hcju/isabelleps huggingface dataset:
image_affinity_atLeastAtMost: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and m :: "'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(\<lambda>x. plus (times m x) c) ` ord.atLeastAtMost less_eq a b = (if ord.atLeastAtMost less_eq a b = {} then {} else if less_eq zero m then ord.atLeastAtMost less_eq (plus (times m a) c) (plus (times m b) c) else ord.atLeastAtMost less_eq (plus (times m b) c) (plus (times m a) c))"
AC_def: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<Squnion> b \<equiv> \<lambda>ab Q s. max (a ab Q s) (b ab Q s)"
bisim1Seq1: fixes P :: "'a prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e2 :: "'addr expr1" assumes "P,e1,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e1;; e2,h \<turnstile> (e';; e2, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
Sel_lower_bound: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and A :: "(symbol \<times> character list) set"   and B :: "(symbol \<times> character list) set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "A \<subseteq> B" shows "A \<subseteq> Sel A B"
vfinite_veqpoll_trans: fixes A :: "V"   and B :: "V" assumes "vfinite A"   and "A \<approx>\<^sub>\<circ> B" shows "vfinite B"
le_sup_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(sup x y \<le> z) = (x \<le> z \<and> y \<le> z)"
hnfInput: fixes a :: "name"   and x :: "name"   and P :: "pi" shows "hnf (a<x>.P)"
perm_vars_def: fixes n :: "nat"   and vs :: "nat list" shows "perm_vars n vs = (distinct vs \<and> set vs = {0..<n})"
non_distinct_low_rank: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "\<not> distinct (cols A)" shows "vec_space.rank n A < n"
exec_move_CallI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and M :: "String.literal"   and v :: "'addr val" assumes "exec_moves ci P t es h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e\<bullet>M(es)) h (stk @ [v], loc, length (compE2 e) + pc, xcp) ta h' (stk' @ [v], loc', length (compE2 e) + pc', xcp')"
integral_empty: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "space M = {}" shows "integral\<^sup>L M f = (0::'d)"
less_imp_le: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "x \<prec>\<^sub>p y" shows "x \<preceq>\<^sub>p y"
induct: fixes P :: "'b rexp \<Rightarrow> 'b rexp \<Rightarrow> bool"   and a0 :: "'b rexp"   and a1 :: "'b rexp" assumes "\<And>r1 r2 s. \<lbrakk>P r2 s; P r1 (ACI_nPlus r2 s)\<rbrakk> \<Longrightarrow> P (Plus r1 r2) s"   and "\<And>s1 s2. (\<lbrakk>Zero \<noteq> s1; \<not> Zero < s1\<rbrakk> \<Longrightarrow> P Zero s2) \<Longrightarrow> P Zero (Plus s1 s2)"   and "\<And>s1 s2. (\<lbrakk>One \<noteq> s1; \<not> One < s1\<rbrakk> \<Longrightarrow> P One s2) \<Longrightarrow> P One (Plus s1 s2)"   and "\<And>v s1 s2. (\<lbrakk>Atom v \<noteq> s1; \<not> Atom v < s1\<rbrakk> \<Longrightarrow> P (Atom v) s2) \<Longrightarrow> P (Atom v) (Plus s1 s2)"   and "\<And>v va s1 s2. (\<lbrakk>Times v va \<noteq> s1; \<not> Times v va < s1\<rbrakk> \<Longrightarrow> P (Times v va) s2) \<Longrightarrow> P (Times v va) (Plus s1 s2)"   and "\<And>v s1 s2. (\<lbrakk>Star v \<noteq> s1; \<not> Star v < s1\<rbrakk> \<Longrightarrow> P (Star v) s2) \<Longrightarrow> P (Star v) (Plus s1 s2)"   and "P Zero Zero"   and "P Zero One"   and "\<And>v. P Zero (Atom v)"   and "\<And>v va. P Zero (Times v va)"   and "\<And>v. P Zero (Star v)"   and "P One Zero"   and "P One One"   and "\<And>v. P One (Atom v)"   and "\<And>v va. P One (Times v va)"   and "\<And>v. P One (Star v)"   and "\<And>v. P (Atom v) Zero"   and "\<And>v. P (Atom v) One"   and "\<And>v va. P (Atom v) (Atom va)"   and "\<And>v va vb. P (Atom v) (Times va vb)"   and "\<And>v va. P (Atom v) (Star va)"   and "\<And>v va. P (Times v va) Zero"   and "\<And>v va. P (Times v va) One"   and "\<And>v va vb. P (Times v va) (Atom vb)"   and "\<And>v va vb vc. P (Times v va) (Times vb vc)"   and "\<And>v va vb. P (Times v va) (Star vb)"   and "\<And>v. P (Star v) Zero"   and "\<And>v. P (Star v) One"   and "\<And>v va. P (Star v) (Atom va)"   and "\<And>v va vb. P (Star v) (Times va vb)"   and "\<And>v va. P (Star v) (Star va)" shows "P a0 a1"
