hcju/isabelleps huggingface dataset:
proves_descend_abs_rule': fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and \<tau>' :: "typ"   and s :: "Core.term"   and t :: "Core.term"   and x :: "variable" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau>' s) (Abs \<tau>' t)"   and "(x, \<tau>') \<notin> FV \<Gamma>"   and "typ_ok \<Theta> \<tau>'"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (subst_bv (Fv x \<tau>') s) (subst_bv (Fv x \<tau>') t)"
sup_inf_implies: fixes x :: "'c"   and y :: "'c" shows "(x \<squnion> y) \<sqinter> (- x \<squnion> y) = y"
assms_3: shows "has_as_binary_product a0 a1 p0 p1"
errorT_eq_iff: fixes a :: "'b\<cdot>('c, 'd) errorT"   and b :: "'b\<cdot>('c, 'd) errorT" shows "(a = b) = (runErrorT\<cdot>a = runErrorT\<cdot>b)"
mirror_elem_eq_conv': fixes I :: "nat set"   and a :: "nat"   and b :: "nat" assumes "finite I"   and "a \<in> I"   and "b \<in> I" shows "(mirror_elem a I = mirror_elem b I) = (a = b)"
restrict_apply': fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "x \<in> A" shows "restrict f A x = f x"
assms: shows "\<forall>(x, m)\<in>collect_clock_pairs g. m \<in> \<int>"
simplicial_subdivision_diff: fixes p :: "nat"   and c1 :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and c2 :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" shows "simplicial_subdivision p (c1 - c2) = simplicial_subdivision p c1 - simplicial_subdivision p c2"
R: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "x \<noteq> y" shows "\<exists>g. arc g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
simps_2: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat" shows "extend S C f (Suc n) = (if extend S C f n \<union> {f n} \<in> C then if \<exists>p. f n = Exists p then extend S C f n \<union> {f n} \<union> {dest_Exists (f n)[App (SOME k. k \<notin> \<Union> (params ` (extend S C f n \<union> {f n}))) []/0]} else if \<exists>p. f n = Neg (Forall p) then extend S C f n \<union> {f n} \<union> {Neg (dest_Forall (dest_Neg (f n))[App (SOME k. k \<notin> \<Union> (params ` (extend S C f n \<union> {f n}))) []/0])} else extend S C f n \<union> {f n} else extend S C f n)"
src_trg: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.src (S.trg \<mu>) = S.trg \<mu>"
assms_2: shows "x \<in>\<^sub>\<circ> r"
star_rd_def: fixes x :: "'a" shows "star_rd x = Sup {proto_dioid.power_rd 1\<^sub>\<sigma> (\<cdot>) sup (Sup {}) x i |i. i \<in> \<nat>}"
characterizationallAllocations: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" shows "(a \<in> allAllocations N G) = (a \<in> injections (Domain a) (Range a) \<and> Range a \<in> all_partitions G \<and> Domain a \<subseteq> N)"
simps_1: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and N :: "'b ltln set" shows "\<phi> W\<^sub>n \<psi>[N]\<^sub>\<Pi>\<^sub>2 = \<phi>[N]\<^sub>\<Pi>\<^sub>2 W\<^sub>n \<psi>[N]\<^sub>\<Pi>\<^sub>2"
assms: shows "(u, v) \<in> set (vwalk_arcs p)"
a_set_attribute_def: fixes ptr :: "(_) element_ptr"   and k :: "char list"   and v :: "char list option" shows "local.a_set_attribute ptr k v = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs_update (if v = None then fmdrop k m else fmupd k (the v) m))"
finite_imageD: fixes f :: "'r \<Rightarrow> 'q"   and A :: "'r set" assumes "finite (f ` A)"   and "inj_on f A" shows "finite A"
singletonI: fixes a :: "'b" shows "(a::'b) \<in> {a}"
