hcju/isabelleps huggingface dataset:
IntE: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
i_nat: shows "real_of_int i = - real (nat (- i))"
mset_tree_merge: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "mset_tree (merge l r) = mset_tree l + mset_tree r"
prems_3: fixes z :: "complex" assumes "z \<in> insert z0 X" shows "f \<in> O[at z](\<lambda>_. 1)"
iT_Mult_mem_iff2: fixes k :: "nat"   and x :: "nat"   and I :: "nat set" assumes "0 < k" shows "(x * k \<in> I \<otimes> k) = (x \<in> I)"
measure_induct_rule: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
src_simps_1: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and F :: "('b bicategorical_language.term, 'b) concrete_category.arr" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "partial_magma.arr (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) F" shows "concrete_category.Dom (strictified_bicategory.src V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B F) = bicategorical_language.Src src\<^sub>B (concrete_category.Dom F)"
spec_impl: fixes G :: "'a list_graph"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_list_graph G" shows "sinvar_spec (list_graph_to_graph G) nP = sinvar_impl G nP"
subseq: fixes s :: "nat \<Rightarrow> nat"   and \<nu> :: "real measure" assumes "strict_mono s"   and "real_distribution \<nu>"   and "weak_conv_m (\<mu> \<circ> s) \<nu>" shows "weak_conv_m (\<mu> \<circ> s) M"
h1: shows "\<tt>\<tt> p \<le> \<tt>\<tt> p'"
dom_def: fixes G :: "nat graph_rec"   and i :: "nat"   and j :: "nat" assumes "cfg_doms G" shows "cfg_doms.dom G i j \<equiv> let res = cfg_doms.dom_kildall G ([] # replicate (length (g_V G) - 1) (rev [0..<length (g_V G)])) ! j in i \<in> set res \<or> i = j"
left_total_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "left_total R = (\<forall>x. \<exists>y. R x y)"
gbinomial_addition_formula: fixes a :: "'a"   and k :: "nat" shows "a gchoose Suc k = a - (1::'a) gchoose Suc k + (a - (1::'a) gchoose k)"
makespan_mono_1: fixes y :: "nat"   and T :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "y \<le> T x" shows "makespan (T(x := y)) \<le> makespan T"
star_conv: fixes x :: "'a" shows "(x\<^sup>\<star>)\<^sup>T = x\<^sup>T\<^sup>\<star>"
mul_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "mul_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; rs1 = get_operand_w5 (op_list ! 1); rd = get_operand_w5 (op_list ! 3) in do operand2 \<leftarrow> gets (get_operand2 op_list); curr_win \<leftarrow> get_curr_win (); rs1_val \<leftarrow> gets (user_reg_val curr_win rs1); psr_val \<leftarrow> gets (cpu_reg_val PSR); result0 \<leftarrow> gets (\<lambda>s. if instr_name \<in> {arith_type UMUL, arith_type UMULcc} then word_of_int (uint rs1_val * uint operand2) else word_of_int (sint rs1_val * sint operand2)); y_val \<leftarrow> gets (\<lambda>s. ucast (result0 >> 32)); write_cpu y_val Y; result \<leftarrow> gets (\<lambda>s. ucast result0); rd_val \<leftarrow> gets (user_reg_val curr_win rd); new_rd_val \<leftarrow> gets (\<lambda>s. if rd \<noteq> 0 then result else rd_val); write_reg new_rd_val curr_win rd; mul_instr_sub1 instr_name result od"
sel_5: fixes x1 :: "((('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) list"   and x2 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x3 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x4 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x5 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x6 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list" shows "transaction_updates (Transaction x1 x2 x3 x4 x5 x6) = x5"
elims: fixes x :: "'c"   and xa :: "('c \<times> nat) list"   and y :: "nat option" assumes "univariate_power_list x xa = y" obtains xb :: "'c" and ya :: "'c" and n :: "nat" where "x = xb" and "xa = [(ya, n)]" and "y = (if xb = ya then Some n else None)"   | uu_ :: "'c" where "x = uu___" and "xa = []" and "y = None"   | uu_ :: "'c" and v :: "'c \<times> nat" and vb :: "'c \<times> nat" and vc :: "('c \<times> nat) list" where "x = uu_" and "xa = v # vb # vc" and "y = None"
HMSet_less: fixes M :: "hmultiset multiset"   and N :: "hmultiset multiset" shows "(HMSet M < HMSet N) = (M < N)"
continuous_closedin_preimage_constant: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b" assumes "continuous_on S f" shows "closedin (top_of_set S) {x \<in> S. f x = a}"
wffp_isState_doo: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl"   and "list_all isState sl" shows "doo (local.getGMState (hd sl)) (map getGMUserCom (tl sl)) = local.getGMState (last sl)"
