hcju/isabelleps huggingface dataset:
simps: fixes var :: "nat"   and x :: "real mpoly" shows "get_coeffs var x = (isolate_variable_sparse x var 2, isolate_variable_sparse x var 1, isolate_variable_sparse x var 0)"
Ord_oexp: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "Ord (\<alpha> \<up> \<beta>)"
is_aligned_no_overflow: fixes ptr :: "'b word"   and sz :: "nat" assumes "is_aligned ptr sz" shows "ptr \<le> ptr + 2 ^ sz - 1"
set_drop_subset: fixes n :: "nat"   and xs :: "'a list" shows "set (drop n xs) \<subseteq> set xs"
before_dist_snd_snd: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "before vs ram1 ram2"   and "distinct vs" shows "snd (splitAt ram2 (snd (splitAt ram1 vs))) = fst (splitAt ram1 (snd (splitAt ram2 vs)))"
more_arith_simps_10: fixes a :: "'a" shows "- (- a) = a"
jordan_matrix_dim_1: fixes n_as :: "(nat \<times> 'a) list" shows "dim_row (jordan_matrix n_as) = sum_list (map fst n_as)"
replace_append: fixes x :: "'c"   and ys :: "'c list"   and as :: "'c list"   and bs :: "'c list" shows "replace x ys (as @ bs) = (if x \<in> set as then replace x ys as @ bs else as @ replace x ys bs)"
affine_independent_card_dim_diffs: fixes S :: "'a set"   and a :: "'a" assumes "\<not> affine_dependent S"   and "a \<in> S" shows "card S = dim ((\<lambda>x. x - a) ` S) + 1"
subcomplexes: fixes A :: "'a set set" assumes "A \<in> \<A>" shows "ChamberSubcomplex A"
set_mset_add_mset_insert: fixes a :: "'c"   and A :: "'c multiset" shows "set_mset (add_mset a A) = insert a (set_mset A)"
incseq_sumI: fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>i. (0::'a) \<le> f i" shows "incseq (\<lambda>i. sum f {..<i})"
subst_introduces_no_lids: fixes e :: "('r, 'l, 'v) expr"   and x :: "'v"   and e' :: "('r, 'l, 'v) expr" shows "LID\<^sub>E (subst e x e') \<subseteq> LID\<^sub>E e \<union> LID\<^sub>E e'"
prime_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 p :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalization_semidom.prime zero times one normalize (normalize p) = comm_semiring_1.prime_elem times one zero p"
distinct_26: fixes x7 :: "CTypeId" shows "CClassT x7 \<noteq> IntgT"
finite_imp_card_of_natLeq_on: fixes A :: "'c set" assumes "finite A" shows "|A| =o natLeq_on (card A)"
parallel_reduction_fvs: fixes A :: "'a trm"   and A' :: "'a trm" assumes "A >> A'" shows "fvs A' \<subseteq> fvs A"
natabs0: fixes x :: "int" shows "(nat \<bar>x\<bar> = 0) = (x = 0)"
insertI: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and D :: "'c set"   and e :: "'c" assumes "x \<notin> A"   and "f x y \<in> D"   and "(A, y) \<in> foldSetD D f e" shows "(insert x A, f x y) \<in> foldSetD D f e"
CHOinitConfig_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and cfg :: "'a \<Rightarrow> 'b"   and coord :: "'a \<Rightarrow> 'a" shows "CHOinitConfig A cfg coord \<equiv> \<forall>p. CinitState A p (cfg p) (coord p)"
agree_plus1: fixes \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and \<nu>' :: "(real, 'c) vec \<times> (real, 'c) vec"   and t1 :: "('d, 'c) trm"   and t2 :: "('d, 'c) trm" assumes "Vagree \<nu> \<nu>' (FVDiff (Plus t1 t2))" shows "Vagree \<nu> \<nu>' (FVDiff t1)"
opt_tag_def: fixes x :: "'b"   and y :: "'b" shows "opt_tag x y \<equiv> x = y"
pdevs_of_list_map_scaleR: fixes r :: "real"   and xs :: "'a list" shows "pdevs_of_list (map ((*\<^sub>R) r) xs) = scaleR_pdevs r (pdevs_of_list xs)"
in_IF1rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d IF1"   and y :: "'e IF1" shows "IF1rel R x y = (\<exists>z. z \<in> IF1in {(x, y). R x y} \<and> IF1map fst z = x \<and> IF1map snd z = y)"
order_le_imp_less_or_eq: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y" shows "x < y \<or> x = y"
offs_num_def: fixes n :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" shows "offs_num n xs index key mi ma i \<equiv> length (filter (\<lambda>x. index key x n mi ma = i) xs)"
