hcju/isabelleps huggingface dataset:
nonzero_mult_divide_mult_cancel_left: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times c b) = divide a b"
size_1_singleton_mset: fixes M :: "'k multiset" assumes "size M = 1" shows "\<exists>a. M = {#a#}"
acyclicI_order: fixes r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>a b. (a, b) \<in> r \<Longrightarrow> f b < f a" shows "acyclic r"
dvd_normalize_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "dvd.dvd times a (normalize b) = dvd.dvd times a b"
iff_def: fixes p :: "fm"   and q :: "fm" shows "Cooper.iff p q = (if p = q then T else if p = not q \<or> not p = q then F else if p = F then not q else if q = F then not p else if p = T then q else if q = T then p else Iff p q)"
trg_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "trg (trg \<mu>) = trg \<mu>"
assoc_1: fixes shows_prec :: "nat \<Rightarrow> 'c \<Rightarrow> char list \<Rightarrow> char list"   and shows_list :: "'c list \<Rightarrow> char list \<Rightarrow> char list"   and d :: "nat"   and x :: "'c"   and r :: "char list"   and s :: "char list" assumes "class.show shows_prec shows_list" shows "shows_prec d x r @ s = shows_prec d x (r @ s)"
fps_integral0_sub: fixes a :: "'a fps"   and b :: "'a fps" shows "fps_integral0 (a - b) = fps_integral0 a - fps_integral0 b"
B_def: shows "B = {x \<in> space M. \<forall>n. f ((T ^^ (n + 1)) x) \<le> f ((T ^^ n) x)}"
dimension_zero: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and E :: "'a set" assumes "ring R"   and "subfield K R"   and "ring.dimension R 0 K E" shows "E = {\<zero>\<^bsub>R\<^esub>}"
v_pi_pj: shows "v mod p_i \<noteq> v mod p_j"
1: shows "insert_weights s m1 w1 = insert_weights s m1 w2"
a_known_ptrs_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and h :: "(_) heap" shows "l_known_ptrs\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t.a_known_ptrs known_ptr h = (\<forall>ptr\<in>fset (object_ptr_kinds h). known_ptr ptr)"
lasso_rel_ext_def: fixes Re :: "('d \<times> 'f) set"   and R :: "('c \<times> 'e) set" shows "\<langle>Re, R\<rangle>lasso_rel_ext = {(\<lparr>lasso_reach = r', lasso_va = va', lasso_cysfx = cysfx', \<dots> = m'\<rparr>, \<lparr>lasso_reach = r, lasso_va = va, lasso_cysfx = cysfx, \<dots> = m\<rparr>) |r' r va' va cysfx' cysfx m' m. (r', r) \<in> \<langle>R\<rangle>list_rel \<and> (va', va) \<in> R \<and> (cysfx', cysfx) \<in> \<langle>R\<rangle>list_rel \<and> (m', m) \<in> Re}"
mkMX_correct: fixes l :: "'a list"   and i :: "nat"   and M :: "mon" assumes "in_carrier l" shows "Imon l (mkMX i M) = Imon l M \<otimes> head l [^] i"
up_to: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and J :: "nat set" assumes "Ionescu_Tulcea P M"   and "finite J" shows "J \<subseteq> {0..<Ionescu_Tulcea.up_to J}"
simps_2: fixes x :: "char" shows "proper_interval None (Some x) = (x \<noteq> CHR 0x00)"
derivs_insertD: fixes G :: "'b triple set"   and t :: "'b triple"   and ts :: "'b triple set" assumes "G||-insert t ts" shows "G|-t \<and> G||-ts"
assms_2: fixes x :: "'a" assumes "x \<in> B" shows "x \<in> A"
