hcju/isabelleps huggingface dataset:
weakParPres: fixes R :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi>'' :: "'b" assumes "\<And>A\<^sub>R \<Psi>\<^sub>R. \<lbrakk>extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>; A\<^sub>R \<sharp>* \<Psi>; A\<^sub>R \<sharp>* P; A\<^sub>R \<sharp>* Q\<rbrakk> \<Longrightarrow> (\<Psi> \<otimes> \<Psi>\<^sub>R, P, Q) \<in> Rel"   and "eqvt Rel"   and "eqvt Rel'"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> S \<leadsto><Rel> T"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (\<Psi>', T, S) \<in> Rel"   and "\<And>\<Psi>' S T \<Psi>'''. (\<Psi>''', S, T) \<in> Rel \<Longrightarrow> (\<Psi>''' \<otimes> \<Psi>'', S, T) \<in> Rel"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> S \<lessapprox><Rel> T"   and "\<And>\<Psi>' S T A\<^sub>U \<Psi>\<^sub>U U. \<lbrakk>(\<Psi>' \<otimes> \<Psi>\<^sub>U, S, T) \<in> Rel; extractFrame U = \<langle>A\<^sub>U, \<Psi>\<^sub>U\<rangle>; A\<^sub>U \<sharp>* \<Psi>'; A\<^sub>U \<sharp>* S; A\<^sub>U \<sharp>* T\<rbrakk> \<Longrightarrow> (\<Psi>', S \<parallel> U, T \<parallel> U) \<in> Rel'"   and "\<And>\<Psi>' S T xvec. \<lbrakk>(\<Psi>', S, T) \<in> Rel'; xvec \<sharp>* \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel'"   and "\<And>\<Psi>' S T \<Psi>''. \<lbrakk>(\<Psi>', S, T) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', S, T) \<in> Rel" shows "\<Psi> \<rhd> P \<parallel> R \<leadsto><Rel'> Q \<parallel> R"
floor_eq3: fixes n :: "nat"   and x :: "real" assumes "real n < x"   and "x < real (Suc n)" shows "nat \<lfloor>x\<rfloor> = n"
right_def: shows "right = (\<lambda>f. \<Sum>x | jumpF f (at_right x) \<noteq> 0 \<and> a \<le> x \<and> x < b. jumpF f (at_right x))"
maxBound_is_length: shows "fromEnum maxBound = length enum_class.enum - 1"
SN_of_FST_of_App: fixes t :: "trm"   and s :: "trm" assumes "SN (App t s)" shows "SN t"
that: fixes C :: "'a set set" assumes "finite C"   and "disjoint C"   and "C \<subseteq> M"   and "a = \<Union> C" shows "thesis"
assms_3: shows "a c \<congruent>\<^sub>K a' c'"
scaleR_le_self_cancel: fixes a :: "real"   and c :: "'a" shows "(a *\<^sub>R c \<le> c) = (1 < a \<and> c \<le> (0::'a) \<or> a < 1 \<and> (0::'a) \<le> c \<or> a = 1)"
integral_Im: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> complex" assumes "complex_integrable M f" shows "LINT x|M. Im (f x) = Im (integral\<^sup>C M f)"
hlp2: fixes vi :: "'a"   and vt :: "'a ifex"   and ve :: "'a ifex"   and "is" :: "'a ifex list"   and a :: "'a" assumes "IF vi vt ve \<in> set is"   and "lowest_tops is = Some a" shows "a \<le> vi"
assms: shows "low_equal s1 s2 \<and> t1 = out_reg_mod w curr_win rd s1 \<and> t2 = out_reg_mod w curr_win rd s2"
comp_null_1: fixes f :: "'j" shows "J.null \<cdot>\<^sub>J f = J.null"
scan_lemma: fixes r :: "nat" shows "rec_exec rec_scan [r] = r mod 2"
rep2_def: shows "rep2 \<equiv> prod.fst (prod.snd (prod.snd (fix_syn (\<lambda>(root2, op2, rep2, left2, right2). (\<Lambda> pat. rep2\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root2\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op2\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. Node\<cdot>usvs\<cdot>(left2\<cdot>pat\<cdot>usvs)\<cdot>(right2\<cdot>pat\<cdot>usvs), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>Null))) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_scons\<cdot>v\<cdot>(\<Lambda> u us. Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>(sfoldl\<cdot>(op2\<cdot>pat)\<cdot>(root2\<cdot>pat)\<cdot>us))))), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> us v. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>Null)) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> us va. match_scons\<cdot>va\<cdot>(\<Lambda> v vs. Fixrec.succeed\<cdot>(rep2\<cdot>pat\<cdot>(us :@ [:v:], vs))))))))))"
