hcju/isabelleps huggingface dataset:
vector_mul_rcancel: fixes a :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec"   and b :: "'a" shows "(a *s x = b *s x) = (a = b \<or> x = 0)"
vec_lambda_beta: fixes g :: "'c \<Rightarrow> 'b"   and i :: "'c" shows "vec_lambda g $ i = g i"
EX_step: fixes x :: "'b"   and y :: "'b"   and f :: "'b set" assumes "x \<rightarrow>\<^sub>i y"   and "y \<in> f" shows "x \<in> EX' f"
distinct_3: fixes x1 :: "'a"   and x3 :: "'c" shows "OpDyn x1 \<noteq> OpUbx2 x3"
cases: fixes a1 :: "'c"   and a2 :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set" assumes "(a1, a2) \<in> rtrancl_on F r" obtains a :: "'c" where "a1 = a__" and "a2 = a__" and "a__ \<in> F"   | a :: "'c" and b :: "'c" and c :: "'c" where "a1 = a" and "a2 = c" and "(a, b) \<in> rtrancl_on F r" and "(b, c) \<in> r" and "c \<in> F"
pred_inject: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" shows "pred_dual P (dual a) = P a"
Enc_keys_clean_valid: fixes I :: "msg set" assumes "I \<subseteq> valid" shows "Enc_keys_clean I"
absorb1: fixes a :: "'b word"   and b :: "'b word" assumes "a \<le>s b" shows "signed.min a b = a"
path_split_snoc: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit_wf.path n as n'"   and "as \<noteq> []" obtains a' :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node" and as' :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list" where "as = as' @ [a']" and "CFGExit_wf.path n as' (src a')" and "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a'" and "n' = trg a'"
less_diff_eq_ennreal: fixes b :: "ennreal"   and c :: "ennreal"   and a :: "ennreal" assumes "b < top \<or> c < top" shows "(a < b - c) = (a + c < b)"
strategyproofness_1: shows "(pmf (sds R10) a < pmf (sds R5) d \<or> pmf (sds R10) a + (pmf (sds R10) c + pmf (sds R10) d) < pmf (sds R5) d + (pmf (sds R5) b + pmf (sds R5) a)) \<or> pmf (sds R10) a = pmf (sds R5) d \<and> pmf (sds R10) a + (pmf (sds R10) c + pmf (sds R10) d) = pmf (sds R5) d + (pmf (sds R5) b + pmf (sds R5) a)"
sumCurry: fixes f :: "'f \<times> 'e \<Rightarrow> 'd"   and x :: "'f"   and Y :: "'e set" shows "sum (curry f x) Y = sum f ({x} \<times> Y)"
OUTfromChCorrect_data11: shows "OUTfromChCorrect data11"
runit_char_2: 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" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "(if partial_magma.arr V (bicategory.runit V H \<a> \<i> src f) then H (bicategory.runit V H \<a> \<i> src f) (src (bicategory.runit V H \<a> \<i> src f)) else partial_magma.null V) = V (H f (\<i> (src f))) (\<a> f (src f) (src f))"
context_le_trans: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and \<Gamma> :: "'b \<Rightarrow> Sec option"   and \<Gamma>' :: "'b \<Rightarrow> Sec option"   and \<Gamma>'' :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "sifum_types.context_le \<Gamma> \<Gamma>'"   and "sifum_types.context_le \<Gamma>' \<Gamma>''" shows "sifum_types.context_le \<Gamma> \<Gamma>''"
conjugate_add_vec: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "conjugate (v + w) = conjugate v + conjugate w"
fresh_3: fixes a :: "name" shows "a \<sharp> tINT = True"
IH: shows "local_time A n x = birkhoff_sum (indicator A) n x"
span_image: fixes S :: "'b set" shows "vs2.span (construct B f ` S) = construct B f ` vs1.span S"
