hcju/isabelleps huggingface dataset:
natToVertexList_pre_subdivFace_face: fixes f :: "face"   and v :: "nat"   and es :: "nat list" assumes "\<not> final f"   and "distinct (vertices f)"   and "v \<in> \<V> f"   and "2 < |es|"   and "incrIndexList es |es| |vertices f|" shows "pre_subdivFace_face f v (natToVertexList v f es)"
new_msg_in_set_implies_occurrence: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and m :: "'m message"   and i :: "nat"   and p :: "'p"   and q :: "'p" assumes "c \<turnstile> ev \<mapsto> c'"   and "m \<notin> set (msgs c i)"   and "m \<in> set (msgs c' i)"   and "channel i = Some (p, q)" shows "occurs_on ev = p"
bij_betw_onto: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "f ` A = B"
tendsto_at_top_translateI: fixes f :: "real \<Rightarrow> 'c"   and l :: "'c"   and t :: "real" assumes "(f \<longlongrightarrow> l) at_top" shows "((\<lambda>x. f (x + t)) \<longlongrightarrow> l) at_top"
ras_set_def: fixes s :: "'c array \<times> nat"   and i :: "nat"   and x :: "'c" shows "ras_set s i x \<equiv> let (a, n) = s in (array_set a (n - (i + 1)) x, n)"
t: shows "t = (p\<^sub>l, is\<^sub>l, xs\<^sub>l, sb\<^sub>l, \<D>\<^sub>l, \<O>\<^sub>l, \<R>\<^sub>l)"
simps_4: fixes x21 :: "'b list set"   and x22 :: "'b rexp"   and x1 :: "'b rexp" shows "Trn x21 x22 \<noteq> Lam x1"
length_polychain_of_eq: fixes p :: "'a"   and qs :: "'a list" shows "length (polychain_of p qs) = length qs"
assms_2: shows "stake (length \<pi>') \<sigma> = \<pi>'"
quotient_is_ring: shows "ring (R Quot (\<Inter>i\<le>n. I (Suc i)))"
dist_last: shows "var_dist (rev [0..<Suc n]) (liftn (Suc n) t 0) = var_dist (rev [1..<Suc n]) (liftn (Suc n) t 0) \<degree> Var 0"
basis_indep: fixes X :: "'a set" assumes "Y_insert.basis X" shows "indep_in (insert x Y) X"
t_wise_balance_axioms: shows "t_wise_balance \<V> \<B> \<t> \<Lambda>\<^sub>t"
setdist_singletons: fixes x :: "'a"   and y :: "'a" shows "setdist {x} {y} = dist x y"
least_squares_approximation: fixes S :: "'a set"   and X :: "'a set"   and v :: "'a"   and y :: "'a" assumes "subspace S"   and "independent X"   and "X \<subseteq> S"   and "S \<subseteq> span X"   and "pairwise orthogonal X"   and "proj_onto v X \<noteq> y"   and "y \<in> S" shows "norm (v - proj_onto v X) < norm (v - y)"
subset_mset_imp_subset_add_mset: fixes A :: "'a multiset"   and B :: "'a multiset"   and x :: "'a" assumes "A \<subseteq># B" shows "A \<subseteq># add_mset x B"
coprime_ideals_def: fixes R :: "('d, 'e) aGroup_scheme"   and A :: "'d set"   and B :: "'d set" shows "coprime_ideals R A B = (A \<minusplus>\<^bsub>R\<^esub> B = carrier R)"
Abs_fset_cases: fixes x :: "'b fset" obtains y :: "'b set" where "x = Abs_fset y" and "y \<in> {A. finite A}"
relation_of: fixes x :: "('c, 'd) action" shows "action.relation_of x \<in> {p. is_CSP_process p}"
term_variants_pred_wf_trms: fixes P :: "('fun, 'atom, 'sets) prot_fun \<Rightarrow> ('fun, 'atom, 'sets) prot_fun list"   and s :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term" assumes "term_variants_pred P s t"   and "\<And>f g. g \<in> set (P f) \<Longrightarrow> arity f = arity g"   and "wf\<^sub>t\<^sub>r\<^sub>m s" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
