hcju/isabelleps huggingface dataset:
path_measure_eq_absc1_new: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and X :: "('s \<times> ('c \<Rightarrow> real) set) stream set"   and Y :: "('s \<times> ('c \<Rightarrow> real)) stream set"   and P :: "('s \<times> ('c \<Rightarrow> real) set) stream \<Rightarrow> bool"   and Q :: "('s \<times> ('c \<Rightarrow> real)) stream \<Rightarrow> bool" assumes "cfg \<in> MDP.valid_cfg"   and "X \<in> sets R_G.St"   and "Y \<in> sets MDP.St"   and "almost_everywhere (R_G.T (absc cfg)) P"   and "almost_everywhere (MDP.T cfg) Q"   and "Measurable.pred R_G.St P"   and "Measurable.pred MDP.St Q"   and "\<And>x y. \<lbrakk>P x; smap abss y = x; x \<in> X\<rbrakk> \<Longrightarrow> y \<in> Y \<and> Q y"   and "\<And>x y. \<lbrakk>Q y; smap abss y = x; y \<in> Y\<rbrakk> \<Longrightarrow> x \<in> X \<and> P x" shows "emeasure (R_G.T (absc cfg)) X = emeasure (MDP.T cfg) Y"
chamber_system_image: shows "g \<turnstile> folding_g.\<C> = folding_g.\<C> \<inter> (g \<turnstile> X)"
flip_subst_trm: fixes y :: "name"   and t :: "trm"   and x :: "name" assumes "atom y \<sharp> t" shows "(x \<leftrightarrow> y) \<bullet> t = subst x (Var y) t"
e1: shows "return_val (expr_sem_rf \<sigma> e1) = expr_sem \<sigma> e1"
simps_2: fixes l :: "'b llist"   and i :: "nat" shows "l \<down> Suc i = (case l of LNil \<Rightarrow> LNil | x ## xs \<Rightarrow> x ## xs \<down> i)"
unwrapS'_def: shows "unwrapS' \<equiv> \<Lambda> f. smap\<cdot>f\<cdot>nats"
merge_on_axioms: shows "merge_on UNIV rha boa ma"
ring_hom_memI: fixes R :: "('a, 'b) ring_scheme"   and h :: "'a \<Rightarrow> 'c"   and S :: "('c, 'd) ring_scheme" assumes "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"   and "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>" shows "h \<in> ring_hom R S"
assms_3: fixes \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" assumes "\<tau> = Tuple \<pi>"   and "strict_subtuple (\<le>) \<pi> \<xi>" shows "P"
sets_PiM_I_countable: fixes I :: "'b set"   and E :: "'b \<Rightarrow> 'c set"   and M :: "'b \<Rightarrow> 'c measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
Union_insert: fixes a :: "'a set"   and B :: "'a set set" shows "\<mu> (insert a B) = a \<union> \<mu> B"
same_values_fo_int: fixes f :: "'a trm \<Rightarrow> 'a trm" assumes "value_is_compatible_with_structure f" shows "fo_interpretation (same_values f)"
run_get_writer: fixes get :: "('d \<Rightarrow> 'a) \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> ('b, 'c, 'a) writerT" shows "run_writer (get_writer get f) = get (\<lambda>s. run_writer (f s))"
induct: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'c list"   and x2 :: "'c list"   and Pa :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "list_emb P x1 x2"   and "\<And>ys. Pa [] ys"   and "\<And>xs ys y. \<lbrakk>list_emb P xs ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa xs (y # ys)"   and "\<And>x y xs ys. \<lbrakk>P x y; list_emb P xs ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa (x # xs) (y # ys)" shows "Pa x1 x2"
inject_2: fixes x2 :: "'n"   and y2 :: "'n" shows "(Inr x2 = Inr y2) = (x2 = y2)"
coboundedI2: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b \<le> c" shows "inf a b \<le> c"