SMP_D': fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "finite_SMP_representation arity Ana \<Gamma> M"   and "s \<in> SMP (set M)"   and "is_Fun s"   and "\<nexists>f. s = Fun f []"   and "t \<in> SMP (set M)"   and "is_Fun t"   and "\<nexists>f. t = Fun f []" obtains \<sigma> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" and s0 :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" and \<theta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" and t0 :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" where "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>" and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)" and "s0 \<in> set M" and "is_Fun s0" and "s = s0 \<cdot> \<sigma>" and "\<Gamma> s = \<Gamma> s0" and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)" and "t0 \<in> set M" and "is_Fun t0" and "t = t0 \<cdot> var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t (set M))) \<cdot> \<theta>" and "\<Gamma> t = \<Gamma> t0"
distinct_3: fixes x3 :: "'a" shows "TrueF \<noteq> fm.Atom x3"
liminf_is_N: shows "Liminf_llist Ns = N"
assms: shows "weak_matrix_match2 A1 B1 B2"
simps: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and v :: "'b"   and w :: "'b" shows "subdivide G (u, v) w = \<lparr>pverts = pverts G \<union> {w}, parcs = parcs G - {(u, v), (v, u)} \<union> {(u, w), (w, u), (w, v), (v, w)}\<rparr>"
f_aggregate_drop_mod: fixes n :: "nat"   and k :: "nat"   and xs :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" assumes "n mod k = 0" shows "f_aggregate (xs \<up> n) k ag = f_aggregate xs k ag \<up> (n div k)"
hyps_1: fixes s :: "('a, 'b) xstate" assumes "\<Gamma>\<turnstile>strip_guards F c1 \<down> s" shows "\<Gamma>\<turnstile>c1 \<down> s"
distinct_2: fixes b' :: "'e"   and a :: "'d" shows "Inr b' \<noteq> Inl a"
Int_lower1: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B \<subseteq> A"
isLubI1: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "leastP (isUb R S) x" shows "isLub R S x"
fres_def: fixes z :: "'a"   and y :: "'a" shows "z \<leftarrow> y = \<Squnion>{x. x \<cdot> y \<le> z}"
conjunct_def: fixes E :: "(('c \<Rightarrow> 'd) \<Rightarrow> bool) set" shows "conjunct E \<equiv> \<lambda>\<sigma>. \<forall>e\<in>E. e \<sigma>"
Let: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and L :: "Vars.var list"   and body :: "exp"   and \<Theta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "atom ` domA \<Delta> \<sharp>* (\<Gamma>, L)"   and "\<Delta> @ \<Gamma> : body \<Down>\<^bsub>L\<^esub> \<Theta> : z" shows "\<Gamma> : Let \<Delta> body \<Down>\<^bsub>L\<^esub> \<Theta> : z"
matrix_to_iarray_echelon_form_of_column_k: fixes k :: "nat"   and A :: "(('d, 'e) vec, 'f) vec"   and i :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "k < ncols A"   and "i \<le> nrows A" shows "matrix_to_iarray (fst (echelon_form_of_column_k bezout (A, i) k)) = fst (echelon_form_of_column_k_iarrays (matrix_to_iarray A, i, bezout) k)"
fps_sum_nth: fixes f :: "'c \<Rightarrow> 'b fps"   and S :: "'c set"   and n :: "nat" shows "sum f S $ n = (\<Sum>k\<in>S. f k $ n)"
cases: fixes a1 :: "pi"   and a2 :: "pi" assumes "a1 \<approx> a2" obtains P :: "pi" and Q :: "pi" where "a1 = P" and "a2 = Q" and "P \<leadsto>\<^sup>^<weakBisim> Q" and "Q \<approx> P"
Arg2pi_le_div_sum_eq: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0" shows "(Arg2pi w \<le> Arg2pi z) = (Arg2pi z = Arg2pi w + Arg2pi (z / w))"
Cons_3: shows "R_lin_independent ((m # ms) @ n # ns)"
mset_pos_diff: fixes A :: "'b zmultiset"   and B :: "'b zmultiset" shows "mset_pos (A - B) = mset_pos A - mset_pos B + (mset_neg B - mset_neg A)"
