hcju/isabelleps huggingface dataset:
full_nat_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<And>n. \<forall>m. Suc m \<le> n \<longrightarrow> P m \<Longrightarrow> P n" shows "P n"
rev_swap: fixes xs :: "'d list"   and ys :: "'d list" shows "(rev xs = ys) = (xs = rev ys)"
not_irreducible_zero: shows "\<not> irreducible (0::'a)"
yun_2: fixes a :: "real poly"   and i :: "nat" assumes "(a, i) \<in> set pis" shows "square_free a \<and> degree a \<noteq> 0"
assms: shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : U \<Leftarrow> T"
sum1: shows "(\<lambda>x. - fds_nth f x * mangoldt x / nat_power x s) abs_summable_on Collect primepow"
order_trans_rules_22: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x < y"   and "y \<le> z" shows "x < z"
iASSERT_def: fixes return :: "unit \<Rightarrow> 'b"   and \<Phi> :: "bool" shows "iASSERT return \<Phi> \<equiv> if \<Phi> then return () else top"
take_prefix_imp_le: fixes a :: "nat"   and xs :: "'b list"   and b :: "nat" assumes "a \<le> length xs"   and "prefix (xs \<down> a) (xs \<down> b)" shows "a \<le> b"
mono_ndet_FD_left: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<le> Q" shows "(P |-| S) \<le> Q"
irc_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "irc_on A f = (\<forall>B\<subseteq>A. \<forall>a\<in>A. a \<notin> f (B \<union> {a}) \<longrightarrow> f (B \<union> {a}) = f B)"
lemScaleNeg: fixes k :: "'a"   and u :: "'a Vector" shows "(- k) ** u = ~ k ** u"
order_refl: fixes x :: "'d DBMEntry" shows "x \<preceq> x"
STA_nonempty: fixes A :: "'b set"   and n :: "nat"   and left :: "(('c, 'b) fm list \<times> 'b) list"   and right :: "(('c, 'b) fm list \<times> 'b) list"   and m :: "nat" assumes "A, n \<turnstile> left @ right" shows "A, Suc m \<turnstile> filter nonempty left @ right"
y_min: fixes y' :: "'a" assumes "y' \<in> set P_k_pre" shows "\<not> hitting_Q_or_new_last y'"
simps_1: fixes x2 :: "'g"   and y2 :: "'g" shows "(Some x2 = Some y2) = (x2 = y2)"
integral_monotone_convergence: fixes M :: "'a measure"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and u :: "'a \<Rightarrow> real"   and x :: "real" assumes "\<And>i. integrable M (f i)"   and "AE x in M. incseq (\<lambda>n. f n x)"   and "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"   and "(\<lambda>i. integral\<^sup>L M (f i)) \<longlonglongrightarrow> x"   and "u \<in> borel_measurable M" shows "integral\<^sup>L M u = x"
suc_underS: fixes a :: "'a" assumes "a \<in> Field r" shows "a = local.suc (local.underS a)"
induced_automorphism_simplex_image_2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and x :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> g \<turnstile> ChamberComplex.chamber_system X"   and "order.greater_eq C x" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g ` x = f ` x"
MultiportPorts_single_rewrite_MatchOr: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and ports :: "ipt_l4_ports"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (Match (MultiportPorts ports)) a p = matches (\<beta>, \<alpha>) (MatchOr (Match (Src_Ports ports)) (Match (Dst_Ports ports))) a p"
wfvarsoccs\<^sub>s\<^sub>s\<^sub>t_append: fixes S :: "('d, 'c) stateful_strand_step list"   and S' :: "('d, 'c) stateful_strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>s\<^sub>t (S @ S') = wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S \<union> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S'"
tensor_one_def: shows "1 = tensor_from_vec [] [1::'b]"
finsertE: fixes a :: "'x"   and b :: "'x"   and A :: "'x fset" assumes "a |\<in>| finsert b A" obtains "a = b"   | "a |\<in>| A"