id_nec: fixes v :: "i"   and \<alpha> :: "'b"   and \<beta> :: "'b" shows "[\<alpha> \<^bold>= \<beta> \<^bold>\<equiv> \<^bold>\<box>\<alpha> \<^bold>= \<beta> in v]"
B: shows "D (lim_proc X) = \<Inter> (D ` X)"
WT_adv: shows "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g \<A> \<surd>"
le_fininf_concat_start: fixes w :: "'item list"   and w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item stream" shows "(w @ w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w @- w\<^sub>2) = (w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2)"
reachable_states_induct: fixes xa :: "gState"   and P :: "program"   and g\<^sub>s' :: "gState"   and Pa :: "gState \<Rightarrow> bool" assumes "xa \<in> reachable_states P g\<^sub>s'"   and "Pa g\<^sub>s'"   and "\<And>g x. \<lbrakk>g \<in> reachable_states P g\<^sub>s'; Pa g; x \<in> ls.\<alpha> (nexts_code P g)\<rbrakk> \<Longrightarrow> Pa x" shows "Pa xa"
b: fixes x :: "real" assumes "b \<le> x" shows "norm (f x) \<le> c * norm (g' x)"
assms_13: shows "\<forall>x. (x \<in> R_I) = (wf_tuple n I x \<and> (\<forall>(A, X)\<in>Q_I_pos. restrict A x \<in> X) \<and> (\<forall>(A, X)\<in>Q_I_neg. restrict A x \<notin> X))"
simps_1: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and u :: "nat"   and v :: "nat" shows "linordered_monoid.len m u v [] = m u v"
add_neg_numeral_simps_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (numeral.numeral one plus m) (uminus (numeral.numeral one plus n)) = neg_numeral.sub minus plus one m n"
allocationGoodsMonotonicity: fixes \<Omega>1 :: "'c set"   and \<Omega>2 :: "'c set"   and N :: "integer set" assumes "\<Omega>1 \<subseteq> \<Omega>2" shows "soldAllocations''' N \<Omega>1 \<subseteq> soldAllocations''' N \<Omega>2"
induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree"   and a1 :: "'b"   and a2 :: "('b \<times> nat) tree" assumes "\<And>A a uu_ B b C. P \<langle>A, (a, uu_), B\<rangle> b C"   and "\<And>b c. P \<langle>\<rangle> b c" shows "P a0 a1 a2"
equals0I: fixes A :: "'g set" assumes "\<And>y. y \<in> A \<Longrightarrow> False" shows "A = {}"
if_cond: fixes cond :: "exp"   and cond' :: "exp"   and thn :: "exp"   and els :: "exp" assumes "cond \<longrightarrow> cond'" shows "EIf cond thn els \<longrightarrow> EIf cond' thn els"
subst_clause_finite: fixes \<Phi> :: "'b ltln fset"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "finite (subst_clause \<Phi> m)"
ran_alt_def: fixes m :: "'c \<Rightarrow> 'b option" shows "ran m = (the \<circ> m) ` dom m"
\<Gamma>_counterex_reducing: fixes N :: "'a literal multiset set"   and D :: "'a literal multiset" assumes "{#} \<notin> N"   and "D \<in> N"   and "\<not> I_of N \<Turnstile> D"   and "\<And>C. \<lbrakk>C \<in> N; \<not> I_of N \<Turnstile> C\<rbrakk> \<Longrightarrow> D \<le> C" shows "\<exists>CC E. set_mset CC \<subseteq> N \<and> I_of N \<Turnstile>m CC \<and> Infer CC D E \<in> \<Gamma> \<and> \<not> I_of N \<Turnstile> E \<and> E < D"
exec_step_ind_NewOOM_Done: fixes sh :: "sheap"   and C :: "char list"   and obj :: "sfields"   and h :: "heap"   and ics :: "init_call_status"   and P :: "jvm_prog"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "sh C = \<lfloor>(obj, Done)\<rfloor>"   and "new_Addr h = None"   and "\<forall>Cs. ics \<noteq> Called Cs" shows "exec_step_ind (StepI (New C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt OutOfMemory\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