nxt_converg: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and y :: "nat" assumes "total_cons s"   and "eval (total_cons.r_search s) [b] \<down>= y" shows "eval (total_cons.r_next s) [b] \<down>= e_append b (Partial_Recursive.list_encode (replicate (y - e_length b) 0 @ [1 - the (\<phi> (the (s b)) y)]))"
apath_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "sdG.apath u p v \<equiv> sdG.awalk u p v \<and> distinct (pawalk_verts u p)"
E: fixes n :: "nat" shows "B (2 * n) \<le> E * B n"
cp\<^sub>P\<^sub>O\<^sub>S\<^sub>T: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2"   and a3 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3"   and res :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'res"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "POST self a1 a2 a3 res \<tau> = POST (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) (\<lambda>_. a2 \<tau>) (\<lambda>_. a3 \<tau>) (\<lambda>_. res \<tau>) \<tau>"
psubsetE: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subset> B" obtains "A \<subseteq> B" and "\<not> B \<subseteq> A"
AXIOMS_imply_SNa_1: fixes a :: "coname"   and M :: "trm"   and B :: "ty" assumes "<a>:M \<in> AXIOMSc B" shows "SNa M"
AAss_\<tau>red1r_xt3: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and a :: "'addr val"   and i :: "'addr val" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (Val a\<lfloor>Val i\<rceil> := e, xs) (Val a\<lfloor>Val i\<rceil> := e', xs')"
assms_4: shows "ccw' (fst x) (snd x) (P + (Pc + (\<Sum>P\<in>set QRRs. e P *\<^sub>R P)))"
app_equality: fixes f :: "hf"   and x :: "hf"   and y :: "hf" assumes "hfunction f"   and "\<langle>x, y\<rangle> \<^bold>\<in> f" shows "app f x = y"
one_side_not_col123: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "A B OS X Y" shows "\<not> Col A B X"
geom_lgeom: fixes u :: "real"   and d :: "real"   and v :: "real"   and n :: "nat"   and w :: "bool stream" shows "geom_rand_walk u d v n w = lgeom_proc u d v (stake n w)"
mult_le_cancel_left1_hmset: fixes y :: "hmultiset"   and x :: "hmultiset" assumes "0 < y" shows "x \<le> x * y"
Collect_case_prod: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q"
IH: assumes "wf\<^sub>s\<^sub>t {} S"   and "simple S"   and "v \<in> wfrestrictedvars\<^sub>s\<^sub>t S" shows "\<exists>S\<^sub>p\<^sub>r\<^sub>e S\<^sub>s\<^sub>u\<^sub>f. S = S\<^sub>p\<^sub>r\<^sub>e @ send\<langle>Var v\<rangle>\<^sub>s\<^sub>t # S\<^sub>s\<^sub>u\<^sub>f \<and> v \<notin> wfrestrictedvars\<^sub>s\<^sub>t S\<^sub>p\<^sub>r\<^sub>e"
bst_iff_sorted_wrt_less: fixes t :: "'b tree" shows "bst t = Sorted_Less.sorted (inorder t)"
size_mset_mono: fixes A :: "'c multiset"   and B :: "'c multiset" assumes "A \<subseteq># B" shows "size A \<le> size B"
csl': shows "cs\<^bsup>\<pi>'\<^esup> l' = xs @ [\<pi> k] @ ys @ [\<pi>' l']"
l_set_tag_name_get_tag_name_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_tag_name_get_tag_name type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs \<equiv> l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_set_tag_name type_wf set_tag_name set_tag_name_locs \<and> l_set_tag_name_get_tag_name_axioms get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs"