open_open_id: fixes i :: "nat"   and j :: "nat"   and t :: "expr"   and t' :: "expr"   and u :: "expr" assumes "i \<noteq> j"   and "open i t (open j t' u) = open j t' u" shows "open i t u = u"
L3_5ii: fixes c1 :: "'a com"   and c2 :: "'a com"   and X :: "'a set" shows "SEM (Seq c1 c2) X = SEM c2 (SEM c1 X)"
indRelLTPO_impl_TRel_is_weak_reduction_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_simulation (indRelLTPO TRel) (STCal Source Target)" shows "weak_reduction_simulation (TRel\<^sup>+) Target"
all_iff_abs': fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and m :: "nat"   and P :: "'a \<Rightarrow> bool" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "(\<forall>x. less x (abs_nat m) \<longrightarrow> P x) = (\<forall>n<m. P (abs_nat n))"
Abs_assn_inverse': fixes y :: "assn_raw" assumes "proper y" shows "Rep_assn (Abs_assn y) = y"
elims_3: fixes x :: "('a, 'b) rbt" assumes "\<not> bd_inv x" obtains l :: "('a, 'b) rbt" and c :: "color" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l c k v r" and "\<not> (bd_inv l \<and> bd_inv r \<and> black_depth l = black_depth r)"
gamma4_def: shows "gamma4 = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
crdt_ops_distinct_fst: fixes xs :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" assumes "crdt_ops xs deps" shows "distinct (map fst xs)"
pair_inj: fixes a :: "'c"   and b :: "'d"   and c :: "'c \<times> 'd" shows "((a, b) = c) = (a = fst c \<and> b = snd c)"
consume_def: fixes trn :: "('state, 'act, 'out) trans"   and vl :: "'value list"   and vl' :: "'value list" shows "consume trn vl vl' \<equiv> if \<phi> trn then vl \<noteq> [] \<and> f trn = hd vl \<and> vl' = tl vl else vl' = vl"
connected_induction_simple: fixes S :: "'b set"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "connected S"   and "a \<in> S"   and "b \<in> S"   and "P a"   and "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. P x \<longrightarrow> P y)" shows "P b"
map_fun_apply: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'c"   and h :: "'e \<Rightarrow> 'f"   and x :: "'d" shows "map_fun f g h x = g (h (f x))"
coplanar_perm_11: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet B D C A"
preorder_reflect: fixes t :: "'c bt" shows "preorder (reflect t) = rev (postorder t)"
in_idealE_homomorphism: fixes B :: "'a set"   and p :: "'a" assumes "B \<subseteq> range h"   and "p \<in> range h"   and "p \<in> ideal B" obtains A :: "'a set" and q :: "'a \<Rightarrow> 'a" where "finite A" and "A \<subseteq> B" and "\<And>b. q b \<in> range h" and "p = (\<Sum>b\<in>A. q b * b)"
munionM_mUnionM2: fixes x :: "'m"   and A :: "'m multiset" shows "munionM x (mUnionM A) = fold_mset munionM x A"
advice_sets_def: fixes \<phi> :: "'b ltln" shows "advice_sets \<phi> = List.product (subseqs (subformulas\<^sub>\<mu>_list \<phi>)) (subseqs (subformulas\<^sub>\<nu>_list \<phi>))"
block_closed: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> P" shows "a \<in> S"
t_BIT_ub2: fixes n :: "nat" assumes "qs ! n \<notin> set init" shows "t_BIT n \<le> real (Suc (length init))"
div_power_2_float: fixes x :: "real"   and d :: "nat" assumes "x \<in> float" shows "x / 2 ^ d \<in> float"
insert_subset_eq_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(a, A \<subseteq># B) = (a \<in># B \<and> A \<subseteq># B - (a, {#}))"
assms_2: shows "a11 * x1 + a12 * x2 = (0::'a)"
wtE_def: fixes \<xi> :: "'b \<Rightarrow> 'univ" shows "wtE \<xi> \<equiv> \<forall>x. D (\<xi> x)"
functor_comp: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> 'f" assumes "functor A B F"   and "functor B C G" shows "functor A C (G \<circ> F)"