inf_chain_is_long: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and X :: "'a set" assumes "[f[x ..]X]" shows "long_ch_by_ord f X \<and> f 0 = x \<and> infinite X"
inv_closed: fixes x :: "'a set" assumes "x \<in> carrier (G Mod N)" shows "inv\<^bsub>G Mod N\<^esub> x \<in> carrier (G Mod N)"
assms_3: fixes x :: "real"   and y :: "real" assumes "x \<in> A"   and "y \<in> A"   and "x \<le> y" shows "f' x \<le> f' y"
post_meas0_bell01_fst: shows "post_meas0 2 |\<beta>\<^sub>0\<^sub>1\<rangle> 0 = |unit_vec 4 1\<rangle>"
ipset_from_netmask_def: fixes addr :: "'a word"   and netmask :: "'a word" shows "ipset_from_netmask addr netmask \<equiv> let network_prefix = addr && netmask in {network_prefix..network_prefix || ~~ netmask}"
prime_multiplicity_le_imp_dvd: fixes x :: "'b"   and y :: "'b" assumes "x \<noteq> (0::'b)"   and "y \<noteq> (0::'b)" shows "(x dvd y) = (\<forall>p. prime p \<longrightarrow> multiplicity p x \<le> multiplicity p y)"
degree_pCons_eq_if: fixes a :: "'c"   and p :: "'c poly" shows "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
q: shows "q = sum_list (map2 (\<cdot>) cs ps)"
assms_2: shows "equivalence_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
pbernpoly_def: fixes n :: "nat"   and x :: "real" shows "pbernpoly n x = bernpoly n (frac x)"
u_servTicket_authentic: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "B \<noteq> Tgs"   and "B \<notin> bad"   and "evs \<in> kerbV" shows "\<exists>Ta authK. Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace> \<in> set evs \<and> Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace> \<in> set evs \<and> servKlife + Ts \<le> authKlife + Ta"
simps_5: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "restricted_subformulas (\<phi> R\<^sub>n \<psi>) = restricted_subformulas \<phi> \<union> restricted_subformulas_inner \<psi>"
set_rev: fixes xs :: "'c list" shows "set (rev xs) = set xs"
B_cost: shows "T\<^sub>p_on_rand' BIT Da qs = bit"
HNatInfinite_is_Suc: fixes x :: "nat star" assumes "x \<in> HNatInfinite" shows "\<exists>y. x = y + 1"
contI: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>Y. chain Y \<Longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)" shows "cont f"
elims_3: fixes x :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "\<not> qGoodAbs x" obtains xs :: "'varSort" and x :: "'var" and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" where "xa__ = qAbs xs x X" and "\<not> qGood X"
assms_1: shows "\<exists>f. f \<stileturn> h"
awp_mono: fixes A :: "'b set"   and B :: "'b set"   and F :: "'b program" assumes "A \<subseteq> B" shows "awp F A \<subseteq> awp F B"
\<phi>\<^sub>y_in_y: shows "\<phi>\<^sub>y \<in> y"
gen_akiyama_tanigawa_code: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and k :: "nat" shows "gen_akiyama_tanigawa f n k = hd (gen_akiyama_tanigawa_row f n k (Suc k))"
Nl_entails_Bl: shows "\<Union> (\<G>_F_L ` Nl) \<Turnstile>G \<Union> (\<G>_F_L ` {Bl})"
p1: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All j (SyntaxN.All j' (SyntaxN.All k' (SeqSubstFormP v a (Var i) (Var j) s (Var k) IMP SeqSubstFormP v a (Var i) (Var j') s' (Var k') IMP Var j' EQ Var j))))(i::=x)"
star_inductl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<cdot> z \<le> y"
a_host_shadow_root_rel_def: fixes get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_host_shadow_root_rel get_shadow_root h = (\<lambda>(x, y). (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y)) ` {(host, shadow_root). host |\<in>| element_ptr_kinds h \<and> |h \<turnstile> get_shadow_root host|\<^sub>r = Some shadow_root}"