jumpF_linear_comp_1: fixes c :: "real"   and f :: "real \<Rightarrow> real"   and b :: "real"   and x :: "real" assumes "c \<noteq> 0" shows "jumpF (f \<circ> (\<lambda>x. c * x + b)) (at_left x) = (if 0 < c then jumpF f (at_left (c * x + b)) else jumpF f (at_right (c * x + b)))"
Let_1: fixes x :: "Vars.var"   and r :: "C"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "x \<in> domA \<Gamma>" shows "(\<N>\<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho>\<^esub>)|\<^bsub>C\<cdot>r\<^esub> = (\<N>\<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>\<^esub>)|\<^bsub>C\<cdot>r\<^esub>"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> otway" shows "\<exists>A. Says A B X \<in> set evs"
is_tangle_right_compose: fixes x :: "wall"   and y :: "wall" assumes "is_tangle_diagram (x \<circ> y)" shows "is_tangle_diagram y"
False: shows "\<not> (topspace X = {} \<or> topspace Y = {})"
abs_eq: fixes x :: "'a" shows "prod_right x = Blinfun ((**) x)"
assms_4: shows "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"
bind_simps_1: fixes f :: "'c \<Rightarrow> 'b list" shows "List.bind [] f = []"
bind_eq_Some_conv: fixes f :: "'e option"   and g :: "'e \<Rightarrow> 'd option"   and x :: "'d" shows "(f \<bind> g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
ipurge_tr_rev_aux_nil: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and xs :: "'c list" shows "(ipurge_tr_rev_aux I D U xs = []) = (\<forall>u\<in>U. \<not> (\<exists>v\<in>D ` set xs. (v, u) \<in> I))"
ide_in_hom: fixes a :: "'a" shows "Cop.ide a = Cop.in_hom a a a"
L: shows "(\<lambda>n. sum (f \<circ> g) {..<n}) \<longlonglongrightarrow> L"
globally_sound_mode_use_def: fixes gc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" shows "abs.globally_sound_mode_use gc \<equiv> \<forall>mdss. mdss \<in> abs.reachable_mode_states gc \<longrightarrow> abs.compatible_modes mdss"
wf_sub_rel_of_2: shows "wf_sub_rel subs"
uint64_msb_test_bit: fixes x :: "uint64" shows "msb x = bit x 63"
bezout_add_nat: fixes a :: "nat"   and b :: "nat" shows "\<exists>d x y. d dvd a \<and> d dvd b \<and> (a * x = b * y + d \<or> b * x = a * y + d)"
holomorphic_fun_eq_const_on_connected: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and z :: "complex"   and w :: "complex" assumes "f holomorphic_on S"   and "open S"   and "connected S"   and "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0"   and "z \<in> S"   and "w \<in> S" shows "f w = f z"
sendmsg_simps_3: fixes P :: "'e \<Rightarrow> bool"   and ip :: "nat"   and m :: "'e" shows "sendmsg P (unicast ip m) = True"
Suc_rw: shows "[0..<Suc k] = [0..<k] @ [k]"
extend_1_eval: fixes 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 e :: "char list exp"   and s :: "(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'' :: "(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' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e'',s''\<rangle>"   and "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
contractible_def: fixes S :: "'a set" shows "contractible S \<equiv> \<exists>a. homotopic_with_canon (\<lambda>x. True) S S id (\<lambda>x. a)"
seq1: fixes c1 :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and m' :: "'id \<Rightarrow> 'val"   and c2 :: "('exp, 'id) MWLsCom" assumes "SRabbr c1 m \<alpha> None m'" shows "SRabbr (c1;c2) m \<alpha> (Some c2) m'"
convex_affine_hull: fixes S :: "'a set" shows "convex (affine hull S)"
add_le_imp_le_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less_eq (plus a c) (plus b c)" shows "less_eq a b"
noFaultStuckn_exec_startD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, bool) com option"   and c :: "('e, 'd, bool) com"   and s :: "('e, bool) xstate"   and n :: "nat"   and t :: "('e, bool) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Fault True, Fault False, Stuck}"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "s \<notin> {Fault True, Fault False, Stuck}"
tree_rank_estimate_lower: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) SkewBinomialTree list" assumes "tree_invar (Node e a r ts)" shows "2 ^ r \<le> size (tree_to_multiset (Node e a r ts))"