dsafe_Const_simps: fixes r :: "real" shows "dsafe (Const r) = True"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and f3 :: "'b val \<Rightarrow> 'a"   and f4 :: "String.literal \<Rightarrow> 'a"   and f5 :: "ty \<Rightarrow> 'a"   and f6 :: "'a"   and f7 :: "'a"   and f8 :: "'a"   and f9 :: "String.literal \<Rightarrow> String.literal \<Rightarrow> 'a"   and f10 :: "String.literal \<Rightarrow> String.literal \<Rightarrow> 'a"   and f11 :: "String.literal \<Rightarrow> String.literal \<Rightarrow> 'a"   and f12 :: "ty \<Rightarrow> 'a"   and f13 :: "ty \<Rightarrow> 'a"   and f14 :: "String.literal \<Rightarrow> nat \<Rightarrow> 'a"   and f15 :: "'a"   and f16 :: "'a"   and f17 :: "'a"   and f18 :: "'a"   and f19 :: "bop \<Rightarrow> 'a"   and f20 :: "int \<Rightarrow> 'a"   and f21 :: "int \<Rightarrow> 'a"   and f22 :: "'a"   and f23 :: "'a"   and f24 :: "'a"   and instr :: "'b instr" shows "P (case instr of Load x \<Rightarrow> f1 x | Store x \<Rightarrow> f2 x | Push x \<Rightarrow> f3 x | New x \<Rightarrow> f4 x | NewArray x \<Rightarrow> f5 x | ALoad \<Rightarrow> f6 | AStore \<Rightarrow> f7 | ALength \<Rightarrow> f8 | Getfield x xa \<Rightarrow> f9 x xa | Putfield x xa \<Rightarrow> f10 x xa | CAS x xa \<Rightarrow> f11 x xa | Checkcast x \<Rightarrow> f12 x | Instanceof x \<Rightarrow> f13 x | Invoke x xa \<Rightarrow> f14 x xa | Return \<Rightarrow> f15 | Pop \<Rightarrow> f16 | Dup \<Rightarrow> f17 | Swap \<Rightarrow> f18 | BinOpInstr x \<Rightarrow> f19 x | Goto x \<Rightarrow> f20 x | IfFalse x \<Rightarrow> f21 x | ThrowExc \<Rightarrow> f22 | MEnter \<Rightarrow> f23 | MExit \<Rightarrow> f24) = (\<not> ((\<exists>x1. instr = Load x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. instr = Store x2 \<and> \<not> P (f2 x2)) \<or> (\<exists>x3. instr = Push x3 \<and> \<not> P (f3 x3)) \<or> (\<exists>x4. instr = New x4 \<and> \<not> P (f4 x4)) \<or> (\<exists>x5. instr = NewArray x5 \<and> \<not> P (f5 x5)) \<or> instr = ALoad \<and> \<not> P f6 \<or> instr = AStore \<and> \<not> P f7 \<or> instr = ALength \<and> \<not> P f8 \<or> (\<exists>x91 x92. instr = Getfield x91 x92 \<and> \<not> P (f9 x91 x92)) \<or> (\<exists>x101 x102. instr = Putfield x101 x102 \<and> \<not> P (f10 x101 x102)) \<or> (\<exists>x111 x112. instr = CAS x111 x112 \<and> \<not> P (f11 x111 x112)) \<or> (\<exists>x12. instr = Checkcast x12 \<and> \<not> P (f12 x12)) \<or> (\<exists>x13. instr = Instanceof x13 \<and> \<not> P (f13 x13)) \<or> (\<exists>x141 x142. instr = Invoke x141 x142 \<and> \<not> P (f14 x141 x142)) \<or> instr = Return \<and> \<not> P f15 \<or> instr = Pop \<and> \<not> P f16 \<or> instr = Dup \<and> \<not> P f17 \<or> instr = Swap \<and> \<not> P f18 \<or> (\<exists>x19. instr = BinOpInstr x19 \<and> \<not> P (f19 x19)) \<or> (\<exists>x20. instr = Goto x20 \<and> \<not> P (f20 x20)) \<or> (\<exists>x21. instr = IfFalse x21 \<and> \<not> P (f21 x21)) \<or> instr = ThrowExc \<and> \<not> P f22 \<or> instr = MEnter \<and> \<not> P f23 \<or> instr = MExit \<and> \<not> P f24))"
exec_assign_local: fixes \<sigma> :: "'a control_state_scheme"   and upd :: "('b list \<Rightarrow> 'b list) \<Rightarrow> 'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and rhs :: "'a control_state_scheme \<Rightarrow> 'b"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> _bind_SE _ (assign_local upd rhs) M) = (\<sigma> \<Turnstile> M)"
euclidean_eqI: fixes x :: "'a"   and y :: "'a" assumes "\<And>b. b \<in> Basis \<Longrightarrow> inner x b = inner y b" shows "x = y"
weakly_bisimilar_transp: shows "transp (\<approx>\<cdot>)"