protocol_ot_def: fixes M :: "'a \<times> 'a"   and \<sigma> :: "bool" shows "protocol_ot M \<sigma> = (let (x0, x1) = M in sample_uniform (order \<G>) \<bind> (\<lambda>r. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>0. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>1. let h0 = \<^bold>g [^] \<alpha>0; h1 = \<^bold>g [^] \<alpha>1; a = \<^bold>g [^] r; b0 = h0 [^] r \<otimes> \<^bold>g [^] (if \<sigma> then 1 else 0); b1 = h1 [^] r \<otimes> \<^bold>g [^] (if \<sigma> then 1 else 0); h = h0 \<otimes> inv h1; b = b0 \<otimes> inv b1 in assert_spmf (a = \<^bold>g [^] r \<and> b = h [^] r) \<bind> (\<lambda>_. sample_uniform (order \<G>) \<bind> (\<lambda>u0. sample_uniform (order \<G>) \<bind> (\<lambda>u1. sample_uniform (order \<G>) \<bind> (\<lambda>v0. sample_uniform (order \<G>) \<bind> (\<lambda>v1. let z0 = b0 [^] u0 \<otimes> h0 [^] v0 \<otimes> x0; w0 = a [^] u0 \<otimes> \<^bold>g [^] v0; e0 = (w0, z0); z1 = (b1 \<otimes> inv \<^bold>g) [^] u1 \<otimes> h1 [^] v1 \<otimes> x1; w1 = a [^] u1 \<otimes> \<^bold>g [^] v1; e1 = (w1, z1) in return_spmf ((), if \<sigma> then z1 \<otimes> inv (w1 [^] \<alpha>1) else z0 \<otimes> inv (w0 [^] \<alpha>0)))))))))))"
simps_6: fixes v :: "'d strand_label"   and vb :: "poscheckvariant"   and vc :: "('e, 'f) Term.term"   and vd :: "('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (v, \<langle>vb: vc \<in> vd\<rangle>) = (v, \<langle>vb: vc \<in> vd\<rangle>)"
catch_option_def: fixes m :: "('a, 'm) optionT"   and h :: "('a, 'm) optionT" shows "local.catch_option m h = OptionT (bind (run_option m) (\<lambda>x. if x = None then run_option h else return x))"
exhaust_sel: fixes step :: "('c, 'd) step" obtains "step = Done"   | "step = Skip (un_Skip step)"   | "step = Yield (un_Yield1 step) (un_Yield2 step)"
assms: shows "t \<^sup>@ m \<cdot> y = t \<^sup>@ k"
invert_side_of_triangle_1: fixes h :: "'a"   and f :: "'a"   and g :: "'a" assumes "arr h"   and "f \<cdot> g = h"   and "local.iso f" shows "seq (local.inv f) h \<and> g = local.inv f \<cdot> h"
omega_complete_INTER: shows "omega_complete (\<lambda>x y. y \<subseteq> x) (\<lambda>f. \<Inter> (range f))"
i_Exec_Stream_Acc_Output_1: fixes output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" shows "i_Exec_Comp_Stream_Acc_Output (Suc 0) output_fun trans_fun input c = output_fun \<circ> i_Exec_Comp_Stream trans_fun input c"
vo_def: shows "vo \<equiv> \<lambda>k. if k = 0 then None else Some (SOME c. v c = k)"
fs_ss: shows "(\<Oplus>\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}\<^esub>v\<in>B. a v \<odot>\<^bsub>vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}\<^esub> v) = (\<Sum>v\<in>B. Polynomial.smult (a v) v)"
assms_1: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd" shows "set (ap gs bs ps hs data) \<subseteq> set ps \<union> set hs \<times> (set gs \<union> set bs \<union> set hs)"
mset_add: fixes a :: "'a"   and A :: "'a multiset" assumes "a \<in># A" obtains B :: "'a multiset" where "A = a, B"
simps_2: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.queue_to_multiset (t # q) = SkewBinomialHeapStruc.tree_to_multiset t + SkewBinomialHeapStruc.queue_to_multiset q"
instant_index_increase: fixes \<rho> :: "'c run"   and \<Gamma> :: "'c constr list"   and n :: "nat"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" assumes "\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g" shows "\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