elims_2: fixes x :: "'g \<times> 'a"   and xa :: "'j \<times> 'd"   and xb :: "'e"   and xc :: "'k \<times> 'f" assumes "check_AND x xa xb xc" obtains x0 :: "'g" and x1 :: "'a" and a0 :: "'j" and a1 :: "'d" and s :: "'e" and z0 :: "'k" and z1 :: "'f" where "x = (x0, x1)" and "xa = (a0, a1)" and "xb = s" and "xc = (z0, z1)" and "check0 x0 a0 s z0 \<and> check1 x1 a1 s z1"
hfset_ord_of: fixes n :: "nat" shows "hfset (ord_of n) = ord_of ` {0..<n}"
shrinking_from_belong_stalk: fixes s :: "('a set set \<times> ('a set \<Rightarrow> ('a \<times> 'a) set)) set"   and t :: "('a set set \<times> ('a set \<Rightarrow> ('a \<times> 'a) set)) set" assumes "s \<in> st.carrier_stalk"   and "t \<in> st.carrier_stalk" obtains U :: "'a set set" and s' :: "'a set \<Rightarrow> ('a \<times> 'a) set" and t' :: "'a set \<Rightarrow> ('a \<times> 'a) set" where "is_zariski_open U" and "\<pp> \<in> U" and "s' \<in> \<O> U" and "s = st.class_of U s'" and "t' \<in> \<O> U" and "t = st.class_of U t'"
mset_subset_eq_single: fixes a :: "'a"   and B :: "'a multiset" assumes "a \<in># B" shows "\<LM> a  \<RM> \<subseteq># B"
span_finite_subset: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "p \<in> phull B" obtains A :: "('b \<Rightarrow>\<^sub>0 'c) set" where "finite A" and "A \<subseteq> B" and "p \<in> phull A"
matrix_vect_scaleR: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and a :: "real"   and v :: "('a, 'c) Finite_Cartesian_Product.vec" shows "A *v a *\<^sub>R v = a *\<^sub>R (A *v v)"
refine2specI: fixes b :: "'a nres"   and a :: "'b nres"   and R :: "('b \<times> 'a) set" assumes "nofail b \<Longrightarrow> a \<le> SPEC (\<lambda>r. \<exists>x. inres b x \<and> (r, x) \<in> R)" shows "a \<le> \<Down> R b"
inf_of_two_greatest: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_lower_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "is_glb L (\<Sqinter>\<^bsub>L\<^esub>{x, y}) {x, y}"
indep_in_iff_subset_basis_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> insert x X" shows "X_insert.indep_in \<E> X = (\<exists>B. X_insert.basis_in \<E> B \<and> X \<subseteq> B)"
reversepath_linepath: fixes a :: "'b"   and b :: "'b" shows "reversepath (linepath a b) = linepath b a"
ovalid_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 "ovalid P f Q"   and "\<And>s. P' s \<Longrightarrow> P s" shows "ovalid P' f Q"
measurable_cong: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and M' :: "'d measure" assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> M \<rightarrow>\<^sub>M M')"
simps_1: fixes s :: "'d \<Rightarrow> bool" shows "val_ifex Trueif s = True"
chain_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set" shows "chain_on P C A = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. P\<^sup>=\<^sup>= x y \<or> P\<^sup>=\<^sup>= y x))"
cmr: fixes n :: "nat" shows "continuous_map (nsphere n) (nsphere n) r"
preserves_reflects_arr: fixes f :: "'j" shows "C.arr (E f) = J.arr f"
pickFT_pred_pickFT: fixes cl :: "('d, 'e, 'f) cmd list" assumes "WtFT cl = 1" shows "pickFT_pred cl (pickFT cl)"
in_code_status_status_rel_iff_4: fixes a :: "'a code_status" shows "((a, SUCCESS) \<in> code_status_status_rel) = (a = CSUCCESS)"
sym_block_intersections_index: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int"   and b1 :: "'a set"   and b2 :: "'a set" assumes "symmetric_bibd point_set block_collection u_block_size index"   and "b1 \<in># block_collection"   and "b2 \<in># remove1_mset b1 block_collection" shows "b1 |\<inter>| b2 = index"
dvd_0_left: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times zero a" shows "a = zero"
echelon_form_upt_k_condition1: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and i :: "'f" assumes "echelon_form_upt_k A k"   and "is_zero_row_upt_k i k A" shows "\<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)"
pure_conj_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and P' :: "bool"   and Q' :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj Q (\<lambda>s. P' \<and> Q' s) = (\<lambda>s. P' \<and> sep_algebra.sep_conj plus sep_disj Q Q' s)"