trcl_rev_uncons: fixes c :: "'f"   and w :: "'g list"   and e :: "'g"   and c' :: "'f"   and T :: "('f \<times> 'g \<times> 'f) set" assumes "(c, w @ [e], c') \<in> trcl T" shows "\<exists>ch. (c, w, ch) \<in> trcl T \<and> (ch, e, c') \<in> T"
coprime_lead_coeff_large_prime: fixes p :: "int"   and f :: "int poly" assumes "prime p"   and "\<bar>lead_coeff f\<bar> < p"   and "f \<noteq> 0" shows "comm_monoid_mult_class.coprime (lead_coeff f) p"
simps_3: fixes I :: "('b, 'c, 'a) interp"   and t1 :: "('b, 'a) trm"   and t2 :: "('b, 'a) trm"   and v :: "(real, 'a) vec" shows "frechet I (Plus t1 t2) v = (\<lambda>v'. frechet I t1 v v' + frechet I t2 v v')"
f_Exec_Stream_Acc_LocalState_nth: fixes k :: "nat"   and n :: "nat"   and xs :: "'d message_af list"   and localState :: "'f \<Rightarrow> 'e"   and trans_fun :: "'d message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f" assumes "0 < k"   and "n < length xs" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c ! n = localState (f_Exec_Comp trans_fun (xs \<down> Suc n \<odot> k) c)"
preserves_arr: fixes f :: "'t" assumes "S'.arr f" shows "S.arr (\<Psi> f)"
induct: fixes P :: "'a\<cdot>llist \<Rightarrow> bool"   and x :: "'a\<cdot>llist" assumes "adm P"   and "P \<bottom>"   and "P LNil"   and "\<And>a app. P app \<Longrightarrow> P (LCons\<cdot>a\<cdot>app)" shows "P x"
boundedByUN: fixes N :: "nat"   and B :: "'c \<Rightarrow> (nat \<times> 'b) set"   and A :: "'c set" shows "boundedBy N (\<Union> (B ` A)) = (\<forall>x\<in>A. boundedBy N (B x))"
partition_main_ref': fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and lo :: "nat"   and hi :: "nat"   and xs :: "'c list" shows "partition_main R h lo hi xs \<le> \<Down> Id (partition_main R h lo hi xs)"
assms_2: shows "Lower_Semicontinuous.convex_on s g"
freshInputTransition: fixes P :: "pi"   and a :: "name"   and u :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> a<u> \<prec> P'"   and "c \<sharp> P"   and "c \<noteq> u" shows "c \<sharp> P'"
ins_ins2_1: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree" assumes "ins2 n d x t = Same2" shows "ins n d x t = Same"
map_sel: fixes f :: "'f \<Rightarrow> 'e"   and a :: "('d, 'f) stateT" shows "run_state (map_stateT f a) = f \<circ> run_state a"
satA_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and at :: "('a, 'b) atm" shows "U.satA \<xi> at = F.satA \<xi> at"
forest_components_increasing: fixes x :: "'a" shows "x \<le> forest_components x"
mod_Suc: fixes m :: "nat"   and n :: "nat" shows "Suc m mod n = (if Suc (m mod n) = n then 0 else Suc (m mod n))"
coeff_of_sum_diff_degree0: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "deg R q < n" shows "(p \<oplus>\<^bsub>UP R\<^esub> q) n = p n"
isNSUCont_def: fixes f :: "'c \<Rightarrow> 'd" shows "isNSUCont f = (\<forall>x y. x \<approx> y \<longrightarrow> (*f* f) x \<approx> (*f* f) y)"
induced_map_unique: fixes w :: "'a"   and w' :: "'a" assumes "h.is_induced_map w"   and "h.is_induced_map w'" shows "w \<cong> w'"
nz: fixes z :: "complex" assumes "z \<in> S" shows "g z \<noteq> 0"
represents_scale: fixes B :: "'b set"   and r :: "'b \<Rightarrow>\<^sub>0 'a"   and x :: "'b"   and c :: "'a" assumes "represents B r x" shows "represents B (monomial c (0::'b) * r) (c *s x)"