map_iterator_abs_rule_P: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and m :: "'h \<Rightarrow> 'f option"   and I :: "'h set \<Rightarrow> 'g \<Rightarrow> bool"   and \<sigma>0 :: "'g"   and c :: "'g \<Rightarrow> bool"   and f :: "'h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g"   and P :: "'g \<Rightarrow> bool" assumes "map_iterator_abs \<alpha> invar iti m"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; invar v; m k = Some (\<alpha> v); it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"
PiE_E: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and x :: "'d" assumes "f \<in> Pi\<^sub>E A B" obtains "x \<in> A" and "f x \<in> B x"   | "x \<notin> A" and "f x = undefined"
full_checker_l_full_checker: fixes A :: "(nat, (char list list \<times> int) list) fmap"   and B :: "(nat, int mpoly) fmap"   and st :: "((char list list \<times> int) list, char list, nat) pac_step list"   and st' :: "(int mpoly, nat, nat) pac_step list"   and spec :: "(char list list \<times> int) list"   and spec' :: "int mpoly" assumes "(A, B) \<in> \<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"   and "(st, st') \<in> \<langle>p2rel (\<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel, var_rel\<rangle>pac_step_rel_raw)\<rangle>list_rel"   and "(spec, spec') \<in> fully_unsorted_poly_rel O mset_poly_rel" shows "full_checker_l spec A st \<le> \<Down> (code_status_status_rel \<times>\<^sub>r \<langle>var_rel\<rangle>set_rel \<times>\<^sub>r \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel) (full_checker spec' B st')"
gauge_def: fixes \<gamma> :: "'b \<Rightarrow> 'b set" shows "gauge \<gamma> = (\<forall>x. x \<in> \<gamma> x \<and> open (\<gamma> x))"
normalize_primitive_extract: fixes m :: "'a match_expr"   and disc_sel :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b)"   and C :: "'b \<Rightarrow> 'a"   and \<gamma> :: "('a \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and f :: "'b negation_type list \<Rightarrow> 'b list"   and a :: "action"   and p :: "'c" assumes "normalized_nnf_match m"   and "wf_disc_sel disc_sel C"   and "\<forall>ml. match_list \<gamma> (map (Match \<circ> C) (f ml)) a p = matches \<gamma> (alist_and (NegPos_map C ml)) a p" shows "match_list \<gamma> (normalize_primitive_extract disc_sel C f m) a p = matches \<gamma> m a p"
lim_sequence_eq: fixes x :: "'a" assumes "x \<in> space M" shows "lim_sequence x = K x \<bind> (\<lambda>y. distr (lim_sequence y) (Pi\<^sub>M UNIV (\<lambda>j. M)) (case_nat y))"
rep_eq: fixes x :: "'state"   and xa :: "('b, 'pred, 'act, 'effect) formula" shows "x \<Turnstile> xa = FL_valid_Tree\<^sub>\<alpha> x (Rep_formula xa)"
WhileTReds: fixes b :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and c :: "char list exp"   and v\<^sub>1 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e :: "char list exp"   and s\<^sub>3 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>b,s\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>2\<rangle>"   and "P \<turnstile> \<langle>while (b) c,s\<^sub>2\<rangle> \<rightarrow>* \<langle>e,s\<^sub>3\<rangle>" shows "P \<turnstile> \<langle>while (b) c,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e,s\<^sub>3\<rangle>"
disjnt: shows "\<D>\<^sub>\<circ> b \<inter>\<^sub>\<circ> \<D>\<^sub>\<circ> c = []\<^sub>\<circ>"
MutuallyDistinct_Insert: fixes S :: "('d, 'e, 'f) seqauto set"   and A :: "('d, 'e, 'f) seqauto" assumes "MutuallyDistinct S"   and "States A \<inter> \<Union> (States ` S) = {}" shows "MutuallyDistinct (insert A S)"