conv_plus_commute: fixes x :: "'a" shows "x\<^sup>+\<^sup>T = x[[x\<^sup>T\<^sup>\<star>]]"
lu_props_4: fixes p :: "real poly"   and x :: "real" assumes "p \<in> set ps"   and "x \<le> l" shows "sgn (poly p x) = poly_neg_inf p"
t_covering_design_axioms: shows "t_covering_design \<V> \<B> \<k> \<t> \<Lambda>\<^sub>t"
insert_4: shows "insert x A \<subseteq> \<Union> (range f)"
simps_1: fixes r :: "'b rexp" shows "slexer r [] = (if nullable r then Some (mkeps r) else None)"
A': fixes kv :: "'c \<times> 'd"   and x :: "'e" shows "flat_ge ((case kv of (x, xa) \<Rightarrow> f x xa) x) ((case kv of (x, xa) \<Rightarrow> f' x xa) x)"
new_f12: shows "f12 \<notin> \<F> g"
subdg_subdigraph_op_dg_op_dg: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>" shows "op_dg \<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> op_dg \<CC>"
simps: fixes instr :: "instr"   and pc :: "nat"   and s :: "integer" shows "exec instr (pc, s) = (case instr of AssI ns bs \<Rightarrow> let bvs = zip ns (map (\<lambda>b. bval b s) bs) in [(pc + 1, foldl (\<lambda>s (x, y). set_bit_class.set_bit s x y) s bvs)] | TestI b d \<Rightarrow> [if bval b s then (pc + 1, s) else (nat (int (pc + 1) + d), s)] | ChoiceI bis \<Rightarrow> let succs = concat (map (\<lambda>(b, i). if bval b s then [(nat (int (pc + 1) + i), s)] else []) bis) in if succs = [] then [(pc, s)] else succs | GotoI d \<Rightarrow> [(nat (int (pc + 1) + d), s)])"
w_simps_4: shows "trg w = src (s\<^sub>0 \<star> p\<^sub>0)"
ag_setfunc_def: fixes n :: "nat"   and B1 :: "nat \<Rightarrow> ('d, 'e) Ring_scheme"   and m :: "nat"   and B2 :: "nat \<Rightarrow> ('d, 'e) Ring_scheme"   and X :: "(nat \<Rightarrow> 'd) set"   and Y :: "(nat \<Rightarrow> 'd) set" shows "ag_setfunc n B1 m B2 X Y = {f. \<exists>g h. g \<in> X \<and> h \<in> Y \<and> f = augm_func n g (Un_carrier {j. j \<le> n} B1) m h (Un_carrier {j. j \<le> m - 1} B2)}"
rb_aux_inv1_D3: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "rb_aux_inv1 bs" shows "sorted_wrt (\<lambda>x y. lt y \<prec>\<^sub>t lt x) bs"
\<delta>: fixes s :: "'state" assumes "s \<in> S" shows "\<delta> s \<subseteq> S"
inner: fixes X :: "'c"   and G :: "('c \<times> ('c, 'd) RHS) list"   and T :: "nat \<times> nat \<Rightarrow> 'c list"   and i :: "nat"   and k :: "nat"   and j :: "nat" shows "(X \<in> set (inner G T i k j)) = (\<exists>l\<ge>k. l < j \<and> X \<in> set (match_prods G (T (i, l)) (T (i + l, j - l))))"
image_constant_conv: fixes c :: "'a"   and A :: "'b set" shows "(\<lambda>x. c) ` A = (if A = {} then {} else {c})"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>"
assms_3: shows "(g \<longlongrightarrow> b) F"
poly_of_focus_def: fixes x :: "'a" shows "poly_of_focus x = poly_of_pm x \<circ> focus {x}"
smult_spvec_cons: fixes y :: "'b"   and a :: "'a \<times> 'b"   and arr :: "('a \<times> 'b) list" shows "smult_spvec y (a # arr) = (fst a, y * snd a) # smult_spvec y arr"
abs_eq: fixes xa :: "'b \<times> 'b"   and x :: "'b \<times> 'b" assumes "fractrel xa xa"   and "fractrel x x" shows "abs_fract xa + abs_fract x = abs_fract (fst xa * snd x + fst x * snd xa, snd xa * snd x)"
arccos_less_mono: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> 1"   and "\<bar>y\<bar> \<le> 1" shows "(arccos x < arccos y) = (y < x)"
measurable_PiM_single: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and N :: "'d measure"   and I :: "'e set"   and M :: "'e \<Rightarrow> 'f measure" assumes "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"   and "\<And>A i. \<lbrakk>i \<in> I; A \<in> sets (M i)\<rbrakk> \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N" shows "f \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
nn_integral_set_ennreal: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real"   and A :: "'a set" shows "\<integral>\<^sup>+x\<in>A. ennreal (f x)\<partial>M = \<integral>\<^sup>+ x. ennreal (f x * indicat_real A x) \<partial>M"
Sup_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. less_eq a b" shows "less_eq (Sup A) (Sup B)"
isoE: fixes f :: "'a" assumes "A.iso f" obtains g :: "'a" where "A.inverse_arrows f g"
dghm_up_ObjMap_vsv: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<CC> :: "V"   and \<phi> :: "V \<Rightarrow> V" shows "vsv (dghm_up I \<AA> \<CC> \<phi>\<lparr>ObjMap\<rparr>)"
iIN_cut_ge1: fixes t :: "nat"   and n :: "nat"   and d :: "nat" assumes "t \<in> [n\<dots>,d]" shows "[n\<dots>,d] \<down>\<ge> t = [t\<dots>,n + d - t]"
gConsPresGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gConsPresGWls MOD \<equiv> gVarPresGWls MOD \<and> gAbsPresGWls MOD \<and> gOpPresGWls MOD"
