hcju/isabelleps huggingface dataset:
valid_states_def: fixes prob :: "(('e, 'g) fmap \<times> ('e, 'h) fmap) set" shows "valid_states prob \<equiv> {s. fmdom' s = prob_dom prob}"
compare_strict_2: fixes x :: "'a" shows "compare\<cdot>x\<cdot>\<bottom> = \<bottom>"
fp_cop_F_worst: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and ds :: "'e set"   and y :: "'d" assumes "ContractsWithUnilateralSubstitutesAndIRC Xd Xh Pd Ch"   and "x \<in> Contracts.CH Ch (Contracts.fp_cop_F Pd Ch ds)"   and "y \<in> Contracts.fp_cop_F Pd Ch ds"   and "Xd y = Xd x" shows "(x, y) \<in> Pd (Xd x)"
acyclic_3c_def: fixes x :: "'a" shows "acyclic_3c x \<equiv> \<forall>y. y \<le> x \<and> x \<le> y\<^sup>+ \<longrightarrow> y = x"
unique_connecting_path_properties_3: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "hd (v \<leadsto> w) = v"
cUn_cempty_right: fixes A :: "'d cset" shows "cUn A cempty = A"
comp: fixes x :: "'a"   and b :: "'b" assumes "x \<in> A"   and "b \<in> Basis" shows "f x \<bullet> b \<le> g x \<bullet> b"
intro: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "N \<subseteq> M"   and "\<And>a b. \<lbrakk>a \<in> N; b \<in> N\<rbrakk> \<Longrightarrow> composition a b \<in> N"   and "unit \<in> N" shows "submonoid_axioms N M composition unit"
intro: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system \<V> \<B>"   and "incidence_system \<V>' \<B>'"   and "incidence_system_isomorphism_axioms \<V> \<B> \<V>' \<B>' bij_map" shows "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"
cod_null: shows "B.VV.cod B.VV.null = B.VV.null"
hom_uminus: fixes x :: "'b poly" shows "map_poly to_fract (- x) = - map_poly to_fract x"
rep_eq: fixes x :: "'a ccsubspace"   and xa :: "'a ccsubspace" shows "space_as_set (x \<squnion> xa) = space_as_set x +\<^sub>M space_as_set xa"
finite_nonempty_carrier_has_minimum: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" assumes "rational_preference carrier relation"   and "finite carrier"   and "carrier \<noteq> {}" shows "\<exists>e\<in>carrier. \<forall>m\<in>carrier. m \<succeq>[relation] e"
rel_compp: fixes R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option (R OO S) = rel_option R OO rel_option S"
simps_7: fixes r :: "'k \<Rightarrow> 'v \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'i"   and k :: "'k"   and \<phi> :: "('a, 'k) aformula" shows "satisfies_gen r \<AA> (FAll k \<phi>) = (\<forall>P. r k P (Length \<AA>) \<longrightarrow> satisfies_gen r (Extend k 0 \<AA> P) \<phi>)"
dynProcModifyReturnSameFaults_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and ModifAbr :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat" assumes "\<And>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P dynCall init p return' c Q,A"   and "\<forall>s\<in>P. \<forall>\<sigma> n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
loc_reach_mem_diff_eq: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and mem :: "'Var \<Rightarrow> 'Val"   and c :: "'Com" assumes "\<forall>x. x \<in> mds AsmNoWrite \<longrightarrow> mem' x = mem x" shows "loc_reach \<langle>c, mds, mem\<rangle> = loc_reach \<langle>c, mds, mem'\<rangle>"
strip_comb_simps_2: fixes t :: "'a" assumes "unapp t = None" shows "strip_comb t = (t, [])"
h7a: shows "derivative x p = (\<Sum>i = 0..MPoly_Type.degree p x - 1. isolate_variable_sparse p x (i + 1) * Var x ^ i * Const (real (i + 1)))"
partition_on_insert': fixes A :: "'b set"   and X :: "'b set"   and P :: "'b set set" assumes "partition_on (A - X) P"   and "X \<subseteq> A"   and "X \<noteq> {}" shows "partition_on A (insert X P)"
is_shadow_root_ptr_ref: fixes n :: "nat" shows "is_shadow_root_ptr (shadow_root_ptr.Ref n)"
closed_map_id: fixes X :: "'c topology" shows "closed_map X X id"