MGT_lemma: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and Z :: "'e"   and c :: "('e, 'd, 'f) com" assumes "\<forall>p\<in>dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)} Call p {t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Abrupt t}" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)} c {t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
skelAbs_vsubstAbs: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var" assumes "goodAbs A" shows "skelAbs (A $[y1 // y2]_ys) = skelAbs A"
lookup_pair_Cons: fixes k :: "'a"   and v :: "'d"   and xs :: "('a \<times> 'd) list"   and k0 :: "'a" assumes "oalist_inv_raw ((k, v) # xs)" shows "lookup_pair ((k, v) # xs) k0 = (if k = k0 then v else lookup_pair xs k0)"
Ptl_not_null: shows "\<not> lnull (ltl P)"
S_sum_lists: shows "S_sum_lists \<equiv> \<Union>ss\<in>lists (S \<union> uminus ` S). {sum_list ss}"
update_accessor_cong_assist_idI: shows "iso_tuple_update_accessor_cong_assist id id"
assms_2: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
ncf5: fixes y :: "real" assumes "0 \<le> y"   and "y \<le> x" shows "0 < numer_cf5 (- y)"
basis_def: fixes A :: "'a vec set" shows "A.basis A = (A.lin_indpt A \<and> A.span A = mat_kernel A \<and> A \<subseteq> mat_kernel A)"
UN_extend_simps_2: fixes A :: "'c \<Rightarrow> 'b set"   and C :: "'c set"   and B :: "'b set" shows "\<Union> (A ` C) \<union> B = (if C = {} then B else \<Union>x\<in>C. A x \<union> B)"
inc_le: fixes i :: "'a word"   and m :: "'a word" assumes "i < m" shows "i + 1 \<le> m"
lc_p'': shows "punit.lc (rep_list p') = punit.lc (rep_list p)"
permute_fun_comp: fixes p :: "perm"   and f :: "'a \<Rightarrow> 'b" shows "p \<bullet> f = permute p \<circ> f \<circ> permute (- p)"
one_complete_def: shows "one_complete = (\<lambda>A. A \<in> ce_sets \<and> (\<forall>B. B \<in> ce_sets \<longrightarrow> one_reducible_to B A))"
elims_3: fixes x :: "'a simple_match" assumes "\<not> empty_match x" obtains uu_ :: "iface" and uv_ :: "iface" and uw_ :: "'a word \<times> nat" and ux_ :: "'a word \<times> nat" and uy_ :: "protocol" and sps1 :: "16 word" and sps2 :: "16 word" and dps1 :: "16 word" and dps2 :: "16 word" where "x = \<lparr>iiface = uu_, oiface = uv_, src = uw_, dst = ux_, proto = uy_, sports = (sps1, sps2), dports = (dps1, dps2)\<rparr>" and "\<not> (sps2 < sps1 \<or> dps2 < dps1)"
dpd5: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and domain_op :: "'b \<Rightarrow> 'b" assumes "class.domain_pre_dioid_one_zerol plus times one zero less_eq less domain_op" shows "domain_op zero = zero"
execn_Normal_elim_cases_14: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Catch c1 c2,Normal s\<rangle> =n\<Rightarrow> t" obtains s' :: "'e" where "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> =n\<Rightarrow> Abrupt s'" and "\<Gamma>\<turnstile> \<langle>c2,Normal s'\<rangle> =n\<Rightarrow> t"   | "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> =n\<Rightarrow> t" and "\<not> isAbr t"
hd_append: fixes xs :: "'b list"   and ys :: "'b list" shows "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
Y_pos_main: fixes y :: "(real, 'n) Finite_Cartesian_Product.vec"   and i :: "'n" assumes "y \<in> pow_A_1 ` X" shows "0 < y $h i"
Gmod1_def: shows "Gmod1 = G\<lparr>carrier := H1 <#> H \<inter> K\<rparr> Mod (H1 <#> H \<inter> K1)"
