hcju/isabelleps huggingface dataset:
monoE: fixes g :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr"   and f' :: "('a S.E.term, 'a) S.arr" assumes "S.mono g"   and "S.seq g f"   and "S.seq g f'"   and "g \<cdot>\<^sub>S f = g \<cdot>\<^sub>S f'" shows "f' = f"
option_rel_id_simp: shows "\<langle>Id\<rangle>option_rel = Id"
commute: fixes a :: "'a DBMEntry"   and b :: "'a DBMEntry" shows "linordered_monoid.max a b = linordered_monoid.max b a"
wf_word_map_project: fixes n :: "nat"   and ws :: "'a list" assumes "wf_word (Suc n) ws" shows "wf_word n (map project ws)"
mplus_fail: fixes m :: "'a match" shows "Fixrec.fail +++ m = m"
name_list_supp: fixes xvec :: "name list" shows "set xvec = supp xvec"
source_determination: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and \<rho> :: "'f"   and v :: "'e" assumes "vwb_lens x"   and "get\<^bsub>x\<^esub> \<sigma> = get\<^bsub>x\<^esub> \<rho>"   and "put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v" shows "\<sigma> = \<rho>"
ParallelEmpty: fixes x :: "('b com option list \<times> 'b) list"   and s :: "'b"   and i :: "nat" assumes "x \<in> par_cp (ParallelCom []) s"   and "Suc i < length x" shows "(x ! i, x ! Suc i) \<notin> par_ctran"
induct: fixes x :: "'a"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> iterates_above a"   and "P a"   and "\<And>x. \<lbrakk>x \<in> iterates_above a; P x\<rbrakk> \<Longrightarrow> P (f x)"   and "\<And>M. \<lbrakk>M \<in> Chains leq; M \<noteq> {}; \<And>x. x \<in> M \<Longrightarrow> x \<in> iterates_above a; \<And>x. x \<in> M \<Longrightarrow> P x\<rbrakk> \<Longrightarrow> P (lub M)" shows "P x"
br_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
irred_0: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set" shows "\<not> punit.is_red F 0"
special_sg_e: fixes G :: "('e, 'f) Group_scheme" assumes "Group G" shows "G \<guillemotright> {\<one>\<^bsub>G\<^esub>} "
l16: fixes x :: "'a"   and y :: "'a" shows "- - x \<le> - (- x \<sqinter> y)"
is_: shows "snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) = rev (take i cs) @ [x] @ rev bs @ [x] @ rev as @ h"
of_bl_max: fixes xs :: "bool list" shows "of_bl xs \<le> mask (length xs)"
sums: shows "(\<lambda>i. Measure.measure M (mkdisjoint A i)) sums Measure.measure M (\<Union> (range (mkdisjoint A)))"
psi_eq_nth_prefix: fixes t :: "nat"   and j :: "nat"   and b :: "nat"   and x :: "nat" assumes "local.prefixes t j \<down>= b"   and "x < e_length b" shows "\<psi> j x \<down>= e_nth b x"
lm078: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'b set" assumes "finite X" shows "sum f X = sum f (X - Y) + sum f (X \<inter> Y)"
prefix_le_fininf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "w\<^sub>1 \<le> w\<^sub>2"   and "traces.le_fininf ind w\<^sub>2 w\<^sub>3" shows "traces.le_fininf ind w\<^sub>1 w\<^sub>3"
represents_plus: fixes A :: "'b set"   and r :: "'b \<Rightarrow>\<^sub>0 'a"   and x :: "'b"   and B :: "'b set"   and s :: "'b \<Rightarrow>\<^sub>0 'a"   and y :: "'b" assumes "represents A r x"   and "represents B s y" shows "represents (A \<union> B) (r + s) (x + y)"