streams_Stream: fixes x :: "'a"   and s :: "'a stream"   and A :: "'a set" shows "(x ## s \<in> streams A) = (x \<in> A \<and> s \<in> streams A)"
outputPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<simeq> Q" shows "a{b}.P \<simeq> a{b}.Q"
fld_simp: fixes C :: "'b"   and f :: "'a decl_scheme" shows "fld (C, f) = f"
lfinite_lrel': fixes ys :: "'c llist"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'d llist" assumes "lfinite ys"   and "lrel R xs ys" shows "lfinite xs"
bij_betw_inv_into_right: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and a' :: "'c" assumes "bij_betw f A A'"   and "a' \<in> A'" shows "f (inv_into A f a') = a'"
SS: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "SS t1 t2 \<in> fmla"
index_conv_takeWhile: fixes xs :: "'b list"   and x :: "'b" shows "index xs x = length (takeWhile ((\<noteq>) x) xs)"
vwalk_joinI_vwalk': fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph"   and q :: "'a list" assumes "vwalk p G"   and "vwalk q G"   and "last p = hd q" shows "vwalk (p \<oplus> q) G"
isCont_tendsto_compose: fixes l :: "'c"   and g :: "'c \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'c"   and F :: "'e filter" assumes "isCont g l"   and "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
assms: assumes "c \<noteq> 0" shows "(f has_integral x) A"
distinct_89: fixes x91 :: "'d"   and x92 :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x93 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "{x91}receive(x92) . x93 \<noteq> call(x10)"
is_proper_suffix_eq_suffix: fixes a :: "'a list"   and b :: "'a list" shows "is_proper_suffix a b = (a \<noteq> b \<and> is_suffix a b)"
size_12: fixes x41 :: "stmt"   and x42 :: "stmt" shows "size (Seq x41 x42) = size x41 + size x42 + Suc 0"
poly_condD_4: fixes p :: "int poly" assumes "poly_cond p" shows "square_free p"
distinct_set_notin: fixes x :: "'c"   and xs :: "'c list" assumes "distinct (x # xs)" shows "x \<notin> set xs"
Hintikka_def: fixes S :: "'b formula set" shows "Hintikka (S::'b formula set) \<equiv> \<bottom> \<notin> S \<and> (\<forall>k::'b. Atom k \<in> S \<longrightarrow> \<^bold>\<not> (Atom k) \<in> S \<longrightarrow> False) \<and> (\<forall>(F::'b formula) G::'b formula. F \<^bold>\<and> G \<in> S \<longrightarrow> F \<in> S \<and> G \<in> S) \<and> (\<forall>(F::'b formula) G::'b formula. F \<^bold>\<or> G \<in> S \<longrightarrow> F \<in> S \<or> G \<in> S) \<and> (\<forall>(F::'b formula) G::'b formula. F \<^bold>\<rightarrow> G \<in> S \<longrightarrow> \<^bold>\<not> F \<in> S \<or> G \<in> S) \<and> (\<forall>F::'b formula. \<^bold>\<not> (\<^bold>\<not> F) \<in> S \<longrightarrow> F \<in> S) \<and> (\<forall>(F::'b formula) G::'b formula. \<^bold>\<not> (F \<^bold>\<and> G) \<in> S \<longrightarrow> \<^bold>\<not> F \<in> S \<or> \<^bold>\<not> G \<in> S) \<and> (\<forall>(F::'b formula) G::'b formula. \<^bold>\<not> (F \<^bold>\<or> G) \<in> S \<longrightarrow> \<^bold>\<not> F \<in> S \<and> \<^bold>\<not> G \<in> S) \<and> (\<forall>(F::'b formula) G::'b formula. \<^bold>\<not> (F \<^bold>\<rightarrow> G) \<in> S \<longrightarrow> F \<in> S \<and> \<^bold>\<not> G \<in> S)"
norm_diff_ineq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm a - norm b \<le> norm (plus a b)"
lex_fun_refl: fixes s :: "'a \<Rightarrow> 'c" shows "rlex_fun s s"
add_nonneg_eq_0_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero x"   and "less_eq zero y" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
map_add_dom_app_simps_1: fixes m :: "'a"   and l2 :: "'a \<Rightarrow> 'b option"   and l1 :: "'a \<Rightarrow> 'b option" assumes "m \<in> dom l2" shows "(l2 \<Oplus> l1) m = l2 m"
