hcju/isabelleps huggingface dataset:
simps: fixes f :: "nat \<Rightarrow> bool"   and np :: "nat"   and ps :: "nat list" shows "find_prime_main f np ps = (case ps of [] \<Rightarrow> let (np', ps') = next_primes np in find_prime_main f np' ps' | p # ps \<Rightarrow> if f p then p else find_prime_main f np ps)"
poly_linear_divides: fixes p :: "'a list"   and a :: "'a" shows "(poly p a = (0::'a)) = (p = [] \<or> (\<exists>q. p = [- a, 1::'a] *** q))"
descendants_bounds_snd: fixes k :: "'c"   and i :: "'c"   and branch :: "(('d, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and v :: "nat"   and v' :: "nat"   and ps :: "('d, 'c) fm list"   and a :: "'c" assumes "descendants k i branch xs"   and "(v, v') \<in> xs"   and "branch !. v = Some (ps, a)" shows "v' < length ps"
extend_basis_superset: fixes B :: "'b set" assumes "vs1.independent B" shows "B \<subseteq> vs1.extend_basis B"
in_var_prod_lens: fixes Y :: "'d \<Longrightarrow> 'e"   and x :: "'f \<Longrightarrow> 'h"   and X :: "'h \<Longrightarrow> 'g" assumes "wb_lens Y" shows "$x ;\<^sub>L X \<times> Y = $X:x"
AE_I: fixes M :: "'a measure"   and P :: "'a \<Rightarrow> bool"   and N :: "'a set" assumes "{x \<in> space M. \<not> P x} \<subseteq> N"   and "emeasure M N = 0"   and "N \<in> sets M" shows "almost_everywhere M P"
le_iff_sup: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x y = (sup x y = y)"
star_of_simps_10: shows "star_of (1::'a) = 1"
TameEnum_tameEnumFilter: fixes p :: "nat"   and t :: "(nat, nat list list list) trie" assumes "tameEnumFilter p = Some t" shows "set_tries t =\<^sub>\<simeq> fgraph ` TameEnum\<^bsub>p\<^esub>"
var: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and x :: "nat"   and T :: "Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type" assumes "\<Gamma> x = T" shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>T `x : T"
max_rank_of_def: fixes \<Sigma> :: "'a set set"   and \<psi> :: "'a ltl" shows "max_rank_of \<Sigma> \<psi> \<equiv> semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<psi>))"
assms_2: shows "h \<turnstile> get_ancestors_si some_ptr \<rightarrow>\<^sub>r ancestors"
absolutely_integrable_measurable_real: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "S \<in> sets lebesgue" shows "(f absolutely_integrable_on S) = (f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (\<lambda>x. \<bar>f x\<bar>))"
satisfies_satisfies_no_lhs: fixes s :: "('a, 'b) state" assumes "\<Turnstile> s" shows "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"
Midpoint_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Midpoint Bet Cong M A B \<equiv> Bet A M B \<and> Cong A M M B"
sigma_finite_measure_count_space_finite: fixes A :: "'c set" assumes "finite A" shows "sigma_finite_measure (count_space A)"
rep_eq: fixes x :: "uint32" shows "Rep_uint32 (- x) = - Rep_uint32 x"
awlast_if_cas: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.cas G u p v" shows "last (pre_digraph.awalk_verts G u p) = v"
le_combine_matrix: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "'c matrix"   and B :: "'c matrix"   and C :: "'d matrix"   and D :: "'d matrix" assumes "f (0::'c) (0::'d) = (0::'b)"   and "\<forall>a b c d. a \<le> b \<and> c \<le> d \<longrightarrow> f a c \<le> f b d"   and "A \<le> B"   and "C \<le> D" shows "combine_matrix f A C \<le> combine_matrix f B D"
assms_2: fixes a :: "'a" assumes "a \<in> H" shows "inv a \<in> H"
minus_def: fixes x :: "'c bit0" shows "- x = Abs_bit0 (- Rep_bit0 x mod int CARD('c bit0))"
