hcju/isabelleps huggingface dataset:
card_lists_length_eq: fixes A :: "'b set"   and n :: "nat" assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs = n} = card A ^ n"
ir_Assign_Assign: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and e :: "aexp"   and x' :: "char list"   and e' :: "aexp" shows "ir_hoare P (x ::= e) (x' ::= e') (\<lambda>t t'. \<exists>v v'. P (t(x := v)) (t'(x' := v')) \<and> t x = aval e (t(x := v)) \<and> t' x' = aval e' (t'(x' := v')))"
mp_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p .\<longrightarrow>. q"   and "A \<turnstile> p" shows "A \<turnstile> q"
abstract_primitive_in_doubt_allow_Allow: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'b common_primitive match_expr"   and p :: "('b, 'c) tagged_packet_scheme"   and disc :: "'b common_primitive negation_type \<Rightarrow> bool" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "matches (\<beta>, in_doubt_allow) m Accept p" shows "matches (\<beta>, in_doubt_allow) (abstract_primitive disc m) Accept p"
borel_singleton: fixes A :: "'c set"   and x :: "'c" assumes "A \<in> sets borel" shows "insert x A \<in> sets borel"
At: fixes x0 :: "'a"   and At :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and Follows :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat" assumes "Chain x0 At Follows" shows "At (Chain.f x0 At Follows n) n"
d: shows "\<forall>C'. C' \<in> Cn A .\<bottom>. \<phi> \<and> Cn A \<div> \<phi> \<subseteq> C' \<longrightarrow> \<psi> \<in> C'"
distinct_mapI: fixes f :: "'f \<Rightarrow> 'e"   and l :: "'f list" assumes "distinct (map f l)" shows "distinct l"
set_awalk_verts_append_cas: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and q :: "'b list"   and w :: "'a" assumes "cas u p v"   and "cas v q w" shows "set (awalk_verts u (p @ q)) = set (awalk_verts u p) \<union> set (awalk_verts v q)"
prime_sum_upto_def: fixes f :: "nat \<Rightarrow> 'b"   and x :: "real" shows "prime_sum_upto f x = (\<Sum>p | prime p \<and> real p \<le> x. f p)"
simps_1: fixes y :: "'b"   and y' :: "'b"   and ys :: "'b list" shows "test_2 (y # (y' # ys =: x'__) =: x__) = x__ @ x'__ @ x'__"
list_all2_lengthD: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list" assumes "list_all2 P xs ys" shows "length xs = length ys"
update_simps_2: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "ip \<in> kD rt"   and "sqn rt ip < \<pi>\<^sub>2 r" shows "update rt ip r = rt(ip \<mapsto> r)"
cong_def: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "[b = c] (mod a) = (b mod a = c mod a)"
affine_hull_affine_Int_open: fixes S :: "'a set"   and T :: "'a set" assumes "affine S"   and "open T"   and "S \<inter> T \<noteq> {}" shows "affine hull (S \<inter> T) = affine hull S"
arr_iff_in_hom: fixes f :: "'b" shows "J.arr f = \<guillemotleft>f : J.dom f \<rightarrow>\<^sub>J J.cod f\<guillemotright>"
sec: shows "simAbs ec = sim_equiv_class a (t \<leadsto> s)"
scale_one: fixes x :: "'c" shows "(1::'a) *b x = x"
rec_sigma_Suc_simp_rewrite: fixes f :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "primerec f (Suc (length xs))" shows "rec_exec (rec_sigma f) (xs @ [Suc x]) = rec_exec (rec_sigma f) (xs @ [x]) + rec_exec f (xs @ [Suc x])"
f: fixes u :: "nat" shows "f u \<subseteq> f (Suc u)"
bdd: shows "bdd_above {\<parallel>A \<tau>\<parallel>\<^sub>o\<^sub>p |\<tau>. \<tau> \<in> cball t (min e1 e2)}"
to_tree_order_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and child :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some parent"   and "parent \<in> set nodes" shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes"
module_hom_iff_linear: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" shows "module_hom s1 s2 f = linear s1 s2 f"
cos_diff_cos: fixes w :: "'a"   and z :: "'a" shows "cos w - cos z = (2::'a) * sin ((w + z) / (2::'a)) * sin ((z - w) / (2::'a))"
insert_mono: fixes C :: "'c set"   and D :: "'c set"   and a :: "'c" assumes "C \<subseteq> D" shows "Set.insert a C \<subseteq> Set.insert a D"