rel_sum_simps_1: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'a"   and b1 :: "'b" shows "rel_sum R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
lemma_2_10_31: fixes b :: "'a"   and a :: "'a" shows "((b r\<rightarrow> a) l\<rightarrow> a) r\<rightarrow> a = b r\<rightarrow> a"
node_total_order_irrefl: fixes e :: "'evt"   and i :: "nat" shows "\<not> e \<sqsubset>\<^sup>i e"
slpa_rv_Low_Use_Low: fixes S :: "'node SDG_node set"   and cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list"   and m :: "'node"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "CFG_node (_Low_) \<in> S"   and "same_level_path_aux cs as"   and "upd_cs cs as = []"   and "same_level_path_aux cs as'"   and "\<forall>c\<in>set cs. valid_edge c"   and "m -as\<rightarrow>* (_Low_)"   and "m -as'\<rightarrow>* (_Low_)"   and "\<forall>i<length cs. \<forall>V\<in>rv S (CFG_node (sourcenode (cs ! i))). fst (s ! Suc i) V = fst (s' ! Suc i) V"   and "\<forall>i<Suc (length cs). snd (s ! i) = snd (s' ! i)"   and "\<forall>V\<in>rv S (CFG_node m). state_val s V = state_val s' V"   and "preds (slice_kinds S as) s"   and "preds (slice_kinds S as') s'"   and "length s = Suc (length cs)"   and "length s' = Suc (length cs)" shows "\<forall>V\<in>Use (_Low_). state_val (transfers (slice_kinds S as) s) V = state_val (transfers (slice_kinds S as') s') V"
coprime_pderiv_imp_squarefree: fixes p :: "'b poly" assumes "coprime p (pderiv p)" shows "squarefree p"
\<O>_sb_simp: fixes p :: "'d"   and instrs :: "'e"   and \<theta> :: "'f"   and sb :: "'c"   and \<D> :: "'g"   and \<O> :: "'b"   and \<R> :: "'h" shows "\<O>_sb (p, instrs, \<theta>, sb, \<D>, \<O>, \<R>) = (\<O>, sb)"
dom_char: fixes f :: "'a" shows "R.dom f = (if R.arr f then local.dom f else null)"
compute_degree: fixes xs :: "(nat, 'd) slist" shows "degree (Pdevs xs) = degree_slist xs"
intro: fixes x :: "'d \<Longrightarrow> 'c" assumes "\<And>\<sigma> v \<rho>. put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v" shows "pbij_lens_axioms x"
sep_implE: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a \<Rightarrow> bool"   and y :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_impl plus sep_disj x y h" obtains "\<forall>h'. sep_disj h h' \<and> x h' \<longrightarrow> y (plus h h')"
periodic_simps_2: fixes x :: "'a"   and na :: "nat" shows "f (x - of_nat na) = f x"
init_impl_def: fixes e :: "'f" shows "init_impl e \<equiv> RETURN \<lparr>ss_stack = [], on_stack = {}, visited = {}, \<dots> = e\<rparr>"
agree_times2: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" assumes "Vagree \<nu> \<nu>' (FVDiff (Times t1 t2))" shows "Vagree \<nu> \<nu>' (FVDiff t2)"
num_ord: shows "0 \<le> 0 \<and> 0 < 1 \<and> 1 < card Y - 1 \<and> card Y - 1 < card Y"
Basis: fixes s :: "'a set"   and S :: "'a set set" assumes "s \<in> S" shows "generate_topology S s"
xs: shows "xs = xs !! 0 ## xs !! 1 ## stl (stl xs)"
bangSC: fixes P :: "pi" shows "!P \<sim>\<^sub>e P \<parallel> !P"
inject_1: fixes x1 :: "bool"   and y1 :: "bool" shows "(Bc x1 = Bc y1) = (x1 = y1)"
sside_wcrossings_ndistinct_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and Cs :: "'b set list"   and D :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "SimplicialComplex.maxsimpchain X (C # Cs @ [D])"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "{f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X} \<in> set (ThinChamberComplex.wall_crossings X (C # Cs @ [D]))" shows "\<not> distinct (ThinChamberComplex.wall_crossings X (C # Cs @ [D]))"
tree_edges_ssE: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme" assumes "DFS_invar G param s" shows "tree_edges s \<subseteq> g_E G"
intro_elim_6_f: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" assumes "[\<phi> \<^bold>\<equiv> \<psi> in v]"   and "[\<phi> \<^bold>\<equiv> \<chi> in v]" shows "[\<chi> \<^bold>\<equiv> \<psi> in v]"
B: shows "mon_w fg e \<inter> mon_c fg cn = {}"