dom_lc3: fixes x :: "'b"   and y :: "'b" shows "ad (ad x) \<cdot> ad (ad (x \<cdot> y)) = ad (ad (x \<cdot> y))"
rcs_fixed2: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "\<forall>h\<in>H. H \<bullet> h = H"
normalize_quot_eq_0_iff: fixes x :: "'b \<times> 'b" shows "(fst (normalize_quot x) = (0::'b)) = (fst x = (0::'b) \<or> snd x = (0::'b))"
eval_def: fixes envVal :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and envObs :: "'c \<Rightarrow> 'b \<Rightarrow> 'e" shows "eval envVal envObs \<equiv> \<lambda>(cec, aec). evalS (clock_evalProp envVal) (clock_knowledge envObs cec) (clock_commonKnowledge envObs cec) aec"
map_nth_conv: fixes f :: "'d \<Rightarrow> 'c"   and ss :: "'d list"   and g :: "'e \<Rightarrow> 'c"   and ts :: "'e list" assumes "map f ss = map g ts" shows "\<forall>i<length ss. f (ss ! i) = g (ts ! i)"
hide_mprefix_distr: fixes B :: "'a set"   and A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "B \<inter> A = {}" shows "Mprefix A P \ B = Mprefix A (\<lambda>x. P x \ B)"
implicit_function_theorem: fixes S :: "('b \<times> 'c) set"   and f :: "'b \<times> 'c \<Rightarrow> 'd"   and f' :: "'b \<times> 'c \<Rightarrow> ('b \<times> 'c) \<Rightarrow>\<^sub>L 'd"   and x :: "'b"   and y :: "'c"   and T :: "'d \<Rightarrow>\<^sub>L 'c" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)"   and "(x, y) \<in> S"   and "open S"   and "DIM('d) \<le> DIM('c)"   and "isCont f' (x, y)"   and "f (x, y) = (0::'d)"   and "T o\<^sub>L (f' (x, y) o\<^sub>L embed2_blinfun) = 1\<^sub>L"   and "f' (x, y) o\<^sub>L embed2_blinfun o\<^sub>L T = 1\<^sub>L" obtains u :: "'b \<Rightarrow> 'c" and e :: "real" and r :: "real" where "f (x, u x) = (0::'d)" and "u x = y" and "\<And>s. s \<in> cball x e \<Longrightarrow> f (s, u s) = (0::'d)" and "continuous_on (cball x e) u" and "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> S" and "0 < e" and "(u has_derivative blinfun_apply (- T o\<^sub>L f' (x, y) o\<^sub>L embed1_blinfun)) (at x)" and "0 < r" and "\<And>U v s. \<lbrakk>v x = y; \<And>s. s \<in> U \<Longrightarrow> f (s, v s) = (0::'d); U \<subseteq> cball x e; continuous_on U v; s \<in> U; (s, v s) \<in> ball (x, y) r\<rbrakk> \<Longrightarrow> u s = v s"
INF_constant: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 bot :: "'b"   and top :: "'b"   and c :: "'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf ((\<lambda>y. c) ` A) = (if A = {} then top else c)"
simps: fixes a1 :: "nat"   and a2 :: "com" shows "(a1 \<turnstile> a2) = ((\<exists>l. a1 = l \<and> a2 = SKIP) \<or> (\<exists>x a l. a1 = l \<and> a2 = x ::= a \<and> sec a \<le> sec x \<and> l \<le> sec x) \<or> (\<exists>l c\<^sub>1 c\<^sub>2. a1 = l \<and> a2 = c\<^sub>1;; c\<^sub>2 \<and> l \<turnstile> c\<^sub>1 \<and> l \<turnstile> c\<^sub>2) \<or> (\<exists>b l c\<^sub>1 c\<^sub>2. a1 = l \<and> a2 = IF b THEN c\<^sub>1 ELSE c\<^sub>2 \<and> max (sec b) l \<turnstile> c\<^sub>1 \<and> max (sec b) l \<turnstile> c\<^sub>2) \<or> (\<exists>b l c. a1 = l \<and> a2 = WHILE b DO c \<and> max (sec b) l \<turnstile> c))"
acc_wfD: fixes r :: "('b \<times> 'b) set"   and x :: "'b" assumes "wf r" shows "x \<in> Wellfounded.acc r"
greatest_unique: fixes x :: "'a"   and A :: "'a set"   and y :: "'a" assumes "greatest L x A"   and "greatest L y A" shows "x = y"