ubx_invertible: fixes opinl :: "opinl"   and xs :: "Unboxed.type option list"   and opubx :: "opubx" assumes "Op_example.ubx opinl xs = Some opubx" shows "deubx opubx = opinl"
r_e2stack_def: shows "r_e2stack \<equiv> r_pdec1"
lstrict_prefix_code_4: fixes x :: "'c"   and xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "lstrict_prefix (LCons x xs) (LCons y ys) = (x = y \<and> lstrict_prefix xs ys)"
map\<^sub>0_def: fixes a :: "'c" shows "map\<^sub>0 a \<equiv> src\<^sub>D (F a)"
2: fixes x :: "real" assumes "x \<in> {a..b} - (s \<union> g -` s')" shows "(g has_real_derivative g' x) (at x within {a..b})"
mod_ring_uminus: shows "(mod_ring_rel ===> mod_ring_rel) (uminus_p p) uminus"
rbt_map_entry_inv2_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and t :: "('a, 'b) rbt" shows "inv2 (ord.rbt_map_entry less k f t) = inv2 t"
succ_ne_self: fixes i :: "V" shows "i \<noteq> ZFC_in_HOL.succ i"
elims: fixes x :: "'a list"   and y :: "nat \<Rightarrow> 'a" assumes "local.coeff x = y" obtains "x = []" and "y = (\<lambda>_. \<zero>)"   | v :: "'a" and va :: "'a list" where "x = v # va" and "y = (\<lambda>i. if i = degree (v # va) then lead_coeff (v # va) else local.coeff (tl (v # va)) i)"
exec_Normal_elim_cases_3: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> \<Rightarrow> t" obtains "t = Abrupt s"
has_fields_rec: fixes P :: "'a prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and FDTs' :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D has_fields FDTs"   and "FDTs' = map (\<lambda>(F, b, T). ((F, C), b, T)) fs @ FDTs" shows "P \<turnstile> C has_fields FDTs'"
c: fixes x :: "complex" assumes "x \<in> ball \<xi> r" shows "exp (h x) / g x = c"
wt_lOfFax: fixes f :: "'fsym" assumes "wtFsym f" shows "TE.wt (lOfFax f)"
nact_active: fixes id :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n'' :: "nat" assumes "\<parallel>id\<parallel>\<^bsub>t n\<^esub>"   and "n \<le> n''"   and "\<not> (\<exists>n'\<ge>n. latestAct_cond id t n'' n')" shows "n = n''"
invertible_inverse_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "local.inverse (local.inverse u) = u"
linear_simps_2: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "bounded_linear f" shows "f (a - b) = f a - f b"
IH2_perm: assumes "Suc p \<le> hi"   and "hi < length xs''" shows "set (WB_Sort.sublist xs''' (Suc p) hi) = set (WB_Sort.sublist xs'' (Suc p) hi)"
avl_def: fixes t :: "'a tree" shows "avl t = (is_bal (erase t) \<and> hinv t)"
plus_rep_def: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "plus_rep p1 p2 = (if is_empty_rep p1 \<or> is_empty_rep p2 then empty_rep else let (l1, h1) = p1; (l2, h2) = p2 in (l1 + l2, h1 + h2))"
simps_5: fixes v :: "nat formula"   and va :: "nat formula" shows "cnf_to_dimacs (v \<^bold>\<or> va) = [disj_to_dimacs (v \<^bold>\<or> va)]"
cs_done_implies_same_snapshots: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and j :: "nat"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "i \<le> j"   and "j \<le> length t"   and "snd (cs (distributed_system.s channel trans send recv init t i) cid) = Done"   and "channel cid = Some (p, q)" shows "cs (distributed_system.s channel trans send recv init t i) cid = cs (distributed_system.s channel trans send recv init t j) cid"
cont_cxh: shows "continuous_on S (complex_of_real \<circ> h)"
comm_monoid_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_monoid_add_hom hom \<equiv> monoid_add_hom hom"
inf_absorb2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "x \<cdot> y = y"
ovalidNF_wp_comb2: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and P' :: "'c \<Rightarrow> bool" assumes "ovalidNF P f Q"   and "\<And>s. P' s \<Longrightarrow> P s" shows "ovalidNF P' f Q"
top_greatest: fixes x :: "'b" shows "x \<sqsubseteq> \<top>"
game2_2a: shows "game2 = map_spmf fst game2_a"
rsp_on_range: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f" shows "rsp_on A f \<subseteq> A \<times> A"