chain_union_closed': fixes f :: "nat \<Rightarrow> 'a set"   and C :: "'a set set"   and S' :: "'a set" assumes "is_chain f"   and "\<forall>n. f n \<in> C"   and "\<forall>S'\<in>C. \<forall>S\<subseteq>S'. S \<in> C"   and "finite S'"   and "S' \<subseteq> \<Union> (range f)" shows "S' \<in> C"
length_fpc: fixes p :: "nat"   and a :: "nat"   and r :: "nat"   and qs :: "nat list" shows "length (build_fpc p a r qs) = length qs + 1"
circ_decompose_8: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<circ> = (x \<squnion> y)\<^sup>\<circ> * x\<^sup>\<circ> * y\<^sup>\<circ>"
drop_take_1: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "xs \<up> n \<down> Suc 0 = [xs ! n]"
resChainAlpha: fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and P :: "('d, 'e, 'f) psi" assumes "(p \<bullet> xvec) \<sharp>* P"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)" shows "\<lparr>\<nu>*xvec\<rparr>P = \<lparr>\<nu>*p \<bullet> xvec\<rparr>p \<bullet> P"
hs_noop_locs_def: shows "hs_noop_locs = prefixed ''hs_noop_''"
span_zero: fixes S :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "0 \<in> phull S"
is_interval_def: fixes s :: "'b set" shows "is_interval s = (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i) \<longrightarrow> x \<in> s)"
S4: shows "c_hd u = hd (nat_to_list u)"
stake_finite_universe_induct: fixes A :: "'b set"   and n :: "nat" assumes "finite A"   and "A \<noteq> {}" shows "stake (Suc n) ` streams A = {s # w |s w. s \<in> A \<and> w \<in> stake n ` streams A}"
degenerate_hexagon_is_pappus: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and Q :: "Points"   and R :: "Points" assumes "distinct6 A B C A' B' C'"   and "col A B C"   and "col A' B' C'"   and "is_a_intersec P A B' A' B"   and "is_a_intersec Q B C' B' C"   and "is_a_intersec R A C' A' C"   and "line A B' = line A' B \<or> line B C' = line B' C \<or> line A C' = line A' C" shows "col P Q R"
fold_fold: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "('b, 'c) RBT.rbt" shows "RBT.fold f t = List.fold (\<lambda>(x, y). f x y) (entries t)"
merge_on_T: fixes A :: "'e set"   and rh :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option" assumes "merge_on A rh bo m" shows "merge_on {x. set1_T\<^sub>m x \<subseteq> A} (root_hash_T rh) (blinding_of_T rh bo) (merge_T rh m)"
test_def: fixes A :: "'s"   and P :: "natural \<Rightarrow> bool" shows "local.test A P = size (filter P A)"
sat_constr_sat_transf_constrs: fixes v :: "nat \<Rightarrow> rat"   and cs :: "constraint" assumes "v \<Turnstile>\<^sub>c cs" shows "v \<Turnstile>\<^sub>c\<^sub>s set (transf_constraint cs)"
app_conv_split_n: fixes es :: "v list"   and es1 :: "v list"   and es2 :: "v list" assumes "es = es1 @ es2" shows "split_n es (length es1) = (es1, es2)"
i_set0_cont_min_def: shows "i_set0_cont_min \<equiv> insert {} i_set_cont_min"
min_signed_def: shows "min_signed \<equiv> - (2 ^ (word_len - 1))"
hbex_triv_one_point1: fixes A :: "hf"   and a :: "hf" shows "(\<exists>x\<^bold>\<in>A. x = a) = (a \<^bold>\<in> A)"
negatively_invariant_complI: fixes M :: "'a set" assumes "positively_invariant M" shows "negatively_invariant (X - M)"
full_chain_ldropn: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist"   and n :: "nat" assumes "full_chain R xs"   and "enat n < llength xs" shows "full_chain R (ldropn n xs)"
coempty: shows "- {} = Denotational_Semantics.allvars"