axioms_3: fixes as\<^sub>0 :: "'b"   and alstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and cs\<^sub>0 :: "'e"   and clstep :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'e \<Rightarrow> bool" assumes "simulation as\<^sub>0 alstep cs\<^sub>0 clstep R" shows "simulation_axioms as\<^sub>0 alstep cs\<^sub>0 clstep R"
cases: fixes a :: "('a, 'b, 'c) hp" assumes "hpsafe a" obtains x :: "'c" where "a = $\<alpha> x__"   | e :: "('a, 'c) trm" and x :: "'c" where "a = (xa__ := e__)" and "dsafe e__"   | e :: "('a, 'c) trm" and x :: "'c" where "a = DiffAssign x e" and "dsafe e"   | P :: "('a, 'b, 'c) formula" where "a = ? P__" and "fsafe P__"   | ODE :: "('a, 'c) ODE" and P :: "('a, 'b, 'c) formula" where "a = EvolveODE ODE P" and "osafe ODE" and "fsafe P"   | aa :: "('a, 'b, 'c) hp" and b :: "('a, 'b, 'c) hp" where "a = (aa__ \<union>\<union> b__)" and "hpsafe aa__" and "hpsafe b__"   | aa :: "('a, 'b, 'c) hp" and b :: "('a, 'b, 'c) hp" where "a = (aaa__ ;; b)" and "hpsafe aaa__" and "hpsafe b"   | aa :: "('a, 'b, 'c) hp" where "a = aa**" and "hpsafe aa"
False: shows "\<not> (m = 0 \<or> n = 0)"
set_selsort: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" shows "set (ccw.selsort x xs) = set xs"
proper_it_def: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and it' :: "('f \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f" shows "proper_it it it' \<equiv> \<exists>l. it = foldli l \<and> it' = foldli l"
simps_2: fixes c :: "'c"   and cs :: "'c list"   and t :: "'d"   and u :: "'c \<Rightarrow> 'd" shows "[c # cs\<rightarrow>t]u = ([cs\<rightarrow>t]u)(c := t)"
is_coext_def: fixes x' :: "'c"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and g :: "'c" shows "x.is_coext x' f g \<equiv> \<guillemotleft>g : x' \<rightarrow>\<^sub>C G (J_C.MkIde D)\<guillemotright> \<and> f = Adj.\<epsilon> (J_C.MkIde D) \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] \<Delta>.map g"
face_cycle_sets: shows "GM.face_cycle_sets = (`) proj_arcs_H ` H.face_cycle_sets"
map_poly_pCons_hom: fixes aa :: "'c"   and a :: "'c poly"   and p :: "'c poly poly" shows "map_poly (\<lambda>p. poly p aa) (pCons a p) = pCons (poly a aa) (map_poly (\<lambda>p. poly p aa) p)"
hyps_1: fixes s :: "'a list"   and v :: "'a val" assumes "s \<in> fst (simp r1) \<rightarrow> v" shows "s \<in> r1 \<rightarrow> snd (simp r1) v"
iso_inclI: fixes A :: "'c ring"   and B :: "'c ring" assumes "id \<in> ring_hom A B" shows "A \<lesssim> B"
compactin_subtopology_imp_compact: fixes X :: "'a topology"   and S :: "'a set"   and K :: "'a set" assumes "compactin (subtopology X S) K" shows "compactin X K"
measure_translation: fixes a :: "'a"   and S :: "'a set" shows "Sigma_Algebra.measure lebesgue ((+) a ` S) = Sigma_Algebra.measure lebesgue S"
cases: fixes x :: "C" obtains "x = C []"   | ns :: "C list" where "x = C (C [] # ns__)"   | va :: "C" and vb :: "C list" and ns :: "C list" where "x = C (C (va # vb) # ns)"
less_SucI: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "m < Suc n"
ex_y: shows "\<forall>x. x \<in># X \<longrightarrow> (\<exists>y. y \<in># Y \<and> x < y)"
execute_trap_result2: fixes s :: "'b sparc_state" assumes "\<not> (reset_trap_val s = False \<and> get_ET (cpu_reg_val PSR s) = 0)" shows "snd (execute_trap () s) = False"
Resolution_complete': fixes S :: "'b formula set"   and F :: "'b formula" assumes "finite S"   and "S \<TTurnstile> F" shows "\<Union> ((cnf \<circ> nnf) ` ({\<^bold>\<not> F} \<union> S)) \<turnstile> \<box>"
less_zeroE: fixes n :: "nat"   and R :: "bool" assumes "n \<sqsubset> 0" shows "R"
prems_3: fixes r :: "'a" assumes "ref = Some r" shows "r \<notin> fst ` set (xs @ [x])"