GSMP_disjoint_if_comp_GSMP_disjoint: fixes A' :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and B' :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and C :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and A :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and B :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) A'"   and "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) B'"   and "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) C"   and "comp_GSMP_disjoint public arity Ana \<Gamma> A' B' A B C" shows "GSMP_disjoint (set A') (set B') ({t \<cdot> \<delta> |t \<delta>. t \<in> set C \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> fv (t \<cdot> \<delta>) = {}} - {m. {} \<turnstile>\<^sub>c m})"
that: fixes n :: "'node"   and ns :: "'node list" assumes "g \<turnstile> n-ns\<rightarrow>defNode g v"   and "old.EntryPath g ns"   and "var g v \<in> oldDefs g n" shows "thesis"
set_iterator_sng_foldli_conv: fixes x :: "'b" shows "set_iterator_sng x = foldli [x]"
bound: fixes F :: "'b \<Rightarrow> ennreal"   and s :: "'a" assumes "F \<in> borel_measurable N" shows "integral\<^sup>N (M s) (f F) < \<infinity>"
arr_eqI: fixes t :: "(bool, 'arr list) concrete_category.arr"   and t' :: "(bool, 'arr list) concrete_category.arr" assumes "arr t"   and "arr t'"   and "Dom t = Dom t'"   and "Cod t = Cod t'"   and "Map t = Map t'" shows "t = t'"
plusl_infty_num: fixes x :: "nat" assumes "0 < x" shows "\<infinity>\<^sub>1 +\<^sub>1 [x]\<^sub>1 = \<top>\<^sub>1"
emeasure_eq_ennreal_measure: fixes M :: "'a measure"   and A :: "'a set" assumes "emeasure M A \<noteq> top" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
spmf_of_pmf_pmf_of_set: fixes A :: "'c set" assumes "finite A"   and "A \<noteq> {}" shows "spmf_of_pmf (pmf_of_set A) = spmf_of_set A"
one_le_power: fixes a :: "'a"   and n :: "nat" assumes "(1::'a) \<le> a" shows "(1::'a) \<le> a ^ n"
A1n_commute: shows "A1n ** A = A ** A1n"
diff_union_single_conv: fixes a :: "'c"   and J :: "'c multiset"   and I :: "'c multiset" assumes "a \<in># J" shows "I + J - {#a#} = I + (J - {#a#})"
PPs_closed_minus: fixes s :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]" shows "s - t \<in> .[X]"
d: shows "integral {0..1} (\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) = integral {0..1} (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b))"
finite_eq_tag_rel: fixes tag :: "'c list \<Rightarrow> 'b" assumes "finite (range tag)" shows "finite (UNIV // =tag=)"
ffUnion_funion_distrib: fixes A :: "'c fset fset"   and B :: "'c fset fset" shows "ffUnion (A |\<union>| B) = ffUnion A |\<union>| ffUnion B"
heterogeneous\<^sub>l\<^sub>s\<^sub>t_def: fixes t :: "('fun, 'var) Term.term"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Secrets :: "('fun, 'var) Term.term set" shows "heterogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Secrets \<equiv> \<exists>l1 l2. \<exists>s1\<in>subterms t. \<exists>s2\<in>subterms t. l1 \<noteq> l2 \<and> proj_specific l1 s1 \<A> Secrets \<and> proj_specific l2 s2 \<A> Secrets"
nat_pow_mult: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier (units_of R)" shows "x [^]\<^bsub>units_of R\<^esub> n \<otimes>\<^bsub>units_of R\<^esub> x [^]\<^bsub>units_of R\<^esub> m = x [^]\<^bsub>units_of R\<^esub> (n + m)"
ndiff_Sucs: fixes i :: "nat"   and f :: "real \<Rightarrow> 'a"   and X :: "real set" shows "ndiff (Suc i) f X = (ndiff i f X \<and> (\<forall>x\<in>X. nderiv i f X differentiable at x within X))"
distinct_3lcomprI: fixes as :: "'b list"   and bs :: "'c list"   and cs :: "'d list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "distinct as"   and "distinct bs"   and "distinct cs"   and "\<And>a b c e i g. f a b c = f e i g \<Longrightarrow> a = e \<and> b = i \<and> c = g" shows "distinct (concat (map (\<lambda>a. concat (map (\<lambda>b. map (f a b) cs) bs)) as))"
