hcju/isabelleps huggingface dataset:
representation_scale: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and v :: "'d"   and r :: "'c" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis" shows "module.representation scale basis (scale r v) = (\<lambda>b. r * module.representation scale basis v b)"
intro: fixes b :: "'a vec"   and m :: "nat"   and c :: "'a vec"   and n :: "nat"   and A :: "'a mat" assumes "b \<in> carrier_vec m"   and "c \<in> carrier_vec n"   and "A \<in> carrier_mat m n" shows "abstract_LP A b c m n"
preImage_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set" shows "preImage f A = {x. f x \<in> A}"
obs_a_convert_extTA: fixes f :: "'i \<Rightarrow> 'f"   and ta :: "('d, 'e, 'i, 'g, 'h, 'c) thread_action" shows "\<lbrace>convert_extTA f ta\<rbrace>\<^bsub>o\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>"
2: shows "enat i < llength (lselect (nth_least s ` t) w)"
float_distinct_1: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_nan a \<and> is_infinity a)"
order_less_le: fixes x :: "'b"   and y :: "'b" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
fresh_4: fixes a :: "name"   and x3 :: "trm"   and x1 :: "name"   and x2 :: "trm" shows "a \<sharp> x3 to x1 in x2 = (a \<sharp> x3 \<and> a \<sharp> [x1].x2)"
dc: shows "density_context (shift_var_set (set vs)) (Suc ` set vs') (case_nat t' \<Gamma>) (\<lambda>x. ennreal (extract_real (cexpr_sem x (map_vars Suc \<delta> *\<^sub>c f))))"
lookup_tensors_ht_l1: fixes j :: "nat"   and r1 :: "nat"   and "is" :: "nat list"   and M :: "nat"   and r0 :: "nat" assumes "j < r1"   and "is \<lhd> [M, M]" shows "lookup (tensors_from_net (witness r1 r0 [M]) $ j) is = (if is ! 0 = is ! 1 \<and> is ! 0 < r0 then 1 else 0)"
diff_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and d :: "'a"   and c :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b"   and "less_eq d c" shows "less_eq (minus a c) (minus b d)"
has_ticket_simps_1: fixes k :: "nat" shows "\<not> has_ticket INIT k"
equivalence_in_bicategory_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" shows "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<equiv> adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<and> equivalence_in_bicategory_axioms V \<eta> \<epsilon>"
supI2: fixes B :: "'a pred"   and x :: "'a"   and A :: "'a pred" assumes "eval B x" shows "eval (A \<squnion> B) x"
zero_reducibility_implies_lt_divisibility': fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and f :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* f 0"   and "f \<noteq> 0" shows "\<exists>h\<in>F. h \<noteq> 0 \<and> local.punit.adds_term (punit.lt h) (punit.lt f)"
is_regular_spairI: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rep_list p \<noteq> 0"   and "rep_list q \<noteq> 0"   and "punit.lt (rep_list q) \<oplus> lt p \<noteq> punit.lt (rep_list p) \<oplus> lt q" shows "is_regular_spair p q"
allDefs_var_disjoint_aux: fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> defs g n" shows "n \<notin> phiDefNodes g v"
Gt_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A B Gt C D \<equiv> C D Lt A B"
idem: fixes a :: "'a\<^sub>m" shows "m a a = Some a"
distrPUTLG1: fixes x :: "'c"   and P :: "'c \<Rightarrow> 'd decision option"   and PL :: "('c \<Rightarrow> 'd decision option) list"   and PUT :: "'c \<Rightarrow> 'd decision option" assumes "x \<in> dom P"   and "list2policy PL x = P x"   and "PUTListGD PUT x PL" shows "PUT x = P x"
mono_monitor: fixes \<pi> :: "(char list \<times> 'a list) prefix"   and \<pi>' :: "(char list \<times> 'a list) prefix" assumes "\<pi> \<le> \<pi>'" shows "M \<pi> \<subseteq> M \<pi>'"