ACI_norm_nullable: fixes r :: "'b rexp" shows "nullable \<guillemotleft>r\<guillemotright> = nullable r"
Num_atrm: fixes n :: "nat" shows "Num n \<in> atrm"
UnOp: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and Q :: "'b assn"   and unop :: "unop" assumes "G,A\<turnstile>{Normal P} e-\<succ> {\<lambda>Val:v:. Q\<leftarrow>In1 (eval_unop unop v)}" shows "G,A\<turnstile>{Normal P} UnOp unop e-\<succ> {Q}"
InvariantVarsMAfterDecide: fixes M :: "(Literal \<times> bool) list"   and F0 :: "Literal list list"   and Vbl :: "nat set"   and literal :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantVarsM M F0 Vbl"   and "var literal \<in> Vbl"   and "M' = M @ [(literal, True)]" shows "InvariantVarsM M' F0 Vbl"
execute_guard_1: fixes P :: "heap \<Rightarrow> bool"   and h :: "heap"   and f :: "heap \<Rightarrow> 'a \<times> heap" assumes "\<not> P h" shows "execute (guard P f) h = None"
below_monic_ValKF: shows "below_monic_cfun ValKF"
downset_prop2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "y \<in> \<down> x"
sub_q63_array: fixes i :: "nat" assumes "i \<in> {..<63}" shows "q63_array !! i = (i \<in> q63)"
hyps_4: fixes i :: "nat"   and \<delta> :: "('a, 'b) ta_rule set"   and \<delta>' :: "('a, 'b) ta_rule set" assumes "i < length qs"   and "\<delta>n = \<delta> \<union> \<delta>'"   and "\<delta>_states \<delta> \<inter> \<delta>_states \<delta>' = {}"   and "qs ! i \<in> \<delta>_states \<delta>" shows "accs \<delta> (ts ! i) (qs ! i)"
neq_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "(x \<noteq> y) = (less x y \<or> less y x)"
min_maxsimpchain_pmaxsimpchain: fixes xs :: "'a set list" assumes "min_gallery xs" shows "pgallery xs"
pref_power: fixes t :: "'b list"   and u :: "'b list"   and k :: "nat" assumes "t \<noteq> \<epsilon>"   and "u \<le>p t \<^sup>@ k" shows "\<exists>m. t \<^sup>@ m \<le>p u \<and> u <p t \<^sup>@ m \<cdot> t"
divisor_sigma_def: fixes x :: "'b"   and n :: "nat" shows "divisor_sigma x n = (\<Sum>d | d dvd n. nat_power d x)"
step_lem: fixes l :: "'a list"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and x1 :: "'a"   and a :: "'a" assumes "l \<noteq> []" shows "tl (map (\<lambda>i. f ((x1 # a # l) ! i) ((a # l) ! i)) [0..<length l]) = map (\<lambda>i. f ((a # l) ! i) (l ! i)) [0..<length l - 1]"
FvarsT_num: fixes n :: "'trm" assumes "n \<in> num" shows "FvarsT n = {}"
ring_n_pd_def: fixes K :: "('c, 'd) Ring_scheme"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set"   and n :: "nat" shows "O\<^bsub>K P n\<^esub> = Sr K {x \<in> carrier K. \<forall>j\<le>n. 0 \<le> (\<nu>\<^bsub>K P j\<^esub>) x}"
comp_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "B.VVV.comp g f = (if B.VVV.arr f \<and> B.VVV.arr g \<and> B.VxVxV.seq g f then B.VxVxV.comp g f else B.VxVxV.null)"
ric: assumes "irreducible (poly_real_alg_1 rai)" shows "root_info_cond ri (poly_real_alg_1 rai)"
augment_edge: fixes G :: "('a, 'b) graph"   and G1 :: "('a, 'b) graph"   and G2 :: "('a, 'b) graph"   and u :: "'a"   and v :: "'a" assumes "valid_graph G"   and "subgraph G1 G"   and "subgraph G2 G"   and "(u, v) \<in> {(a, b) |a b. nodes_connected G1 a b}"   and "(u, v) \<notin> {(a, b) |a b. nodes_connected G2 a b}" shows "\<exists>a b e. (a, b) \<notin> {(a, b) |a b. nodes_connected G2 a b} \<and> e \<notin> edges G2 \<and> e \<in> edges G1 \<and> (case e of (aa, w, ba) \<Rightarrow> a = aa \<and> b = ba \<or> a = ba \<and> b = aa)"
uJ_def: shows "uJ = (\<lambda>n x. x 0 n)"
intros_10: shows "Real \<sqsubset>\<^sub>B OclAny"