distinct_5: fixes lam1 :: "lam"   and lam2 :: "lam"   and name' :: "name"   and lam' :: "lam" shows "App lam1 lam2 \<noteq> Lam [name'].lam'"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "I.satL \<xi> (Pos a) = I.satA \<xi> a"
simps: fixes A :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)" shows "On_Off.embed A = (\<lambda>s. return_pmf (fst A s), \<lambda>s r. return_pmf (snd A s r))"
least_simps_1: fixes x :: "'c option" shows "least None x = x"
n_add_ub2: fixes y :: "'a"   and x :: "'a" shows "n y \<sqsubseteq> n x \<oplus> n y"
map_sel'_axioms: shows "map_sel' ws_\<alpha> ws_invar ws_sel"
mpt_factor_projection: fixes f :: "'a \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "sigma_finite_measure (distr M borel (\<lambda>x n. f ((T ^^ n) x)))" shows "mpt_factor (\<lambda>x n. f ((T ^^ n) x)) (distr M borel (\<lambda>x n. f ((T ^^ n) x))) nat_left_shift"
is_aligned_shift: fixes k :: "'b word"   and m :: "nat" shows "is_aligned (k << m) m"
add_le_less_mono: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset"   and d :: "'a multiset" assumes "a \<subseteq># b"   and "c \<subset># d" shows "a + c \<subset># b + d"
15: shows "j < length (A1 \<circ> A2) * length (B1 \<circ> B2)"
order_unique_lemma: fixes a :: "'a"   and n :: "nat"   and p :: "'a poly" assumes "[:- a, 1::'a:] ^ n dvd p"   and "\<not> [:- a, 1::'a:] ^ Suc n dvd p" shows "Polynomial.order a p = n"
UNIV_I: fixes x :: "'b" shows "x \<in> UNIV"
preflect_solution: fixes t0 :: "real"   and T :: "real set"   and x :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" assumes "t0 \<in> T"   and "((\<lambda>t. x (preflect t0 t)) solves_ode (\<lambda>t x. - f (preflect t0 t) x)) (preflect t0 ` T) X" shows "(x solves_ode f) T X"
p: fixes s :: "'a" assumes "guard s" shows "p \<le> spmf (map_spmf guard (body s)) False"
assms: shows "finite (supp_rel (=\<^sub>\<alpha>) t)"
monic_degree_0: fixes p :: "'c poly" assumes "monic p" shows "(degree p = 0) = (p = 1)"
VLambda_vid_on: fixes A :: "V" shows "(\<lambda>a\<in>\<^sub>\<circ>A. a) = vid_on A"
pr_ideal_axioms_def: fixes R :: "'c set"   and I :: "'c set"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "pr_ideal_axioms R I multiplication \<equiv> I \<noteq> R \<and> (\<forall>x y. x \<in> R \<longrightarrow> y \<in> R \<longrightarrow> multiplication x y \<in> I \<longrightarrow> x \<in> I \<or> y \<in> I)"
Gpr: shows "\<guillemotleft> G \<guillemotright> \<tturnstile> wp b \<guillemotleft> G \<guillemotright>"
exp_conj_1_right: fixes A :: "'a \<Rightarrow> real" assumes "nneg A" shows "A && (\<lambda>_. 1) = A"
subgroup_is_comm_group: fixes H :: "'a set" assumes "subgroup H G" shows "comm_group (G\<lparr>carrier := H\<rparr>)"
eWlsAbs_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and us :: "'varSort"   and s :: "'sort" shows "eWlsAbs MOD (us, s) ERR = False"
Probabilistic_Timed_Automaton_axioms: shows "Probabilistic_Timed_Automaton A"
simps_3: fixes v :: "'b" shows "l_sorted [v] = True"
prob_space_exponential: fixes l :: "real" assumes "0 < l" shows "prob_space (exponential l)"
exp_cf5_pos: fixes x :: "real" assumes "0 < numer_cf5 x"   and "0 < numer_cf5 (- x)" shows "0 < exp_cf5 x"