weight_le_def: fixes w :: "'c \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'c \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" shows "weight_fun.weight_le w w0 scf s t \<equiv> vars_term_ms (scf_term scf s) \<subseteq># vars_term_ms (scf_term scf t) \<and> weight_fun.weight w w0 scf s \<le> weight_fun.weight w w0 scf t"
dma_consistent: fixes mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and x\<^sub>A :: "'Var\<^sub>A" shows "dma\<^sub>A (\<lambda>x\<^sub>A. mem\<^sub>C (var\<^sub>C_of x\<^sub>A)) x\<^sub>A = dma\<^sub>C mem\<^sub>C (var\<^sub>C_of x\<^sub>A)"
distrib_lattice_def: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.distrib_lattice inf less_eq less sup \<equiv> class.lattice inf less_eq less sup \<and> class.distrib_lattice_axioms inf sup"
fin_R: shows "finite {(x, y). x \<in> A \<and> y \<in> A \<and> (x, y) \<in> r}"
\<b>_2: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat" assumes "(h, U) \<in> set ps"   and "i \<le> card U" shows "poly_deg h < \<b> ps i"
3: shows "is_left_adjoint (h \<star> p\<^sub>0)"
borel_measurable_nn_cond_exp2: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "nn_cond_exp M F f \<in> borel_measurable M"
dim_span: fixes S :: "('c, 'd) vec set" shows "vec.dim (vec.span S) = vec.dim S"
fdia_star_unfold_var: fixes y :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set" shows "rdom y \<union> relfdia x (relfdia (x\<^sup>*) y) = relfdia (x\<^sup>*) y"
simps_2: fixes n :: "nat" shows "deeptree (Suc n) = Branching (deeptree n) (deeptree n)"
max_Var_floatarith_subst_floatarith_le: fixes fa :: "floatarith"   and xs :: "floatarith list"   and z :: "nat" assumes "max_Var_floatarith fa \<le> length xs"   and "max_Var_floatariths xs \<le> z" shows "max_Var_floatarith (subst_floatarith ((!) xs) fa) \<le> z"
append_Cons: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "(x # xs) \<cdot> ys = x # xs \<cdot> ys"
of_int_mod_ring_to_int_mod_ring: fixes x :: "'a mod_ring" shows "of_int_mod_ring (to_int_mod_ring x) = x"
fmupd_lookup: fixes a :: "'b"   and b :: "'a"   and m :: "('b, 'a) fmap"   and a' :: "'b" shows "fmlookup (fmupd a b m) a' = (if a = a' then Some b else fmlookup m a')"
CNV_meqD: fixes x :: "'a"   and y :: "'a" assumes "CNV x y" shows "x \<equiv> y"
monitorable_formula_code: fixes \<phi> :: "'b MFOTL.formula" shows "mmonitorable \<phi> = mmonitorable_exec \<phi>"
std_normal_moment_even: shows "has_bochner_integral lborel (\<lambda>x. std_normal_density x * x ^ (2 * k)) (fact (2 * k) / (2 ^ k * fact k))"
allDefs_disjoint': fixes n :: "'node"   and g :: "'g"   and m :: "'node"   and v :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "m \<in> set (\<alpha>n g)"   and "v \<in> uninst_code.ssa.allDefs phis g n"   and "v \<in> uninst_code.ssa.allDefs phis g m" shows "n = m"
distinct_7: shows "SizeOp \<noteq> ToBooleanOp"
pwd_spy_enc_mult_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_enc_mult n)"
MGT_alternD: fixes G :: "state triple set"   and c :: "com" assumes "state_not_singleton"   and "G|-{\<lambda>Z s0. \<forall>s1. <c,s0> -c-> s1 \<longrightarrow> Z = s1}. c .{(=)}" shows "G|-{=}.c.{->}"
k: shows "continuous_map (prod_topology (top_of_set {0..1}) X2) Y2 k"
of_list_inv: fixes x :: "'a list" shows "inv (of_list x)"