OppositeCategory_def: fixes C :: "('d, 'e, 'f) Category_scheme" shows "Op C \<equiv> \<lparr>Obj = obj\<^bsub>C\<^esub>, Mor = mor\<^bsub>C\<^esub>, Dom = Cod C, Cod = Dom C, Id = Category.Id C, Comp = \<lambda>f g. g ;;\<^bsub>C\<^esub> f, \<dots> = more C\<rparr>"
inject: fixes x :: "'c"   and ya :: "'c" shows "(dual x = dual ya) = (x = ya)"
admissible_not_below: fixes y :: "'a" shows "ccpo.admissible Sup (\<le>) (\<lambda>x. \<not> x \<le> y)"
Infi_greatest: fixes X :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" shows "z \<le> \<Sqinter> X"
igSubstIPresIGWls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSubstPresGWls MOD" shows "igSubstIPresIGWls (fromMOD MOD)"
elements_mat_def: fixes A :: "'b mat" shows "elements_mat A = set (concat (map (\<lambda>i. map (\<lambda>j. A $$ (i, j)) [0..<dim_col A]) [0..<dim_row A]))"
REMOVE_INTERNAL_def: fixes x :: "'a" shows "REMOVE_INTERNAL x \<equiv> x"
map_poly_inj_idom_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "map_poly_inj_idom_hom hom \<equiv> inj_idom_hom hom"
valid: shows "G,{}\<Turnstile>{P} t\<succ> {Q}"
a: shows "\<exists>u. vNM_utility outcomes \<R> u"
simplex_like_pow_above_in: fixes A :: "'b set"   and X :: "'b set" assumes "finite A"   and "order.greater_eq A X" shows "dual_order.simplex_like (dual_order.below_in (Pow A) X)"
lowner_le_smult: fixes c :: "real"   and A :: "complex mat"   and B :: "complex mat"   and n :: "nat" assumes "0 \<le> c"   and "A \<le>\<^sub>L B"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "complex_of_real c \<cdot>\<^sub>m A \<le>\<^sub>L complex_of_real c \<cdot>\<^sub>m B"
simps_5: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e error_result \<Rightarrow> 'c"   and x1 :: "'d" shows "(case Rval x1 of Rval x \<Rightarrow> f1 x | Rerr x \<Rightarrow> f2 x) = f1 x1"
disjI1: fixes s :: "Store"   and x :: "Value"   and y :: "Value" assumes "\<And>l. s\<turnstile> l reachable_from x \<Longrightarrow> \<not> s\<turnstile> l reachable_from y" shows "StoreProperties.disj x y s"
prod_simps_3: fixes f0 :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f1 :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f0"   and "arr f1" shows "cod (HF'.prod f0 f1) = HF'.prod (cod f0) (cod f1)"
arg: fixes f :: "'b"   and m :: "nat"   and ts :: "'a list"   and t :: "'a"   and P :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and s :: "'a" assumes "(f, m) \<in> F"   and "length ts = m"   and "\<forall>t\<in>set ts. t \<in> trees"   and "t \<in> set ts"   and "emb P s t" shows "emb P s (mk f ts)"
step: fixes x :: "'a" assumes "x \<in> Field r"   and "\<And>y. y \<in> aboveS r x \<Longrightarrow> P y" shows "P x"
ZObisT_def: shows "ZObisT \<equiv> bis ZOretrT"
subspace_def: fixes S :: "'b set" shows "m1.subspace S = ((0::'b) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *a x \<in> S))"
simps_6: fixes f :: "'e \<Rightarrow> 'd"   and x1 :: "'e"   and x2 :: "'e stream" shows "smap f (x1 ## x2) = f x1 ## smap f x2"
scale: fixes x :: "'a"   and u :: "real" assumes "x \<in> S"   and "0 \<le> u"   and "u < 1" shows "u *\<^sub>R x \<in> S - frontier S"
closure_sequential: fixes l :: "'a"   and S :: "'a set" shows "(l \<in> closure S) = (\<exists>x. (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> l)"
that: fixes A\<^sub>F' :: "name list" assumes "\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle> = \<langle>A\<^sub>F', \<Psi>\<^sub>F\<rangle>"   and "distinct A\<^sub>F'"   and "A\<^sub>F' \<sharp>* C" shows "thesis"
assms: fixes n :: "nat" shows "norm (f n) < 1 / real (Suc n)"
