hcju/isabelleps huggingface dataset:
split_tuple_list_by_id: fixes a :: "'c"   and b :: "'d"   and c :: "'e"   and xs :: "('c \<times> 'd \<times> 'e) list" assumes "(a, b, c) \<in> set xs"   and "distinct (map fst xs)" shows "\<exists>pre suf. xs = pre @ (a, b, c) # suf \<and> (\<forall>y\<in>set pre. fst y \<noteq> a)"
FINITE_valid_states: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set" assumes "finite PROB" shows "finite (valid_states PROB)"
imp_def: fixes p :: "fm"   and q :: "fm" shows "imp p q = (if p = F \<or> q = T \<or> p = q then T else if p = T then q else if q = F then not p else Imp p q)"
nth_the_index: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "xs ! the_index xs x = x"
rbt_lookup_rbt_interwk: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t1 :: "('c, 'd) rbt"   and t2 :: "('c, 'd) rbt"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and k :: "'c" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t1"   and "ord.rbt_sorted less t2" shows "ord.rbt_lookup less (ord.rbt_inter_with_key less f t1 t2) k = (case ord.rbt_lookup less t1 k of None \<Rightarrow> None | Some v \<Rightarrow> case ord.rbt_lookup less t2 k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f k v w))"
preserves_iso: fixes f :: "'b" assumes "local.iso f" shows "local.iso (\<ll> f)"
valid_single: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and s :: "'f list" shows "valid fg {#s#}"
genPrefix_refl: fixes r :: "('a \<times> 'a) set"   and l :: "'a list" assumes "refl r" shows "(l, l) \<in> genPrefix r"
Ker_Im_iff: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FGModuleHom G smult V smult' T" shows "(ker T \<inter> V = V) = (T ` V = 0)"
P: fixes f :: "'c \<Rightarrow> 'b" assumes "continuous_on U f"   and "f ` U \<subseteq> t"   and "Q f" shows "P (k \<circ> f)"
insertion_add: fixes f :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly"   and q :: "'a mpoly" shows "insertion f (p + q) = insertion f p + insertion f q"
fixespointwise_subset_and_diff_imp_eq_on: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "order.greater_eq B A"   and "fixespointwise f A"   and "fixespointwise f (B - A)" shows "fixespointwise f B"
ZassenhausTr6_3: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and H1 :: "'e set"   and K :: "'e set"   and K1 :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>\<^bsub>G\<^esub>H \<triangleright> H1"   and "\<natural>\<^bsub>G\<^esub>K \<triangleright> K1" shows "Group (\<natural>\<^bsub>G\<^esub>K1 \<diamondop>\<^bsub>G\<^esub> K \<inter> H / (K1 \<diamondop>\<^bsub>G\<^esub> K \<inter> H1))"
list_of_map: fixes subs :: "'a \<Rightarrow> 'b option" assumes "finite (dom subs)" shows "\<exists>l. map_of l = subs"
uniqAppendElement: fixes l :: "'a list"   and e :: "'a" assumes "uniq l" shows "(e \<notin> set l) = uniq (l @ [e])"
insert_remove: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "finite A" shows "sum g (insert x A) = g x + sum g (A - {x})"
cycle_free_no_edges: fixes g :: "'b ugraph" assumes "edges g = {}" shows "cycle_free g"
borel_measurable_iff_halfspace_less: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i < a} \<in> sets M)"
frequently_rev_mp: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and Q :: "'a \<Rightarrow> bool" assumes "frequently P F"   and "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x" shows "frequently Q F"
matrix_vector_mul_assoc: fixes A :: "(('c, 'e) vec, 'd) vec"   and B :: "(('c, 'f) vec, 'e) vec"   and x :: "('c, 'f) vec" shows "A *v (B *v x) = A ** B *v x"