interleavings_filter_eq2: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "interleavings xs ys zs"   and "(\<forall>x\<in>set xs. \<not> P x) \<and> (\<forall>y\<in>set ys. P y)" shows "filter P zs = ys"
undo_factorize_polys_def: fixes ls :: "(rat \<times> (nat \<times> nat) list) list"   and signs :: "rat list" shows "undo_factorize_polys ls signs = map (\<lambda>l. undo_factorize l signs) ls"
PiE_eq_singleton: fixes I :: "'e set"   and S :: "'e \<Rightarrow> 'f set"   and f :: "'e \<Rightarrow> 'f" shows "(Pi\<^sub>E I S = {restrict f I}) = (\<forall>i\<in>I. S i = {f i})"
rcp_specD4: fixes rcp :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "rcp_spec rcp" shows "component_of_term ` Keys (fst ` set (fst (rcp gs bs ps sps data))) \<subseteq> component_of_term ` Keys (args_to_set (gs, bs, sps))"
simps: fixes s1 :: "('a, 'e, 'f, 'g) SLin_state_scheme"   and s2 :: "('a, 'h, 'i, 'j) SLin_state_scheme" shows "P9 (s1, s2) = (initialized s2 \<longrightarrow> dstate s1 \<preceq> dstate s2)"
language_state_for_inputs_empty: fixes V :: "'d list set"   and M :: "('d, 'e, 'f) FSM"   and q :: "'f" assumes "[] \<in> V" shows "[] \<in> LS\<^sub>i\<^sub>n M q V"
less_omega_exp: fixes x :: "ordinal" assumes "x < \<epsilon>\<^sub>0" shows "x < \<omega> ** x"
genby_uminus_closed: fixes g :: "'a"   and S :: "'a set" assumes "g \<in> \<langle>S\<rangle>" shows "- g \<in> \<langle>S\<rangle>"
hyps_2: shows "\<not> (\<nexists>a b. a ^ 4 + b ^ 4 = x\<^sup>2 \<and> a * b * x \<noteq> 0 \<and> odd a \<and> coprime a b)"
funcset_mem: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and B :: "'h set"   and x :: "'g" assumes "f \<in> A \<rightarrow> B"   and "x \<in> A" shows "f x \<in> B"
hyps: shows "let prefixes = {t. \<exists>t2. x = t @ t2} in finite prefixes \<and> card prefixes = length x + 1"
group_size_lt_v: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and G :: "'b set" assumes "group_design point_set block_collection groups"   and "G \<in> groups" shows "int (card G) < int (card point_set)"
of_rat_val_eval: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> rat" shows "p \<lbrace> of_rat_val v \<rbrace> = real_of_rat p \<lbrace> v \<rbrace>"
hom_one: shows "hom (1::'a) = (1::'b)"
assms_1: shows "A1 = B1 \<union> C1"
d_test_inf_idempotent: fixes x :: "'a" shows "d x \<sqinter> d x = d x"
assoc: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" shows "signed.max (signed.max a b) c = signed.max a (signed.max b c)"
execute_parallel_operator_effect_1: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list"   and op :: "'b strips_operator"   and v :: "'b" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops" shows "op \<in> set ops \<and> v \<in> set (add_effects_of op) \<longrightarrow> execute_parallel_operator s ops v = Some True"
subalgebra_in_carrier: fixes K :: "'a list set set"   and V :: "'a list set set" assumes "subalgebra K V (Rupt A P)" shows "V \<subseteq> carrier (Rupt A P)"
same_sgn_sgn_add: 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "sgn b = sgn a" shows "sgn (plus a b) = sgn a"
distrib: fixes g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "F (\<lambda>x. g x \<^bold>* h x) A = F g A \<^bold>* F h A"