iso_is_section: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.iso f" shows "S.section f"
def_1: fixes n :: "nat"   and upw :: "bool"   and i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "LLL_Impl.basis_reduction_add_rows n upw i state = (if upw then LLL_Impl.basis_reduction_add_rows_loop n state i i (small_fs_state state) else state)"
wts_same_types_typesconf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and es :: "expr list"   and Ts :: "ty list" assumes "P,E,h \<turnstile> es [:] Ts" shows "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts"
NoSmallSeparationsInduct_axioms_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and sep_size :: "nat" shows "NoSmallSeparationsInduct_axioms G v0 v1 sep_size \<equiv> (\<forall>S. Separation G v0 v1 S \<longrightarrow> Suc sep_size \<le> card S) \<and> (\<forall>G'. (\<forall>S. Separation G' v0 v1 S \<longrightarrow> sep_size \<le> card S) \<longrightarrow> v0_v1_Digraph G' v0 v1 \<longrightarrow> (\<exists>paths. DisjointPaths G' v0 v1 paths \<and> card paths = sep_size))"
fcomp_fid_on_rel: fixes A :: "V"   and r :: "V" shows "fid_on A \<circ>\<^sub>\<bullet> r = r \<restriction>\<^sup>r\<^sub>\<bullet> A"
conjl2_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<lhd> x \<le> z \<lhd> y"
backflows_filternew_flows_state: fixes \<T> :: "'b stateful_policy" shows "backflows (filternew_flows_state \<T>) = backflows (flows_state \<T>) - flows_fix \<T>"
sel_3: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset" shows "old_concl_of (old_Infer x1 x2 x3) = x3"
cases: fixes x :: "object ast_effect" obtains a :: "object atom formula list" and d :: "object atom formula list" where "x = Effect a d"
quadratic_sub_fm: fixes var :: "nat"   and L :: "real list"   and Dv :: "real"   and Cv :: "real"   and c :: "real mpoly"   and a :: "real mpoly"   and Av :: "real"   and b :: "real mpoly"   and Bv :: "real"   and d :: "real mpoly"   and F :: "atom fm" assumes "var < length L"   and "Dv \<noteq> 0"   and "0 \<le> Cv"   and "var \<notin> vars c"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) a = Av"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) b = Bv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) c = Cv"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) d = Dv" shows "eval F (L[var := (Av + Bv * sqrt Cv) / Dv]) = eval (quadratic_sub_fm var a b c d F) (L[var := sqrt Cv])"
smult_dvd_iff: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "(smult a p dvd q) = (if a = (0::'b) then q = 0 else p dvd q)"
assms: shows "\<forall>x. case A\<^sub>C x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>1\<^sub>C x \<noteq> v \<or> mem\<^sub>2\<^sub>C x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written mds\<^sub>C x"
preserves_reflects_arr: fixes f :: "'a" shows "S.arr (R f) = S.arr f"
heap_ordered_single: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree" shows "BinomialHeapStruc.heap_ordered t = (\<forall>x\<in>#BinomialHeapStruc.tree_to_multiset t. BinomialHeapStruc.prio t \<le> snd x)"
map_scale_one_left: fixes t :: "'b \<Rightarrow>\<^sub>0 'c" shows "(1::'c) \<cdot> t = t"
left_neutral: fixes a :: "'a" shows "(0::'a) + a = a"
loc_reach_mem_diff_eq: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and mem :: "'Var \<Rightarrow> 'Val"   and c :: "'Com" assumes "\<forall>x. var_asm_not_written mds x \<longrightarrow> mem' x = mem x \<and> dma mem' x = dma mem x" shows "loc_reach \<langle>c, mds, mem\<rangle> = loc_reach \<langle>c, mds, mem'\<rangle>"
semilat_le_err_Err_plus: fixes x :: "'b err"   and A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b err \<Rightarrow> 'b err \<Rightarrow> 'b err" assumes "x \<in> err A"   and "semilat (err A, le r, f)" shows "Err \<squnion>\<^bsub>f\<^esub> x = Err"
has_field_derivative_higher_deriv: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and x :: "complex"   and n :: "nat" assumes "f holomorphic_on S"   and "open S"   and "x \<in> S" shows "((deriv ^^ n) f has_field_derivative (deriv ^^ Suc n) f x) (at x)"