pfxm_mask_def: fixes x :: "'b prefix_match" shows "pfxm_mask x \<equiv> mask (LENGTH('b) - pfxm_length x)"
prems_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s T"
lookup_eq_None_iff: fixes M :: "('b, 'a) mapping"   and x :: "'b" shows "(Mapping.lookup M x = None) = (x \<notin> Mapping.keys M)"
Var_is_action: fixes v :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and A :: "('c, 'd) action" shows "(R (true \<turnstile> \<lambda>(A, A'). \<exists>a. tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = increase v a (alpha_rp.more A)) ;; action.relation_of A ;; R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A' = decrease v (alpha_rp.more A))) \<in> {p. is_CSP_process p}"
self_finance_updated_suc_suc: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market"   and asset :: "'b"   and w :: "'c"   and v :: "real"   and n :: "nat" assumes "portfolio pf"   and "\<forall>n. prices Mkt asset n w \<noteq> 0" shows "cls_val_process Mkt (self_finance Mkt v pf asset) (Suc (Suc n)) w = cls_val_process Mkt pf (Suc (Suc n)) w + prices Mkt asset (Suc (Suc n)) w / prices Mkt asset (Suc n) w * (cls_val_process Mkt (self_finance Mkt v pf asset) (Suc n) w - val_process Mkt pf (Suc n) w)"
X_in: shows "X \<in> UNIV // \<approx>A"
coloured_def: fixes b :: "nat" shows "coloured b = {(i, j). (i + j) mod 2 = b}"
prime_elem_int_minus_iff: fixes p :: "int" shows "prime_elem (- p) = prime_elem p"
is_parser_manyof: fixes cs :: "char list" shows "is_parser (manyof cs)"
emeasure_lborel_ball_finite: fixes c :: "'a"   and r :: "real" shows "emeasure lborel (ball c r) < \<infinity>"
that: fixes N :: "nat" assumes "\<forall>n>N. f $$ - int n = (0::'a)" shows "thesis"
coreflexive_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "coreflexive x" shows "coreflexive (x \<sqinter> y)"
bl: shows "butlast (cs\<^bsup>\<pi>\<^esup> i) = butlast (cs\<^bsup>\<pi>\<^esup> j)"
v_notin_Op: shows "State k (index (\<Pi>\<^sub>\<V>) v) \<notin> {Operator k (index (\<Pi>\<^sub>\<O>) op) |k op. k \<in> {0..<length \<pi>} \<and> op \<in> set (\<Pi>\<^sub>\<O>)}"
arr_tensor: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "arr (MC.tensor f g)"
permute_hash_def: fixes \<pi> :: "perm"   and h :: "hash" shows "\<pi> \<bullet> h = h"
prime_sum_upto_altdef2: fixes f :: "nat \<Rightarrow> 'b"   and x :: "real" shows "prime_sum_upto f x = (\<Sum>p | prime p \<and> p \<le> nat \<lfloor>x\<rfloor>. f p)"
IH_1: fixes d :: "nat"   and t' :: "'a tree" assumes "ins n d x l = Bal t'"   and "bal_i n (height l + d)" shows "bal_i (n + 1) (height t' + d)"
degree_eq_zeroE: fixes p :: "'a poly" assumes "Polynomial.degree p = 0" obtains a :: "'a" where "p = [:a:]"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
homotopic_circlemaps_imp_homotopic_loops: fixes S :: "'a set"   and f :: "complex \<Rightarrow> 'a"   and g :: "complex \<Rightarrow> 'a" assumes "homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f g" shows "homotopic_loops S (f \<circ> exp \<circ> (\<lambda>t. 2 * complex_of_real pi * complex_of_real t * \<i>)) (g \<circ> exp \<circ> (\<lambda>t. 2 * complex_of_real pi * complex_of_real t * \<i>))"
F_prop: shows "[\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lbrace>x\<^sup>P,H\<rbrace> in v]"
llast_LCons2: fixes x :: "'c"   and y :: "'c"   and xs :: "'c llist" shows "llast (LCons x (LCons y xs)) = llast (LCons y xs)"
web_of_network_sel_4: fixes \<Delta> :: "('a, 'b) network_scheme" shows "B (web_of_network \<Delta>) = {(x, sink \<Delta>) |x. edge \<Delta> x (sink \<Delta>)}"