sup_inf_distrib_2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<cdot> z + x = (y + x) \<cdot> (z + x)"
sdrop_if_suntil: fixes p :: "'b stream \<Rightarrow> bool"   and q :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" assumes "(p suntil q) \<omega>" shows "\<exists>j. q (sdrop j \<omega>) \<and> (\<forall>k<j. p (sdrop k \<omega>))"
Nmlize_Nmlize: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
five: fixes xs :: "('b com \<times> 'b set \<times> ('b \<times> 'b) set \<times> ('b \<times> 'b) set \<times> 'b set) list"   and rely :: "('b \<times> 'b) set"   and pre :: "'b set"   and post :: "'b set"   and x :: "('b com option list \<times> 'b) list"   and s :: "'b" assumes "xs \<noteq> []"   and "\<forall>i<length xs. rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j)) \<subseteq> Rely (xs ! i)"   and "pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i))"   and "(\<Inter>i\<in>{i. i < length xs}. Post (xs ! i)) \<subseteq> post"   and "\<forall>i<length xs. \<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)]"   and "x \<in> par_cp (ParallelCom xs) s"   and "x \<in> par_assum (pre, rely)"   and "All_None (fst (last x))" shows "snd (last x) \<in> post"
all_final_except_mono': fixes ts :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and A :: "'b set"   and B :: "'b set" assumes "all_final_except ts A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "all_final_except ts B"
isomorphic_group_reduced_homology_of_contractible: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "contractible_space X"   and "topspace X \<inter> S \<noteq> {}" shows "reduced_homology_group p (subtopology X S) \<cong> relative_homology_group (p + 1) X S"
1: shows "FvarsT (rawpsubstT \<chi> (zip (map fst txs) us)) = FvarsT \<chi> - set us \<union> \<Union> {if u \<in> FvarsT \<chi> then FvarsT t else {} |t u. (t, u) \<in> set (zip (map fst txs) us)}"
convex_polychain_half_segments_of_aform: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" shows "convex_polychain (half_segments_of_aform X)"
prod_mOp_mem_i: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and X :: "'c \<Rightarrow> 'd"   and i :: "'c" assumes "\<forall>j\<in>I. aGroup (A j)"   and "X \<in> carr_prodag I A"   and "i \<in> I" shows "prod_mOp I A X i = -\<^sub>a\<^bsub>A i\<^esub> X i"
null_measure_def: fixes M :: "'b measure" shows "null_measure M = sigma (space M) (sets M)"
case_3: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Gt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f3"
subseqs_refl: fixes xs :: "'b list" shows "xs \<in> ssubseqs xs"
e: shows "x = y \<otimes> (inv y \<otimes> x)"
distinct_1: fixes x1 :: "nat"   and x21 :: "'b"   and x22 :: "'b form list" shows "At x1 \<noteq> Compound x21 x22"
prime_elem_multiplicity_prod_distrib: fixes p :: "'a"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "prime_elem p"   and "(0::'a) \<notin> f ` A"   and "finite A" shows "multiplicity p (prod f A) = (\<Sum>x\<in>A. multiplicity p (f x))"
alice_result: fixes \<phi> :: "complex Matrix.mat"   and \<alpha> :: "complex"   and \<beta> :: "complex" assumes "state 1 \<phi>"   and "\<alpha> = \<phi> $$ (0, 0)"   and "\<beta> = \<phi> $$ (1, 0)" shows "alice \<phi> = Tensor.mat_of_cols_list 8 [[\<alpha> / 2, \<beta> / 2, \<beta> / 2, \<alpha> / 2, \<alpha> / 2, - \<beta> / 2, - \<beta> / 2, \<alpha> / 2]]"
spoly_repI_zero: fixes g1 :: "'t \<Rightarrow>\<^sub>0 'c"   and g2 :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "spoly g1 g2 = 0" shows "spoly_rep d m G g1 g2"
nil_less2: fixes t :: "'c list"   and a :: "'c" shows "[] < t @ [a]"
assms: shows " N \<longrightarrow>\<^sub>1 N'"