A_of_mds\<^sub>0_is_mds\<^sub>s: shows "mds\<^sub>A_of mds\<^sub>0 = mds\<^sub>s"
pred_envir_conv: fixes p :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c" shows "pred_envir p f = (\<forall>x. p (f x))"
TS_index_2: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<in> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs" shows "vs @ xs \<in> C M2 M1 \<Omega> V m (Suc (length xs))"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> TargetTerm T2"
f_measurable: shows "set_borel_measurable lborel (einterval a b) f"
fls_inverse_base: fixes f :: "'a fls" assumes "f \<noteq> 0" shows "fls_nth (inverse f) (- fls_subdegree f) = inverse (fls_nth f (fls_subdegree f))"
trace_rest_eqI: fixes IA :: "'a set"   and IU :: "'d set"   and rest1 :: "('f, 'c, 'a, 'd, 'b, 'e, 'g) rest_scheme"   and p :: "'f spmf"   and rest2 :: "('h, 'c, 'a, 'd, 'b, 'e, 'i) rest_scheme"   and q :: "'h spmf" assumes "\<And>tr ia. \<lbrakk>set tr \<subseteq> IA \<times> UNIV <+> IU \<times> UNIV; ia \<in> IA\<rbrakk> \<Longrightarrow> fst (trace_rest' rest1 p tr) ia = fst (trace_rest' rest2 q tr) ia"   and "\<And>tr iu. \<lbrakk>set tr \<subseteq> IA \<times> UNIV <+> IU \<times> UNIV; iu \<in> IU\<rbrakk> \<Longrightarrow> snd (trace_rest' rest1 p tr) iu = snd (trace_rest' rest2 q tr) iu" shows "trace_rest_eq rest1 rest2 IA IU p q"
zero_divisor_smult_left: fixes a :: "'b"   and f :: "'b poly" assumes "zero_divisor a" shows "zero_divisor (smult a f)"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "S.in_hom f a b = (S.arr a \<and> S.arr b \<and> S.arr f \<and> \<guillemotleft>f : a \<Rightarrow> b\<guillemotright>)"
strict_monoD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a" assumes "OrdinalInduct.strict_mono f"   and "A < B" shows "f A < f B"
simps_1: fixes ht :: "('a, 'b) hashtable" shows "hm_it_adjust 0 ht = return 0"
lc_constant0: fixes T :: "Unary_PCF.type" shows "lc (constant0 T)"
openin_subtopology_Int2: fixes X :: "'b topology"   and T :: "'b set"   and S :: "'b set" assumes "openin X T" shows "openin (subtopology X S) (S \<inter> T)"
of_rat_1_def: fixes x :: "rat" shows "of_rat_1 x \<equiv> (poly_rat x, x, x)"
new_element_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'ag" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "ptr \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
prv_cnj_imp_monoR: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp \<phi> \<chi>) (imp (imp \<phi> \<psi>) (imp \<phi> (cnj \<chi> \<psi>))))"
t2: shows "\<Gamma> \<turnstile>\<^sub>c e2 : BOOL"
distproj_in_proj_set_3: fixes x :: "'c"   and A :: "'c set" assumes "proj_set x A \<noteq> {}" shows "dist x (distproj x A) = infdist x A"
nontriv_solution_def: fixes D :: "nat" assumes "pell D" shows "pell.nontriv_solution D = (\<lambda>(a, b). (a, b) \<noteq> (1::'a, 0::'a) \<and> a\<^sup>2 = (1::'a) + of_nat D * b\<^sup>2)"
rel_interior_affine: fixes S :: "'c set" assumes "affine S" shows "rel_interior S = S"
b_def: shows "b \<equiv> rightrec u v leftcut rightcut"
mult_minus_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = uminus (times a b)"
SpecialR_def: shows "SpecialR \<equiv> {(x, y). x \<noteq> Empty \<and> y = Empty}"
simps: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set" assumes "getIJ getIJ" shows "New_max.filterQuery V Q = Set.filter (\<lambda>(s, uu_). \<not> Set.is_empty (s \<inter> V)) Q"
weak_pref_mix_with_my_U_weak_pref: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<succeq>[\<R>] q" shows "mix_pmf (local.my_U p) local.some_best local.some_worst \<succeq>[\<R>] mix_pmf (local.my_U q) local.some_best local.some_worst"
Si_neg: fixes T :: "real" assumes "0 \<le> T" shows "Si (- T) = - Si T"
monom_mult_def: fixes n :: "nat"   and f :: "'b poly" shows "monom_mult n f = monom (1::'b) n * f"
execute_instr_sub1_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" shows "snd (execute_instr_sub1 i s) = False"