epath_decompose: fixes G :: "'b uprod set"   and v :: "'b"   and p :: "'b uprod list"   and v' :: "'b"   and a :: "'b"   and b :: "'b" assumes "epath G v p v'"   and "Upair a b \<in> set p" shows "\<exists>u u' p' p''. u \<in> {a, b} \<and> u' \<in> {a, b} \<and> epath G v p' u \<and> epath G u' p'' v' \<and> length p' < length p \<and> length p'' < length p"
finite_cartesian_productD1: fixes A :: "'b set"   and B :: "'c set" assumes "finite (A \<times> B)"   and "B \<noteq> {}" shows "finite A"
contour_integrable_linepath_same_Im_iff: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "Im a = Im b"   and "Re a < Re b" shows "(f contour_integrable_on linepath a b) = ((\<lambda>x. f (complex_of_real x + complex_of_real (Im a) * \<i>)) integrable_on {Re a..Re b})"
isom4: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and H :: "'e set" assumes "Group G"   and "G \<triangleright> N"   and "G \<guillemotright> H " shows "(\<natural>\<^bsub>G\<^esub>H / (H \<inter> N)) \<cong> (\<natural>\<^bsub>G\<^esub>N \<diamondop>\<^bsub>G\<^esub> H / N)"
simps_49: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a"   and x1 :: "bool" shows "(case boolV x1 of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = f1 x1"
weakSimI: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "eqvt Rel"   and "\<And>\<Psi>' \<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; distinct (bn \<alpha>); \<alpha> \<noteq> \<tau>\<rbrakk> \<Longrightarrow> \<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>', P', Q') \<in> Rel)"   and "\<And>Q'. \<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q' \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
scalar_prod_add_distrib: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec"   and v\<^sub>3 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (v\<^sub>2 + v\<^sub>3) = v\<^sub>1 \<bullet> v\<^sub>2 + v\<^sub>1 \<bullet> v\<^sub>3"
card_not_QuadRes_set_prime: fixes p :: "int" assumes "prime p"   and "odd p" shows "card {x. \<not> QuadRes p x \<and> x \<in> {0..<p}} = nat (p - 1) div 2"
mod: fixes m :: "nat" shows "\<^bold>g [^] m = \<^bold>g [^] (m mod Suc n)"
f_def: fixes n :: "nat" shows "f n = [x, y, z] ! n"
assms_3: shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
mat_annil: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow> 'c" shows "\<delta> \<otimes> f = \<delta>"
BRBang: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> Rel" shows "(!P, !Q) \<in> bangRel Rel"
cqt_basic_10: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>"   and \<chi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>) in v]"
dom_cod: fixes f :: "'b" assumes "D.arr f" shows "D.dom (D.cod f) = D.cod f"
abc_list_crsp_simp: fixes lma :: "nat list"   and lmb :: "nat list"   and m :: "nat"   and n :: "nat" assumes "abc_list_crsp lma lmb"   and "m < length lma"   and "m < length lmb" shows "abc_list_crsp (lma[m := n]) (lmb[m := n])"
normalize_quot_proj_1: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "fst (normalize_quot (a, b)) = a div (gcd a b * unit_factor b)"
ac_fProd_ProdTr1: fixes n :: "nat"   and B :: "nat \<Rightarrow> ('d, 'e) Ring_scheme" assumes "\<forall>k\<le>Suc n. Ring (B k)" shows "ag_setfunc n B (Suc 0) (compose {0} B (slide (Suc n))) (carr_prodag {i. i \<le> n} B) (carr_prodag {0} (compose {0} B (slide (Suc n)))) \<subseteq> carr_prodag {i. i \<le> Suc n} B"