tranclp_unfold: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" shows "r\<^sup>+\<^sup>+ a b \<equiv> (a, b) \<in> {(x, y). r x y}\<^sup>+"
lift2_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b option"   and a :: "'c option"   and b :: "'d option" shows "lift2 f a b \<equiv> case (a, b) of (None, ba) \<Rightarrow> None | (Some a, None) \<Rightarrow> None | (Some a, Some b) \<Rightarrow> f a b"
compactinD: fixes X :: "'c topology"   and S :: "'c set"   and \<U> :: "'c set set" assumes "compactin X S"   and "\<And>U. U \<in> \<U> \<Longrightarrow> openin X U"   and "S \<subseteq> \<Union> \<U>" shows "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> S \<subseteq> \<Union> \<F>"
set_check_generate: fixes a :: "nat list"   and b :: "nat list" shows "set (check_generate a b) = {(x, y). (x, y) \<noteq> (zeroes (length a), zeroes (length b)) \<and> length x = length a \<and> length y = length b \<and> a \<bullet> x = b \<bullet> y \<and> (\<forall>i<length a. x ! i \<le> Max (set b)) \<and> (\<forall>j<length b. y ! j \<le> Max (set a))}"
lin_tss': shows "linearize tss' = linearize tss @ [nt]"
H2list_list2H_inv: shows "H2list oo list2H = ID"
gauss_sum_nonzero_noncoprime_necessary_condition_3: fixes k :: "nat"   and a :: "nat" assumes "gauss_sum k \<noteq> 0"   and "\<not> coprime k n"   and "0 < k"   and "coprime a n"   and "[a = 1] (mod n div gcd k n)" shows "\<chi> a = 1"
full_gb_is_full_pmdl: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "is_full_pmdl (fst ` set (full_gb bs))"
find_first_some_strong: fixes x :: "'b"   and n :: "nat"   and xs :: "'b list" assumes "x \<in> set (take n xs)"   and "n \<le> length xs" obtains i :: "nat" where "find_first x xs = Some i" and "i < n"
matched_Append: fixes n'' :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node"   and ns :: "'node SDG_node list" assumes "matched n'' ns' n'"   and "matched n ns n''" shows "matched n (ns @ ns') n'"
subsumes_list_filterI: fixes Ls :: "('a, 'b) Term.term literal list"   and P :: "('a, 'c) Term.term literal \<Rightarrow> bool"   and Ks :: "('a, 'c) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term option" assumes "\<And>L K \<sigma> \<tau>. \<lbrakk>L \<in> set Ls; match_term_list [(atm_of L, atm_of K)] \<sigma> = Some \<tau>; is_pos L = is_pos K\<rbrakk> \<Longrightarrow> P K"   and "subsumes_list Ls Ks \<sigma>" shows "subsumes_list Ls (filter P Ks) \<sigma>"
Abs_bit0_inverse: fixes y :: "int" assumes "y \<in> {0..<2 * int CARD('a)}" shows "Rep_bit0 (Abs_bit0 y) = y"
pls: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "pls t1 t2 \<in> trm"
H2': fixes A :: "('a, int mpoly) fmap"   and x11 :: "'a"   and x12 :: "'a"   and r :: "int mpoly"   and Aa :: "'a multiset" assumes "the (fmlookup A x11) + the (fmlookup A x12) - r \<in> More_Modules.ideal polynomial_bool"   and "B = add_mset (the (fmlookup A x11)) {#the (fmlookup A x). x \<in># Aa#}"   and "the (fmlookup A x11) + the (fmlookup A x12) - r \<in> More_Modules.ideal (insert (the (fmlookup A x11)) ((\<lambda>x. the (fmlookup A x)) ` set_mset Aa \<union> polynomial_bool)) \<Longrightarrow> - r \<in> More_Modules.ideal (insert (the (fmlookup A x11)) ((\<lambda>x. the (fmlookup A x)) ` set_mset Aa \<union> polynomial_bool))" shows "r \<in> pac_ideal (insert (the (fmlookup A x11)) ((\<lambda>x. the (fmlookup A x)) ` set_mset Aa))"
E'_prop2: shows "E' = backflows (filternew_flows_state \<T>) - E"