integrable_mult_indicator: fixes A :: "'d set"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "A \<in> sets M"   and "integrable M f" shows "integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
scaleR: fixes r :: "real"   and b :: "'a" shows "f'' (r *\<^sub>R b) = r *\<^sub>R f'' b"
zero_right: fixes a :: "'a" shows "a ** (0::'b) = (0::'c)"
combine_common_factor: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and e :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring plus times" shows "plus (times a e) (plus (times b e) c) = plus (times (plus a b) e) c"
psubstT_eq_rawpsubstT: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "distinct (map snd txs)"   and "\<And>i j. \<lbrakk>i < j; j < length txs\<rbrakk> \<Longrightarrow> snd (txs ! j) \<notin> FvarsT (fst (txs ! i))" shows "psubstT r txs = rawpsubstT r txs"
inext_nth_cut_ge_inext_nth: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" assumes "I \<noteq> {}" shows "I \<down>\<ge> (I \<rightarrow> a) \<rightarrow> b = I \<rightarrow> (a + b)"
simps_4: fixes r :: "'b mrexp3"   and s :: "'b mrexp3"   and uu :: "'b set"   and uv :: "bool" shows "strip3 (Plus3 r s uu uv) = Plus (strip3 r) (strip3 s)"
finite_fvi: fixes b :: "nat"   and \<phi> :: "'b formula" shows "finite (fvi b \<phi>)"
genby_genset_subset: fixes S :: "'a set" shows "order.greater_eq \<langle>S\<rangle> S"
lset_lappend1: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lset xs \<subseteq> lset (lappend xs ys)"
distinct_formal_outs: fixes p :: "'f"   and ins :: "'c list"   and outs :: "'c list" assumes "(p, ins, outs) \<in> set procs" shows "distinct outs"
HPT_Source_Target: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and SA :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "States SA \<inter> Source TS = {}"   and "SA \<in> SAs (HA ST)" shows "States SA \<inter> Target TS = {}"
fixp_def: fixes f :: "'a \<Rightarrow> 'a" shows "ccpo_class.fixp f = \<Squnion> ccpo_class.iterates f"
map_fst_zip: fixes xs :: "'c list"   and ys :: "'d list" assumes "length xs = length ys" shows "map fst (zip xs ys) = xs"
int_cong: fixes T :: "('fsym, 'tp) efsym trm"   and \<xi>1 :: "Preliminaries.var \<Rightarrow> univ"   and \<xi>2 :: "Preliminaries.var \<Rightarrow> univ" assumes "\<And>x. x \<in> vars T \<Longrightarrow> \<xi>1 x = \<xi>2 x" shows "TE.int \<xi>1 T = TE.int \<xi>2 T"
rel_imp_elem: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "CLF cl f"   and "(x, y) \<in> order cl" shows "x \<in> pset cl"
subcocycle_avg_ereal_Tinv: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "I.subcocycle u"   and "I.invertible_qmpt" shows "I.subcocycle_avg_ereal (\<lambda>n x. u n ((I.Tinv ^^ n) x)) = I.subcocycle_avg_ereal u"
evnodd_Un: fixes A :: "(nat \<times> nat) set"   and B :: "(nat \<times> nat) set"   and b :: "nat" shows "evnodd (A \<union> B) b = evnodd A b \<union> evnodd B b"
assms_1: shows "(red F)\<^sup>*\<^sup>* f 0"
trm_simp_3: fixes A :: "'b trm"   and B :: "'b trm"   and C :: "'b trm"   and D :: "'b trm" assumes "App A B = App C D" shows "B = D"
a: shows "Seed\<^bsub>p\<^esub> [next_plane0\<^bsub>p\<^esub>]\<rightarrow>* g"
items_insert: fixes X :: "msg"   and H :: "msg set" shows "insert X (items H) \<subseteq> items (insert X H)"
dagger_plus_one: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<dagger> = x\<^sup>\<dagger>"
degree_eq_univariate_degree: fixes p :: "'b mpoly"   and v :: "nat" shows "MPoly_Type.degree p v = (if p = 0 then 0 else Polynomial.degree (mpoly_to_nested_poly p v))"
I: shows "(drop (length ws) xs @ ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys Y) \<in> failures R"
ideal_generated_singleton: fixes a :: "'a" shows "ideal_generated {a} = {k * a |k. k \<in> UNIV}"
in_charts_in_atlas: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and x :: "('c, 'd) chart" assumes "c_manifold charts k"   and "x \<in> charts" shows "x \<in> c_manifold.atlas charts k"
mersenne_mod_nonneg_strong: fixes p :: "nat"   and a :: "int" assumes "- (2 ^ p) + 1 < a" shows "0 \<le> mersenne_mod a p"