in_rel_def: fixes R :: "('b \<times> 'c) set"   and x :: "'b"   and y :: "'c" shows "in_rel R x y \<equiv> (x, y) \<in> R"
SourcesA31_L1: shows "Sources level1 sA31 = {sA11, sA22, sA23, sA31, sA32, sA41}"
indRelR_cond_preservation_iff_indRelL_cond_reflection: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_preserves_pred Rel Pred) = (\<exists>Rel. (\<forall>S. (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> rel_reflects_pred Rel Pred)"
Gcd_insert: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Gcd (insert a A) = gcd a (Gcd A)"
composite_simps_4: shows "cod (\<nu> \<star> \<pi>) = cod \<nu> \<star> cod \<pi>"
impl_OrdSum_first_second: fixes x :: "'b"   and y :: "'c" assumes "x \<noteq> (1::'b)" shows "Abs_OrdSum (x, 1::'c) l\<rightarrow> Abs_OrdSum (1::'b, y) = 1"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "color \<Rightarrow> ('c, 'd) RBT_Impl.rbt \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> ('c, 'd) RBT_Impl.rbt \<Rightarrow> 'b"   and rbt :: "('c, 'd) RBT_Impl.rbt" shows "P (case rbt of rbt.Empty \<Rightarrow> f1 | Branch x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd) = ((rbt = rbt.Empty \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23 x24 x25. rbt = Branch x21 x22 x23 x24 x25 \<longrightarrow> P (f2 x21 x22 x23 x24 x25)))"
mprop_2: shows "trading_strategy p \<and> AEeq M (cls_val_process Mkt2 p 0) (\<lambda>w. 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt2 p m w) \<and> 0 < prob {w \<in> space M. 0 < cls_val_process Mkt2 p m w}"
finite_positions_in_row: fixes n :: "nat"   and x :: "('a \<times> bool list) list" assumes "0 < n" shows "finite (positions_in_row (x @- sconst (any, replicate n False)) 0)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>n. (\<not> n < 2 \<Longrightarrow> P (n div 2)) \<Longrightarrow> P n" shows "P a0"
R'_cases: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "(x, y) \<in> R' n" obtains "x \<le> n" and "y \<le> n" and "(x, y) \<in> R"   | x' :: "nat" where "x = n + 1" and "y \<le> n" and "n < x'__" and "(x'__, y) \<in> R"   | y' :: "nat" where "x \<le> n" and "y = n + 1" and "n < y'__" and "(x, y'__) \<in> R"   | x' :: "nat" and y' :: "nat" where "x = n + 1" and "y = n + 1" and "n < x'" and "n < y'" and "(x', y') \<in> R"
lift_by1: fixes x :: "V" shows "lift x 1 = set {x}"
simps: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" shows "a1 cc-a2\<rightarrow>\<^sub>d* a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> valid_SDG_node n) \<or> (\<exists>n ns n'' n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> n cc-ns\<rightarrow>\<^sub>d* n'' \<and> n'' \<longrightarrow>\<^bsub>cd\<^esub> n') \<or> (\<exists>n ns n'' p n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> n cc-ns\<rightarrow>\<^sub>d* n'' \<and> n'' -p\<rightarrow>\<^bsub>call\<^esub> n'))"
E: shows "0 < offs_num (length ns) (x # xs) index key mi ma i"
minarc_bot_iff: fixes x :: "'a" shows "(minarc x = bot) = (x = bot)"
intro: fixes U :: "'b set"   and V :: "'b set" assumes "U \<noteq> {}"   and "U \<subseteq> V"   and "\<And>x y. \<lbrakk>x \<in> U; y \<in> U\<rbrakk> \<Longrightarrow> x + y \<in> U"   and "\<And>x a. x \<in> U \<Longrightarrow> a \<cdot> x \<in> U" shows "Subspace.subspace U V"
antimonoD: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "class.order less_eq less"   and "order.antimono less_eq f"   and "less_eq x y" shows "f y \<le> f x"
