows "A B Le A B"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "M.satL \<xi> (Neg a) = (\<not> M.satA \<xi> a)"
minus_mod_self1: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and uminus :: "'d \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus b a) b = modulo (uminus a) b"
group_sum_ker_image: fixes f :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and g :: "'d \<Rightarrow> 'g"   and K :: "('g, 'h) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H K"   and "(g \<circ> f) ` carrier G = carrier K"   and "Group.group G"   and "Group.group H"   and "Group.group K" shows "kernel H K g <#>\<^bsub>H\<^esub> f ` carrier G = carrier H"
abs_eq: fixes xa :: "int"   and x :: "int" shows "Float xa x = float_of (real_of_int xa * 2 powr real_of_int x)"
sum_list_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'c list" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero (map (\<lambda>x. zero) xs) = zero"
distinct_1: fixes x11 :: "name"   and x12 :: "name" shows "OutputR_Rep x11 x12 \<noteq> TauR_Rep"
ord_strict_canc_left: fixes u :: "'a"   and s :: "'a"   and t :: "'a" assumes "u + s \<prec> u + t" shows "s \<prec> t"
tom_Nil: fixes ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" assumes "length ms = length s" shows "S,f \<turnstile> (ms,s) =[]\<Rightarrow>* (ms,s)"
W_consts_4: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n false\<^sub>n = w \<Turnstile>\<^sub>n G\<^sub>n \<phi>"
Suc_double_inj_on: fixes A :: "nat set" shows "inj_on (\<lambda>i. Suc (2 * i)) A"
CoxeterSystem: fixes S :: "'b set" assumes "PreCoxeterSystemWithDeletion S" shows "CoxeterSystem S"
assms: fixes x :: "'a"   and y :: "'b" shows "(F *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)"
1: fixes y :: "real_alg_2_list" assumes "(y, xs) \<in> measure real_alg_2_list_length" shows "real_alg_2_list_convert y = real_alg_2_list_convert_id y"
rel_setD1: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and A :: "'e set"   and B :: "'f set"   and x :: "'e" assumes "rel_set R A B"   and "x \<in> A" shows "\<exists>y\<in>B. R x y"
fixp_spmf_parametric: fixes F :: "('d \<Rightarrow> 'e spmf) \<Rightarrow> 'd \<Rightarrow> 'e spmf"   and G :: "('f \<Rightarrow> 'g spmf) \<Rightarrow> 'f \<Rightarrow> 'g spmf"   and A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>x. spmf.mono_body (\<lambda>f. F f x)"   and "\<And>x. spmf.mono_body (\<lambda>f. G f x)"   and "rel_fun (rel_fun A (rel_spmf R)) (rel_fun A (rel_spmf R)) F G" shows "rel_fun A (rel_spmf R) (spmf.fixp_fun F) (spmf.fixp_fun G)"
valid_sharing_axioms: shows "valid_sharing \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
that: fixes x0 :: "real" assumes "t0 < x0"   and "x0 \<le> tg"   and "w' x0 \<le> v' x0"   and "v x0 = w x0"   and "\<And>t. \<lbrakk>t0 < t; t < x0\<rbrakk> \<Longrightarrow> v t < w t" shows "thesis"
left_plus_circ: fixes x :: "'a" shows "x\<^sup>+\<^sup>\<star> = x\<^sup>\<star>"
u_3: shows "t \<in> set (u \<leadsto> s)"
impl_model_check_correct: fixes sysi :: "'sai"   and sys :: "('s, 'prop set, 'sa_more) sa_rec_scheme"   and \<phi> :: "'prop ltlc"   and cfg :: "'cfg_l2b \<times> 'cfg_int \<times> 'cfg_ce" assumes "(sysi, sys) \<in> sa_rel"   and "sa sys"   and "finite ((g_E sys)\<^sup>* `` g_V0 sys)" shows "case impl_model_check cfg sysi \<phi> of None \<Rightarrow> sa.lang sys \<subseteq> language_ltlc \<phi> | Some None \<Rightarrow> \<not> sa.lang sys \<subseteq> language_ltlc \<phi> | Some (Some ri) \<Rightarrow> \<exists>r. (ri, r) \<in> mce_rel \<and> graph_defs.is_run sys r \<and> sa_L sys \<circ> r \<notin> language_ltlc \<phi>"
chine_assoc_in_hom: shows "\<guillemotleft>fgh.chine_assoc : HHfgh.chine \<rightarrow>\<^sub>C HfHgh.chine\<guillemotright>"
