hcju/isabelleps huggingface dataset:
gr_zeroI: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "n = zero \<Longrightarrow> False" shows "less zero n"
wa_ref_refl: fixes WA :: "'a while_algo" shows "wa_refine WA WA id"
intro\<^sub>1: 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> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<turnstile> \<Gamma> {c} \<Gamma>'" shows "\<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"
cross3_simps_8: fixes a :: "'c"   and b :: "'c" shows "a + b = b + a"
fps_deriv_nth: fixes f :: "'a fps"   and n :: "nat" shows "fps_nth (fps_deriv f) n = of_nat (n + 1) * fps_nth f (n + 1)"
B2_LLL_def: fixes f :: "int poly" shows "B2_LLL f = 2 ^ (2 * degree f) * \<parallel>f\<parallel>\<^sup>2"
ae: shows "AE x in M1. AE y in M2. P (x, y)"
domains_def: fixes f :: "'a" shows "H.domains f \<equiv> {a. H.ide a \<and> f \<star> a \<noteq> H.null}"
decseq_emb_PiE: fixes J :: "nat \<Rightarrow> 'd set"   and I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'e set" assumes "incseq J" shows "decseq (\<lambda>i. prod_emb I M (J i) (Pi\<^sub>E (J i) X))"
commutative_squares_2: shows "C.commutative_square \<nu>.leg0 \<pi>\<rho>.leg1 \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<^sub>1 \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0"
f_shrink_last_nth': fixes n :: "nat"   and xs :: "'d list"   and k :: "nat" assumes "n < length xs div k" shows "xs \<div>\<^sub>l k ! n = xs ! (Suc n * k - Suc 0)"
poly_normalized_nil: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b list"   and x :: "'b" assumes "class.semiring_0 plus zero times" shows "semiring_0.pnormalize zero p = [] \<longrightarrow> semiring_0.poly plus zero times p x = zero"
w_simps_3: shows "src THfgh_HHfgh.chine = src (tab\<^sub>0 h \<star> THfgh.p\<^sub>0)"
is_factorization_of_order: fixes p :: "'a poly"   and a :: "'a"   and fctrs :: "('a \<times> nat) list"   and c :: "'a"   and n :: "nat" assumes "p \<noteq> 0"   and "is_factorization_of (a, fctrs) p"   and "(c, n) \<in> set fctrs" shows "order c p = Suc n"
f: shows "f \<in> N \<rightarrow>\<^sub>M stream_space M"
Seed_def: fixes p :: "nat" shows "Seed\<^bsub>p\<^esub> \<equiv> graph (maxGon p)"
Min_gr_iff: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Min less_eq A) = (\<forall>a\<in>A. less x a)"
Res: fixes T :: "pi"   and U :: "pi"   and x :: "name" assumes "(T, U) \<in> Rel''" shows "(<\<nu>x>T, <\<nu>x>U) \<in> Rel''"
group_ow_axioms: shows "group_ow U (\<^bold>*\<^sub>o\<^sub>w) \<^bold>1\<^sub>o\<^sub>w inverse"
degree_monom_eq: fixes a :: "'c"   and n :: "nat" assumes "a \<noteq> (0::'c)" shows "degree (monom a n) = n"
rep_eq: fixes x :: "linear_poly" shows "fmlookup (linear_poly_map x) = (\<lambda>xa. if Rep_linear_poly x xa = 0 then None else Some (Rep_linear_poly x xa))"
BinOpThrow\<^sub>2\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp"   and e :: "nat exp"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and bop :: "bop" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>"
simps_4: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<^bold>\<lfloor>t \<^bold>\<cdot> u\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>"
BInv_def: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "BInv ic q oc \<equiv> List.last<$oc # $q> = $ic \<and> noreps<$oc # $q>"
approx_scaleR2: fixes a :: "'a star"   and b :: "'a star"   and c :: "real" assumes "a \<approx> b" shows "c *\<^sub>R a \<approx> c *\<^sub>R b"
TESL_interp_right_idem: fixes \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> (\<Phi>\<^sub>1 @ \<Phi>\<^sub>2) @ \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi>\<^sub>1 @ \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