comm_neutral: fixes a :: "'c multiset" shows "a \<union># {#} = a"
assms_1: shows "p = {l}send(fmsg) . qq"
ladder_ix_cut: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and n :: "nat" assumes "0 < index"   and "index < length L" shows "ladder_ix (ladder_cut L n) index = ladder_ix L index"
simps_1: fixes xs :: "'a list" shows "nonzero xs [] = True"
simps_5: fixes x :: "'a" shows "norm_instr (Ubx.instr.ILoad x) = Inca.instr.ILoad x"
u: shows "S \<inter> {x. a' \<bullet> x = b'} face_of S"
r_maxletr_prim: shows "prim_recfn 2 r_maxletr"
erase2jumpover2: fixes q :: "nat"   and list :: "nat list"   and n :: "nat"   and RR :: "bool" assumes "q < length list"   and "\<forall>rn. <drop q list> @ Bk # Bk \<up> n \<noteq> Oc # Oc \<up> list ! q @ Bk # <drop (Suc q) list> @ Bk \<up> rn" shows "RR"
simps_4: fixes x2 :: "num" shows "num.Bit0 x2 \<noteq> num.One"
scc_reach_scc_root: fixes r :: "'a"   and scc :: "'a set"   and v :: "'a" assumes "scc_root s r scc"   and "v \<in> scc" shows "(v, r) \<in> E\<^sup>*"
lcoeff_monic_mult: fixes p :: "'b poly"   and q :: "'b poly" assumes "monic p" shows "coeff (p * q) (degree p + degree q) = lead_coeff q"
r_into_equivclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "r x y" shows "equivclp r x y"
zero_le_Fract_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(0 \<le> Fract a b) = (0 \<le> a)"
stutter_reduced_stutter_free: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "stutter_free (\<natural>\<sigma>)"
viewLn_inv: fixes t :: "('c, 'd) FingerTreeStruc"   and n :: "nat"   and nd :: "('c, 'd) Node"   and s :: "('c, 'd) FingerTreeStruc" assumes "is_measured_ftree t"   and "is_leveln_ftree n t"   and "viewLn t = Some (nd, s)" shows "is_measured_ftree s \<and> is_measured_node nd \<and> is_leveln_ftree n s \<and> is_leveln_node n nd"
sum_upto_cong': fixes x :: "real"   and f :: "nat \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'b"   and x' :: "real" assumes "\<And>n. \<lbrakk>0 < n; real n \<le> x\<rbrakk> \<Longrightarrow> f n = f' n"   and "x = x'" shows "sum_upto f x = sum_upto f' x'"
pell_mul_trivial_left_nat: fixes z :: "nat \<times> nat" shows "pell_mul (Suc 0, 0) z = z"
ln_exp_cf3_lower_bound_neg: fixes x :: "real" assumes "x \<le> 0"   and "0 < numer_cf3 x" shows "ln (exp_cf3 x) \<le> x"
elims_1: fixes x :: "'edge list"   and xa :: "'edge list"   and y :: "bool" assumes "valid_path_aux x xa = y" obtains cs :: "'edge list" where "x = csb__" and "xa = []" and "y"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> valid_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> valid_path_aux [] as | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> valid_path_aux cs' as | _ \<Rightarrow> valid_path_aux cs as)"
cases: fixes a :: "'t"   and 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" assumes "a \<in> active_threads s" obtains t :: "'t" and x :: "'x" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x'm' :: "'x \<times> 'm" and ln :: "'l \<Rightarrow>f nat" where "a = t__" and "thr s t__ = \<lfloor>(x__, ln__)\<rfloor>" and "ln__ = no_wait_locks" and "t__ \<turnstile> (x__, shr s) -ta\<rightarrow> x'm'" and "actions_ok s t__ ta"   | t :: "'t" and x :: "'x" and ln :: "'l \<Rightarrow>f nat" where "a = t" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "ln \<noteq> no_wait_locks" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln"