step_elim_cases_10: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "('b, 'c) xstate"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Throw, s) \<rightarrow> u" obtains f :: "'c" where "s = Fault f__" and "u = (Skip, Fault f__)"   | "s = Stuck" and "u = (Skip, Stuck)"   | f :: "'b" where "s = Abrupt f" and "u = (Skip, Abrupt f)"
exhaust: fixes y :: "val" obtains "y = Unit"   | x2 :: "bool" where "y = Bool x2"   | x3 :: "int" where "y = Intg x3"   | "y = Null"   | x5 :: "loc" where "y = Addr x5"
some_in_set_if_set_neq_vempty: fixes A :: "V" assumes "A \<noteq> []\<^sub>\<circ>" shows "(SOME x. x \<in>\<^sub>\<circ> A) \<in>\<^sub>\<circ> A"
cpoly_of_decompose: fixes p :: "complex poly" shows "p = cpoly_of (map_poly Re p) (map_poly Im p)"
Pi_mono: fixes A :: "'g set"   and B :: "'g \<Rightarrow> 'h set"   and C :: "'g \<Rightarrow> 'h set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi A B \<subseteq> Pi A C"
sum_list_nlex_eq_zero_iff: fixes xs :: "(real \<times> real) list" assumes "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "(sum_list xs = 0) = list_all ((=) 0) xs"
fun_ord_def: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b" shows "fun_ord ord f g = (\<forall>x. ord (f x) (g x))"
OclIsModifiedOnly_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> Set('e)" shows "X->oclIsModifiedOnly() \<equiv> \<lambda>(\<sigma>, \<sigma>'). let X' = oid_of ` \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X (\<sigma>, \<sigma>'))\<rceil>\<rceil>; S = dom (heap \<sigma>) \<inter> dom (heap \<sigma>') - X' in if (\<delta> X) (\<sigma>, \<sigma>') = true (\<sigma>, \<sigma>') \<and> (\<forall>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X (\<sigma>, \<sigma>'))\<rceil>\<rceil>. x \<noteq> null) then \<lfloor>\<lfloor>\<forall>x\<in>S. heap \<sigma> x = heap \<sigma>' x\<rfloor>\<rfloor> else invalid (\<sigma>, \<sigma>')"
cond_spmf_fst_bind: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> ('b \<times> 'a) spmf"   and x :: "'b" shows "cond_spmf_fst (p \<bind> f) x = cond_bind_spmf_fst p (map_spmf fst \<circ> f) x \<bind> (\<lambda>y. cond_spmf_fst (f y) x)"
assms_2: shows "g \<in> dgrad_sig_set' (length fs) d"
wp_strengthen_modset: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "wp \<pi> c Q s" shows "wp \<pi> c (\<lambda>s'. Q s' \<and> modifies (lhsv \<pi> c) s' s) s"
phantom_rel_unop_def: fixes fid :: "'f \<Rightarrow> 'e"   and x :: "'f option" shows "phantom_rel_unop fid x = (case x of None \<Rightarrow> None | Some x \<Rightarrow> Some (fid x))"
iprefix_length_le_imp_prefix: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "nat \<Rightarrow> 'b" assumes "xs \<sqsubseteq> ys \<frown> f"   and "length xs \<le> length ys" shows "prefix xs ys"
coeff_poly_shift: fixes n :: "nat"   and p :: "'b poly"   and i :: "nat" shows "coeff (poly_shift n p) i = coeff p (i + n)"
e_in: shows "E \<in> concl_of ` F.Inf_between N {C}"
map_add_dom_app_simps_3: fixes m :: "'c"   and l2 :: "'c \<Rightarrow> 'd option"   and l1 :: "'c \<Rightarrow> 'd option" assumes "m \<notin> dom l2" shows "(l1 ++ l2) m = l1 m"
simps: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list"   and a2 :: "char list" shows "subcls1p P a1 a2 = (\<exists>C Bs rest D. a1 = C \<and> a2 = D \<and> class P C = \<lfloor>(Bs, rest)\<rfloor> \<and> D \<in> baseClasses Bs)"
Output: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<Longrightarrow>\<^sup>^a[b] \<prec> P"
refl_trans_closure_of_indRelT: fixes TRel :: "('procT \<times> 'procT) set" assumes "refl TRel" shows "indRelTEQ TRel = (indRelT TRel)\<^sup>*"