c_def: shows "c = to_fun (shift 2 (T\<^bsub>b\<^esub> g)) (a \<ominus> b)"
exec_meth_append: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and s :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and ins' :: "'a instr list" assumes "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h s ta h' s'" shows "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P (ins @ ins') xt t h s ta h' s'"
mirror_elem_less_conv: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" assumes "finite I"   and "a \<in> I"   and "b \<in> I" shows "(mirror_elem b I < mirror_elem a I) = (a < b)"
balance_left_inv1: fixes a :: "('a, 'b) rbt"   and b :: "('a, 'b) rbt"   and k :: "'a"   and x :: "'b" assumes "inv1l a"   and "inv1 b"   and "color_of b = B" shows "inv1 (balance_left a k x b)"
edge_bipartite_webI1: fixes \<Gamma> :: "('a, 'b) web_scheme"   and u :: "'a"   and v :: "'a" assumes "edge \<Gamma> u v" shows "edge (bipartite_web_of \<Gamma>) (Inl u) (Inr v)"
filtermap_at_shift: fixes d :: "'a"   and a :: "'a" shows "filtermap (\<lambda>x. x - d) (at a) = at (a - d)"
multiplicity_diff_self_pow: fixes p :: "nat"   and x :: "int"   and y :: "int" assumes "prime p"   and "int p dvd x - y"   and "\<not> int p dvd x"   and "\<not> int p dvd y"   and "2 < p"   and "x \<noteq> y" shows "multiplicity (int p) (x ^ p - y ^ p) = Suc (multiplicity (int p) (x - y))"
top_admS_def: shows "top \<equiv> mklr UNIV"
action_by_conjugation_on_power_set: shows "group_action G {H. H \<subseteq> carrier G} (\<lambda>g. \<lambda>H\<in>{H. H \<subseteq> carrier G}. g <# H #> inv g)"
h_sub_rewrite: fixes h :: "'a"   and x :: "nat"   and z :: "nat"   and c :: "nat" assumes "h = \<^bold>g [^] x"   and "z < order \<G>" shows "\<^bold>g [^] (z + order \<G> * x * c - x * c) = \<^bold>g [^] z \<otimes> inv (h [^] c)"
set_def: fixes ba :: "'c \<times> 'c" shows "local.set ba \<equiv> \<phi> (fst ba, snd ba) ` C.hom (fst ba) (snd ba)"
keys_add: fixes f :: "'c \<Rightarrow>\<^sub>0 'd"   and g :: "'c \<Rightarrow>\<^sub>0 'd" shows "Poly_Mapping.keys (f + g) \<subseteq> Poly_Mapping.keys f \<union> Poly_Mapping.keys g"
cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) character_data_ptr" shows "cast ptr = node_ptr.Ext (Inr (Inl ptr))"
extend_state_oracle2_def: fixes callee :: "'h \<Rightarrow> 'j \<Rightarrow> ('k \<times> 'h) spmf" shows "callee\<dagger> = (\<lambda>(s, s') x. map_spmf (\<lambda>(y, s). (y, s, s')) (callee s x))"
evt_def: fixes \<gamma> :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.evt \<gamma> \<equiv> \<lambda>t n. \<exists>n'\<ge>n. \<gamma> t n'"
prems_8: shows "(ag_, a'g_) \<in> \<langle>R\<rangle>ltlr_rel \<rightarrow> \<langle>R\<rangle>ltlr_rel \<rightarrow> Rv \<rightarrow> Rv \<rightarrow> Rv"
\<Gamma>_reductive: fixes \<Gamma> :: "'b inference set"   and \<gamma> :: "'b inference" assumes "reductive_inference_system \<Gamma>"   and "\<gamma> \<in> \<Gamma>" shows "concl_of \<gamma> < main_prem_of \<gamma>"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "inj_comm_semiring_hom hom" shows "map_poly_inj_comm_semiring_hom hom"
notin_alphabet\<^sub>F_imp_splitLeaf\<^sub>F_id: fixes a :: "'a"   and ts :: "'a tree list"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat"   and b :: "'a" assumes "a \<notin> alphabet\<^sub>F ts" shows "splitLeaf\<^sub>F ts w\<^sub>a a w\<^sub>b b = ts"
viterbi_final_def: fixes s :: "'s"   and os :: "'t list" shows "viterbi_final s os = fst (argmax snd (map (\<lambda>t. viterbi s t os) state_list))"
bal_full_node: fixes k :: "nat"   and a :: "'b"   and h :: "nat" shows "bal (full_node k a h)"
induct: fixes P :: "nat list \<Rightarrow> bool"   and a0 :: "nat list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (x # xs)" shows "P a0"