strict_mono_less_eq: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x \<le> f y) = less_eq x y"
StarLR_Times_Plus_One: fixes r :: "'a language"   and s :: "'a language" shows "StarLR (Times r (Plus One s)) s = StarLR r s"
find_first_correct_2: fixes x :: "'b"   and xs :: "'b list"   and i :: "nat" assumes "find_first x xs = Some i" shows "xs ! i = x"
wm_to_state_def: fixes P :: "ast_problem"   and M :: "object atom formula set" assumes "wf_ast_problem P" shows "wm_to_state M = (\<lambda>(p, xs). Atom (predAtm p xs) \<in> M)"
W_def: fixes i :: "nat" shows "W i = inv_chart (\<psi> i) ` ball (0::'b) 3"
radj_adj: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "g = radj f"
base: shows "(rel_chain_fun 0 a b r, rel_chain_fun (Suc 0) a b r) \<in> r"
intros_2: fixes a :: "aexp"   and x :: "char list"   and ps :: "char list \<Rightarrow> int option"   and v :: "int"   and ps' :: "char list \<Rightarrow> int option" assumes "vars a \<union> {x} \<subseteq> dom ps"   and "paval a ps = v"   and "ps' = ps(x \<mapsto> v)" shows "(x ::= a, ps) \<Rightarrow>\<^sub>A Suc 0 \<Down> ps'"
ereal_mult_eq_MInfty: fixes a :: "ereal"   and b :: "ereal" shows "(a * b = - \<infinity>) = (a = \<infinity> \<and> b < 0 \<or> a < 0 \<and> b = \<infinity> \<or> a = - \<infinity> \<and> 0 < b \<or> 0 < a \<and> b = - \<infinity>)"
up_int_preservation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry" assumes "dbm_int M n" shows "dbm_int (up M) n"
elims_3: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "\<not> SkewBinomialHeapStruc.rank_skew_invar x" obtains t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # t' # bq" and "\<not> (SkewBinomialHeapStruc.rank t \<le> SkewBinomialHeapStruc.rank t' \<and> SkewBinomialHeapStruc.rank_invar (t' # bq))"
mk_diagonal_def: fixes as :: "'b list" shows "mk_diagonal as = diag_block_mat (map (\<lambda>a. mat (Suc 0) (Suc 0) (\<lambda>_. a)) as)"
HEndPhase2_valueChosen2: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and D :: "Disk set"   and b :: "nat"   and p :: "Proc" assumes "HEndPhase2 s s' q"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))" shows "\<forall>d\<in>D. b \<le> bal (disk s' d p) \<and> (\<forall>q. phase s' q = 1 \<and> b \<le> mbal (dblock s' q) \<and> hasRead s' q d p \<longrightarrow> (\<exists>br\<in>blocksRead s' q d. b \<le> bal (block br)))"
measurable_lift_IntVal': fixes f :: "int \<Rightarrow> int" shows "lift_IntVal f \<in> stock_measure INTEG \<rightarrow>\<^sub>M stock_measure INTEG"
SUP_bot: fixes U\<^sub>2 :: "'c set"   and f :: "'c \<Rightarrow> 'al"   and A :: "'c set" assumes "\<forall>x\<in>U\<^sub>2. f x = \<bottom>\<^sub>o\<^sub>w"   and "A \<subseteq> U\<^sub>2" shows "\<Squnion>\<^sub>o\<^sub>w (f ` A) = \<bottom>\<^sub>o\<^sub>w"
lex_pp_alt: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "lex_pp s t = (s = t \<or> (\<exists>x. lookup_pp s x < lookup_pp t x \<and> (\<forall>y<x. lookup_pp s y = lookup_pp t y)))"
is_done_impl_def: fixes SBIP :: "'c list \<times> nat list \<times> ('c \<Rightarrow> node_state option) \<times> (nat \<times> 'c set) list"   and v :: "'c" shows "GS.is_done_impl SBIP v \<equiv> case GS.I SBIP v of None \<Rightarrow> False | Some (STACK nat) \<Rightarrow> False | Some DONE \<Rightarrow> True"
simps_6: fixes prec :: "nat"   and p :: "float interval poly"   and I :: "float interval list"   and a :: "float interval list" shows "compute_bound_poly prec (Neg p) I a = - compute_bound_poly prec p I a"
finprod_minus: fixes a :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "a \<in> A"   and "f \<in> A \<rightarrow> carrier G"   and "finite A" shows "finprod G f A = f a \<otimes> finprod G f (A - {a})"
pre: fixes s :: "char list \<Rightarrow> int" shows "pre C (\<lambda>a. enat k * Q a) s \<le> enat k * P s"
allRdBlks_def: fixes s :: "state"   and p :: "Proc" shows "allRdBlks s p = \<Union> (range (blocksRead s p))"
strict_trans2: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "a \<subset># b"   and "b \<subseteq># c" shows "a \<subset># c"