paths: fixes xs :: "'a list" assumes "xs \<in> Q" shows "v0 \<leadsto>xs\<leadsto>\<^bsub>H_x\<^esub> v1"
lincomb_0coeffs: fixes ms :: "'v list"   and rs :: "'f list" assumes "set ms \<subseteq> V"   and "\<forall>s\<in>set rs. s = (0::'f)" shows "rs \<bullet>\<cdot> ms = (0::'v)"
S: shows "maximal S (\<lambda>T. T \<subseteq> set (cols A\<^sup>H) \<and> \<not> module.lin_dep class_ring (module_vec TYPE('a) nc) T)"
Inequality: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list" assumes "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop6 n (l, r) = (inv_loop6_loop n (l, r) \<or> inv_loop6_exit n (l, r))"
exhaust: fixes y :: "'b Maybe" obtains "y = \<bottom>"   | "y = Nothing"   | a :: "'b" where "y = Just\<cdot>a"
eq: shows "(\<Sum>i | i dvd order (mult_of R). card {a \<in> carrier (mult_of R). G.ord a = i}) = (\<Sum>i | i dvd order (mult_of R). phi' i)"
dWHILEI_def: shows "dWHILEI \<equiv> iWHILEI (\<bind>) dRETURN"
fps_compose_0_right: fixes a :: "'a fps" shows "a oo 0 = fps_const (fps_nth a 0)"
not_le_less: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "(\<not> a \<preceq> b) = b \<prec> a"
insert: fixes A :: "'a set"   and x :: "'a"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "x \<notin> A" shows "prod g (insert x A) = g x * prod g A"
2: shows "Well_order r'' \<and> Field r'' = f ` A"
multiplicity_prime_prime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q" shows "factorial_semiring.multiplicity times one p q = (if p = q then 1 else 0)"
cases: fixes a :: "('a, 'b, 'c) psi" assumes "quiet a" obtains P :: "('a, 'b, 'c) psi" where "a = P" and "\<forall>\<Psi>. insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F \<langle>\<epsilon>, \<Psi>\<rangle> \<and> (\<forall>Rs. \<Psi> \<rhd> P \<longmapsto> Rs \<longrightarrow> (\<exists>P'. Rs = \<tau> \<prec> P' \<and> quiet P'))"
zero_cblinfun_image: fixes S :: "'a ccsubspace" shows "0 *\<^sub>C S = \<bottom>"
arr_cod_iff_arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C" shows "partial_magma.arr C (partial_magma.cod C f) = partial_magma.arr C f"
measurable_Pair: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and M1 :: "'e measure"   and g :: "'d \<Rightarrow> 'f"   and M2 :: "'f measure" assumes "f \<in> M \<rightarrow>\<^sub>M M1"   and "g \<in> M \<rightarrow>\<^sub>M M2" shows "(\<lambda>x. (f x, g x)) \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
ord0_Lim_ord0rel: fixes f :: "nat \<Rightarrow> ord0"   and g :: "nat \<Rightarrow> ord0" assumes "\<forall>n. (f n, g n) \<in> ord0rel" shows "(ord0_Lim f, ord0_Lim g) \<in> ord0rel"
state_dannach: shows "s_TS init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i + 1) = step (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) (cs ! i) (index (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) (cs ! i) - index (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) z, [])"
stream_pred_coinduct_shift: fixes R :: "'b stream \<Rightarrow> bool"   and w :: "'b stream"   and P :: "'b \<Rightarrow> bool" assumes "R w"   and "\<And>w. R w \<Longrightarrow> \<exists>u v. w = u @- v \<and> u \<noteq> [] \<and> list_all P u \<and> R v" shows "pred_stream P w"