update_convs_5: fixes getConflictClause' :: "nat \<Rightarrow> nat"   and getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getConflictClause_update getConflictClause' \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause' getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr>"
i_drop_0: fixes f :: "nat \<Rightarrow> 'b" shows "f \<Up> 0 = f"
axioms_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and w :: "'b" assumes "arrow_of_spans_of_maps V H \<a> \<i> src trg r\<^sub>0 r\<^sub>1 s\<^sub>0 s\<^sub>1 w" shows "bicategory_of_spans V H \<a> \<i> src trg"
coprime_imp_gcd_eq_1: fixes a :: "'a"   and b :: "'a" assumes "coprime a b" shows "gcd a b = (1::'a)"
simps_7: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Abs a) vs = approx_floatarith p a vs \<bind> (\<lambda>r. let ivl = ivl_of_aform_err p r; i = lower ivl; s = upper ivl in if 0 < i then Some r else if s < 0 then Some (apfst uminus_aform r) else Some (ivl_err (real_interval (abs_interval ivl))))"
simps_1: fixes t :: "'a tree" shows "huffman [t] = t"
bind_spmf_const_return_None: fixes p :: "'d spmf" shows "p \<bind> (\<lambda>_. return_pmf None) = return_pmf None"
elem_order_zmset_exists_foundation: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and M :: "'a zmultiset" assumes "class.order less_eq less"   and "x \<in>#\<^sub>z M" shows "\<exists>s\<in>#\<^sub>zM. less_eq s x \<and> (\<forall>u\<in>#\<^sub>zM. \<not> less u s)"
iMOD_plus_divisor: fixes x :: "nat"   and r :: "nat"   and m :: "nat" assumes "x \<in> [ r, mod m ]" shows "x + m \<in> [ r, mod m ]"
plus_monomial_less_def: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and c :: "'d"   and u :: "'c" shows "plus_monomial_less p c u = p + monomial c u"
functor_axioms: shows "functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) (\<lambda>f. f \<otimes>\<^sub>D \<I>\<^sub>D)"
exhaust: fixes y :: "'e process_name" obtains x1 :: "'e" where "y = mutator x1"   | "y = gc"   | "y = sys"
mono_def: fixes f :: "'a \<Rightarrow> 'b" shows "local.mono f = (\<forall>x y. y \<le> x \<longrightarrow> f x \<le> f y)"
locate1_def: fixes f :: "'a \<Rightarrow> nat"   and al :: "'a list"   and i :: "nat" shows "locate1 f al i \<equiv> fst (locate f al i)"
lin_data_prev': shows "linearize data_prev' = filter (\<lambda>(t, X). nt - t < left I) (linearize data_prev)"
iso_in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VV.in_hom f a b"   and "VV.iso f" shows "VV.iso_in_hom f a b"
finite_total_preorder_on_axioms_def: fixes carrier :: "'a set" shows "finite_total_preorder_on_axioms carrier \<equiv> finite carrier"
hyps: shows "(xs @ pi2) \<bullet> c = xs \<bullet> pi2 \<bullet> c"
h1_is_pr: shows "h1 \<in> PrimRec1"
append_carrier_vec: fixes v :: "'a vec"   and n1 :: "nat"   and w :: "'a vec"   and n2 :: "nat" assumes "v \<in> carrier_vec n1"   and "w \<in> carrier_vec n2" shows "v @\<^sub>v w \<in> carrier_vec (n1 + n2)"
30: fixes l :: "nat" assumes "l \<in> A" shows "least A \<le> l"
foldri_not_cond: fixes c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and xs :: "'d list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<not> c \<sigma>" shows "foldri xs c f \<sigma> = \<sigma>"
sprefI1: fixes v :: "'b list"   and u :: "'b list"   and z :: "'b list" assumes "v = u \<cdot> z"   and "z \<noteq> \<epsilon>" shows "u <p v"
add_mset_remove_trivial_iff: fixes N :: "'a multiset"   and a :: "'a"   and b :: "'a" shows "(N = add_mset a (remove1_mset b N)) = (a \<in># N \<and> a = b)"
v_in_set_vs: shows "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"
ClassD: fixes b :: "'a"   and a :: "'a" assumes "b \<in> Class a"   and "a \<in> G" shows "(b, a) \<in> E"