atomize_poly_def: fixes p :: "'k \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "atomize_poly p = Abs_poly_mapping (\<lambda>v. lookup (lookup p (component_of_term v)) (pp_of_term v))"
digraph_iso_swap: fixes G :: "('e, 'f) pre_digraph"   and H :: "('g, 'h) pre_digraph" assumes "wf_digraph G"   and "digraph_iso G H" shows "digraph_iso H G"
permutes_not_in: fixes f :: "'d \<Rightarrow> 'd"   and S :: "'d set"   and x :: "'d" assumes "f permutes S"   and "x \<notin> S" shows "f x = x"
nb\<^sub>1: fixes op :: "'variable strips_operator" assumes "is_valid_operator_strips \<Pi> op" shows "are_operator_effects_consistent op op"
exec_step_ind_cases_13: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap"   and \<sigma> :: "jvm_state" assumes "exec_step_ind (StepI Pop) P h stk loc C M pc ics frs sh \<sigma>" obtains "\<sigma> = (None, h, (tl stk, loc, C, M, Suc pc, ics) # frs, sh)"
that: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> 'n"   and g :: "'n \<Rightarrow> nat \<Rightarrow> real" assumes "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) f g"   and "\<And>x. x \<in> topspace (Euclidean_space n) \<Longrightarrow> (norm (f x))\<^sup>2 = (\<Sum>i<n. (x i)\<^sup>2)" shows "thesis"
islimptE: fixes x :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "x islimpt S"   and "x \<in> T"   and "open T" obtains y :: "'b" where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
fw_upd_canonical_subs_id: fixes n :: "nat"   and k :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "canonical_subs n {k} M"   and "i \<le> n"   and "j \<le> n" shows "fw_upd M k i j = M"
inj_funpow: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "inj f" shows "inj (f ^^ n)"
cases: fixes a1 :: "'a"   and a2 :: "'b"   and r :: "('b \<times> 'a) set" assumes "(a1, a2) \<in> r\<inverse>" obtains a :: "'b" and b :: "'a" where "a1 = b" and "a2 = a" and "(a, b) \<in> r"
inf_sup_distrib2: fixes y :: "'d multiset"   and z :: "'d multiset"   and x :: "'d multiset" shows "y \<union># z \<inter># x = y \<inter># x \<union># (z \<inter># x)"
weakEqWeakTransitionAppend: fixes P :: "pi"   and P' :: "pi"   and \<alpha> :: "freeRes"   and P'' :: "pi" assumes "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'"   and "P' \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P''" shows "P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P''"
AM_def: shows "AM \<equiv> {l \<in> A. list.set l \<subseteq> M}"
snd_fst_flip: fixes xy :: "'d \<times> 'c" shows "snd xy = (fst \<circ> (\<lambda>(x, y). (y, x))) xy"
Ints_of_int: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus z \<in> ring_1.Ints one plus zero minus"
emeasure_lborel_box: fixes l :: "'b"   and u :: "'b" assumes "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b" shows "emeasure lborel (box l u) = ennreal (prod ((\<bullet>) (u - l)) Basis)"
ide_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.arr f" shows "CCC.ide (CCC.dom f)"
star_rtc1: fixes x :: "'a" shows "1' + x + x\<^sup>\<star> ; x\<^sup>\<star> \<le> x\<^sup>\<star>"
abc_only_cba_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "\<not> [[b a c]]"
ZOretr_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "ZOretr theta \<equiv> {(c, d). matchC_ZO theta c d \<and> matchT_ZO c d}"
sel: fixes \<Gamma> :: "('k, 'l) web_scheme"   and x3 :: "'m" assumes "countable_bipartite_web \<Gamma>" shows "bi.inner (bi.Inner x3) = x3"
ag_eq_sol1: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<in> carrier A"   and "x \<in> carrier A"   and "b \<in> carrier A"   and "a \<plusminus> x = b" shows "x = -\<^sub>a a \<plusminus> b"