GreatestI_ex: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (Greatest P)"
conjunctive_def: fixes inf_b :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf_c :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and times_abc :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "conjunctive.conjunctive inf_b inf_c times_abc = {x. \<forall>y z. times_abc x (inf_b y z) = inf_c (times_abc x y) (times_abc x z)}"
inf_top_def: fixes x :: "'a top"   and y :: "'a top" shows "inf x y = (case x of top.Value v \<Rightarrow> case y of top.Value v' \<Rightarrow> top.Value (inf v v') | Top \<Rightarrow> x | Top \<Rightarrow> y)"
R1a1_def: shows "R1a1 \<equiv> {(s, t). s = med1a1 t}"
simps_2: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
span_is_subspace: fixes S :: "rat vec set" assumes "S \<subseteq> Rn" shows "VectorSpace.subspace class_ring (gs.span S) gs.V"
elementLevelAppend: fixes a :: "'b"   and M :: "('b \<times> bool) list"   and M' :: "('b \<times> bool) list" assumes "a \<in> set (elements M)" shows "elementLevel a M = elementLevel a (M @ M')"
assm: shows "left (ext v) \<le> right (space ts v c)"
cylinder_def: fixes S :: "'a set"   and A :: "'a set"   and B :: "'a set" shows "cylinder S A B = {T \<in> Pow S. A \<subseteq> T \<and> B \<inter> T = {}}"
elims: fixes x :: "'trm"   and xa :: "('trm \<times> 'var) list"   and y :: "'trm" assumes "rawpsubstT x xa = y" obtains t :: "'trm" where "x = ta__" and "xa = []" and "y = ta__"   | t :: "'trm" and t1 :: "'trm" and x1 :: "'var" and txs :: "('trm \<times> 'var) list" where "x = t" and "xa = (t1, x1) # txs" and "y = rawpsubstT (substT t t1 x1) txs"
assms_3: shows "Esigma m = Esigma n \<and> Esigma m = m + n"
indets_def: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "indets p = \<Union> (keys ` keys p)"
subprob_measurableD_1: fixes N :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and S :: "'c measure"   and x :: "'b" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"   and "x \<in> space M" shows "space (N x) = space S"
semAssignSC: fixes x :: "'id"   and e :: "'exp"   and \<iota> :: "nat" assumes "AssignSideCondition x e \<iota>" shows "e \<equiv>\<^bsub>DA x,htchLoc \<iota>\<^esub> e \<and> (\<forall>m m' d \<iota>'. m \<sim>\<^bsub>d,htchLoc \<iota>'\<^esub> m' \<and> \<lbrakk>x:=\<^bsub>\<iota>\<^esub> e\<rbrakk>(m) =\<^bsub>d\<^esub> \<lbrakk>x:=\<^bsub>\<iota>\<^esub> e\<rbrakk>(m') \<longrightarrow> \<lbrakk>x:=\<^bsub>\<iota>\<^esub> e\<rbrakk>(m) \<sim>\<^bsub>d,htchLoc \<iota>'\<^esub> \<lbrakk>x:=\<^bsub>\<iota>\<^esub> e\<rbrakk>(m'))"
unlabel_snoc_inv: fixes A :: "('b strand_label \<times> 'a) list"   and B :: "'a list"   and b :: "'a" assumes "unlabel A = B @ [b]" shows "\<exists>A'. (\<exists>n. A = A' @ [(ln n, b)]) \<or> A = A' @ [(\<star>, b)]"
beta_star_preserves_typ_of1: fixes r :: "Core.term"   and s :: "Core.term"   and Ts :: "typ list"   and T :: "typ" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* s"   and "typ_of1 Ts r = Some T" shows "typ_of1 Ts s = Some T"