ordinal_numberTr1: fixes X :: "'c Order set" assumes "X \<in> carrier ODnods" shows "\<exists>D. Worder D \<and> D \<in> X"
res_lc1: fixes x :: "'a"   and y :: "'a" shows "x \<le> x \<cdot> y \<leftarrow> y"
preserves_seq: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "seq g f" shows "seq (src g) (src f)"
klift_def: fixes f :: "'a \<Rightarrow> 'b set" shows "bd\<^sub>\<F> f = \<mu> \<circ> \<P> f"
CH_irc_idem: 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 A :: "'d set" assumes "ContractsWithIRC Xd Xh Pd Ch" shows "Contracts.CH Ch (Contracts.CH Ch A) = Contracts.CH Ch A"
naturality: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"   and "C.ide f" shows "\<tau>\<^sub>1 f \<cdot>\<^sub>D (G f \<star>\<^sub>D \<Delta> a) = (\<Delta> b \<star>\<^sub>D F f) \<cdot>\<^sub>D \<sigma>\<^sub>1 f"
igFreshInp_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) model_scheme"   and ys :: "'c"   and y :: "'f"   and inp :: "'j \<Rightarrow> 'g option" shows "igFreshInp MOD ys y inp \<equiv> liftAll (igFresh MOD ys y) inp"
Height_Bounded_Labeling_axioms: shows "Height_Bounded_Labeling c s t f l"
Gauss_Jordan_column_k_iarrays_def: fixes A' :: "nat \<times> 'd iarray iarray"   and k :: "nat" shows "Gauss_Jordan_column_k_iarrays A' k = (let A = snd A'; i = fst A' in if vector_all_zero_from_index (i, column_iarray k A) \<or> i = nrows_iarray A then (i, A) else (Suc i, Gauss_Jordan_in_ij_iarrays A i k))"
IH: fixes S :: "('b, 'd) Term.term list" assumes "\<forall>i<length T. \<exists>s. P (T ! i) s \<and> S ! i = s \<cdot> \<delta>"   and "length T = length S" shows "\<exists>U. length T = length U \<and> (\<forall>i<length T. P (T ! i) (U ! i)) \<and> S = U \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>"
trans_equiv: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b = Equiv"   and "cmp b c = Equiv" shows "cmp a c = Equiv"
l8_18_uniqueness: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and C :: "'p"   and Y :: "'p" assumes "Col A B X"   and "A B Perp C X"   and "Col A B Y"   and "A B Perp C Y" shows "X = Y"
properfactor_imp_properfactor_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R - {\<zero>}"   and "b \<in> carrier R"   and "properfactor R b a" shows "properfactor (Ring_Divisibility.mult_of R) b a"
simps_1: fixes x :: "'b \<Rightarrow> 'c \<times> 'b"   and ya :: "'b \<Rightarrow> 'c \<times> 'b" shows "(State x = State ya) = (x = ya)"
space_dens_ctxt_measure: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and \<rho> :: "nat \<Rightarrow> val" shows "space (dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho>) = space (state_measure (V \<union> V') \<Gamma>)"
zero: fixes bs :: "bit list" assumes "w = \<zero> # bs" shows "P w"
modular_1_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> z \<le> x ; (y \<cdot> x\<^sup>T ; z)"
polyadd_different_degreen: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "degreen p m \<noteq> degreen q m"   and "m \<le> min n0 n1" shows "degreen (p +\<^sub>p q) m = max (degreen p m) (degreen q m)"
sub: fixes i :: "name"   and t :: "trm"   and u :: "trm"   and x :: "trm" assumes "atom i \<sharp> tms" shows "(P t u)(i::=x) = P (subst i x t) (subst i x u)"
last_ConsL: fixes xs :: "'b list"   and x :: "'b" assumes "xs = []" shows "last (x # xs) = x"
subst_closed_in': fixes T :: "ty"   and X :: "tyvrs"   and S :: "ty"   and \<Gamma> :: "binding list"   and U :: "ty" assumes "T closed_in (TVarB X S # \<Gamma>)"   and "U closed_in \<Gamma>" shows "T[X \<mapsto> U]\<^sub>\<tau> closed_in \<Gamma>"
