hcju/isabelleps huggingface dataset:
Cons: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and Q :: "'b assn"   and es :: "expr list"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} e-\<succ> {Q}"   and "\<forall>v. G,A\<turnstile>{Q\<leftarrow>In1 v} es\<doteq>\<succ> {\<lambda>Vals:vs:. R\<leftarrow>In3 (v # vs)}" shows "G,A\<turnstile>{Normal P} e # es\<doteq>\<succ> {R}"
PCF_consts_relI: fixes R :: "('a \<Rightarrow> ValD) set" assumes "\<bottom> \<in> R"   and "(\<lambda>i. ValTT) \<in> R"   and "(\<lambda>i. ValFF) \<in> R"   and "\<And>n. (\<lambda>i. ValN\<cdot>n) \<in> R" shows "PCF_consts_rel R"
Inf_fin_Inf: fixes A :: "'a::type set" assumes "finite (A::'a::type set)"   and "(A::'a::type set) \<noteq> {}" shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>n (A::'a::type set) = \<Sqinter> A"
HPhase1or2Write_allBlocks: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase1or2Write s s' p d" shows "allBlocks s' \<subseteq> allBlocks s"
hyperplane_facet_of_halfspace_le: fixes a :: "'b"   and b :: "real" assumes "a \<noteq> (0::'b)" shows "{x. a \<bullet> x = b} facet_of {x. a \<bullet> x \<le> b}"
spec_refines_impl: assumes "finite SYN" shows "SYSTEM \<sqsubseteq> COPY"
cBexE: fixes A :: "'f cset"   and P :: "'f \<Rightarrow> bool" assumes "cBex A P" obtains x :: "'f" where "cin x A" and "P x"
subset_Collect_iff: fixes B :: "'a set"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "B \<subseteq> A" shows "(B \<subseteq> {x \<in> A. P x}) = (\<forall>x\<in>B. P x)"
div_mod_fold_coeffs: fixes p :: "'a poly"   and q :: "'a poly" shows "(p div q, p mod q) = (if q = 0 then (0, p) else fold_coeffs (\<lambda>a (s, r). let b = coeff (pCons a r) (degree q) / lead_coeff q in (pCons b s, pCons a r - smult b q)) p (0, 0))"
singleton_insert_inj_eq': fixes a :: "'a"   and A :: "'a set"   and b :: "'a" shows "(insert a A = {b}) = (a = b \<and> order.greater_eq {b} A)"
inconsistent_parts: fixes ps :: "('a, 'b) fm list"   and S :: "('a, 'b) fm set" assumes "ps \<turnstile> \<^bold>\<bottom>"   and "set ps \<subseteq> S" shows "\<not> consistent S"
weakly_logically_equivalent_eqvt: fixes P :: "'state"   and Q :: "'state"   and p :: "perm" assumes "P \<equiv>\<cdot> Q" shows "p \<bullet> P \<equiv>\<cdot> p \<bullet> Q"
list_fast_rec_term: fixes us :: "'b tree list"   and ts :: "'b tree list" assumes "us \<noteq> []"   and "us = filter (\<lambda>t. t \<noteq> \<langle>\<rangle>) ts" shows "sum_list (map (size \<circ> left) us) + sum_list (map (size \<circ> right) us) < sum_list (map size ts)"
ladder_cut_prefix_length: fixes d :: "nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "length (ladder_cut_prefix d L) = length L"
is_red_reduced_GB_monomial_dgrad_set: fixes S :: "'t set"   and m :: "nat" assumes "finite (component_of_term ` S)"   and "pp_of_term ` S \<subseteq> dgrad_set d m" shows "is_red (reduced_GB (monomial (1::'c) ` S)) = is_red (monomial (1::'c) ` S)"
case_prod_conv: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
shift1_def: fixes p :: "'b poly" shows "shift1 p = CN 0\<^sub>p 0 p"
bisim1NewThrow: fixes P :: "'b prog"   and h :: "'heap"   and C :: "String.literal"   and xs :: "'addr val list" shows "P,new C,h \<turnstile> (Throw (addr_of_sys_xcpt OutOfMemory), xs) \<leftrightarrow> ([], xs, 0, \<lfloor>addr_of_sys_xcpt OutOfMemory\<rfloor>)"
ttree_restr_is_empty: fixes t :: "'a ttree"   and S :: "'a set" assumes "carrier t \<inter> S = {}" shows "ttree_restr S t = TTree.empty"
lm091: fixes P :: "('b \<times> 'a) set"   and X :: "'b set" assumes "runiq (P\<inverse>)" shows "P `` (Domain P - X) \<inter> P `` X = {}"
simps_2: fixes \<Sigma> :: "exesignature"   and uu :: "variable"   and S :: "String.literal set" shows "exetyp_ok_sig \<Sigma> (Tv uu S) = exewf_sort (fst (exesorts \<Sigma>)) S"