simps: fixes A_ll :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat" shows "local.mute A_ll k l (r, A) = (let p = A $$ (k, l) in if p = (0::'a) then (r, A) else case mf A_ll p of (q', p', g) \<Rightarrow> (r * q', addrow (- p') k l (multrow k q' A)))"
diagonal_to_Smith_PQ_JNF_internalized: fixes A :: "'f mat"   and S :: "'f mat"   and P :: "'f mat"   and Q :: "'f mat"   and bezout :: "'f \<Rightarrow> 'f \<Rightarrow> 'f \<times> 'f \<times> 'f \<times> 'f \<times> 'f" assumes "class.nontriv TYPE('d)"   and "class.nontriv TYPE('e)"   and "A \<in> carrier_mat CARD('e) CARD('d)"   and "S \<in> carrier_mat CARD('e) CARD('d)"   and "P \<in> carrier_mat CARD('e) CARD('e)"   and "Q \<in> carrier_mat CARD('d) CARD('d)"   and "diagonal_to_Smith_PQ_JNF A bezout = (P, S, Q)"   and "isDiagonal_mat A"   and "is_bezout_ext bezout" shows "S = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S"
eq: shows "\<Union> (\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int i}) = \<Union> \<F>"
inner_mult_left: fixes m :: "real"   and a :: "'c"   and b :: "'c" shows "inner (of_real m * a) b = m * inner a b"
rel_upd2: fixes a :: "'b"   and b :: "'b"   and r :: "'b \<Rightarrow> 'b ref"   and q :: "'b"   and t :: "'b ref" assumes "(a, b) \<notin> rel r"   and "(a, b) \<in> rel (r(q := t))" shows "a = q"
cons2: shows "evaluate_dec ck_ mn_ env_ s1_ d_ (s2_, Rval new_env_) \<and> evaluate_decs ck_ mn_ (extend_dec_env new_env_ env_) s2_ ds_ (s3_, r_) \<and> (\<forall>s' ra extra. (s3_, r_) = (s', ra) \<and> ck_ = True \<and> ra \<noteq> Rerr (Rabort Rtimeout_error) \<longrightarrow> evaluate_decs True mn_ (extend_dec_env new_env_ env_) (update_clock (\<lambda>_. clock s2_ + extra) s2_) ds_ (update_clock (\<lambda>_. clock s' + extra) s', ra))"
assms_4: shows "v \<in> nodes g\<^sub>2"
intPF_def: shows "M.M.intPF \<equiv> local.M.intPF"
assms_1: shows "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"
rev_hbexI: fixes x :: "hf"   and A :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "x \<^bold>\<in> A"   and "P x" shows "HBex A P"
divide_powr_uminus: fixes a :: "real"   and b :: "real"   and c :: "real" shows "a / b powr c = a * b powr - c"
permutes_comp_preserves_cyclic1: fixes g :: "'b \<Rightarrow> 'b"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and A :: "'b set" assumes "g permutes B"   and "cyclic_on f C"   and "A \<inter> B = {}"   and "C \<subseteq> A" shows "cyclic_on (f \<circ> g) C"
atLeastAtMost_iff: fixes i :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and u :: "'c" shows "(i \<in> ord.atLeastAtMost less_eq l u) = (less_eq l i \<and> less_eq i u)"
subst_false: fixes \<sigma> :: "'a \<Rightarrow> 'b" shows "\<sigma> \<dagger> false = false"
assms: shows "\<forall>f\<in>set fs. is_literal_formula f"
disc_1: fixes x1 :: "'b" shows "is_pos (Pos x1)"
fres_one: fixes x :: "'b \<times> 'c" shows "less_eq_prod one_prod (prod.fres x x)"
inf_bot_left: fixes x :: "'b" shows "\<bottom> \<sqinter> x = \<bottom>"
SbisL_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and cl :: "('d, 'c, 'a) cmd list"   and dl :: "('d, 'c, 'a) cmd list" assumes "PL_Indis cval indis" shows "PL_Indis.SbisL aval tval cval indis cl dl \<equiv> length cl = length dl \<and> (\<forall>n<length cl. (cl ! n, dl ! n) \<in> PL_Indis.Sbis aval tval cval indis)"