union_subseteqIl: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and A' :: "'a set" assumes "A \<union> B \<subseteq> C"   and "A' \<subseteq> A" shows "A' \<union> B \<subseteq> C"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "TE.wtE \<xi> \<equiv> \<forall>x. eintT (TE.tpOfV x) (\<xi> x)"
of_weak_ranking_weak_ranking: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite_total_preorder_on carrier le" shows "of_weak_ranking (weak_ranking le) = le"
sums_If_finite_set: fixes A :: "nat set"   and f :: "nat \<Rightarrow> 'a" assumes "finite A" shows "(\<lambda>r. if r \<in> A then f r else (0::'a)) sums sum f A"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow> Undecided"
pOp_assocTr41: fixes A :: "('h, 'i) aGroup_scheme"   and a :: "'h"   and b :: "'h"   and c :: "'h"   and d :: "'h" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c \<plusminus>\<^bsub>A\<^esub> d = a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> (c \<plusminus>\<^bsub>A\<^esub> d)"
partial_equiv_def: fixes A :: "'b set"   and R :: "('b \<times> 'b) set" shows "partial_equiv A R = (R \<subseteq> A \<times> A \<and> sym R \<and> trans R)"
finite_CBasis: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "finite CBasis"
JM: fixes n_as :: "(nat \<times> 'b) list" shows "jordan_matrix n_as \<in> carrier_mat (sum_list (map fst n_as)) (sum_list (map fst n_as))"
lappend_ltake_ldrop: fixes n :: "enat"   and xs :: "'b llist" shows "lappend (ltake n xs) (ldrop n xs) = xs"
h_def: fixes x :: "'a"   and y :: "'b" shows "h x y = ennreal (if x \<in> A \<and> y \<in> B then h' (to_nat_on A x) (to_nat_on B y) else 0)"
right_total_mod_ring_rel_integer: shows "right_total mod_ring_rel_integer"
edge_succ_permutes: shows "edge_succ M permutes arcs (with_proj G)"
quotient_ring_def: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "Comm_Ring.quotient_ring S R addition multiplication zero unit \<equiv> comm_ring R addition multiplication zero unit \<and> submonoid S R multiplication unit"
top_greatest: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and top :: "'k"   and a :: "'k" assumes "class.order_top less_eq less top" shows "less_eq a top"
both_mono2: fixes t :: "'b ttree"   and t' :: "'b ttree"   and t'' :: "'b ttree" assumes "paths t \<subseteq> paths t'" shows "paths (t'' \<otimes>\<otimes> t) \<subseteq> paths (t'' \<otimes>\<otimes> t')"
it_dfs_eq_spec: shows "it_dfs = SPEC (\<lambda>s. DFS_invar G param s \<and> \<not> cond s)"
kat_4: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "d x \<cdot> y = d x \<cdot> y \<cdot> d z" shows "d x \<cdot> y \<le> y \<cdot> d z"
the_apartment_isoD_2: fixes A :: "'a set set"   and A' :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> A'" shows "fixespointwise (the_apartment_iso A A') (\<Union> (A \<inter> A'))"
bernoulli_1: shows "bernoulli 1 = - 1 / 2"
lm069: fixes x :: "'c"   and X :: "'c set"   and f :: "('c \<times> 'd) set" assumes "x \<in> X \<inter> Domain f"   and "runiq (f || X)" shows "(f || X) ,, x = f ,, x"
AlgSimIncrEnvironment_axioms: shows "AlgSimIncrEnvironment jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit KBP.k_succs simAction"
\<a>'_def: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" shows "\<a>\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] \<equiv> \<alpha>' (\<mu>, \<nu>, \<tau>)"