append_rsp2_pre: fixes x :: "'b list list"   and x' :: "'b list list"   and z :: "'b list list"   and z' :: "'b list list" assumes "list_all2 (\<approx>) x x'"   and "list_all2 (\<approx>) z z'" shows "list_all2 (\<approx>) (x @ z) (x' @ z')"
orderI: fixes a :: "'d"   and b :: "'d" assumes "a = min a b" shows "a \<le> b"
make_map_def: fixes ks :: "'c fset"   and v :: "'d" shows "make_map ks v \<equiv> \<lambda>k. if k \<in> fset ks then Some v else None"
disj_matrices_x_add: fixes A :: "'a matrix"   and B :: "'a matrix"   and C :: "'a matrix" assumes "disj_matrices A B"   and "disj_matrices A C" shows "disj_matrices A (B + C)"
assms: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A"
se_cast: fixes e :: "exp"   and C :: "nat" shows "e \<in> isubexprs(Cast C e)"
take_bit_nat_def: fixes n :: "nat"   and m :: "nat" shows "take_bit n m = m mod 2 ^ n"
global_var_approx_def: fixes prog :: "'a program_scheme" shows "global_var_approx prog \<equiv> set (global_vars prog)"
simps_22: fixes v :: "'b rexp"   and vb :: "'b" shows "le_rexp (Star v) (Atom vb) = False"
card_of_Un_Plus_ordIso1: fixes A :: "'a set"   and B :: "'a set" shows "|A \<union> B| =o |A <+> B - A|"
flip_bit_nonnegative_int_iff: fixes n :: "nat"   and k :: "int" shows "(0 \<le> flip_bit n k) = (0 \<le> k)"
finite_neq_0': fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite {x. f x \<noteq> (0::'b)}"   and "finite {x. g x \<noteq> (0::'c)}"   and "h (0::'b) (0::'c) = (0::'d)" shows "finite {x. h (f x) (g x) \<noteq> (0::'d)}"
minus_unique: fixes G :: "('a, 'b) ring_scheme"   and y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "abelian_monoid G"   and "y \<oplus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"   and "x \<oplus>\<^bsub>G\<^esub> y' = \<zero>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "y' \<in> carrier G" shows "y = y'"
divide_self: fixes a :: "'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>" shows "a \<oslash> a = \<one>"
mult_nat_right_at_top: fixes c :: "nat" assumes "0 < c" shows "LIM x sequentially. x * c :> sequentially"
collapse_2: fixes tm :: "('c, 'd) tm" assumes "is_App tm" shows "App (fun tm) (arg tm) = tm"
segment_bound1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> {a--b}" shows "norm (x - a) \<le> norm (b - a)"
G6: fixes v :: "'v" assumes "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)" shows "range_of \<Psi> v \<noteq> None"
sup_left_commute: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
ys: shows "h \<turnstile> map_M (\<lambda>x. return (the x)) ys_no_opts \<rightarrow>\<^sub>r ys"
size_pass\<^sub>2: fixes hp :: "'b tree" shows "size (pass\<^sub>2 hp) = size hp"
Rep_take_rows: fixes A :: "'b matrix"   and r :: "nat"   and j :: "nat"   and i :: "nat" shows "Rep_matrix (take_rows A r) j i = (if j < r then Rep_matrix A j i else (0::'b))"
id_cblinfun_apply: fixes \<psi> :: "'a" shows "id_cblinfun *\<^sub>V \<psi> = \<psi>"
Un_Int_eq_3: fixes S :: "'e set"   and T :: "'e set" shows "S \<inter> (S \<union> T) = S"
eq_iff: fixes x :: "'c list"   and y :: "'c list" shows "(x = y) = (subseq x y \<and> subseq y x)"
nfa_of_dfa_reach: fixes M :: "nat bdd list \<times> bool list"   and l :: "nat"   and bss :: "bool list list"   and n1 :: "bool list"   and q :: "nat"   and n2 :: "bool list"   and p :: "nat" assumes "wf_dfa M l"   and "list_all (is_alph l) bss"   and "n1 = (replicate (length (fst M)) False)[q := True]"   and "n2 = (replicate (length (fst M)) False)[p := True]"   and "dfa_is_node M q" shows "nfa_reach (nfa_of_dfa M) n1 bss n2 = dfa_reach M q bss p"