cpu_reg_mod_asr_privilege: fixes s' :: "'b sparc_state"   and w :: "word32"   and r :: "word5"   and s :: "'b sparc_state" assumes "s' = cpu_reg_mod w (ASR r) s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
simps_4: fixes x3 :: "'a" shows "SOURCE \<noteq> \<langle>x3\<rangle>"
simps_4: fixes p :: "perm"   and f :: "'e"   and \<alpha> :: "'d"   and t :: "('b, 'c, 'd, 'e) Tree" shows "p \<bullet> tAct f \<alpha> t = tAct (p \<bullet> f) (p \<bullet> \<alpha>) (p \<bullet> t)"
cases: fixes x :: "'b tree" obtains l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>"   | "x = \<langle>\<rangle>"
not_in_\<pi>': shows "\<nexists>i'. is_kth_obs \<pi>' k i'"
pell_valuation_solution_gt1D: fixes z :: "int \<times> int" assumes "solution z"   and "1 < pell_valuation z" shows "0 < fst z \<and> 0 < snd z"
non_empty_graph_def: fixes G :: "('a, 'b) pre_digraph" shows "Designs_And_Graphs.non_empty_graph G \<equiv> graph G \<and> non_empty_digraph G"
map_butlast: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map f (butlast xs) = butlast (map f xs)"
InvariantImpliedLiteralsAfterDecide: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and literal :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantImpliedLiterals F M"   and "var literal \<notin> vars (elements M)"   and "M' = M @ [(literal, True)]" shows "InvariantImpliedLiterals F M'"
matchC_ZOC_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchC_ZOC theta c d \<equiv> \<forall>s t c' s'. s \<approx> t \<and> (c, s) \<rightarrow>c (c', s') \<longrightarrow> s' \<approx> t \<and> (c', d) \<in> theta \<or> (\<exists>d' t'. (d, t) \<rightarrow>c (d', t') \<and> s' \<approx> t' \<and> (c', d') \<in> theta)"
XQx: shows "\<forall>X\<in>Q. \<forall>x\<in>X. X = [x]\<^sub>\<sim>"
1: shows "local.map (gf' \<odot> gf) = S.mkArr (local.set (CopxC.dom gf)) (local.set (CopxC.cod gf')) (\<phi> (CopxC.cod gf') \<circ> (\<lambda>h. snd (gf' \<odot> gf) \<cdot> h \<cdot> fst (gf' \<odot> gf)) \<circ> \<psi> (CopxC.dom gf))"
simps_1: fixes n :: "int"   and s :: "char list \<Rightarrow> int" shows "aval (N n) s = n"
coeff_0_snd_quot_of_ratfps_nonzero: fixes x :: "'a ratfps" shows "constant_term (snd (local.quot_of_ratfps x)) \<noteq> (0::'a)"
int_via_rat_eqI: fixes x :: "int"   and y :: "int" assumes "rat_of_int x = rat_of_int y" shows "x = y"
Un_carrier_def: fixes I :: "'i set"   and A :: "'i \<Rightarrow> ('h, 'j) aGroup_scheme" shows "Un_carrier I A = \<Union> {X. \<exists>i\<in>I. X = carrier (A i)}"
that: fixes l :: "'a list"   and m :: "'a list" assumes "w = l \<cdot> m"   and "Lyndon l"   and "Lyndon m"   and "l <lex m" shows "thesis"
b: shows "b = r \<or> b \<in> B \<and> b \<noteq> p"
measurable_stream_filtration2: fixes f :: "'b \<Rightarrow> 'c stream"   and N :: "'b measure"   and M :: "'c measure"   and n :: "enat" assumes "f \<in> space N \<rightarrow> streams (space M)"   and "\<And>i. enat i \<le> n \<Longrightarrow> (\<lambda>x. f x !! i) \<in> N \<rightarrow>\<^sub>M M" shows "f \<in> N \<rightarrow>\<^sub>M stream_filtration M n"
Par2F: fixes Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<alpha> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<alpha> \<prec> P \<parallel> Q'"
update_trie_def: fixes ks :: "'c list"   and v :: "'d" shows "update_trie ks v = update_with_trie ks (\<lambda>_. v)"