h': fixes e' :: "'v \<times> 'v" shows "h_plus (Suc i) e' - h_minus i e' \<noteq> \<top>"
inv_rule_basic: fixes T :: "('e, 'f) TS_scheme"   and P :: "'e set" assumes "init T \<subseteq> P"   and "{P} TS.trans T {> P}" shows "reach T \<subseteq> P"
distinct_6: fixes x21 :: "'b"   and x22 :: "'b form list" shows "ff \<noteq> Compound x21 x22"
returns_result_def: fixes h :: "'a"   and p :: "('a, 'b, 'c) prog"   and r :: "'c" shows "h \<turnstile> p \<rightarrow>\<^sub>r r = (case h \<turnstile> p of Inl x \<Rightarrow> False | Inr (r', x) \<Rightarrow> r = r')"
a: shows "real_cond_exp M F f \<in> space\<^sub>N (\<LL> p (restr_to_subalg M F))"
assms: fixes c :: "real"   and t :: "real" shows "lSol c t \<equiv> exp ((c * (t - t\<^sub>0)) *\<^sub>R A)"
det_nfa_accepts: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and bss :: "bool list list" assumes "wf_nfa A n"   and "list_all (is_alph n) bss" shows "dfa_accepts (det_nfa A) bss = nfa_accepts A bss"
prj_simps_12: shows "C.seq \<nu>\<pi>.prj\<^sub>0 \<p>\<^sub>0[\<mu>.leg0, \<nu>.leg1 \<cdot> \<nu>\<pi>.prj\<^sub>1]"
queue_invar_simps_2: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.queue_invar (t # q) = (BinomialHeapStruc.tree_invar t \<and> BinomialHeapStruc.queue_invar q)"
w_defs_4: shows "w4 = fst (bv_chop (snd (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)) k)"
elgamal: fixes \<eta> :: "nat" shows "elgamal (\<G> \<eta>)"
rank_member: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and v :: "nat \<times> 'a" assumes "finite (nodes A)"   and "w \<notin> language A"   and "v \<in> gunodes A w" shows "v \<in> graph A w (rank A w v)"
map_pair: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and A :: "'c pmf"   and B :: "'d pmf" shows "map_pmf (\<lambda>(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)"
matricize_cprank_max1: fixes A :: "'b tensor"   and I :: "nat set" assumes "cprank_max1 A" shows "mrank (matricize I A) \<le> 1"
chine_simps_1: shows "C.arr \<mu>'.chine"
induct: fixes P :: "('a, 'b) strand_step list \<Rightarrow> bool"   and a0 :: "('a, 'b) strand_step list" assumes "P []"   and "\<And>x S. P S \<Longrightarrow> P (x # S)" shows "P a0"
less_QF: fixes j' :: "nat"   and j :: "nat"   and k :: "nat"   and i' :: "nat"   and i :: "nat" assumes "j' < j"   and "j \<le> k"   and "i' < m"   and "i < m" shows "QF k (j', i') \<lless> QF k (j, i)"
h_IN: fixes y :: "'v" shows "d_IN h y = (if y \<in> \<^bold>E `` X then weight \<Gamma> y else 0)"
sets_stream_space_cong: fixes M :: "'b measure"   and N :: "'b measure" assumes "sets M = sets N" shows "sets (stream_space M) = sets (stream_space N)"
case_4: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case D of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = f4"
a: shows "nat_of_ipv4addr ((ipv4addr_of_nat (d + 256 * c + 65536 * b + 16777216 * a) >> 24) && mask 8) = a"
mono_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat"   and A :: "'b set" shows "SemilatAlg.mono r step n A = (\<forall>\<tau> p \<tau>'. \<tau> \<in> A \<and> p < n \<and> \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>' \<longrightarrow> set (step p \<tau>) {\<sqsubseteq>\<^bsub>r\<^esub>} set (step p \<tau>'))"
arrI: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "J.in_hom f a b" shows "J.arr f"
limit_iff_frequent: fixes a :: "'c"   and x :: "nat \<Rightarrow> 'c" shows "(a \<in> limit x) = (\<exists>\<^sub>\<infinity>n. x n = a)"