s1_correct: fixes rt :: "'b routing_rule list"   and p :: "('b, 'c) simple_packet_scheme" assumes "valid_prefixes rt"   and "has_default_route rt" shows "\<exists>rm ra. generalized_sfw (lr_of_tran_s1 rt) p = Some (rm, ra) \<and> ra = output_iface (routing_table_semantics rt (p_dst p))"
sigma_finite_iff_density_finite': fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (density M h)" shows "sigma_finite_measure (density (density M h) f) = (AE x in density M h. f x \<noteq> \<infinity>)"
clock_simVal_def2: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and t :: "'e Trace" assumes "Environment jkbp" shows "Environment.clock_simVal envVal (Environment.clock_sim jkbp envInit envAction envTrans envVal envObs t) = envVal (tLast t)"
safe\<^sub>0_def: fixes s :: "event list"   and r :: "room" shows "safe\<^sub>0 s r = (\<exists>s\<^sub>1 s\<^sub>2 s\<^sub>3 g c. s = s\<^sub>3 @ [Enter g r c] @ s\<^sub>2 @ [Check_in g r c] @ s\<^sub>1 \<and> no_Check_in (s\<^sub>3 @ s\<^sub>2) r)"
fun_dual2: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f = \<partial> \<circ> g) = (\<partial> \<circ> f = g)"
assms: shows "toProcess sidx p = (ss, a, n, l, idx, start, r)"
inj_vrat_of_rat: shows "inj vrat_of_rat"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C FG.map FG.cmp"
sdrop_simps_1: fixes n :: "nat"   and s :: "'d stream" shows "shd (sdrop n s) = s !! n"
prems_1: shows "add_independentS ((A # As) @ Bs)"
sup_def: fixes P :: "'b prog"   and mxs :: "nat"   and mxl :: "nat" shows "JVM_SemiType.sup P mxs mxl \<equiv> snd (snd (JVM_SemiType.sl P mxs mxl))"
deriv_cmult_at: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and c :: "complex" assumes "f analytic_on {z}" shows "deriv (\<lambda>w. c * f w) z = c * deriv f z"
mismatch_lexord: fixes u :: "'b list"   and v :: "'b list"   and r :: "('b \<times> 'b) set" assumes "\<not> u \<bowtie> v"   and "mismatch_pair u v \<in> r" shows "(u, v) \<in> lexord r"
gen_g_impl_rel_ext_def: fixes Rm :: "('l \<times> 'n) set"   and Rv :: "('i \<times> 'm set) set"   and Re :: "('j \<times> ('m \<times> 'm) set) set"   and Rv0 :: "('k \<times> 'm set) set" shows "\<langle>Rm, Rv, Re, Rv0\<rangle>gen_g_impl_rel_ext \<equiv> {(\<lparr>gi_V = Vi, gi_E = Ei, gi_V0 = V0i, \<dots> = mi\<rparr>, \<lparr>g_V = V, g_E = E, g_V0 = V0, \<dots> = m\<rparr>) |Vi Ei V0i mi V E V0 m. (Vi, V) \<in> Rv \<and> (Ei, E) \<in> Re \<and> (V0i, V0) \<in> Rv0 \<and> (mi, m) \<in> Rm}"
alpha_Tree_eqvt': fixes t1 :: "('a, 'b, 'c) Tree"   and t2 :: "('a, 'b, 'c) Tree"   and p :: "perm" shows "(t1 =\<^sub>\<alpha> t2) = (p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2)"
pred_list_it_correct: fixes g :: "('a, 'c, 'b) gen_cfg_wf"   and v :: "'a" assumes "True" shows "set_iterator (foldri (gen_wf_inEdges' g v)) (pred (gen_wf.\<alpha> g) v)"
bit_yxyx: fixes x :: "'b"   and y :: "'b"   and init :: "'b list"   and qs :: "'b list"   and r :: "'b list" assumes "x \<noteq> y"   and "init \<in> {[x, y], [y, x]}"   and "qs \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x))])" shows "T\<^sub>p_on_rand' BIT (type0 init x y) (qs @ r) = 75 / 10\<^sup>2 * real (length qs) + T\<^sub>p_on_rand' BIT (type1 init x y) r \<and> Partial_Cost_Model.config'_rand BIT (type0 init x y) qs = type1 init x y"