comp_arr_inv: fixes f :: "'d"   and g :: "'d" assumes "D.inverse_arrows f g" shows "f \<cdot>\<^sub>D g = D.dom g"
insert_ops_memb_ref_older: fixes op_list :: "('b \<times> 'b option) list"   and oid :: "'b"   and ref :: "'b" assumes "insert_ops op_list"   and "(oid, Some ref) \<in> set op_list" shows "ref < oid"
assms_1: fixes \<sigma> :: "nat \<Rightarrow> ('a, 'b, 'c, 'd, 'e) system_state" assumes "prerun sys \<sigma>" shows "\<lceil>I\<rceil> \<sigma>"
simps_2: fixes h1 :: "'b"   and t1 :: "'b list"   and l2 :: "'a list"   and P :: "'b \<Rightarrow> bool" shows "lookup2 (h1 # t1) l2 P = (if P h1 then Some (hd l2) else lookup2 t1 (tl l2) P)"
tm_mul_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t1 :: "taylor_model"   and y :: "real"   and t2 :: "taylor_model"   and a :: "float interval list"   and I :: "float interval list"   and prec :: "nat"   and ord :: "nat" assumes "x \<in>\<^sub>i range_tm e t1"   and "y \<in>\<^sub>i range_tm e t2"   and "develops_at_within e a I"   and "num_params (tm_poly t1) \<le> length I"   and "num_params (tm_poly t2) \<le> length I" shows "x * y \<in>\<^sub>i range_tm e (tm_mul prec ord I a t1 t2)"
eq_iff_3: fixes fm1 :: "fm"   and fm2 :: "fm"   and fm1a :: "fm"   and fm2a :: "fm" shows "(fm1 OR fm2 = fm1a OR fm2a) = (fm1 = fm1a \<and> fm2 = fm2a)"
dual_filtered: shows "filtered = ord.directed (\<lambda>x y. y \<le> x)"
v: shows "v = (\<lambda>i. if i < dim_vec x then x $ i else 0)"
assms_2: shows "theta \<subseteq> ZOretr (theta \<union> ZObis)"
assms_2: fixes oid :: "'a"   and op1 :: "'b"   and op2 :: "'b" assumes "(oid, op1) \<in> ops"   and "(oid, op2) \<in> ops" shows "op1 = op2"
that: fixes x :: "'a" assumes "x \<in> S"   and "Inf (f ` S) = f x" shows "thesis"
assms_2: fixes u :: "real" assumes "x0 \<in> X"   and "u \<in> existence_ivl t0 x0" shows "flow t0 x0 u = flow s0 x0 (g u)"
exp_cf4_pos: fixes x :: "real" shows "0 < exp_cf4 x"
bset_8: fixes D :: "int"   and t :: "int"   and B :: "int set" assumes "0 < D"   and "t - 1 \<in> B" shows "\<forall>x. (\<forall>j\<in>{1..D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> t \<le> x \<longrightarrow> t \<le> x - D"
2: fixes x :: "'b" shows "cont lubb ordb Sup (\<le>) (f x)"
folding: fixes C :: "'a set" assumes "chamber C"   and "C \<in> f \<turnstile> X" shows "\<exists>!D. chamber D \<and> D \<notin> f \<turnstile> X \<and> f ` D = C"
rt_strictly_fresher_trans: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubset>\<^bsub>dip\<^esub> rt3" shows "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt3"
pseudonatural_equivalence_def: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" shows "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<and> pseudonatural_equivalence_axioms V\<^sub>C src\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<tau>\<^sub>0"
nn_integral_bound_simple_function: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> f x < \<infinity>"   and "simple_function M f"   and "emeasure M {x \<in> space M. f x \<noteq> 0} < \<infinity>" shows "integral\<^sup>N M f < \<infinity>"
distinct_concat: fixes xs :: "'g list list" assumes "distinct xs"   and "\<And>ys. ys \<in> set xs \<Longrightarrow> distinct ys"   and "\<And>ys zs. \<lbrakk>ys \<in> set xs; zs \<in> set xs; ys \<noteq> zs\<rbrakk> \<Longrightarrow> set ys \<inter> set zs = {}" shows "distinct (concat xs)"
map_M_pure_I: fixes xs :: "'l list"   and f :: "'l \<Rightarrow> ('m, 'n, 'o) prog"   and h :: "'m" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "pure (map_M f xs) h"