repv_def: fixes \<omega> :: "variable \<Rightarrow> real"   and x :: "variable"   and r :: "real" shows "repv \<omega> x r = \<omega>(x := r)"
put_env_def: fixes s :: "'s"   and m :: "('c, 'm) envT" shows "local.put_env s m = EnvT (\<lambda>r. put s (run_env m r))"
sequentially_imp_eventually_within: fixes "open" :: "'a set \<Rightarrow> bool"   and s :: "'a set"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.first_countable_topology open"   and "\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> filterlim f (topological_space.nhds open a) sequentially \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n))" shows "eventually P (topological_space.at_within open a s)"
tiTable_ack_init: fixes req :: "nat \<Rightarrow> reqType list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat" assumes "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) b ack i1 vc st"   and "ts lose"   and "msg (Suc 0) a1"   and "msg (Suc 0) stop"   and "\<forall>t1\<le>t. req t1 = []" shows "ack t = [init_state]"
SUBSEQ_complement: fixes A :: "'b list set" shows "- SUBSEQ A = SUPSEQ (- SUBSEQ A)"
p2pp_def: fixes P :: "'b \<Rightarrow> bool" shows "\<lceil>P\<rceil> = {Node s |s. P s}"
Norm_triangular_ineq: fixes x :: "'c"   and N :: "'c quasinorm"   and y :: "'c" assumes "x \<in> space\<^sub>N N" shows "Norm N (x + y) \<le> defect N * Norm N x + defect N * Norm N y"
simps_16: fixes sh :: "sheap"   and b :: "'a exp"   and e :: "'a exp" shows "iconf sh (while (b) e) = (\<not> sub_RI b \<and> \<not> sub_RI e)"
literalTrue_def: fixes literal :: "Literal"   and valuation :: "Literal list" shows "literalTrue literal valuation \<equiv> literal el valuation"
card_of_Sigma_mono: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set"   and J :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and B :: "'b \<Rightarrow> 'd set" assumes "inj_on f I"   and "f ` I \<subseteq> J"   and "\<forall>j\<in>J. |A j| \<le>o |B j|" shows "|SIGMA i:I. A (f i)| \<le>o |Sigma J B|"
ccHeap_Heap: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e' :: "exp"   and e :: "exp"   and a :: "Arity"   and a' :: "Arity" assumes "map_of \<Delta> x = Some e'"   and "(Aheap \<Delta> e\<cdot>a) x = up\<cdot>a'" shows "\<G>\<^bsub>a'\<^esub> e' \<sqsubseteq> ccHeap \<Delta> e\<cdot>a"
le_iff_sup: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "less_eq_prod x y = (sup_prod x y = y)"
eq: shows "Re (\<Sum>d | d dvd k. \<chi> d) = (\<Prod>p\<in>prime_factors k. Re (\<Sum>d | d dvd p ^ multiplicity p k. \<chi> d))"
nonzero_divide_eq_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero" shows "(divide b c = a) = (b = times a c)"
simps_3: fixes t :: "ternaryformula" shows "ternary_ternary_eval (TernaryNot t) = eval_ternary_Not (ternary_ternary_eval t)"
pred_mono_strong: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d list"   and Pa :: "'d \<Rightarrow> bool" assumes "list_all P x"   and "\<And>z. \<lbrakk>z \<in> set x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "list_all Pa x"
exhaust: fixes y :: "'a val" obtains "y = Unit"   | "y = Null"   | x3 :: "bool" where "y = Bool x3"   | x4 :: "32 word" where "y = Intg x4"   | x5 :: "'a" where "y = Addr x5"
is_ok_sp_generalize: fixes F :: "'n \<Rightarrow> 'o fundef option"   and xs :: "('c, 'p, 'n, 'e, 'f, 'g, 'h, 'i) instr list"   and \<Sigma> :: "'q list" assumes "is_ok (sp F xs (map Map.empty \<Sigma>))" shows "is_ok (sp F (map generalize_instr xs) (map Map.empty \<Sigma>))"
lowner_is_lub_dim: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and M :: "complex mat" assumes "matrix_seq dim f"   and "matrix_seq.lowner_is_lub f M" shows "M \<in> carrier_mat dim dim"
dist_cases: fixes v :: "nat"   and d :: "nat"   and v' :: "nat" assumes "dist v d v'" obtains "v = v'" and "d = 0"   | vh :: "nat" and dd :: "nat" where "d = Suc dd" and "dist v dd vh" and "(vh, v') \<in> E"