set_eq_trans_aux: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and a :: "'a" assumes "A \<subseteq> carrier S"   and "B \<subseteq> carrier S"   and "C \<subseteq> carrier S"   and "A {.=} B"   and "B {.=} C"   and "a \<in> A" shows "a .\<in> C"
fz: fixes n :: "nat" shows "\<xi> n = f (z n)"
intros_1: fixes n :: "'n" assumes "n \<in> sources" shows "n \<in> slice"
word_enum_prefix: fixes x :: "'b word"   and y :: "'b word"   and as :: "'b word list"   and a :: "'b word"   and bs :: "'b word list" assumes "[x .e. y] = as @ a # bs" shows "as = (if x < a then [x .e. a - 1] else [])"
Times_insert_left: fixes A :: "'b multiset"   and x :: "'c"   and B :: "'c multiset" shows "A \<times># add_mset x B = A \<times># B + {#(a, x). a \<in># A#}"
list_all_map: fixes \<phi> :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and al :: "'b list" shows "list_all \<phi> (map f al) = list_all (\<phi> \<circ> f) al"
partition_on_mset_def: fixes A :: "'a multiset"   and P :: "'a multiset multiset" shows "partition_on_mset A P = (\<Sum>\<^sub># P = A \<and> {#} \<notin># P)"
lexordp_append_rightI: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys \<noteq> []" shows "lexordp xs (xs @ ys)"
elims_3: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal set" assumes "\<not> validate_ground_clause x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and C :: "'b literal set" where "x = I" and "xa = C" and "\<nexists>L. L \<in> C \<and> validate_ground_lit I L"
least_power_dvd: fixes p :: "'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "permutation p" shows "(least_power p a dvd n) = ((p ^^ n) a = a)"
hyps_2: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList (applyUnitPropagate state')) (getF (applyUnitPropagate state'))"   and "InvariantWatchListsUniq (getWatchList (applyUnitPropagate state'))"   and "InvariantWatchListsCharacterization (getWatchList (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesEl (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesDiffer (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantConflictClauseCharacterization (getConflictFlag (applyUnitPropagate state')) (getConflictClause (applyUnitPropagate state')) (getF (applyUnitPropagate state')) (getM (applyUnitPropagate state'))" shows "let state' = exhaustiveUnitPropagate (applyUnitPropagate state') in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
RSadj_fst: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "RSadj x y (fst \<nu>) = fst (Radj x y \<nu>)"
rtranclp_idemp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "r\<^sup>*\<^sup>*\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
is_interval_existence_ivl: fixes x0 :: "'c" shows "is_interval (existence_ivl0 x0)"
pnet_node_init: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and i :: "nat"   and R :: "nat set" assumes "s \<in> init (pnet np \<langle>i; R\<rangle>)" shows "s \<in> {NodeS i s R |s. s \<in> init (np i)}"
distinct_1: fixes x11 :: "char list"   and x12 :: "fterm list"   and x2 :: "char list" shows "Fun x11 x12 \<noteq> \<epsilon> x2"
assms: shows "all_subfrmls (fst n_ns) \<subseteq> B \<and> (\<forall>nd\<in>snd n_ns. all_subfrmls nd \<subseteq> B)"
suntil_Stream: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and x :: "'a"   and s :: "'a stream" shows "(\<phi> suntil \<psi>) (x ## s) = (\<psi> (x ## s) \<or> \<phi> (x ## s) \<and> (\<phi> suntil \<psi>) s)"
set_pmf_closed: fixes s :: "'s"   and D :: "'s pmf"   and t :: "'s" assumes "s \<in> S"   and "D \<in> K s"   and "t \<in> set_pmf D" shows "t \<in> S"