f: shows "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>"
rel_gpv''_Grp: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'k"   and B :: "'i set"   and g :: "'i \<Rightarrow> 'l"   and h :: "'m \<Rightarrow> 'j" shows "rel_gpv'' (BNF_Def.Grp A f) (BNF_Def.Grp B g) (BNF_Def.Grp UNIV h)\<inverse>\<inverse> = BNF_Def.Grp {x. results_gpv (\<I>_uniform UNIV (range h)) x \<subseteq> A \<and> outs_gpv (\<I>_uniform UNIV (range h)) x \<subseteq> B} (map_gpv' f g h)"
no_out_VWrite\<^sub>s\<^sub>b: shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"
simps: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and xs :: "('a \<times> 'b) list"   and ko :: "'c" shows "oalist_raw.sort_oalist_raw rep_key_order (xs, ko) = (comparator.sort_oalist (key_compare (rep_key_order ko)) xs, ko)"
differentiable_const: fixes a :: "'c"   and F :: "'b filter" shows "(\<lambda>z. a) differentiable F"
assms_2: fixes a :: "'a \<times> 'b" assumes "snd (f a) = (0::'c)" shows "snd a = (0::'b)"
Derives1_keep_first_terminal: fixes x :: "symbol"   and u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and y :: "symbol"   and v :: "symbol list" assumes "Derives1 (x # u) i r (y # v)"   and "is_terminal x" shows "x = y"
composable: shows "src (\<nu> \<star> \<pi>) = trg \<rho>"
qo_on_imp_transp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "qo_on P A" shows "transp_on P A"
is_lrun_def: fixes l :: "nat \<Rightarrow> 'b"   and s :: "nat \<Rightarrow> 'a" shows "is_lrun l s \<equiv> s 0 = s\<^sub>0 \<and> (\<forall>i. lstep (l i) (s i) (s (Suc i)))"
simCases: fixes Q :: "pi"   and P :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "\<And>a y Q'. \<lbrakk>Q \<longmapsto> a<\<nu>y> \<prec> Q'; y \<sharp> P\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto> a<\<nu>y> \<prec> P' \<and> (P', Q') \<in> Rel"   and "\<And>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>[Rel] Q"
map_of_eq_dom: fixes xs :: "('d \<times> 'e) list"   and ys :: "('d \<times> 'e) list" assumes "map_of xs = map_of ys" shows "fst ` set xs = fst ` set ys"
normal_form_with_flatten_pi_2_length_def: fixes \<phi> :: "'b ltln" shows "normal_form_with_flatten_pi_2_length \<phi> \<equiv> \<Sum>(M, N)\<in>{(M, N) |M N. M \<subseteq> subformulas\<^sub>\<mu> \<phi> \<and> N \<subseteq> subformulas\<^sub>\<nu> \<phi>}. normal_form_disjunct_with_flatten_pi_2_length \<phi> M N"
domintros: fixes xs :: "'a list"   and k :: "nat" assumes "\<And>aa ab ba xe ac bb. \<lbrakk>\<not> length xs \<le> 20; (aa, ab, ba) = partition3 (mom_select (((length xs + 4) div 5 - Suc 0) div 2) (map slow_median (chop 5 xs))) xs; (xe, ac, bb) = partition3 (mom_select (((length xs + 4) div 5 - Suc 0) div 2) (map slow_median (chop 5 xs))) xs; k < length xe\<rbrakk> \<Longrightarrow> T_mom_select_dom (k, xe)"   and "\<And>aa ab ba xe ac bb. \<lbrakk>\<not> length xs \<le> 20; (aa, ab, ba) = partition3 (mom_select (((length xs + 4) div 5 - Suc 0) div 2) (map slow_median (chop 5 xs))) xs; (xe, ac, bb) = partition3 (mom_select (((length xs + 4) div 5 - Suc 0) div 2) (map slow_median (chop 5 xs))) xs; \<not> k < length xe + length ac\<rbrakk> \<Longrightarrow> T_mom_select_dom (k - (length xe + length ac), bb)"   and "\<And>aa ab ba xe ac bb. \<lbrakk>\<not> length xs \<le> 20; (aa, ab, ba) = partition3 (mom_select (((length xs + 4) div 5 - Suc 0) div 2) (map slow_median (chop 5 xs))) xs; (xe, ac, bb) = partition3 (mom_select (((length xs + 4) div 5 - Suc 0) div 2) (map slow_median (chop 5 xs))) xs\<rbrakk> \<Longrightarrow> T_mom_select_dom (((length xs + 4) div 5 - Suc 0) div 2, map slow_median (chop 5 xs))" shows "T_mom_select_dom (k, xs)"