multiplicative_function_mult: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f"   and "multiplicative_function g" shows "multiplicative_function (\<lambda>n. f n * g n)"
fds_nth_one_not_Suc_0: fixes n :: "nat" assumes "n \<noteq> Suc 0" shows "fds_nth 1 n = (0::'a)"
moebius_mu_formula: fixes n :: "nat" shows "moebius_mu n = dirichlet_inverse (\<lambda>_. 1::'b) (1::'b) n"
Sup_lim: fixes b :: "nat \<Rightarrow> 'a"   and s :: "'a set"   and a :: "'a" assumes "\<And>n. b n \<in> s"   and "b \<longlonglongrightarrow> a" shows "a \<le> \<Squnion> s"
gfp_exp_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "gfp_exp t = Sup_exp {P. unitary P \<and> P \<tturnstile> t P}"
increasing_outer_measure: fixes f :: "'a set \<Rightarrow> ennreal" shows "increasing (Pow \<Omega>) (outer_measure M f)"
isomorphic_implies_hpar_3: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong>\<^sub>B f'" shows "src f = src f'"
wf_PLUS: fixes n :: "nat"   and xs :: "'b rexp list" shows "local.wf n (PLUS xs) = (\<forall>r\<in>set xs. local.wf n r)"
assms_1: shows "prim_invar1 (A Q \<pi>)"
shift_Int: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I1 :: "nat set"   and I2 :: "nat set" shows "(local.shift cl n I1 \<inter> local.shift cl n I2 = {}) = (I1 \<inter> I2 = {})"
c6: fixes x :: "('a \<times> 'a set) set" shows "x \<cdot> 1\<^sub>\<pi> \<subseteq> 1\<^sub>\<pi>"
prefix_match_semantics_ipset_from_netmask: shows "prefix_to_wordset pfx = ipset_from_netmask (pfxm_prefix pfx) (~~ pfxm_mask pfx)"
iT_Div_card_ge_div: fixes I :: "nat set"   and d :: "nat" shows "card I div d \<le> card (I \<oslash> d)"
traps_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "traps s1 = traps s2"
obs_prefix_def: fixes a :: "nat \<Rightarrow> 'out option"   and b :: "nat \<Rightarrow> 'out option" shows "a \<lesssim> b \<equiv> \<forall>i. a i \<noteq> None \<longrightarrow> a i = b i"
summable_dcharacter_antimono: fixes x0 :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "\<chi> \<noteq> \<chi>\<^sub>0"   and "\<And>x. x0 \<le> x \<Longrightarrow> (f has_real_derivative f' x) (at x)"   and "\<And>x. x0 \<le> x \<Longrightarrow> 0 \<le> f x"   and "\<And>x. x0 \<le> x \<Longrightarrow> f' x \<le> 0"   and "(f \<longlongrightarrow> 0) at_top" shows "summable (\<lambda>n. \<chi> n * complex_of_real (f (real n)))"
A: shows "A \<equiv> (P' \<union> Pw' \<union> R' \<union> Rw') \<inter> Ms"
pB: fixes i :: "nat" assumes "i < n" shows "0 \<le> B $$ (i, i)"
assms_8: shows "Q_J_neg = filterQuery J (Qn - Q_I_neg)"
minim_in: fixes r :: "('c \<times> 'c) set"   and B :: "'c set" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "B \<noteq> {}" shows "minim r B \<in> B"
Seq_def: fixes f :: "unit \<Rightarrow> 'b seq" shows "Seq f = pred_of_seq (f ())"
take_norm_stack: fixes n :: "nat"   and xs :: "('dyn, 'ubx1, 'ubx2) unboxed list" shows "take n (norm_stack xs) = norm_stack (take n xs)"
BinOpRed2: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v\<^sub>1 :: "val"   and bop :: "bop" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> e,s\<rangle> \<rightarrow> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> e',s'\<rangle>"
simps: fixes s :: "'b + 'c"   and a :: "'c" shows "setrp s a = (\<exists>x. a = x \<and> s = Inr x)"
intros_3: fixes uv :: "Syntax.var"   and uw :: "Syntax.term" shows "value (Lam uv uw)"
assms_1: shows "\<forall>n'''>n'. n''' \<le> n'' \<longrightarrow> Option.is_none (devBC t n nid n''')"
cfun_eqI: fixes f :: "'d \<rightarrow> 'e"   and g :: "'d \<rightarrow> 'e" assumes "\<And>x. f\<cdot>x = g\<cdot>x" shows "f = g"
s_def: fixes n :: "nat" shows "s n = t n + \<tau> (flow0 x (t n))"