data_prev'_def: shows "data_prev' \<equiv> dropWhile_queue (\<lambda>(t, X). left I \<le> nt - t) data_prev"
set_spmf_of_set_finite: fixes A :: "'l set" assumes "finite A" shows "set_spmf (spmf_of_set A) = A"
permutes_inj_on: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and A :: "'e set" assumes "f permutes S" shows "inj_on f A"
simp: shows "denote_while_n M0 M1 DS n \<rho> + matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n - matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n = denote_while_n M0 M1 DS n \<rho>"
stock_portfolio_def: fixes Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "stock_portfolio Mkt p = (portfolio p \<and> support_set p \<subseteq> stocks Mkt)"
table_in: shows "table (args_n args) (args_R args) (Mapping.keys tuple_in')"
col_permutation_4: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C" shows "Col B A C"
l2: fixes s :: "'a" assumes "s \<in> S" shows "l2 s \<le> measure_pmf.expectation (K s) l2 + c s"
l_neg: fixes x :: "'a" shows "- x + x = (0::'a)"
monotoneI: fixes orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i" assumes "\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)" shows "monotone orda ordb f"
parCasesB': fixes A :: "pi"   and B :: "pi"   and b :: "name"   and y :: "name"   and A' :: "pi" assumes "A \<parallel> B \<longmapsto> b<\<nu>y> \<prec> A'" obtains P :: "pi" and a :: "name" and x :: "name" and P' :: "pi" and Q :: "pi" where "A \<parallel> B = Pa__ \<parallel> Q__" and "b<\<nu>y> \<prec> A' = a__<\<nu>x__> \<prec> (P' \<parallel> Q__)" and "Pa__ \<longmapsto> a__<\<nu>x__> \<prec> P'" and "x__ \<sharp> Pa__" and "x__ \<sharp> Q__" and "x__ \<noteq> a__"   | Q :: "pi" and a :: "name" and x :: "name" and Q' :: "pi" and P :: "pi" where "A \<parallel> B = P \<parallel> Q" and "b<\<nu>y> \<prec> A' = a<\<nu>x> \<prec> (P \<parallel> Q')" and "Q \<longmapsto> a<\<nu>x> \<prec> Q'" and "x \<sharp> P" and "x \<sharp> Q" and "x \<noteq> a"
zero_in_Field: fixes r :: "('d \<times> 'd) set" assumes "wo_rel r"   and "Field r \<noteq> {}" shows "zero r \<in> Field r"
measurable_poisson: shows "poisson \<in> stock_measure REAL \<rightarrow>\<^sub>M subprob_algebra (stock_measure INTEG)"
prod_rule: fixes p :: "'c \<times> 'd"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> 'e nres"   and \<Phi> :: "'e \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> S a b \<le> SPEC \<Phi>" shows "(case p of (x, xa) \<Rightarrow> S x xa) \<le> SPEC \<Phi>"
D: shows "infinite {y. [(x, y)] \<bullet> M[[(x, y)] \<bullet> xvec::=[(x, y)] \<bullet> Tvec] \<noteq> M[xvec::=Tvec]}"
sorted_sorted_list_of_set: fixes A :: "'a word set" shows "signed.sorted (signed.sorted_list_of_set A)"
is_arg_min_on_Min: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "finite A"   and "is_arg_min_on f A a" shows "Min (f ` A) = f a"
last_message_append_one: fixes xs :: "'b message_af list"   and m :: "'b message_af" shows "last_message (xs @ [m]) = (if m = NoMsg then last_message xs else m)"
assms_1: shows "ys \<noteq> xs \<or> y \<noteq> x"
isPrefixElements: fixes a :: "('b \<times> bool) list"   and b :: "('b \<times> bool) list" assumes "isPrefix a b" shows "isPrefix (elements a) (elements b)"
virtual_memop_step_simulates_direct_memop_step: fixes "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and x :: "unit"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and x' :: "unit"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "(is, \<theta>, x, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is', \<theta>', x', m', \<D>', \<O>', \<R>', \<S>')" shows "(is, \<theta>, x, m, \<D>, \<O>, (), \<S>) \<rightarrow>\<^sub>v (is', \<theta>', x', m', \<D>', \<O>', (), \<S>')"
lookup7: fixes x :: "name"   and a :: "coname"   and \<Gamma> :: "(name \<times> ty) list"   and b :: "coname" shows "lookupc x a (idn \<Gamma> b) = Ax x a"