assms_2: shows "abs_conv_abscissa f < ereal (s \<bullet> (1::'a))"
abs_eq: fixes xa :: "'d"   and x :: "'d" shows "dual xa - dual x = dual (- (x - xa))"
that: fixes \<sigma>' :: "heap option"   and r :: "'a" assumes "run c \<sigma> \<sigma>' r" shows "thesis"
sorted_wrt_Nil: fixes R :: "('b \<times> 'b) set" shows "Linorder_Relations.sorted_wrt R []"
Step_axiom: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and a :: "axiom" shows "step_result (SG, C) (i, Axiom a) = (closeI SG i, C)"
simps_2: fixes disc :: "'c \<Rightarrow> bool"   and neg :: "bool"   and a :: "'c" shows "has_disc_negated disc neg (Match a) = (if disc a then neg else False)"
not_sum_squares_lt_zero: fixes 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 abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "\<not> less (plus (times x x) (times y y)) zero"
powr_one_eq_one: fixes a :: "'c" shows "(1::'c) powr a = (1::'c)"
elims: fixes x :: "key"   and y :: "key" assumes "invKey x = y" obtains A :: "agent" where "x = pubK A__" and "y = priK A__"   | A :: "agent" where "x = priK A" and "y = pubK A"   | v :: "fresh_t" where "x = sesK v" and "y = sesK v"   | va :: "agent" where "x = shrK va" and "y = shrK va"
j_def: shows "j = (LEAST j'. \<b> ps j' \<le> poly_deg h)"
intro: fixes G :: "('d, 'c) pre_digraph" assumes "\<And>e. e \<in> arcs G \<Longrightarrow> tail G e \<in> verts G"   and "\<And>e. e \<in> arcs G \<Longrightarrow> head G e \<in> verts G" shows "wf_digraph G"
exhaust: fixes y :: "\<langle>'a\<rangle>\<^sub>\<bottom>" obtains "y = \<bottom>"   | x2 :: "'a" where "y = \<lfloor>x2\<rfloor>"
abs_eq: fixes xa :: "'a \<Rightarrow> 'a"   and x :: "'a \<Rightarrow> 'a" assumes "eq_onp bij xa xa"   and "eq_onp bij x x" shows "Abs_permutation xa - Abs_permutation x = Abs_permutation (xa \<circ> the_inv x)"
circ_left_unfold: fixes x :: "'c" shows "Rf x (x\<^sup>\<star>) = x\<^sup>\<star>"
assms_1: shows "\<NN> : \<HH> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>' : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
psubst_def: shows "psubst \<equiv> \<lambda>\<theta>. trm_rec Unit (lookup \<theta>) (\<lambda>x t. Lam x) (\<lambda>t\<^sub>1 t\<^sub>2. App) Const"
Cauchy_fst: fixes X :: "nat \<Rightarrow> 'd \<times> 'e" assumes "Cauchy X" shows "Cauchy (\<lambda>n. fst (X n))"
cond_action_oks_append: fixes final :: "'c \<Rightarrow> bool"   and s :: "('d, 'e) locks \<times> (('e \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option) \<times> 'f) \<times> ('e \<Rightarrow> 'g wait_set_status option) \<times> 'e set"   and t :: "'e"   and cts :: "'e conditional_action list"   and cts' :: "'e conditional_action list" shows "final_thread.cond_action_oks final s t (cts @ cts') = (final_thread.cond_action_oks final s t cts \<and> final_thread.cond_action_oks final s t cts')"