safe_step_preserves_simple_ownership_distinct: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "computation.concurrent_step direct_memop_step empty_storebuffer_step program_step (\<lambda>p p' is sb. sb) (ts, m, \<S>) (ts', m', \<S>')"   and "safe_delayed (ts, m, \<S>)"   and "simple_ownership_distinct ts" shows "simple_ownership_distinct ts'"
ring_hom_a_inv: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and b :: "'c" assumes "ring R"   and "ring S"   and "h \<in> ring_hom R S"   and "b \<in> carrier R" shows "h (\<ominus>\<^bsub>R\<^esub> b) = \<ominus>\<^bsub>S\<^esub> h b"
monom_mult_monomial: fixes c :: "'d"   and s :: "'c"   and v :: "'c"   and d :: "'d" shows "punit.monom_mult c s (monomial d v) = monomial (c * d) (punit.splus s v)"
lst_of_sp_is_vert: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "SubRel.subpath g v1 es v2 subs" shows "v2 \<in> Graph.vertices g"
dist_at1: fixes vs :: "'a list"   and a :: "'a list"   and r :: "'a"   and b :: "'a list"   and c :: "'a list"   and d :: "'a list" assumes "distinct vs"   and "vs = a @ r # b"   and "vs = c @ r # d" shows "a = c"
K_block_design_axioms: shows "K_block_design \<V> \<B> {\<k>}"
mult_left_mono_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq b a"   and "less_eq c zero" shows "less_eq (times c a) (times c b)"
inverse_arrow_unique: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "category C"   and "category.inverse_arrows C f g"   and "category.inverse_arrows C f g'" shows "g = g'"
vector_component_1: fixes x :: "'a"   and i :: "'b" shows "vec x $ i = x"
insert_def: fixes a :: "'b"   and B :: "'b set" shows "Set.insert a B = {x. x = a} \<union> B"
assms: shows "take (max_Var_floatariths ea) ys = take (max_Var_floatariths ea) zs"
costsMoney_aux: fixes j :: "state stream" assumes "\<exists>p r i. j = nxt (make_full_observation drinks (Some 1) r p) i" shows "alw (\<lambda>xs. nxt (state_eq (Some 2)) xs \<longrightarrow> check_exp (Ge (V (Rg 2)) (L (Num 100))) xs) j"
\<G>_finite: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl" assumes "\<G> \<subseteq> \<^bold>G \<phi>" shows "finite \<G>"
simps_1: fixes vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and uu :: "'t" shows "sym_preproc_addnew [] vs fs uu = (vs, fs)"
opt_def: fixes A :: "'a set" shows "opt A = insert None {\<lfloor>y\<rfloor> |y. y \<in> A}"
hermitean_congruence: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" assumes "hermitean H" shows "hermitean (congruence M H)"
trm_simp_7: fixes P :: "'b trm"   and Q :: "'b trm" assumes "Fst P = Fst Q" shows "P = Q"
fix_ind2: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b \<rightarrow> 'b" assumes "adm P"   and "P \<bottom>"   and "P (F\<cdot>\<bottom>)"   and "\<And>x. \<lbrakk>P x; P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (F\<cdot>(F\<cdot>x))" shows "P (fix\<cdot>F)"
special_soundness_def: shows "Schnorr_\<Sigma>.special_soundness \<equiv> \<forall>h e e' a z z'. h \<in> valid_pub \<longrightarrow> e \<in> challenge_space \<longrightarrow> e' \<in> challenge_space \<longrightarrow> e \<noteq> e' \<longrightarrow> check h a e z \<longrightarrow> check h a e' z' \<longrightarrow> lossless_spmf (ss_adversary h (a, e, z) (a, e', z')) \<and> (\<forall>w'\<in>set_spmf (ss_adversary h (a, e, z) (a, e', z')). (h, w') \<in> R_DL)"
zero_Nat_def: shows "0 \<equiv> box\<cdot>(0::'a)"
in_wp_iff: fixes a :: "'b"   and act :: "('b \<times> 'b) set"   and B :: "'b set" shows "(a \<in> wp act B) = (\<forall>x. (a, x) \<in> act \<longrightarrow> x \<in> B)"
rep_eq: fixes x :: "'a permutation"   and xa :: "'a permutation" shows "(\<rightarrow>) (x - xa) = (\<rightarrow>) x \<circ> the_inv ((\<rightarrow>) xa)"
scale_right_imp_eq: fixes x :: "'c"   and a :: "real"   and b :: "real" assumes "x \<noteq> (0::'c)"   and "a *\<^sub>R x = b *\<^sub>R x" shows "a = b"