Greater_unbounded_aux: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and c :: "'a"   and t :: "real" assumes "finite X"   and "Regions.valid_region X k I r"   and "c \<in> X"   and "I c = Regions.intv.Greater (k c)" shows "\<exists>u\<in>Regions.region X I r. t < u c"
prefI: fixes A :: "'a \<Rightarrow> bool" assumes "\<And>w. A w" shows "|~ A"
split_all_acquired_in: fixes a :: "nat"   and xs :: "'b memref list" assumes "a \<in> all_acquired xs" shows "(\<exists>sop a' v ys zs A L R W. xs = ys @ Write\<^sub>s\<^sub>b True a' sop v A L R W # zs \<and> a \<in> A) \<or> (\<exists>A L R W ys zs. xs = ys @ Ghost\<^sub>s\<^sub>b A L R W # zs \<and> a \<in> A)"
rep_eq: fixes x :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and xa :: "('c, 'e) fmap" shows "fmlookup (fmmap_keys x xa) = (\<lambda>a. map_option (x a) (fmlookup xa a))"
ord_of_nat_vone: shows "1 = 1\<^sub>\<nat>"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'a"   and f2 :: "'a"   and x :: "enat" shows "P (case x of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = ((\<forall>nat. x = enat nat \<longrightarrow> P (f1 nat)) \<and> (x = \<infinity> \<longrightarrow> P f2))"
pref_rev_suf_iff: fixes u :: "'b list"   and v :: "'b list" shows "(u \<le>p v) = rev u \<le>s rev v"
lookup_pair_update_by_fun_pair: fixes xs :: "('a \<times> 'c) list"   and k1 :: "'a"   and f :: "'c \<Rightarrow> 'c"   and k2 :: "'a" assumes "oalist_inv_raw xs" shows "lookup_pair (update_by_fun_pair k1 f xs) k2 = (if k1 = k2 then f else id) (lookup_pair xs k2)"
min_elt_minimal: fixes B :: "'a set"   and y :: "'a" assumes "B \<subseteq> A"   and "B \<noteq> {}"   and "y \<in> A"   and "P y (min_elt B)" shows "y \<notin> B"
transfer_FOREACHci_plain: fixes iterate :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'d set"   and fi :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool" assumes "set_iterator iterate s"   and "\<And>x \<sigma>. RETURN (fi x \<sigma>) \<le> f x \<sigma>" shows "RETURN (iterate c fi \<sigma>) \<le> FOREACH\<^sub>C\<^bsup>I\<^esup> s c f \<sigma>"
T_def: fixes P :: "'c process" shows "T P = TRACES (Rep_process P)"
system_automaton_simps_2: fixes prog :: "program" shows "li.sa.E prog = li'.s1.step"
exists_base: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set" assumes "subfield K (L\<lparr>carrier := A\<rparr>)"   and "Hom.dimension n K E" shows "\<exists>Vs. set Vs \<subseteq> carrier (L\<lparr>carrier := A\<rparr>) \<and> Hom.independent K Vs \<and> length Vs = n \<and> Hom.Span K Vs = E"
neg: shows "A $$ (i, i) \<noteq> A $$ (j, j)"
dec_I2_Extend_FO: fixes i :: "nat"   and \<AA> :: "interp"   and P :: "nat fset"   and bs2 :: "WS1S_Nameful.so list" shows "dec_I2 (Extend FO i \<AA> P) bs2 = dec_I2 \<AA> bs2"
assms: shows "h \<turnstile> f \<rightarrow>\<^sub>e e"
domains_simp: fixes f :: "('obj, 'obj) concrete_category.arr" assumes "arr f" shows "domains f = {local.dom f}"
gcd_lcm_distrib: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "gcd x (lcm y z) = lcm (gcd x y) (gcd x z)"
simps_2: fixes k :: "nat" shows "exH_k (Suc k) = exH_k k * hadamard_on_i (Suc k)"
dual_star_least: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "less_eq (sup (times x z) y) z" shows "less_eq (times (dual_star x) y) z"
add_edge1_def: shows "add_edge1 \<equiv> \<lambda>(u, v) d g. M_update u ((v, d) # the_default [] (M_lookup g u)) g"