mod_ring_eq32: shows "(mod_ring_rel32 ===> mod_ring_rel32 ===> (=)) (=) (=)"
cSum1: assumes "P \<longmapsto> Rs" shows "Prop"
Gamma_integral_real': fixes x :: "real" assumes "0 < x" shows "((\<lambda>t. t powr (x - 1) / exp t) has_integral Gamma x) {0<..}"
strong_induct: fixes P :: "'a \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a"   and trm :: "trm" assumes "\<And>name z. P z (Var name)"   and "\<And>trm1 trm2 z. \<lbrakk>\<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (App trm1 trm2)"   and "\<And>name trm z. \<lbrakk>name \<sharp> z; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (\<Lambda> name . trm)"   and "\<And>trm1 name trm2 z. \<lbrakk>name \<sharp> z; name \<sharp> trm1; \<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (trm1 to name in trm2)"   and "\<And>trm z. (\<And>z. P z trm) \<Longrightarrow> P z [trm]" shows "P z trm"
finsum_closed: fixes f :: "'b \<Rightarrow> 'a vec"   and A :: "'b set" assumes "f \<in> A \<rightarrow> carrier_vec n" shows "finsum V f A \<in> carrier_vec n"
simulation_\<tau>Runs_table1: fixes s1 :: "'s1"   and s2 :: "'s2"   and stlsss1 :: "('tl1 \<times> 's1, 's1 option) tllist" assumes "s1 \<approx> s2"   and "trsys1.\<tau>Runs_table s1 stlsss1" shows "\<exists>stlsss2. trsys2.\<tau>Runs_table s2 stlsss2 \<and> tllist_all2 (\<lambda>(tl1, s1'') (tl2, s2''). tl1 \<sim> tl2 \<and> s1'' \<approx> s2'') (rel_option bisim) stlsss1 stlsss2"
rel_gpv''_map_gpv'1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'f \<Rightarrow> 'e"   and gpv :: "('a, 'c, 'e) gpv"   and gpv' :: "('b, 'd, 'f) gpv" assumes "rel_gpv'' A C (BNF_Def.Grp UNIV h)\<inverse>\<inverse> gpv gpv'" shows "rel_gpv'' A C (=) (map_gpv' id id h gpv) gpv'"
Diff_iff: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
is_mgu_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "is_mgu \<sigma> E = (\<sigma> \<in> unifiers E \<and> (\<forall>\<tau>\<in>unifiers E. \<sigma> \<preceq>\<^sub>\<circ> \<tau>))"
Abs_kripke_inverse: fixes y :: "'d set \<times> 'd set \<times> ('d \<times> 'd) set \<times> ('d \<Rightarrow> 'e set option)" assumes "y \<in> kripke" shows "Rep_kripke (Abs_kripke y) = y"
UN_ball_bex_simps_1: fixes A :: "'c set set"   and P :: "'c \<Rightarrow> bool" shows "(\<forall>x\<in>\<Union> A. P x) = (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
assms_3: shows "summable (\<lambda>n. Sigma_Algebra.measure M (A n))"
prod_relE: fixes p :: "'b \<times> 'd"   and p' :: "'e \<times> 'f"   and R1 :: "('b \<times> 'e) set"   and R2 :: "('d \<times> 'f) set" assumes "(p, p') \<in> R1 \<times>\<^sub>r R2" obtains a :: "'b" and b :: "'d" and a' :: "'e" and b' :: "'f" where "p = (a, b)" and "p' = (a', b')" and "(a, a') \<in> R1" and "(b, b') \<in> R2"
inf_Sup: fixes a :: "'a"   and B :: "'a set" shows "inf a (Sup B) = Sup (inf a ` B)"
Cons_eq_append_conv: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(x # xs = ys @ zs) = (ys = [] \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' @ zs))"
option_assn_pure_conv: fixes R :: "('b \<times> 'a) set" shows "option_assn (pure R) = pure (\<langle>R\<rangle>option_rel)"
artanh_tanh_real: fixes x :: "real" shows "artanh (tanh x) = x"
fin_support_Field_osum: fixes f :: "'d + 'e \<Rightarrow> 'f"   and z :: "'f"   and A :: "'d set"   and B :: "'e set" shows "(f \<in> fin_support z (Inl ` A \<union> Inr ` B)) = (f \<circ> Inl \<in> fin_support z A \<and> f \<circ> Inr \<in> fin_support z B)"
history_order_def: fixes x :: "('msgid \<times> 'op) event"   and i :: "nat"   and z :: "('msgid \<times> 'op) event" shows "x \<sqsubset>\<^sup>i z \<equiv> \<exists>xs ys zs. xs @ x # ys @ z # zs = history i"
update_def: fixes f :: "'c \<Rightarrow> 'c" shows "State_Monad.update f = State_Monad.get \<bind> State_Monad.set \<circ> f"