assms_1: shows "wf\<^sub>s\<^sub>t\<^sub>s \<S>\<^sub>0"
closure_of_subset: fixes S :: "'b set"   and X :: "'b topology" assumes "S \<subseteq> topspace X" shows "S \<subseteq> X closure_of S"
to_nat_on_inj: fixes A :: "'a set"   and a :: "'a"   and b :: "'a" assumes "countable A"   and "a \<in> A"   and "b \<in> A" shows "(to_nat_on A a = to_nat_on A b) = (a = b)"
assms_3: shows "SPEC Q \<le> \<Down> (S\<inverse> O R) (SPEC P)"
uJ_int2: fixes n :: "nat" shows "J.expectation (uJ n) = expectation (u n)"
assms_3: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "length xs = length ys"   and "P xs ys" shows "P (xs @ [x]) (ys @ [y])"
T_pre: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) (u @ v) = Partial_Cost_Model.T_on' (rTS h0) ([x, y], rev u @ h) v"
comp_image: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "(`) f \<circ> (`) g = (`) (f \<circ> g)"
subsequence_append: fixes w :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" shows "prefix (i + j) w = prefix i w @ w [i \<rightarrow> i + j]"
path_split_Cons: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'"   and "as \<noteq> []" obtains a' :: "'edge" and as' :: "'edge list" where "as = a' # as'" and "n = sourcenode a'" and "valid_edge a'" and "targetnode a' -as'\<rightarrow>* n'"
lookup_append_none: fixes env :: "('a, 'b, 'c) env"   and xs :: "'c list"   and ys :: "'c list" assumes "lookup env xs = None" shows "lookup env (xs @ ys) = None"
invc2_joinL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invc l"   and "invc r"   and "bheight l \<le> bheight r" shows "invc2 (joinL l x r) \<and> (bheight l \<noteq> bheight r \<and> color r = Black \<longrightarrow> invc (joinL l x r))"
monic_0_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "(monic p = 0) = (p = 0)"
at_prm_fresh: fixes c :: "'a"   and pi :: "('a \<times> 'a) list" assumes "at TYPE('a)"   and "c \<sharp> pi" shows "pi \<bullet> c = c"
assms: shows "list_all P as \<and> list_all P as2"
ceuclidean_eqI: fixes x :: "'a"   and y :: "'a" assumes "\<And>b. b \<in> CBasis \<Longrightarrow> \<langle>x, b\<rangle> = \<langle>y, b\<rangle>" shows "x = y"
sel_2: fixes x1 :: "'a"   and x2 :: "'a stream" shows "stl (x1 ## x2) = x2"
sint_range': fixes x :: "'c word" shows "- (2 ^ (LENGTH('c) - Suc 0)) \<le> sint x \<and> sint x < 2 ^ (LENGTH('c) - Suc 0)"
add_cancel_left_right: fixes a :: "'b"   and b :: "'b" shows "(a + b = a) = (b = (0::'b))"
WT: shows "\<I>1, \<I>2 \<turnstile>\<^sub>C CNV callee s \<surd>"
split_aforms_lemma: fixes XS :: "(real \<times> real pdevs) list"   and i :: "nat"   and YS :: "(real \<times> real pdevs) list"   and ZS :: "(real \<times> real pdevs) list"   and xs :: "real list" assumes "split_aforms XS i = (YS, ZS)"   and "xs \<in> Joints XS" shows "xs \<in> Joints YS \<union> Joints ZS"
hurwitz_zeta_conv_suminf: fixes a :: "real"   and s :: "complex" assumes "0 < a"   and "1 < Re s" shows "hurwitz_zeta a s = (\<Sum>n. (of_nat n + complex_of_real a) powr - s)"
Nset2_prep2: fixes x :: "'c"   and A :: "'c set" assumes "x \<in> A" shows "A - {x} \<union> {x} = A"
inner_prod_distrib_left: fixes a :: "complex Matrix.vec"   and b :: "complex Matrix.vec"   and c :: "complex" assumes "dim_vec a = dim_vec b" shows "\<langle>c \<cdot>\<^sub>v a|b\<rangle> = cnj c * \<langle>a|b\<rangle>"