\<Phi>o_preserves_ide: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and a :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "partial_magma.ide S a" shows "partial_magma.ide S' (two_set_categories_bij_betw_Univ.\<Phi>o S S' \<AA> \<phi> a)"
le_LCons: fixes x :: "'b"   and xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" shows "(x ## xs \<le> y ## ys) = (x = y \<and> xs \<le> ys)"
scaleR: fixes r :: "real"   and b :: "'b" shows "f' y x (r *\<^sub>R b) = r *\<^sub>R f' y x b"
hyps_2: fixes t :: "nat" assumes "valid_sops_expr t e\<^sub>2" shows "valid_sop (eval_expr t e\<^sub>2)"
prv_scnj1_imp: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "prv (imp (scnj {\<phi>}) \<phi>)"
e_lam_intro3: fixes x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list" shows "VFun {||} \<in> E (ELam x e) \<rho>"
has_domain_iff_has_codomain: fixes f :: "'b" shows "(\<phi>.B.domains f \<noteq> {}) = (\<phi>.B.codomains f \<noteq> {})"
trans: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a dvd b"   and "b dvd c" shows "a dvd c"
ssubst: fixes t :: "'b"   and s :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "t = s"   and "P s" shows "P t"
assms_2: shows "exp l = fds_nth f (Suc 0)"
indRelRTPO_impl_TRel_weakly_preserves_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelRTPO TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs TRel TWB"
comp_fun_commute: fixes y :: "'a"   and x :: "'a" shows "insort y \<circ> insort x = insort x \<circ> insort y"
prems_3: shows "\<Union> (set (s # ss)) \<subseteq> \<Union> (set ts)"
pd_refl: fixes x :: "'n" assumes "x \<in> local.nodes" shows "x pd\<rightarrow> x"
horner_sum_eq_sum: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'b list" shows "horner_sum f a xs = (\<Sum>n = 0..<length xs. f (xs ! n) * a ^ n)"
ipresIGSubstAbs_def: fixes h :: "'u \<Rightarrow> 'v"   and hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'varSort, 'sort, 'aa, 'ab, 'u, 'w, 'ac) model_scheme"   and MOD' :: "('ad, 'ae, 'varSort, 'af, 'ag, 'ab, 'v, 'x, 'ah) model_scheme" shows "ipresIGSubstAbs h hA MOD MOD' \<equiv> \<forall>ys Y y us s A. igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A \<longrightarrow> hA (igSubstAbs MOD ys Y y A) = igSubstAbs MOD' ys (h Y) y (hA A)"
sort_snoc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) (xs @ [x]) = linorder.insort_key less_eq (\<lambda>x. x) x (linorder.sort_key less_eq (\<lambda>x. x) xs)"
atMost_subset_iff: fixes x :: "'a"   and y :: "'a" shows "({..x} \<subseteq> {..y}) = (x \<le> y)"
qpk: fixes k :: "nat" assumes "k < n" shows "is_quantum_predicate (P k)"
dagger_prod_unfold: fixes x :: "'a"   and y :: "'a" shows "(1::'a) + x \<cdot> (y \<cdot> x)\<^sup>\<star> \<cdot> y = (x \<cdot> y)\<^sup>\<star>"
abel_semigroup_ow_axioms: shows "abel_semigroup_ow U (+\<^sub>o\<^sub>w)"
smalleraux2a: fixes a :: "'b"   and b :: "'b"   and l :: "'b set list"   and c :: "'b"   and d :: "'b"   and p :: "'c" assumes "{a, b} \<in> set l"   and "{c, d} \<in> set l"   and "{a, b} \<noteq> {c, d}"   and "smaller (DenyAllFromTo a b) (AllowPortFromTo c d p) l" shows "\<not> smaller (AllowPortFromTo c d p) (DenyAllFromTo a b) l"
zeta_bigo_at_1: fixes A :: "complex set" shows "zeta \<in> O[at 1 within A](\<lambda>x. 1 / (x - 1))"