left_cancel: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a + b = a + c) = (b = c)"
map_simp_ide: fixes a :: "'a" assumes "ide a" shows "\<rr>.map a = \<r>[a]"
insort_insert_key_def: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "insort_insert_key f x xs = (if f x \<in> f ` set xs then xs else insort_key f x xs)"
LCons: fixes x :: "'a"   and xs :: "'a llist" assumes "x \<notin> lset xs"   and "ldistinct xs" shows "ldistinct (LCons x xs)"
ABa: shows "\<forall>a\<in>A. \<forall>b\<in>B. a \<sqsubseteq> b \<longrightarrow> b \<sqsubseteq> a \<longrightarrow> a = b"
parent_child_rel_parent: fixes h :: "(_) heap"   and child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent" shows "(parent, 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_node) \<in> parent_child_rel h"
assms: shows "chain_on (LEX P) (eq_upto C f i) (SEQ A)"
of_nat_monom: shows "of_nat = monom 0 \<circ> of_nat"
zero_im_sheaf_def: shows "imS.zero_im_sheaf \<equiv> \<lambda>V. \<zero>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
step_pres_bounded: shows "\<forall>(q, \<tau>)\<in>set (step (hd b) (a ! hd b)). q < length a \<and> \<tau> \<in> A"
restrict_def: fixes f :: "'ag \<Rightarrow> 'ah"   and A :: "'ag set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
isDERIV: shows "isDERIV n f (vs[n := x])"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "\<And>n. (\<And>x y. (x, y) = prod_decode n \<Longrightarrow> P y) \<Longrightarrow> P (Suc n)" shows "P a0"
WHILEIT_def: shows "WHILEIT \<equiv> iWHILEIT (\<bind>) RETURN"
LT33: fixes P :: "bool"   and F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> (#P \<and> F \<leadsto> G) = (#P \<longrightarrow> F \<leadsto> G)"
suffix_0: fixes x :: "nat \<Rightarrow> 'b" shows "suffix 0 x = x"
path_split_1: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and a :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node"   and as' :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit_wf.path n (as @ a # as') n'" shows "CFGExit_wf.path n as (src a)"
width_spec_width_spec: shows "(width_spec, width_spec) \<in> \<langle>lv_rel\<rangle>set_rel \<rightarrow> \<langle>rnv_rel\<rangle>nres_rel"
nodes_target_elim: fixes q :: "'state" assumes "q \<in> R.nodes" obtains r :: "'action list" and p :: "'state" where "init p" and "R.path r p" and "q = target r p"
assms: shows "\<exists>q\<in>ctermsl p. r \<in> ctermsl q"
valid_cfg_cont: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> MDP.valid_cfg"   and "s \<in> set_pmf (action cfg)" shows "cont cfg s \<in> MDP.valid_cfg"
exhaust_sel: fixes fundef :: "'b fundef" obtains "fundef = Fundef (body fundef) (arity fundef)"
assms_2: shows "f \<in> A \<rightarrow> A"
Tensor_null: fixes xs :: "'a list list" shows "xs \<otimes> [] = []"
boundedE: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "A \<noteq> {}"   and "Max A \<le> x" obtains "\<And>a. a \<in> A \<Longrightarrow> a \<le> x"
glob_def: fixes \<gamma> :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.glob \<gamma> \<equiv> \<lambda>t n. \<forall>n'\<ge>n. \<gamma> t n'"
vec1_index: fixes j :: "nat"   and n :: "nat"   and ze :: "'c"   and on :: "'c"   and i :: "nat" assumes "j < n" shows "vec1I ze on n i ! j = (if i = j then on else ze)"
m_cf_pol_coeff: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "pol_coeff S (m_cf S c)"
set_sorted_list_of_set: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
gen_division: shows "gen_division s (cubeImage ` twoChain)"
phis_wf: fixes g :: "'g"   and n :: "'node"   and v :: "'var \<times> 'node \<times> Def"   and args :: "('var \<times> 'node \<times> Def) list" assumes "phis' g (n, v) = Some args" shows "length (predecessors g n) = length args"
cinner_canonical_basis: fixes i :: "nat"   and j :: "nat" assumes "i < length canonical_basis"   and "j < length canonical_basis" shows "canonical_basis ! i \<bullet>\<^sub>C canonical_basis ! j = (if i = j then 1 else 0)"