ipresIGWlsAll_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'd, 'e, 'k, 'l, 'g, 'h, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGWlsAll h hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresWlsAll Delta h hA MOD"
single_zero: fixes k :: "'d" shows "monomial (0::'e) k = 0"
discr0_coind: fixes phi :: "('test, 'atom) com \<Rightarrow> bool"   and c :: "('test, 'atom) com" assumes "phi c"   and "\<And>c s c' s'. \<lbrakk>mustT c s; phi c; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> s \<approx> s' \<and> (phi c' \<or> discr0 c')"   and "\<And>c s s'. \<lbrakk>mustT c s; phi c; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> s \<approx> s'" shows "discr0 c"
step_diff: 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 i :: "nat"   and x :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "i < k"   and "x\<^sub>1 \<le> x" shows "bs ! i * x + (hs ! i) x + 1 < x"
Suc_6: shows "\<S>\<bullet>\<C> \<turnstile> LI : ts _> ts'"
psi_def: shows "psi = Cn 2 r_phi [Cn 2 r_ifz [Cn 2 d [recf.Id 2 0], r_dummy 1 (r_const j\<^sub>2), r_dummy 1 (r_const j\<^sub>1)], recf.Id 2 1]"
g_4: shows "{x--y} = g ` {0..dist x y}"
compare_EQ_iff_eq_TT: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" assumes "class.Ord_linear eq compare" shows "(compare\<cdot>x\<cdot>y = EQ) = (eq\<cdot>x\<cdot>y = TT)"
range_pm_of_idx_pm_subset: fixes xs :: "'d list"   and f :: "nat \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.range (pm_of_idx_pm xs f) \<subseteq> lookup f ` {0..<length xs} - {0::'c}"
map_inv_def: fixes f :: "'d \<Rightarrow> 'c option" shows "map_inv f \<equiv> \<lambda>y. if y \<in> ran f then Some (SOME x. f x = Some y) else None"
fundfold_comp_fixespointwise_C0: fixes ss :: "'a permutation list" assumes "order.greater_eq (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) (set ss)" shows "fixespointwise (fold fst (map Spair ss)) C0"
dbm_le_def: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "a \<preceq> b \<equiv> a \<prec> b \<or> a = b"
rel_envTE: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and M :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and m :: "('d, 'f) envT"   and m' :: "('e, 'g) envT" assumes "Monomorphic_Monad.rel_envT R M m m'" obtains f :: "'d \<Rightarrow> 'f" and g :: "'e \<Rightarrow> 'g" where "m = EnvT f" and "m' = EnvT g" and "(R ===> M) f g"
im_d_gchainsTr1_2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "d_gchain G n f"   and "i \<le> n"   and "f i \<in> f ` {i. i \<le> n} - {f 0}" shows "(LEAST j. f j \<in> f ` {i. i \<le> n} - {f 0}) \<le> i"
induced_function_integral_restr_nonergodic_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "conservative_mpt M T"   and "A \<in> sets M"   and "integrable M f" shows "integrable (restrict_space M A) (qmpt.induced_function T A f)"
wf_weaken: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and T :: "POPLmark.type"   and B :: "binding" assumes "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f T" shows "\<up>\<^sub>e (Suc 0) 0 \<Delta> @ B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f \<up>\<^sub>\<tau> (Suc 0) \<parallel>\<Delta>\<parallel> T"
degree_m_mult_le: fixes f :: "int poly"   and g :: "int poly"   and h :: "int poly" assumes "f =m g * h" shows "degree_m f \<le> degree_m g + degree_m h"
assms: shows "a{b}.P \<longmapsto> c\<guillemotleft>x\<guillemotright> \<prec> P'"
Sup_bot_conv_2: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup A) = (\<forall>x\<in>A. x = bot)"