hext_refl: fixes h :: "'heap" shows "h \<unlhd> h"
assms_3: shows "(i, Create i e) \<in> set (node_deliver_messages xs)"
assms_4: shows "perp_foot b l = perp_foot c l"
simps_1: fixes e :: "'a"   and a :: "'b" shows "FingerTreeStruc.n_unwrap (Tip e a) = (e, a)"
oconf_init_fields: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P \<turnstile> C has_fields FDTs" shows "P,h \<turnstile> (C, init_fields FDTs) \<surd>"
u_shared: shows "\<forall>a. a \<notin> u_owns \<longrightarrow> a \<notin> owned (ts ! i) \<longrightarrow> u_shared a = \<S> a"
orderE: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" obtains "a = a \<sqinter> b"
R_bound': fixes x :: "real" assumes "0 \<le> x" shows "norm (R x) \<le> 1 * norm (sqrt x)"
objT_def: shows "objT \<equiv> map_of (objects P) ++ constT"
redundant_code_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g" shows "uninst_code.ssa.redundant_code p g = (uninst_code.ssa.trivial_phis p g \<noteq> {})"
chain_ncritE: fixes data :: "nat \<times> 'e"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and q_in_bs :: "bool"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" assumes "chain_ncrit data gs bs hs q_in_bs ps p q"   and "snd ` set ps \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"   and "unique_idx (gs @ bs @ hs) data"   and "p \<in> set hs"   and "q \<in> set gs \<union> set bs \<union> set hs" obtains r :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" where "r \<in> set gs \<union> set bs \<union> set hs" and "fst r \<noteq> 0" and "r \<noteq> p" and "r \<noteq> q" and "lt (fst r) adds\<^sub>t term_of_pair (lcs (lp (fst p)) (lp (fst q)), component_of_term (lt (fst p)))" and "(p, r) \<in>\<^sub>p snd ` set ps" and "r \<in> set gs \<union> set bs \<and> q_in_bs \<or> (q, r) \<in>\<^sub>p snd ` set ps"
discI_2: fixes status :: "status" assumes "status = SUCCESS" shows "is_success status"
stable_pair_on_Cd_match: 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 ds :: "'e set"   and XD_XH :: "'d set \<times> 'd set"   and d :: "'e" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_pair_on Pd Ch ds XD_XH"   and "d \<in> ds" shows "Contracts.Cd Pd d (fst XD_XH \<inter> snd XD_XH) = Contracts.Cd Pd d (fst XD_XH)"
set_mset_poly_roots: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "set_mset (poly_roots p) = {x. poly p x = (0::'b)}"
contradict_dnf_def: fixes ass :: "atom list list" shows "contradict_dnf ass = (\<exists>css. list_all2 contradict ass css)"
bfs_impl_fr_rule: shows "(uncurry2 (bfs_impl (\<lambda>(am, cf). succ_imp N am cf)), uncurry2 (PR_CONST bfs.op_bfs)) \<in> (is_am \<times>\<^sub>a asmtx_assn N int_assn)\<^sup>k *\<^sub>a nat_assn\<^sup>k *\<^sub>a nat_assn\<^sup>d \<rightarrow>\<^sub>a option_assn (list_assn (nat_assn \<times>\<^sub>a nat_assn))"
P_k_def: shows "P_k \<equiv> SOME P_k. P_k \<in> paths_with_new \<and> hd (tl P_k) \<notin> Q.second_vertices"
point_mult_add: fixes a :: "int"   and b :: "int"   and p :: "int point"   and m :: "nat"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p"   and "nonsingular a b" shows "point_mult a (m + n) p = local.add a (point_mult a m p) (point_mult a n p)"
properfactor_mult_r: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G (a \<otimes> c) (b \<otimes> c) = properfactor G a b"
select_convs_2: fixes carrier :: "'c set"   and pop :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mop :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and more :: "'d" shows "mop \<lparr>carrier = carrier, pop = pop, mop = mop, zero = zero, \<dots> = more\<rparr> = mop"
size_2: fixes x21 :: "fmla"   and x22 :: "fmla" shows "size (x21 OR x22) = size x21 + size x22 + Suc 0"