prod_dfs_def: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list"   and x :: "nat \<times> nat" shows "prod_dfs A B x = gen_dfs (prod_succs A B) prod_ins prod_memb (prod_empt A B) [x]"
assms: shows "T \<in> GRepHomSet (\<star>) W"
powr_bigo_iff: fixes g :: "'a \<Rightarrow> real"   and F :: "'a filter"   and p :: "real"   and q :: "real" assumes "filterlim g at_top F"   and "F \<noteq> bot" shows "((\<lambda>x. g x powr p) \<in> O[F](\<lambda>x. g x powr q)) = (p \<le> q)"
size_4: fixes x2 :: "'b" shows "size (Inr x2) = Suc 0"
one_matrix_mult_right: fixes A :: "'b matrix"   and n :: "nat" assumes "ncols A \<le> n" shows "A * one_matrix n = A"
set_intros_6: fixes yl :: "'d"   and x23 :: "('e, 'd) ctxt"   and x21 :: "'e"   and x22 :: "('e, 'd) Term.term list"   and x24 :: "('e, 'd) Term.term list" assumes "yl \<in> vars_ctxt x23" shows "yl \<in> vars_ctxt (More x21 x22 x23 x24)"
DerivProp_Aux: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "(X, c, A) \<in> Deriv" shows "DProp A"
assms: fixes i :: "nat" assumes "i < count_weights s m" shows "w1 i = w2 i"
streams_stl: fixes s :: "'a stream"   and A :: "'a set" assumes "s \<in> streams A" shows "stl s \<in> streams A"
div_mult_twopow_eq: fixes a :: "nat"   and n :: "nat"   and b :: "nat" shows "a div 2 ^ n div b = a div (b * 2 ^ n)"
aND_RS2: fixes p :: "('b set set, 'c) Combinators list" assumes "singleCombinators p"   and "allNetsDistinct p" shows "allNetsDistinct (removeShadowRules2 p)"
FL_logically_equivalent_def: fixes L :: "'b \<times> 'c fs_set \<times> 'c \<Rightarrow> 'c fs_set"   and satisfies :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and transition :: "'d \<Rightarrow> ('b, 'd) residual \<Rightarrow> bool"   and effect_apply :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c fs_set"   and P :: "'d"   and Q :: "'d" assumes "indexed_effect_nominal_ts TYPE('a) L satisfies transition effect_apply" shows "indexed_effect_nominal_ts.FL_logically_equivalent TYPE('a) L satisfies transition effect_apply F P Q \<equiv> \<forall>x. effect_nominal_ts.is_FL_formula L F x \<longrightarrow> effect_nominal_ts.FL_valid satisfies transition effect_apply P x = effect_nominal_ts.FL_valid satisfies transition effect_apply Q x"
ZMod_def: fixes k :: "int"   and r :: "int" shows "ZMod k r = Idl\<^bsub>\<Z>\<^esub> {k} +>\<^bsub>\<Z>\<^esub> r"
blinded_blinds_embedded: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and e :: "'c \<Rightarrow> 'a"   and st :: "'c rose_tree" shows "blinding_of_tree h bo (blind_source_tree (h \<circ> e) st) (embed_source_tree e st)"
eq_numeral_simps_4: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "numeral.numeral one plus n \<noteq> zero"
eval_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and V :: "'b \<Rightarrow> 'd"   and t :: "'b local.term"   and u :: "'b local.term" assumes "evaluation_map C D T \<alpha> \<iota> V"   and "monoidal_language.Arr C t \<and> monoidal_language.Arr C u \<and> monoidal_language.Dom C t = monoidal_language.Dom C u \<and> monoidal_language.Cod C t = monoidal_language.Cod C u"   and "monoidal_language.Diagonalize C t = monoidal_language.Diagonalize C u" shows "evaluation_map.eval D T \<alpha> \<iota> V t = evaluation_map.eval D T \<alpha> \<iota> V u"