sorted_list_of_set_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
swap_triple: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "a \<noteq> c"   and "b \<noteq> c" shows "Fun.swap a b (Fun.swap b c (Fun.swap a b f)) = Fun.swap a c f"
set_of_intervalI: fixes y :: "'a"   and Y :: "'a interval" assumes "y \<in>\<^sub>i Y" shows "set_of (interval_of y) \<subseteq> set_of Y"
exi_ren: fixes y :: "name"   and \<phi> :: "fmla"   and x :: "name" assumes "y \<notin> Fvars \<phi>" shows "exi x \<phi> = exi y (\<phi>(x::=Var y))"
List_list: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and ps :: "ref list" assumes "List p h ps" shows "list p h = ps"
lim_1_over_n: shows "(\<lambda>n. (1::'a) / of_nat n) \<longlonglongrightarrow> (0::'a)"
f_nxt_empty: fixes f :: "'b \<Rightarrow> 'b ttree"   and x :: "'b"   and T :: "'b set"   and x' :: "'b" assumes "f x = TTree.empty" shows "f_nxt f T x' x = TTree.empty"
sigma_sets_superset_generator: fixes A :: "'b set set"   and X :: "'b set" shows "A \<subseteq> sigma_sets X A"
hom_relboundary_def: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set"   and T :: "'a set" shows "hom_relboundary p X S T = hom_induced (p - 1) (subtopology X S) {} (subtopology X S) T id \<circ> hom_boundary p X S"
mult_smult_right: fixes p :: "'a poly"   and a :: "'a"   and q :: "'a poly" shows "p * smult a q = smult a (p * q)"
equivp_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "equivp R = (\<forall>x y. R x y = (R x = R y))"
der_g: fixes x :: "real" assumes "x \<in> S" shows "(g has_vector_derivative g' x) (at x within S)"
morE1: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and z :: "'d" assumes "mor B1 B2 s1 s2 B1' B2' s1' s2' f g"   and "z \<in> B1" shows "F1map id f g (s1 z) = s1' (f z)"
finite_empty_induct: fixes A :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite A"   and "P A"   and "\<And>a A. \<lbrakk>finite A; a \<in> A; P A\<rbrakk> \<Longrightarrow> P (A - {a})" shows "P {}"
pell_square_solution_nat_iff: fixes D :: "nat"   and x :: "nat"   and y :: "nat" assumes "is_square D" shows "(x\<^sup>2 = 1 + D * y\<^sup>2) = (x = 1 \<and> (D = 0 \<or> y = 0))"
id_2: shows "((Fun f (bef @ s # aft), Fun f (bef @ t # aft)) \<in> NS) = True"
embedS_Field: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "Well_order r"   and "embedS r r' f" shows "f ` Field r \<subset> Field r'"
that: fixes h' :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U' :: "'x set" assumes "(h', U') \<in> set ps"   and "poly_deg h' = d"   and "card U \<le> card U'" shows "thesis"
inf_greatest: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (z::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (y::'b::type) (z::'b::type))"
prems_5: shows "finite (varacom ({A} WHILE b DO C))"
maxR_absorb1: fixes y :: "'a"   and x :: "'a" assumes "(y, x) \<in> r" shows "maxR x y = x"
GuardsCons: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and f :: "'f"   and g :: "'e set"   and gs :: "('f \<times> 'e set) list"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Guard f g (guards gs c) Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P guards ((f, g) # gs) c Q,A"
Cfun_app_eqvt: fixes \<pi> :: "perm"   and f :: "'b \<rightarrow> 'a"   and x :: "'b" shows "\<pi> \<bullet> f\<cdot>x = (\<pi> \<bullet> f)\<cdot>(\<pi> \<bullet> x)"
that: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set"   and c :: "'a" assumes "(h, U) \<in> set ps"   and "c \<noteq> (0::'a)"   and "monomial c t \<in> cone (h, U)" shows "thesis"
suminf_geometric: fixes c :: "'b" assumes "norm c < 1" shows "(\<Sum>n. c ^ n) = (1::'b) / ((1::'b) - c)"
presheaf_of_rings_axioms: shows "presheaf_of_rings Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec"
power_Suc2: fixes a :: "'b hrel"   and n :: "nat" shows "power.power II (;;) a (Suc n) = power.power II (;;) a n ;; a"