bit_0_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit zero = bot"
Liminf_llist_def: fixes Xs :: "'c set llist" shows "Liminf_llist Xs = (\<Union>i\<in>{i. enat i < llength Xs}. \<Inter> (lnth Xs ` {j. i \<le> j \<and> enat j < llength Xs}))"
length_induct: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "\<And>xs. \<forall>ys. |ys| < |xs| \<longrightarrow> P ys \<Longrightarrow> P xs" shows "P xs"
strip_while_coeffs: fixes p :: "'a poly" shows "strip_while ((=) (0::'a)) (coeffs p) = coeffs p"
assms_7: shows "\<Phi>' Start clinit = start_\<phi>\<^sub>m"
D_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "D x y = (fst x = fst y \<and> D (snd x) (snd y))"
rel_refl_strong: fixes x :: "('a, 'b) hd"   and R1a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2a :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> syms_hd x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> vars_hd x \<Longrightarrow> R2a z2 z2" shows "rel_hd R1a R2a x x"
iT_Div_card_ge__is_maximal_lower_bound: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and I :: "nat set"   and d :: "nat" assumes "\<forall>I d. card I div d + (if card I mod d = 0 then 0 else Suc 0) \<le> f (card I) d \<and> f (card I) d \<le> card (I \<oslash> d)" shows "f (card I) d = card I div d + (if card I mod d = 0 then 0 else Suc 0)"
simps_6: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "formula" shows "sat \<sigma> V v i (Neg \<phi>) = (\<not> sat \<sigma> V v i \<phi>)"
max_eq: shows "max (IArray.length A) (IArray.length 0) = IArray.length A"
Rep_inverse': fixes a :: "'b word"   and r :: "int" assumes "sint a = r" shows "word_of_int r = a"
wf_subst_properties: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "(\<theta> \<in> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_set) = wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
HCP_adv_def: fixes A :: "nat \<times> nat \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> (bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf) \<Rightarrow> nat \<Rightarrow> bool spmf"   and \<sigma> :: "bool"   and b\<^sub>\<sigma> :: "bool"   and D :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "etp_rsa.HCP_adv A \<sigma> b\<^sub>\<sigma> D = \<bar>spmf (etp_rsa.HCP_game A \<sigma> b\<^sub>\<sigma> D) True - 1 / 2\<bar>"
list_of_vec_def: shows "list_of_vec \<equiv> map_fun Rep_vec id (\<lambda>(n, v). map v [0..<n])"
ih_26: shows "InvariantVarsQ (getQ state') F0 Vbl"
preserves_cod: fixes f :: "'j" assumes "J.arr f" shows "C.cod (A'.map f) = A'.map (J.cod f)"
minus_Mp_1: fixes f :: "int poly"   and g :: "int poly" shows "q.eq_m (q.Mp f - g) (f - g)"
h_le_k: fixes e :: "'v \<times> 'v" shows "h e \<le> k e"
establish_invarI_ND: fixes I :: "'v tarjan_state \<Rightarrow> bool" assumes "\<And>u v s. on_discover tarjan_params u v s = on_new_root tarjan_params v s"   and "on_init tarjan_params \<le>\<^sub>n SPEC (\<lambda>x. I (empty_state x))"   and "\<And>s s' v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; v \<notin> dom (discovered s); discovered s' = discovered s(v \<mapsto> counter s); finished s' = finished s; counter s' = Suc (counter s); stack s' = v # stack s; back_edges s' = back_edges s; cross_edges s' = cross_edges s; tree_edges s \<subseteq> tree_edges s'; state.more s' = state.more s\<rbrakk> \<Longrightarrow> on_new_root tarjan_params v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; u = hd (stack s); pending s `` {u} = {}; s' = finish u s\<rbrakk> \<Longrightarrow> on_finish tarjan_params u s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); v \<in> dom (finished s); s' = cross_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_cross_edge tarjan_params u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"   and "\<And>s s' u v. \<lbrakk>DFS_invar G tarjan_params s; I s; cond s; \<not> is_break tarjan_params s; stack s \<noteq> []; (u, v) \<in> pending s; u = hd (stack s); v \<in> dom (discovered s); v \<notin> dom (finished s); s' = back_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)\<rbrakk> \<Longrightarrow> on_back_edge tarjan_params u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G tarjan_params (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))" shows "is_invar I"
R: fixes x :: "'a"   and y :: "'a" assumes "x \<in> topspace X"   and "y \<in> topspace X" shows "\<exists>S. connectedin X S \<and> x \<in> S \<and> y \<in> S"
rows_to_keep_lem: fixes ell :: "nat"   and A :: "'a mat" assumes "ell \<in> set (rows_to_keep A)" shows "ell < dim_row A"
is_prefix_take: fixes n :: "nat"   and a :: "'b list" shows "is_prefix (take n a) a"
2: shows "E.Ide (Dom (fst fg) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom (snd fg))"
circ_circ_mult_1: fixes x :: "'a" shows "x\<^sup>\<circ> * (1::'a)\<^sup>\<circ> = x\<^sup>\<circ>\<^sup>\<circ>"
extension_valid: fixes v :: "'a" assumes "v \<rightarrow> w0" shows "valid_path (LCons v P)"
commute: fixes a :: "'b word"   and b :: "'b word" shows "signed.min a b = signed.min b a"
simps_9: fixes I :: "\<I>"   and \<psi> :: "'c MFOTL.formula" shows "gen_unique (formula.Prev I \<psi>) = gen_unique \<psi>"
float_distinct_finite_1: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_nan a \<and> is_finite a)"
1: shows "enat i < llength (lselect t (lselect s w))"
pell_mul_eq_trivial_nat_iff: fixes z1 :: "nat \<times> nat"   and z2 :: "nat \<times> nat" shows "(pell_mul z1 z2 = (Suc 0, 0)) = (z1 = (Suc 0, 0) \<and> z2 = (Suc 0, 0))"
exec_gpv_def: fixes callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf" shows "exec_gpv callee \<equiv> curry (spmf.fixp_fun (\<lambda>exec_gpv (c, s). the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y))))"
sig_inv_set_closed_monom_mult: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and c :: "'b"   and t :: "'a" assumes "r \<in> sig_inv_set' j" shows "monom_mult c t r \<in> sig_inv_set' j"
dis_log_break': fixes y :: "nat"   and m' :: "nat"   and m :: "nat"   and d :: "nat"   and d' :: "nat" assumes "y < order \<G>"   and "\<not> m' < m"   and "m \<noteq> m'"   and "m' < order \<G>"   and "\<^bold>g [^] d \<otimes> (\<^bold>g [^] y) [^] m = \<^bold>g [^] d' \<otimes> (\<^bold>g [^] y) [^] m'" shows "y = nat ((int d - int d') * Number_Theory_Aux.inverse (m' - m) (order \<G>) mod int (order \<G>))"
left_zero_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "0\<^sub>v n + v = v"
bex2I: fixes a :: "'e"   and b :: "'f"   and S :: "('e \<times> 'f) set"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "(a, b) \<in> S"   and "(a, b) \<in> S \<Longrightarrow> P a b" shows "\<exists>a b. (a, b) \<in> S \<and> P a b"
compos_def: fixes A :: "('c, 'e) aGroup_scheme"   and g :: "'f \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'f" shows "compos A g f = compose (carrier A) g f"
simps: 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 a1 :: "ty"   and a2 :: "val"   and a3 :: "val" shows "(P \<turnstile> a1 casts a2 to a3 ) = ((\<exists>T v. a1 = T \<and> a2 = v \<and> a3 = v \<and> (\<forall>C. T \<noteq> Class C)) \<or> (\<exists>C. a1 = Class C \<and> a2 = Null \<and> a3 = Null) \<or> (\<exists>Cs C Cs' Ds a. a1 = Class C \<and> a2 = Ref (a, Cs) \<and> a3 = Ref (a, Ds) \<and> P \<turnstile> Path last Cs to C via Cs'  \<and> Ds = Cs @\<^sub>p Cs'))"