preserves_dom: fixes f :: "'c" assumes "C.arr f" shows "C.dom (\<epsilon> f) = (F \<circ> G) (C.dom f)"
TESL_interpretation_stepwise_zero': fixes \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> 0\<^esup>"
assms_1: shows "Bisimulation_Invariant A B sim PA PB"
rel_intros_1: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R None None"
assms: shows "\<tau> \<star> \<sigma> \<noteq> null"
prefix_prefix: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "prefix xs ys" shows "prefix xs (ys @ zs)"
recurrent_subset_of_recurrent_subset: fixes A :: "'a set" shows "recurrent_subset_infty (recurrent_subset_infty A) = recurrent_subset_infty A"
U0_in_NUM: shows "U\<^sub>0 \<in> NUM"
runit_commutes_with_R: fixes f :: "'a" assumes "B.ide f" shows "B.runit (B.R f) = B.R (B.runit f)"
wtE_transE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "I.wtE \<xi>" shows "Ik.wtE (transE \<xi>)"
select_convs_1: fixes access :: "acc_modi"   and imethods :: "(sig \<times> mhead) list"   and more :: "'b" shows "imethods \<lparr>access = access, imethods = imethods, \<dots> = more\<rparr> = imethods"
cinner_diff_right: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (minus y z) = cinner x y - cinner x z"
rep_eq: fixes x :: "('a, 'b) pp"   and xa :: "('a, 'b) pp" shows "drlex_pp_strict x xa = drlex_pm_strict (pp.mapping_of x) (pp.mapping_of xa)"
assms_3: shows "vectorspace.dim K (vs (local.module.span S)) = card S"
f_simps_3: fixes n :: "nat"   and x :: "'v" shows "f n (\<langle>x\<rangle>, vertex.SINK) = (if x \<in> A \<Gamma> then d_OUT (h n) x - d_OUT (h 0) x else 0)"
vec_of_basis_enum_scaleC: fixes c :: "complex"   and b :: "'b" shows "vec_of_basis_enum (c *\<^sub>C b) = c \<cdot>\<^sub>v vec_of_basis_enum b"
subS: fixes n :: "nat" shows "A n \<subseteq> S"
simps: fixes a :: "'b"   and fs :: "'b list"   and rs :: "'b list" shows "first (a # fs, rs) = a"
assms_1: fixes x :: "'a"   and y :: "'a" assumes "P x" shows "P (app x y)"
tm_sub_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t1 :: "taylor_model"   and y :: "real"   and t2 :: "taylor_model" assumes "x \<in>\<^sub>i range_tm e t1"   and "y \<in>\<^sub>i range_tm e t2" shows "x - y \<in>\<^sub>i range_tm e (tm_sub t1 t2)"
power_add: fixes a :: "'b hrel"   and m :: "nat"   and n :: "nat" shows "power.power II (;;) a (m + n) = power.power II (;;) a m ;; power.power II (;;) a n"
sem\<^sub>e\<^sub>s\<^sub>t_d_eq_sem_st: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<A> :: "('fun, 'var) extstrand_step list" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> \<A> = \<lbrakk>M\<^sub>0; to_st \<A>\<rbrakk>\<^sub>d' \<I>"
se_star_sat_imp_sat: fixes c :: "('c, 'd) conf"   and ls :: "('c, 'd) label list"   and c' :: "('c, 'd) conf" assumes "se_star c ls c'"   and "Conf.sat c'" shows "Conf.sat c"
prrf: shows "primerec (Cn (Suc (Suc (length xs))) rec_not [Cn (Suc (Suc (length xs))) rf (get_fstn_args (Suc (Suc (length xs))) (length xs) @ [recf.id (Suc (Suc (length xs))) (Suc (length xs))])]) (Suc (length (xs @ [Suc w]))) \<and> primerec (recf.id (Suc (length xs)) (length xs)) (length (xs @ [Suc w]))"
IC_bound: shows "\<bar>IC l i (max (te l i) (te m i)) - IC m i (max (te l i) (te m i))\<bar> \<le> \<delta>S"
continuous_def: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "continuous F f = (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
wcode_double_case_le_def: shows "wcode_double_case_le \<equiv> inv_image Recursive.lex_pair wcode_double_case_measure"