chainI: fixes Y :: "nat \<Rightarrow> 'b" assumes "\<And>i. Y i \<sqsubseteq> Y (Suc i)" shows "chain Y"
FactGroup_hom: shows "(\<lambda>X. the_elem (h ` X)) \<in> hom (G Mod kernel G H h) H"
ffold_ord_cons: fixes h :: "'b"   and t :: "'b list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c" assumes "sorted (h # t)"   and "distinct (h # t)" shows "ffold_ord f (fset_of_list (h # t)) b = ffold_ord f (fset_of_list t) (f h b)"
P_func1: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "\<P> (f \<circ> g) = \<P> f \<circ> \<P> g"
BC: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q = resChain [] (P \<parallel> Q)"
d: shows "dom h\<^sub>0 = dom h\<^sub>0'"
subgraph_preserves_hom: fixes A :: "('a, 'b) labeled_graph"   and B :: "('a, 'b) labeled_graph"   and X :: "('a, 'c) labeled_graph"   and h :: "('c \<times> 'b) set" assumes "subgraph A B"   and "graph_homomorphism X A h" shows "graph_homomorphism X B h"
simps_2: fixes l :: "'a tree"   and x :: "'a"   and r :: "'a tree" shows "inorder \<langle>l, x, r\<rangle> = inorder l @ [x] @ inorder r"
summable_ereal: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>" shows "summable f"
inject_5: fixes x51 :: "'b gexp"   and x52 :: "'b gexp"   and y51 :: "'b gexp"   and y52 :: "'b gexp" shows "(Nor x51 x52 = Nor y51 y52) = (x51 = y51 \<and> x52 = y52)"
"10_2": shows "set s\<^sub>3 \<subseteq> invisible"
rank_match: shows "vec_space.rank (dim_row (take_rows_from_matrix (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))))) (take_rows_from_matrix (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))))) = dim_row (take_rows_from_matrix (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))))"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "\<GG>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<GG>\<lparr>ArrMap\<rparr>)"
foldl_conv_foldr: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'b list" shows "foldl f a xs = foldr (\<lambda>x y. f y x) (rev xs) a"
inf_assoc: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" shows "x \<inter># y \<inter># z = x \<inter># (y \<inter># z)"
Abs_uint16_inverse: fixes y :: "16 word" assumes "y \<in> UNIV" shows "Rep_uint16 (Abs_uint16 y) = y"
is_UP_monom_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "is_UP_monom p"   and "is_UP_monom q" shows "is_UP_monom (p \<otimes>\<^bsub>P\<^esub> q)"
valid_def: fixes P :: "assn"   and c :: "stmt"   and Q :: "assn" shows "\<Turnstile> {P} c {Q} \<equiv> \<forall>s t. P s \<longrightarrow> (\<exists>n. s -c-n\<rightarrow> t) \<longrightarrow> Q t"
odiff_le_iff: fixes z :: "V"   and x :: "V"   and y :: "V" assumes "z \<unlhd> x" shows "(odiff x z \<le> y) = (x \<le> z + y)"
step_5: shows "\<exists>DD\<subseteq>CC. finite DD \<and> D \<in> saturate DD"
RS3subset: fixes p :: "((int \<times> int) set set, int) Combinators list" shows "set (rm_MT_rules C p) \<subseteq> set p"
sprefI2: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>p v"   and "\<^bold>|u\<^bold>| < \<^bold>|v\<^bold>|" shows "u <p v"
sum_SucD: fixes f :: "'c \<Rightarrow> nat"   and A :: "'c set"   and n :: "nat" assumes "sum f A = Suc n" shows "\<exists>a\<in>A. 0 < f a"
MGFNormalI: fixes G :: "prog"   and A :: "state triple set"   and t :: "Term.term" assumes "G,A\<turnstile>{Normal \<doteq>} t\<succ> {G\<rightarrow>}" shows "G,A\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}"
sdrop_add: fixes n :: "nat"   and m :: "nat"   and s :: "'d stream" shows "sdrop n (sdrop m s) = sdrop (m + n) s"
distinct_entries: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "distinct (map fst (RBT_Impl.entries t))"
dmdInit_stp: fixes F :: "state \<Rightarrow> bool" shows "\<diamond>F \<equiv> \<diamond>Init F"