amd_update_def: fixes dflt :: "'b"   and k :: "nat"   and v :: "'b"   and a :: "'b array" shows "amd_update dflt k v a \<equiv> array_set_dyn dflt a k v"
distinct_56: fixes pi1' :: "pi"   and pi2' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "pi1' \<parallel> pi2' \<noteq> name1<name2>.pi"
assms_2: fixes Tl :: "'a trm list"   and f :: "'a" assumes "list_all \<phi> Tl" shows "\<phi> (Fn f Tl)"
f_flow_continuous: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "isCont (\<lambda>t. f (flow0 x0 t)) t"
cf_get_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity"   and e :: "nat \<times> nat" shows "cf_get cf e \<equiv> assert (case e of (u, v) \<Rightarrow> u \<in> V \<and> v \<in> V) \<bind> (\<lambda>_. return (cf e))"
run_env_def: fixes envT :: "('d, 'e) envT" shows "run_env envT = (case envT of EnvT x \<Rightarrow> x)"
rconf_In2_no_LVar: fixes va :: "Term.var"   and G :: "prog"   and L :: "lenv"   and s :: "st"   and vf :: "vvar"   and T :: "ty" assumes "\<forall>n. va \<noteq> LVar n" shows "G,L,s\<turnstile>In2 va\<succ>In2 vf\<Colon>\<preceq>Inl T = (G,s\<turnstile>fst vf\<Colon>\<preceq>T \<and> s\<le>|snd vf\<preceq>T\<Colon>\<preceq>(G, L))"
card_line_intersect_S: fixes T :: "proj2 set" assumes "T \<subseteq> S"   and "proj2_set_Col T" shows "card T \<le> 2"
wf_def: fixes r :: "('g \<times> 'g) set" shows "wf r = (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
simps_1: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" shows "scan f [] a = []"
lq: shows "isrlfm (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))"
compose_chart_inv_chart: fixes p :: "'c \<Rightarrow> 'c"   and p' :: "'c \<Rightarrow> 'c"   and c :: "('d, 'c) chart" assumes "homeomorphism UNIV UNIV p p'" shows "inv_chart (compose_chart p p' c) = inv_chart c \<circ> p'"
insert_absorb_fset: fixes S :: "'b fset"   and x :: "'b" assumes "x |\<in>| S" shows "insert_fset x S = S"
right_hom_axioms: shows "right_hom (\<cdot>) (\<star>) (src f)"
HMSet_alt: shows "HMSet = Hereditary_Multiset.Abs_hmultiset \<circ> MSet \<circ> image_mset Hereditary_Multiset.Rep_hmultiset"
homeomorphic_space_def: fixes X :: "'a topology"   and Y :: "'b topology" shows "X homeomorphic_space Y \<equiv> \<exists>f g. homeomorphic_maps X Y f g"
adjugate_def: fixes A :: "(('c, 'd) vec, 'd) vec" shows "adjugate A = Finite_Cartesian_Product.transpose (cofactorM A)"
game\<^sub>3: fixes x :: "nat"   and y :: "nat" shows "game\<^sub>3 (\<lambda>b m\<^sub>1 m\<^sub>2. map_spmf (([\<oplus>]) (if b then m\<^sub>1 else m\<^sub>2)) (spmf_of_set (nlists UNIV len_plain))) x y = game\<^sub>3 (\<lambda>b m\<^sub>1 m\<^sub>2. spmf_of_set (nlists UNIV len_plain)) x y"
induct: fixes P :: "'a simple_rule list \<Rightarrow> bool"   and a0 :: "'a simple_rule list" assumes "P []"   and "\<And>m uu_. P [SimpleRule m uu_]"   and "\<And>uv_ v va. P (v # va) \<Longrightarrow> P (uv_ # v # va)" shows "P a0"
intros_2: fixes xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c"   and oper :: "'d" assumes "crdt_ops xs deps"   and "oid \<notin> set (map fst xs)"   and "\<forall>ref\<in>deps oper. ref \<in> set (map fst xs) \<and> ref < oid" shows "crdt_ops (xs @ [(oid, oper)]) deps"
2: shows "(\<lambda>\<phi>. subst \<phi> t x) ` F = F"
size_lt_imp_ex_count_lt: fixes M :: "'b multiset"   and N :: "'b multiset" assumes "size M < size N" shows "\<exists>x\<in>#N. count M x < count N x"