degree_eq: fixes p :: "real mpoly"   and q :: "real mpoly"   and x :: "nat" assumes "p = q" shows "MPoly_Type.degree p x = MPoly_Type.degree q x"
Mf_factorization_m: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "factorization_m f (Mf cfs) = factorization_m f cfs"
root_main'_newton_pos: fixes x :: "int"   and n :: "int"   and y :: "int"   and b :: "bool" assumes "0 \<le> x"   and "0 \<le> n" shows "root_main' x n = root_newton_int_main x n \<and> (root_main' x n = (y, b) \<longrightarrow> 0 \<le> y \<and> y ^ p \<le> n \<and> b = (y ^ p = n))"
closed_imp_locally_compact: fixes S :: "'b set" assumes "closed S" shows "locally compact S"
yield\<^sub>C_def: fixes f :: "'a \<Rightarrow> 'b" shows "yield\<^sub>C f \<equiv> \<lambda>\<sigma>. Some (f \<sigma>, \<sigma>)"
simps_8: shows "pratt1_max \<infinity>\<^sub>1 \<infinity>\<^sub>1 = \<infinity>\<^sub>1"
db\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('e, 'f) stateful_strand_step list"   and B :: "('e, 'f) stateful_strand_step list"   and I :: "'f \<Rightarrow> ('e, 'f) Term.term"   and D :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list" shows "db'\<^sub>s\<^sub>s\<^sub>t (A @ B) I D = db'\<^sub>s\<^sub>s\<^sub>t B I (db'\<^sub>s\<^sub>s\<^sub>t A I D)"
induct: fixes P :: "'b \<Rightarrow> 'b poly \<Rightarrow> 'b poly \<Rightarrow> 'b poly \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "'b poly"   and a2 :: "'b poly"   and a3 :: "'b poly"   and a4 :: "nat"   and a5 :: "nat" assumes "\<And>lc q r d dr n. (\<And>x xa xb xc. \<lbrakk>x = smult lc r; xa = coeff r dr; xb = x - monom xa n * d; xc = smult lc q + monom xa n\<rbrakk> \<Longrightarrow> P lc xc xb d (dr - 1) n) \<Longrightarrow> P lc q r d dr (Suc n)"   and "\<And>lc q r d dr. P lc q r d dr 0" shows "P a0 a1 a2 a3 a4 a5"
pair_term: fixes p :: "'a \<times> nat" shows "pair_of_term (term_of_pair p) = p"
lookup_pp_of_fun: fixes f :: "'b \<Rightarrow> 'c" assumes "finite {x. f x \<noteq> (0::'c)}" shows "lookup_pp (pp_of_fun f) = f"
product_nn_integral_insert: fixes M :: "'a \<Rightarrow> 'b measure"   and I :: "'a set"   and i :: "'a"   and f :: "('a \<Rightarrow> 'b) \<Rightarrow> ennreal" assumes "product_sigma_finite M"   and "finite I"   and "i \<notin> I"   and "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)" shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x(i := y)) \<partial>M i \<partial>Pi\<^sub>M I M"
ord_inj_func: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Order D"   and "Order E"   and "ord_inj D E f" shows "f \<in> carrier D \<rightarrow> carrier E"
2: shows "b1 \<union> b2 \<subseteq> subspace_sum W1 W2"
simps_1: fixes t :: "'a tree23" shows "T_join_all (T t) = 1"
l'_p: shows "\<forall>l\<in>L. l \<cdot>\<^sub>l \<sigma> = l'"
simps_7: fixes f :: "nat \<Rightarrow> atom \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and var :: "nat" shows "liftmap f (ExQ \<phi>) var = ExQ (liftmap f \<phi> (var + 1))"
nth_first_ind: fixes xs :: "'b list"   and x :: "'b"   and i :: "nat" assumes "distinct xs"   and "x \<in> set xs" shows "xs ! (first_ind xs x i - i) = x"
inj_on_map_eq_map: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and ys :: "'b list" assumes "inj_on f (set xs \<union> set ys)" shows "(map f xs = map f ys) = (xs = ys)"
simps_1: fixes Y :: "nat" shows "deep_model' Y [] = Input Y"
ipoly_add_rat: fixes r :: "rat"   and p :: "int poly"   and x :: "'b" shows "ipoly (poly_add_rat r p) x = of_int (snd (quotient_of r)) ^ degree p * ipoly p (x - of_rat r)"
q: fixes y :: "'b" assumes "y \<in> set_pmf q" shows "(\<Sum>\<^sup>+ x\<in>set_pmf p. h x y) = ennreal (pmf q y)"