exhaust_sel: fixes tree :: "'d tree" obtains "tree = \<langle>\<rangle>"   | "tree = \<langle>un_Node1 tree, value tree, un_Node3 tree\<rangle>"
exists_worst_degenerate: shows "\<exists>x\<in>local.degenerate_lotteries. \<forall>y\<in>local.degenerate_lotteries. y \<succeq>[\<R>] x"
exH_n_dim: shows "exH_n \<in> carrier_mat N N"
rat_to_int_factor_content_1: fixes p :: "int poly"   and g :: "rat poly"   and h :: "rat poly"   and r :: "rat"   and rg :: "int poly"   and s :: "rat"   and sh :: "int poly" assumes "content p = 1"   and "of_int_poly p = g * h"   and "rat_to_normalized_int_poly g = (r, rg)"   and "rat_to_normalized_int_poly h = (s, sh)"   and "p \<noteq> 0" shows "p = rg * sh"
H_def: shows "H = (\<lambda>N. {x \<in> space M. \<forall>n\<ge>N. real (card ({n. P x n} \<inter> {..<n})) < a * real n})"
singleton_quotient: fixes x :: "'a"   and r :: "('a \<times> 'a) set" shows "{x} // r = {r `` {x}}"
power_Suc0_right: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.monoid_mult one times" shows "power.power one times a (Suc 0) = a"
L3_def: fixes x :: "nat"   and y :: "nat" shows "L3 x y = {(x, y), (x + 1, y), (x + 1, y + 1)}"
Suc: shows "Digamma (of_nat (Suc n)) = harm n - euler_mascheroni"
minus_float: fixes x :: "real"   and y :: "real" assumes "x \<in> float"   and "y \<in> float" shows "x - y \<in> float"
p_pp_comp: fixes x :: "'a"   and y :: "'a" shows "- (- - x * y) = - (x * y)"
mono_neutral_cong_left: fixes T :: "'d set"   and S :: "'d set"   and h :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = \<^bold>1"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g S = F h T"
size_9: fixes x3 :: "'b formula" shows "size (\<^bold>\<not> x3) = size x3 + Suc 0"
fundamental_theorem_of_calculus_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<forall>x\<in>{a..b}. (f has_real_derivative f' x) (at x within {a..b})" shows "(f' has_integral f b - f a) {a..b}"
ipurge_ref_eq: fixes D :: "'c \<Rightarrow> 'b"   and x :: "'c"   and I :: "('b \<times> 'b) set"   and u :: "'b"   and xs :: "'c list"   and X :: "'c set" assumes "D x \<in> sinks I D u (xs @ [x])" shows "ipurge_ref I D u (xs @ [x]) X = ipurge_ref I D u xs {x' \<in> X. (D x, D x') \<notin> I}"
Int_UN_distrib: fixes B :: "'a set"   and A :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "B \<inter> \<Union> (A ` I) = (\<Union>\<^bsub>i\<in>I\<^esub> B \<inter> A i)"
inference_computation_2: shows "N = mset_set ((\<lambda>D. (D, n)) ` concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) (set_mset (image_mset fst Q)) C))"
lossless: fixes \<eta> :: "nat" assumes "lossless" shows "plossless_gpv (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) (\<A> \<eta>)"
all_sub_consts_conjoin: fixes e :: "nat \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f list \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f list \<Rightarrow> bool"   and cs :: "'d list"   and G :: "('d, 'e) form list" shows "list_all (eval e f g) (map (sub_consts cs) G) = eval e f g (sub_consts cs (conjoin G))"
component_diff_connected: fixes S :: "'b set"   and U :: "'b set"   and C :: "'b set" assumes "connected S"   and "connected U"   and "S \<subseteq> U"   and "C \<in> components (U - S)" shows "connected (U - C)"
elims: fixes x :: "char list"   and y :: "char list" assumes "upChar x = y" obtains Y :: "char list" where "x = Y" and "y = (if Y \<noteq> [] \<and> 97 \<le> of_char (last Y) \<and> of_char (last Y) < 122 then butlast Y @ [char_of (of_char (last Y) + 1)] else Y @ ''a'')"
foldl_mono: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and l :: "'a list" assumes "\<And>x. mono (\<lambda>s. f s x)" shows "mono (\<lambda>s. foldl f s l)"
A4: shows "weak_reduction_bisimulation Rel (STCal Source Target)"
strict_implies_order: fixes b :: "'c list"   and a :: "'c list" assumes "strict_prefix b a" shows "prefix b a"