distinct_split2: fixes xs :: "'c list"   and y :: "'c list"   and r :: "'c"   and z :: "'c list" assumes "distinct xs"   and "xs = y @ [r] @ z" shows "r \<notin> set z"
fresh_nat: fixes a :: "'d"   and n :: "nat" shows "a \<sharp> n"
zeros: shows "\<forall>k\<in>{1..n} - {k \<in> {1..n}. coprime k n}. f k n * dirichlet_prod_neutral (gcd k n) = 0"
generic_Assert_axioms: shows "generic_Assert abind areturn aASSERT aASSUME"
simps_29: fixes vb :: "'a"   and vc :: "bexp"   and vd :: "'a acom"   and ve :: "'a"   and v :: "'a acom"   and va :: "'a acom" shows "({vb} WHILE vc DO vd {ve} \<le> v;; va) = False"
computatition_positive_flip: shows "rat_poly_plus (rat_poly_inv (rat_poly_times A (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) A))) (rat_poly_inv (rat_poly_times B (rat_poly_times A B))) = rat_poly_inv (rat_poly_times A (rat_poly_times A A))"
z_def: shows "z = get_lower_bound (fst x) (fst y)"
connected_inV_iff: fixes u :: "nat"   and v :: "nat" assumes "cf.connected u v" shows "(v \<in> cf.V) = (u \<in> cf.V)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. T (f, \<I>)) local.map \<rho>.map"
SumPres: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<equiv>\<^sub>e Q" shows "P \<oplus> R \<equiv>\<^sub>e Q \<oplus> R"
simps_3: fixes x21 :: "'e"   and x22 :: "'e list" shows "x21 # x22 \<noteq> []"
simps_5: fixes test :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and r :: "'d regex" shows "lpd test i (Star r) = TimesR (lpd test i r) (Star r)"
permutation_eq_image: fixes a :: "'b permutation"   and A :: "'b set"   and B :: "'b set" assumes "a `\<rightarrow> A = a `\<rightarrow> B" shows "A = B"
trefoil_polynomial_def: shows "trefoil_polynomial \<equiv> rat_poly_plus (rat_poly_times (rat_poly_times A A) (rat_poly_plus (rat_poly_times B (rat_poly_times B (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) (rat_poly_times A A)))) (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) (rat_poly_plus (rat_poly_times B (rat_poly_times B (rat_poly_times A A))) (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) (rat_poly_times A A))))))) (rat_poly_plus (rat_poly_times 2 (rat_poly_times A (rat_poly_times A (rat_poly_times A (rat_poly_times A (rat_poly_times A (rat_poly_times B B))))))) (rat_poly_times (rat_poly_times B B) (rat_poly_times B (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) (rat_poly_times B (rat_poly_times B B))))))"
lnull_ldrop: fixes n :: "enat"   and xs :: "'b llist" shows "lnull (ldrop n xs) = (llength xs \<le> n)"
flow_in_ivp_sols_ex_ivl: fixes s :: "'a" assumes "s \<in> S" shows "flow 0 s \<in> Sols (\<lambda>t. f) ex_ivl S 0 s"
rep_eq: fixes x :: "CoCalls" shows "ccField x = Field (Rep_CoCall x)"
neighborhoods_eq: shows "neighborhoods x = {U. is_open U \<and> x \<in> U}"
hyps: shows "Q \<in> sets (M1 \<Otimes>\<^sub>M M)"
ParPres2: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "(\<Psi>', S \<parallel> S, T \<parallel> T) \<in> Rel"
poly_dvd_1: fixes p :: "'b poly" shows "(p dvd 1) = (degree p = 0 \<and> coeff p 0 dvd (1::'b))"
open_superdiagonal: shows "open {(x, y) |x y. y < x}"
items_le_split_via_eq: fixes k :: "nat"   and J :: "item set" shows "items_le (Suc k) J = items_le k J \<union> items_eq (Suc k) J"
dis_elim: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'"   and "Shallow.dis \<phi> \<psi> \<pi>l" obtains "\<phi> \<pi>l"   | "\<psi> \<pi>l"
pCons: fixes a :: "'a"   and p :: "'a poly" assumes "a \<noteq> (0::'a) \<or> p \<noteq> 0"   and "P p" shows "P (pCons a p)"