exec_mbindFStop_success: fixes ioprog :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and a :: "'f"   and \<sigma> :: "'e"   and b :: "'d"   and \<sigma>' :: "'e"   and S :: "'f list"   and M :: "'d list \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "ioprog a \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p (a # S) ioprog) M) = (\<sigma>' \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p S ioprog) (M (b # s)))"
image_eqvt: fixes p :: "perm"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "p \<bullet> f ` A = (p \<bullet> f) ` (p \<bullet> A)"
Un_left_absorb: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> (A \<union> B) = A \<union> B"
vars_mset_subst_subseteq: fixes s :: "('b, 'a) tm"   and t :: "('b, 'a) tm"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm" assumes "vars_mset s \<subseteq># vars_mset t" shows "vars_mset (subst \<rho> s) \<subseteq># vars_mset (subst \<rho> t)"
step_weight: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat"   and \<pi> :: "nat \<Rightarrow> nat"   and \<phi> :: "nat \<Rightarrow> nat" assumes "f \<in> local.F_inj"   and "g \<in> local.F"   and "\<pi> permutes {0..<n}"   and "\<phi> permutes {0..<n}"   and "\<forall>x\<in>{0..<n}. f x = g (\<phi> x)" shows "local.weight f \<pi> = signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) * signof (\<pi> \<circ> inv \<phi>) * (\<Prod>i = 0..<n. B $$ (g i, (\<pi> \<circ> inv \<phi>) i))"
carrier_vs_is_self: fixes W :: "'e set" shows "carrier (W.vs W) = W"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "monoid_add_hom hom" shows "group_add_hom hom"
intro: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state" assumes "\<And>t. \<triangle> t \<Longrightarrow> \<triangle> (\<T> (init t))"   and "\<And>t v. \<triangle> t \<Longrightarrow> v \<Turnstile>\<^sub>t t = v \<Turnstile>\<^sub>t \<T> (init t)"   and "\<And>t. \<triangle> t \<Longrightarrow> \<not> \<U> (init t) \<longrightarrow> \<Turnstile> (init t)"   and "\<And>t. \<triangle> t \<Longrightarrow> \<U> (init t) \<longrightarrow> minimal_unsat_state_core (init t)"   and "\<And>t. \<triangle> t \<Longrightarrow> {} \<doteq> \<B> (init t)"   and "\<And>t. \<triangle> t \<Longrightarrow> {} \<Turnstile>\<^sub>i \<B>\<I> (init t)" shows "Init' init"
snapshot_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" shows "snapshot PROB s = {a |a. a \<in> PROB \<and> agree (fst a) s \<and> agree (snd a) s}"
plusU_assoc: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c"   and plusU :: "udom\<cdot>'c \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c"   and a :: "udom\<cdot>'c"   and b :: "udom\<cdot>'c"   and c :: "udom\<cdot>'c" assumes "class.functor_plus fmapU plusU" shows "plusU\<cdot>(plusU\<cdot>a\<cdot>b)\<cdot>c = plusU\<cdot>a\<cdot>(plusU\<cdot>b\<cdot>c)"
add_lattice: fixes x :: "'a"   and y :: "'a" shows "x + y = max x y + min x y"
lexord_append_left_rightI: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and u :: "'b list"   and x :: "'b list"   and y :: "'b list" assumes "(a, b) \<in> r" shows "(u @ a # x, u @ b # y) \<in> lexord r"
lv_1: shows "(\<lambda>n. v n / real n) \<longlonglongrightarrow> lv"
cupcake_single_preserve0: fixes env :: "v sem_env"   and e :: "exp0"   and res :: "(v, v) result" assumes "cupcake_evaluate_single env e res"   and "is_cupcake_all_env env"   and "is_cupcake_exp e" shows "if_rval is_cupcake_value res"
simps_15: fixes v :: "MIR.num" shows "qfree (NEq v) = True"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. S \<in>S a1 \<and> S \<in>S a2) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> Q \<and> Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> R))"