rhs_eq_val: fixes rhs_eq_val :: "(nat, 'a) mapping \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a"   and v :: "(nat, 'a) mapping"   and e :: "nat \<times> linear_poly"   and x :: "nat"   and c :: "'a" assumes "RhsEqVal rhs_eq_val"   and "\<langle>v\<rangle> \<Turnstile>\<^sub>e e" shows "rhs_eq_val v x c e = rhs e \<lbrace> \<langle>v\<rangle>(x := c) \<rbrace>"
remap_polys_spec: fixes spec :: "int mpoly"   and \<V> :: "nat set"   and A :: "(nat, int mpoly) fmap" shows "remap_polys spec \<V> A \<le> remap_polys_polynomial_bool spec \<V> A"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Final of Final \<Rightarrow> f1 | Nonfinal \<Rightarrow> f2) = f1"
equals_dom_if_value_is_ide: assumes "C.ide a" shows "\<alpha>.map = \<alpha>.F.map"
sub_complement_antitone: fixes x :: "'a"   and y :: "'a" assumes "- x \<le> - y" shows "- - y \<le> - - x"
dom_map_of_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "length xs = length ys" shows "dom (map_of (zip xs ys)) = set xs"
aff_dim_open: fixes S :: "'b set" assumes "open S"   and "S \<noteq> {}" shows "aff_dim S = int DIM('b)"
scale_minus_right: fixes a :: "rat"   and x :: "'a" shows "a *R - x = - (a *R x)"
star_one_def: shows "1 \<equiv> star_of (1::'a)"
unifier\<^sub>l\<^sub>s_def: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" shows "unifier\<^sub>l\<^sub>s \<sigma> L = (\<exists>l'. \<forall>l\<in>L. l \<cdot>\<^sub>l \<sigma> = l')"
distinct_297: fixes x151 :: "floatarith"   and x152 :: "nat"   and x17 :: "nat" shows "x151 ^\<^sub>e x152 \<noteq> Var x17"
eSwapBinp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'd option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "igWlsBinp MOD delta binp" shows "eSwapBinp MOD zs z1 z2 (OKI binp) = OKI (igSwapBinp MOD zs z1 z2 binp)"
star_inductr': fixes z :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" assumes "less_eq_prod (sup_prod z (times_prod' y x)) y" shows "less_eq_prod (times_prod' z (prod.qstar x)) y"
distinct_concat_map_E_1: fixes f :: "'y \<Rightarrow> 'x list"   and xs :: "'y list"   and x :: "'y"   and y :: "'y" assumes "distinct (concat (map f xs))"   and "x \<in> set xs"   and "y \<in> set xs"   and "x \<noteq> y" shows "set (f x) \<inter> set (f y) = {}"
ordered_map_def: fixes S :: "('d \<times> 'e) set" shows "ordered_map S = map (get_map S) (sorted_list_of_set (fst ` S))"
simple_function_density_eq: fixes Mu :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and u :: "'b \<Rightarrow> 'c" shows "simple_function (density Mu f) u = simple_function Mu u"
owner_v_state: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "(Owner n, Crypt (SesK SK) (Pwd n)) \<in> s" shows "(Owner n, SesKey SK) \<in> s \<and> (\<exists>A B C. Token n A B C SK \<in> used s \<and> B \<in> fst (snd SK))"
exec: shows "(ta, xcp', h', [(stk', loc', undefined, undefined, pc')]) \<in> exec_instr (ins ! pc) P t h stk loc undefined undefined pc []"
relax_outgoing'_def: fixes u :: "'v"   and du :: "nat"   and V :: "'v \<Rightarrow> 'b option"   and Q :: "'v \<Rightarrow> nat option"   and v :: "'v" shows "relax_outgoing' u du V Q v \<equiv> case w (u, v) of enat d \<Rightarrow> case Q v of None \<Rightarrow> if v \<in> dom V then None else Some (du + d) | Some d' \<Rightarrow> Some (min d' (du + d)) | \<infinity> \<Rightarrow> Q v"
f0: shows "\<phi> $$ (0, 0) \<noteq> 0 \<or> \<phi> $$ (1, 0) \<noteq> 0"
octo_mult_components_1: fixes x :: "octo"   and y :: "octo" shows "Ree (x * y) = Ree x * Ree y - Im1 x * Im1 y - Im2 x * Im2 y - Im3 x * Im3 y - Im4 x * Im4 y - Im5 x * Im5 y - Im6 x * Im6 y - Im7 x * Im7 y"
swap_def: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "Fun.swap a b f = f(a := f b, b := f a)"