assms_2: shows "\<forall>y\<in>cball x e. f y \<le> b"
set_zmset_single: fixes b :: "'b" shows "set_zmset {#b#}\<^sub>z = {b}"
abs_summable_on_cong: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and B :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "A = B" shows "(f abs_summable_on A) = (g abs_summable_on B)"
acc_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "acc r = wf {(y, x). x \<sqsubset>\<^bsub>r\<^esub> y}"
vseq_implies_hpar_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.seq \<nu> \<mu>" shows "trg\<^sub>B \<nu> = trg\<^sub>B \<mu>"
wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('b, 'a) stateful_strand_step list" shows "wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
Un_upper1: fixes A :: "'b set"   and B :: "'b set" shows "order.greater_eq (A \<union> B) A"
d1_sum_var: fixes x :: "'a"   and y :: "'a" shows "x + y \<le> (d x + d y) \<cdot> (x + y)"
Im_exp: fixes z :: "complex" shows "Im (exp z) = exp (Re z) * sin (Im z)"
pdevs_apply_pdevs_inner: fixes p :: "'b pdevs"   and a :: "'b"   and i :: "nat" shows "pdevs_apply (pdevs_inner p a) i = pdevs_apply p i \<bullet> a"
domD: fixes h' :: "'b \<Rightarrow> 'b"   and b :: "'b"   and Q :: "'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "recursive h' b"   and "tailrec.dom Q h x"   and "\<not> Q x" shows "tailrec.dom Q h (h x)"
LeftDerivation_from_in_between: fixes \<alpha> :: "symbol list"   and u :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and \<beta> :: "symbol list"   and v :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivation \<alpha> (take u D) \<beta>"   and "LeftDerivation \<alpha> (take v D) \<gamma>"   and "u \<le> v" shows "LeftDerivation \<beta> (drop u (take v D)) \<gamma>"
ennreal_SUP_add_left: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and c :: "ennreal" assumes "I \<noteq> {}" shows "(SUP i\<in>I. f i + c) = Sup (f ` I) + c"
similar_matrix_wit_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and P :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and Q :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "similar_matrix_wit A B P Q = (P ** Q = mat (1::'a) \<and> Q ** P = mat (1::'a) \<and> A = P ** B ** Q)"
prod_mset_replicate_mset: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and a :: "'a" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one (replicate_mset n a) = power.power one times a n"
2: shows "\<forall>i<2. \<forall>j<2. brickmat vert ! i ! j = (if i = j then 1 else 0)"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_discard return bind" shows "monad_discard_axioms bind"
assms: shows "e \<turnstile>\<^sub>R t : T"
open_sub: fixes i :: "'a"   and S :: "'a \<Rightarrow> 'b set" assumes "i \<in> domain x"   and "open (S i)"   and "(x)\<^sub>F i \<in> S i" shows "\<exists>a\<in>A i. a \<subseteq> S i"
lookup_default: fixes x :: "'b"   and d :: "'a"   and m :: "('b, 'a) mapping" shows "Mapping.lookup (Mapping.default x d m) x = Some (Mapping.lookup_default d m x)"
Join_absorb2: fixes G :: "'c program"   and F :: "'c program" assumes "G \<le> F" shows "F \<squnion> G = F"
zcf_ltrm_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "0 < deg R p" shows "zcf (monom P (p (deg R p)) (deg R p) \<otimes>\<^bsub>P\<^esub> q) = \<zero>"
some_eq_imp: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Eps P = a"   and "P b" shows "P a"
continuous_on_inner: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x \<bullet> g x)"
tpOfV_tpick: fixes V :: "Preliminaries.var set"   and \<sigma> :: "'tp" assumes "tinfinite V" shows "tpOfV (tpick \<sigma> V) = \<sigma>"
bounded_measure: fixes p :: "'c spmf"   and A :: "'c set" shows "Sigma_Algebra.measure (measure_spmf p) A \<le> weight_spmf p"