MatchOr: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m1 :: "'b match_expr"   and m2 :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (MatchOr m1 m2) a p = (matches \<gamma> m1 a p \<or> matches \<gamma> m2 a p)"
dim_span: fixes S :: "'b set" shows "vs1.dim (vs1.span S) = vs1.dim S"
v_simple_fusion: fixes G :: "'b::complete_lattice \<Rightarrow> 'b::complete_lattice"   and F :: "'b::complete_lattice \<Rightarrow> 'c::refinement_lattice"   and H :: "'c::refinement_lattice \<Rightarrow> 'c::refinement_lattice" assumes "mono (G::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)"   and "\<forall>x::'b::complete_lattice. ((F::'b::complete_lattice \<Rightarrow> 'c::refinement_lattice) \<circ> (G::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)) x \<sqsubseteq> ((H::'c::refinement_lattice \<Rightarrow> 'c::refinement_lattice) \<circ> F) x" shows "(F::'b::complete_lattice \<Rightarrow> 'c::refinement_lattice) (gfp (G::'b::complete_lattice \<Rightarrow> 'b::complete_lattice)) \<sqsubseteq> gfp (H::'c::refinement_lattice \<Rightarrow> 'c::refinement_lattice)"
iso_cancel_left: fixes f :: "'d"   and g :: "'d"   and g' :: "'d" assumes "D.iso f"   and "f \<cdot>\<^sub>D g = f \<cdot>\<^sub>D g'"   and "D.seq f g" shows "g = g'"
MakeFtorObj: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and X :: "'a" assumes "PreFunctor F"   and "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "MakeFtor F @@ X = F @@ X"
map_simp: fixes f :: "'j" assumes "J.arr f" shows "\<chi>.A.map f = x"
simps_22: fixes x41 :: "nat"   and x42 :: "nat"   and x21 :: "nat"   and x22 :: "nat" shows "connect(x41, x42) \<noteq> x21:deliver(x22)"
subtype_refl_1: fixes \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T" shows "\<Gamma> \<turnstile> T <: T"
intro: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "symmetric A equiv"   and "reflexive A equiv"   and "transitive A equiv" shows "equivalence A equiv"
split_relation_length: fixes xs :: "'b list"   and ls :: "'b list"   and rs :: "'b list" shows "split_relation xs (ls, rs) (length xs) = (ls = xs \<and> rs = [])"
assms_1: fixes x :: "nat \<Rightarrow> 'a" shows "dims (T x) = ds"
ccsubspace_leI: fixes A :: "'a ccsubspace"   and B :: "'a ccsubspace" assumes "space_as_set A \<subseteq> space_as_set B" shows "A \<le> B"
li_trans_l: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset" assumes "locally_irrefl R (A + B)" shows "locally_irrefl R A"
dom_simp: fixes f :: "'a" assumes "Right_a.arr f" shows "Right_a.dom f = local.dom f"
type_definition_natural: shows "type_definition nat_of_natural natural_of_nat UNIV"
domain_eq: shows "Domainp (pcr_restr (=) (=)) = (\<lambda>x. x \<in> {(xs, ys). length xs = length ys})"
the_elem_hom: shows "(\<lambda>x. the_elem (f ` x)) \<in> hom (G Mod H) (image_group f G)"
BVG_elem: fixes x :: "variable"   and \<alpha> :: "game" shows "(x \<in> BVG \<alpha>) = (\<exists>I \<omega> X. \<omega> \<in> game_sem I \<alpha> X \<and> \<omega> \<notin> game_sem I \<alpha> (selectlike X \<omega> {x}))"
poincare_line_inversion_1: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "on_circline (poincare_line z w) (inversion z)"
continuous_on_compose2: fixes t :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and s :: "'d set"   and f :: "'d \<Rightarrow> 'b" assumes "continuous_on t g"   and "continuous_on s f"   and "f ` s \<subseteq> t" shows "continuous_on s (\<lambda>x. g (f x))"