pdevs_apply_msum_slist: fixes m :: "nat"   and xs :: "(nat, 'd) slist"   and ys :: "(nat, 'd) slist"   and i :: "nat" shows "pdevs_apply (Pdevs (msum_slist m xs ys)) i = (if i < m then pdevs_apply (Pdevs xs) i else pdevs_apply (Pdevs ys) (i - m))"
ot14_base: fixes n :: "nat"   and R1_12_0T :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT121 spmf"   and R2_12OT :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT122 spmf" shows "ot14_base (S1_OT12 n) (R1_12_0T n) (adv_OT12 n) (S2_OT12 n) (R2_12OT n) protocol_OT12"
L1_space: fixes M :: "'b measure" shows "space\<^sub>N (\<LL> 1 M) = {f. integrable M f}"
fold_simps_1: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c" shows "fold f [] s = s"
prod_root_unity_def: fixes ns :: "nat list" shows "prod_root_unity ns = prod_list (map root_unity ns)"
rep_eq: fixes x :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b pdevs" shows "pdevs_apply (filter_pdevs x xa) = filter_pdevs_raw x (pdevs_apply xa)"
orbit_disjoint: fixes f :: "'b perm"   and a :: "'b"   and b :: "'b" assumes "orbit f a \<noteq> orbit f b" shows "orbit f a \<inter> orbit f b = {}"
FG_suffix: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'a set"   and \<psi> :: "'a ltln" shows "suffix i w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)"
assms_2: shows "p $ subdegree p * inverse (q $ 0) \<noteq> (0::'a)"
five_segment: fixes d :: "'p"   and d' :: "'p" shows "\<forall>a b c a' b' c'. a \<noteq> b \<and> Bet a b c \<and> Bet a' b' c' \<and> Cong a b a' b' \<and> Cong b c b' c' \<and> Cong a d a' d' \<and> Cong b d b' d' \<longrightarrow> Cong c d c' d'"
inj_on_exp_pi: fixes z :: "complex" shows "inj_on exp (ball z pi)"
nat_pow_one: fixes n :: "nat" shows "[n] \<cdot>\<^bsub>V\<^esub> 0\<^sub>v n = 0\<^sub>v n"
1: shows "igSwapInpIPresIGWlsInpSTR (errMOD MOD) \<and> igSwapBinpIPresIGWlsBinpSTR (errMOD MOD)"
prems_1: shows "{a. case a of (t, X) \<Rightarrow> (t, X) \<in> set (x # xs) \<and> as \<in> Z X t} \<noteq> {}"
stateSpace_def: fixes ss :: "('c, 'd) fmap set"   and vs :: "'c set" shows "stateSpace ss vs \<equiv> \<forall>s. s \<in> ss \<longrightarrow> fmdom' s = vs"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_locate_n_b (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 tn m ml mr rn. lm @ 0 \<up> tn = lm1 @ [m] @ lm2 \<and> length lm1 = s \<and> m + 1 = ml + mr \<and> ml = 1 \<and> tn = s + 1 - length lm \<and> (if lm1 = [] then l = Oc \<up> ml @ Bk # Bk # ires else l = Oc \<up> ml @ Bk # <rev lm1> @ Bk # Bk # ires) \<and> (r = Oc \<up> mr @ [Bk] @ <lm2> @ Bk \<up> rn \<or> lm2 = [] \<and> r = Oc \<up> mr))"
not_final_thread_iff: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "r1.not_final_thread s t = (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (\<not> final1 x \<or> ln \<noteq> no_wait_locks \<or> (\<exists>w. wset s t = \<lfloor>w\<rfloor>)))"
e1: shows "\<guillemotleft>e1 : local.dom e0 \<rightarrow> local.dom f1\<guillemotright>"
inv_character: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" assumes "character G \<chi>" shows "character G (inv_character \<chi>)"
genPolyTame_notame: fixes g' :: "graph"   and n :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph" assumes "g' \<in> set (generatePolygon n v f g)"   and "g' \<notin> set (generatePolygonTame n v f g)"   and "Invariants.inv g"   and "3 \<le> n" shows "notame7 g'"