sum_nonneg_leq_bound: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset"   and B :: "'c multiset"   and i :: "'b" assumes "finite s"   and "\<And>i. i \<in> s \<Longrightarrow> {#} \<subseteq># f i"   and "comm_monoid_add.sum (+) {#} f s = B"   and "i \<in> s" shows "f i \<subseteq># B"
a_star_OrR1: fixes M :: "trm"   and M' :: "trm"   and a :: "coname"   and b :: "coname" assumes "M \<longrightarrow>\<^sub>a* M'" shows "OrR1 <a>.M b \<longrightarrow>\<^sub>a* OrR1 <a>.M' b"
trivial_group_def: fixes G :: "('c, 'd) monoid_scheme" shows "trivial_group G \<equiv> Group.group G \<and> carrier G = {\<one>\<^bsub>G\<^esub>}"
map_T_parametric: fixes L1 :: "'n \<Rightarrow> 'p \<Rightarrow> bool"   and L1' :: "'o \<Rightarrow> 'q \<Rightarrow> bool"   and Co1 :: "'r \<Rightarrow> 'aa \<Rightarrow> bool"   and Co1' :: "'s \<Rightarrow> 'ab \<Rightarrow> bool"   and Co2 :: "'t \<Rightarrow> 'ac \<Rightarrow> bool"   and Co2' :: "'u \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra1' :: "'v \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra1 :: "'w \<Rightarrow> 'af \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra2 :: "'y \<Rightarrow> 'ah \<Rightarrow> bool" shows "((L1 ===> L1') ===> (Co1 ===> Co1') ===> (Co2 ===> Co2') ===> (Contra1' ===> Contra1) ===> (Contra2' ===> Contra2) ===> rel_T L1 Co1 Co2 Contra1 Contra2 ===> rel_T L1' Co1' Co2' Contra1' Contra2') map_T map_T"
numsubst0_I: fixes b :: "int"   and bs :: "int list"   and a :: "Cooper.num"   and t :: "Cooper.num" shows "Inum (b # bs) (numsubst0 a t) = Inum (Inum (b # bs) a # bs) t"
assms_2: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<cong> Q" shows "F P \<leadsto><weakBisimulation> F Q"
shift_append_preserves_T_aux: fixes \<tau> :: "ty"   and \<rho>1 :: "val set list"   and \<rho>3 :: "val set list"   and \<rho>2 :: "val set list" shows "T \<tau> (\<rho>1 @ \<rho>3) = T (shift (length \<rho>2) (length \<rho>1) \<tau>) (\<rho>1 @ \<rho>2 @ \<rho>3)"
assms_3: shows "(c, s) \<rightarrow>*c (c', s')"
tdghm_dghm_comp_def: fixes \<NN> :: "V"   and \<HH> :: "V" shows "\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> = [\<lambda>a\<in>\<^sub>\<circ>\<HH>\<lparr>HomDom\<rparr>\<lparr>Obj\<rparr>. \<NN>\<lparr>NTMap\<rparr>\<lparr>\<HH>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr>, \<NN>\<lparr>NTDom\<rparr> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>, \<NN>\<lparr>NTCod\<rparr> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>, \<HH>\<lparr>HomDom\<rparr>, \<NN>\<lparr>NTDGCod\<rparr>]\<^sub>\<circ>"
simps: fixes s :: "'sequent"   and S :: "'structure"   and r :: "'rule"   and s' :: "'sequent"   and S' :: "'structure" shows "descent (s, S) r (s', S') = (\<forall>v v' b. (v, b, v') \<in> \<delta> s r s' \<longrightarrow> (\<sigma> (v', S'), \<sigma> (v, S)) \<in> Ord \<and> (b \<longrightarrow> \<sigma> (v', S') \<noteq> \<sigma> (v, S)))"
Domainp_is_poly: shows "Domainp poly_rel = is_poly ops"
cases: fixes a :: "'a sequent list \<times> 'a sequent"   and R :: "('a sequent list \<times> 'a sequent) set" assumes "a \<in> R*" obtains r :: "'a sequent list \<times> 'a sequent" and seq :: "'a sequent" where "a = extendRule seq r" and "r \<in> R"
subgraph_imp_subverts: fixes H :: "('a, 'b) pre_digraph"   and G :: "('a, 'b) pre_digraph" assumes "subgraph H G" shows "verts H \<subseteq> verts G"
dge: shows "derivation_ge (drop n D) (length \<beta>)"