ordLess_imp_ordLeq: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set" assumes "r <o r'" shows "r \<le>o r'"
simps_2: fixes mag :: "'d"   and dim :: "(int, 'c) dimvec"   and more :: "'e" shows "Quantity.dim \<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = dim"
Sigma_fm_Ex: fixes A :: "fm"   and B :: "fm"   and i :: "name" assumes "Sigma_fm A"   and "Sigma_fm B" shows "Sigma_fm (SyntaxN.Ex i A)"
ATOM_axioms: shows "ATOM neg\<^sub>d\<^sub>l\<^sub>o (\<lambda>a. True) I\<^sub>d\<^sub>l\<^sub>o depends\<^sub>d\<^sub>l\<^sub>o decr\<^sub>d\<^sub>l\<^sub>o"
isomorphic_is_subgraph_isomorphic: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set" assumes "G\<^sub>1 \<simeq> G\<^sub>2" shows "G\<^sub>1 \<sqsubseteq> G\<^sub>2"
strategyproofness_1: shows "pmf (sds R2) b + pmf (sds R2) a \<le> pmf (sds R27) c + pmf (sds R27) a"
intros_2: fixes A :: "fmla"   and H :: "fmla set" assumes "A \<in> Q_axioms" shows "H \<turnstile> A"
assms_4: shows "(_Entry_) -as\<rightarrow>\<^sub>\<surd>* (_Exit_)"
select_convs_1: fixes nodes :: "'d set"   and edges :: "('d \<times> 'e \<times> 'd) set"   and more :: "'f" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
joinableI: fixes x :: "'a"   and z :: "'b"   and R :: "('a \<times> 'b) set"   and y :: "'a" assumes "(x, z) \<in> R"   and "(y, z) \<in> R" shows "(x, y) \<in> joinable R"
scaleR: fixes r :: "real"   and b :: "'a" shows "frechet_derivative g (at x) (r *\<^sub>R b) = r *\<^sub>R frechet_derivative g (at x) b"
BinOpRedsThrow1: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 bop :: "bop"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
rref_pivot_positions: fixes R :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" assumes "row_echelon_form R"   and "R \<in> carrier_mat nr nc"   and "(i, j) \<in> set (pivot_positions R)" shows "i < nr \<and> j < nc"
h_def: shows "h = Cn 1 (r_smn 1 1) [r_const (encode r_univuniv), recf.Id 1 0]"
tfl_some: shows "\<forall>P x. P x \<longrightarrow> P (Eps P)"
that: fixes h :: "V"   and g :: "V"   and f :: "V" assumes "hgf = [h, g, f]\<^sub>\<circ>"   and "h \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "g \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"   and "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "thesis"
matrix_same: shows "M_mat_R (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) = kronecker_product (get_matrix_R (calculate_data_R p qs1)) (get_matrix_R (calculate_data_R p qs2))"
s_Inj2_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Inj2 e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Inj2 e'a" and "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"
scaleC_one: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC 1 x = x"
is_LTLGlobal_def: fixes ltl :: "'a ltl" shows "is_LTLGlobal ltl = (\<exists>x8. ltl = G x8)"
Vagree_and: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set"   and W :: "variable set" shows "(Vagree \<nu> \<nu>' V \<and> Vagree \<nu> \<nu>' W) = Vagree \<nu> \<nu>' (V \<union> W)"
BIT_Off_def: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list" shows "BIT_Off acts qs init \<equiv> distinct init \<and> length acts = length qs"
homogeneous_setI: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>a n. a \<in> A \<Longrightarrow> hom_component a n \<in> A" shows "homogeneous_set A"
in_interval_supI1: fixes x :: "'a"   and a :: "'a interval"   and b :: "'a interval" assumes "x \<in>\<^sub>i a" shows "x \<in>\<^sub>i sup a b"
fminus_disjoint: fixes A :: "'b fset"   and B :: "'b fset" shows "A |\<inter>| (B |-| A) = {||}"
well_def_subsets1: fixes l :: "nat list"   and i :: "nat" assumes "l \<in> set (get_subsets (calculate_data p qs1))"   and "i \<in> set l" shows "i < length qs1"
assms_2: fixes x :: "real" shows "f x \<le> u"