run_owner_ii_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_owner_ii n s \<equiv> SOME x. pred_owner_ii n (run_asset_i n s) x"
containID: fixes i :: "nat"   and xs :: "NominalSequents.var list"   and \<Gamma> :: "form multiset"   and \<Delta> :: "form multiset"   and R :: "(sequent list \<times> sequent) set" assumes "At i xs \<in># \<Gamma> \<and> At i xs \<in># \<Delta>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* \<Delta>, 0) \<in> derivable  R*"
foldseq_zerotail: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and m :: "nat" assumes "f (0::'b) (0::'b) = (0::'b)"   and "\<forall>i\<ge>n. s i = (0::'b)"   and "n \<le> m" shows "foldseq f s n = foldseq f s m"
elims: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and xa :: "'c"   and xb :: "'b list"   and y :: "'b option" assumes "applied_rule x xa xb = y" obtains C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option" and a :: "'c" and xc :: "'b" and xs :: "'b list" where "x = C__" and "xa = a__" and "xb = xc # xs" and "y = (if a__ \<in> dom (C__ xc) then \<lfloor>xc\<rfloor> else applied_rule C__ a__ xs)"   | C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option" and a :: "'c" where "x = C" and "xa = a" and "xb = []" and "y = \<bottom>"
greaterThanAtMost_empty_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and l :: "'b" assumes "class.preorder less_eq less" shows "(ord.greaterThanAtMost less_eq less k l = {}) = (\<not> less k l)"
dim_vec_of_list: fixes x :: "'b list" shows "dim_vec (vec_of_list x) = length x"
cong: fixes p :: "nat" shows "GAUSS.A p \<equiv> GAUSS.A p"
perms_imp_bij: fixes f :: "'b \<Rightarrow> 'b" assumes "f \<in> perms" shows "bij f"
E: shows "(map (inv p) (filter (\<lambda>x. x \<in> range p) (xs @ zs)), inv p ` Z) \<in> failures P"
partIps_equi: fixes s :: "'a wordinterval"   and ts :: "'a wordinterval list" shows "map wordinterval_to_set (partIps s ts) = partList4 (wordinterval_to_set s) (map wordinterval_to_set ts)"
Greatest_bounded_Suc: fixes P :: "nat \<Rightarrow> nat"   and n :: "nat" shows "(if 0 < P n then Suc n else if \<exists>j<n. 0 < P j then Suc (GREATEST j. j < n \<and> 0 < P j) else 0) = (if \<exists>j<Suc n. 0 < P j then Suc (GREATEST j. j < Suc n \<and> 0 < P j) else 0)"
list_all2_all_nthI: fixes a :: "'g list"   and b :: "'h list"   and P :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "length a = length b"   and "\<And>n. n < length a \<Longrightarrow> P (a ! n) (b ! n)" shows "list_all2 P a b"
pdg_tiny_digraph_dg_prod': shows "tiny_digraph (\<alpha> + \<omega>) (dg_prod I \<AA>)"
inc_predict_support_trading_strat': fixes pf1 :: "'b \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and asset :: "'b"   and B :: "'b set" assumes "trading_strategy pf1"   and "asset \<in> support_set pf1 \<union> B" shows "borel_predict_stoch_proc G (pf1 asset)"
wf_less: shows "wf {(x, y). x <\<^sub>v y}"
DDH_\<A>'_bound: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "advantage (DDH_\<A>' \<A>) = \<bar>spmf (ddh_1 \<A>) True - spmf (DDH1 \<A>) True\<bar>"
simps_1: fixes e :: "'a"   and a :: "'b" shows "SkewBinomialHeapStruc.insert e a [] = [SkewBinomialHeapStruc.Node e a 0 []]"
g1: shows "R_Chop(r,Abs_real_int (left r1, right r3),r4)"
le_ref_def: fixes P :: "'a process"   and Q :: "'a process" shows "P \<le> Q \<equiv> D Q \<subseteq> D P \<and> F Q \<subseteq> F P"