cong_exp_iff_simps_13: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit1 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 q))) = (modulo (numeral.numeral one plus m) (numeral.numeral one plus q) = modulo (numeral.numeral one plus n) (numeral.numeral one plus q))"
ParCL_MtransC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and c1' :: "('a, 'c) com"   and s' :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.MtransC tval aval (c1, s) (c1', s')" shows "PL.MtransC tval aval (c1 | c2, s) (c1' | c2, s')"
sorted_drop: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and n :: "nat" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (drop n xs)"
assms_2: fixes \<phi> :: "'a fm"   and xs :: "'b list" assumes "nqfree \<phi>" shows "I (qe \<phi>) xs = (\<exists>x. I \<phi> (x # xs))"
refl: fixes x :: "'a" assumes "x \<in> Y" shows "x \<sqsubseteq> x"
assms_2: shows "WT_rest \<I>_adv_rest \<I>_usr_rest I_rest rest"
tendsF: fixes x :: "'a" assumes "x \<notin> N" shows "(\<lambda>n. F n x) \<longlonglongrightarrow> f x"
ranI: fixes m :: "'c \<Rightarrow> 'b option"   and a :: "'c"   and b :: "'b" assumes "m a = Some b" shows "b \<in> ran m"
IdNatTransNatTrans': fixes F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor" assumes "Functor F" shows "NatTransP (IdNatTrans' F)"
primorial_ge: fixes x :: "real" shows "2 powr primes_pi x \<le> real (primorial x)"
allDefs_def: fixes g :: "'g"   and n :: "'node" shows "uninst_code.ssa.allDefs phis g n \<equiv> defs g n \<union> uninst_code.ssa.phiDefs phis g n"
minf_8: fixes t :: "'a" shows "\<exists>z. \<forall>x<z. (t \<le> x) = False"
pmf_of_alist_support_aux_2: fixes \<mu> :: "('c \<times> real) list" assumes "\<forall>(uu_, p)\<in>set \<mu>. 0 \<le> p"   and "sum_list (map snd \<mu>) = 1"   and "distinct (map fst \<mu>)" shows "\<integral>\<^sup>+ x. ennreal (case map_of \<mu> x of None \<Rightarrow> 0 | Some p \<Rightarrow> p) \<partial>count_space UNIV = 1"
OneIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E OneIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<tau> = Boolean[1]" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<rho>)" and "length its \<le> Suc 0" and "\<rho> \<le> Boolean[?]"
not_equal_is_parallel: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<noteq> ys"   and "length xs = length ys" shows "xs \<parallel> ys"
add_ac_2: fixes a :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool" shows "(a \<and>* b) = (b \<and>* a)"
eq_iff: fixes x :: "'c list"   and y :: "'c list" shows "(x = y) = (sublist x y \<and> sublist y x)"
Field_square: fixes x :: "'a set" shows "Field (x \<times> x) = x"
associated_iff_dvd: fixes a :: "'a"   and b :: "'a" shows "(normalize a = normalize b) = (a dvd b \<and> b dvd a)"
representsD: fixes T :: "'s Trace set" assumes "represents T" shows "jkbpT T = T"
select_object_any\<^sub>S\<^sub>e\<^sub>t_def: fixes f :: "'f \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> 'e"   and s_set :: "'f list" shows "select_object_any\<^sub>S\<^sub>e\<^sub>t f s_set = (let s = select_object\<^sub>S\<^sub>e\<^sub>t f s_set in if s->size\<^sub>S\<^sub>e\<^sub>t() \<triangleq> \<one> then s->any\<^sub>S\<^sub>e\<^sub>t() else \<bottom> endif)"
comm: fixes x :: "'b list"   and y :: "'b list" shows "(x \<cdot> y = y \<cdot> x) = (\<exists>t m k. x = t \<^sup>@ k \<and> y = t \<^sup>@ m)"
apply_cm_is_cm: fixes c :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c loc) \<and> t' \<le> t"   and "n \<noteq> 0" shows "next_change_multiplicity' c (apply_cm c loc t n) loc t n"
Syntax_with_Connectives_False_axioms: shows "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls"
almost_full_on_imp_no_antichain_on: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and f :: "nat \<Rightarrow> 'd" assumes "almost_full_on P A" shows "\<not> antichain_on P f A"