contraction: fixes A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A (insert A H) \<turnstile> B" shows "insert A H \<turnstile> B"
le_fin_reflp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a list" assumes "traces ind" shows "traces.le_fin ind w w"
mono_onI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<le> y\<rbrakk> \<Longrightarrow> f x \<le> f y" shows "Basis.mono_on A f"
bound_le: fixes \<eta> :: "nat" shows "bound_outer \<eta> * max (bound_sim \<eta>) 1 \<le> bound_inner \<eta>"
AE_in_set_eq_1: fixes A :: "'a set" assumes "A \<in> events" shows "(AE x in M. x \<in> A) = (prob A = 1)"
induct: fixes x1 :: "'node SDG_node set"   and x2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and x3 :: "'node list"   and x4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and x5 :: "'edge"   and x6 :: "'node list"   and x7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and P :: "'node SDG_node set \<Rightarrow> ('edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind) \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> 'edge \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> bool" assumes "x1,x2 \<turnstile> (x3,x4) -x5\<rightarrow>\<^sub>\<tau> (x6,x7)"   and "\<And>f a s s' ms S ms'. \<lbrakk>pred (f a) s; transfer (f a) s = s'; valid_edge a; intra_kind (kind a); (\<exists>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>) \<or> hd ms \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>; Ball (set (tl ms)) return_node; length s' = length s; length ms = length s; hd ms = sourcenode a; ms' = targetnode a # tl ms\<rbrakk> \<Longrightarrow> P S f ms s a ms' s'"   and "\<And>f a s s' Q r p fs a' ms S ms'. \<lbrakk>pred (f a) s; transfer (f a) s = s'; valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; valid_edge a'; a' \<in> get_return_edges a; (\<exists>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>) \<or> hd ms \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>; Ball (set (tl ms)) return_node; length ms = length s; length s' = Suc (length s); hd ms = sourcenode a; ms' = targetnode a # targetnode a' # tl ms\<rbrakk> \<Longrightarrow> P S f ms s a ms' s'"   and "\<And>f a s s' Q p f' ms S ms'. \<lbrakk>pred (f a) s; transfer (f a) s = s'; valid_edge a; kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f'; \<exists>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>; Ball (set (tl ms)) return_node; length ms = length s; length s = Suc (length s'); s' \<noteq> []; hd ms = sourcenode a; hd (tl ms) = targetnode a; ms' = tl ms\<rbrakk> \<Longrightarrow> P S f ms s a ms' s'" shows "P x1 x2 x3 x4 x5 x6 x7"
replace_comp_cls_val_process_Suc: fixes Mkt :: "('c, 'd) discrete_market"   and x :: "'d"   and pf2 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "\<forall>n w. prices Mkt x (Suc n) w = cls_val_process Mkt pf2 (Suc n) w"   and "portfolio pf1"   and "portfolio pf2" shows "\<forall>n w. cls_val_process Mkt (qty_replace_comp pf1 x pf2) (Suc n) w = cls_val_process Mkt pf1 (Suc n) w"
fib_lowerbound: fixes n :: "nat" shows "((1 + sqrt 5) / 2) ^ n \<le> real (fib (n + 2))"
terminates_sequence_appD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and xs :: "('e, 'd, 'f) com list"   and ys :: "('e, 'd, 'f) com list"   and s :: "'e" assumes "\<Gamma>\<turnstile>sequence Seq (xs @ ys) \<down> Normal s" shows "\<Gamma>\<turnstile>sequence Seq xs \<down> Normal s \<and> (\<forall>s'. \<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>sequence Seq ys \<down> s')"
source_stability: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c" assumes "wb_lens x" shows "\<exists>v. put\<^bsub>x\<^esub> \<sigma> v = \<sigma>"
adjunction_data_in_span_bicategory_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and \<eta> :: "'b arrow_of_spans_data"   and \<epsilon> :: "'b arrow_of_spans_data" shows "adjunction_data_in_span_bicategory C prj0 prj1 f g \<eta> \<epsilon> \<equiv> span_bicategory C prj0 prj1 \<and> adjunction_data_in_bicategory (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) (span_vertical_category.trg C) f g \<eta> \<epsilon>"
negGorbit_list_Cons0: fixes T :: "'a \<Rightarrow> 'w"   and as :: "'a list"   and gs :: "'g list" assumes "T ` set as \<subseteq> W" shows "HRep.negGorbit_list ((0::'g) # gs) T as = map T as # HRep.negGorbit_list gs T as"
tfr\<^sub>s\<^sub>s\<^sub>t_if_comp_tfr\<^sub>s\<^sub>s\<^sub>t: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and S :: "('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step list" assumes "comp_tfr\<^sub>s\<^sub>s\<^sub>t arity Ana \<Gamma> Pair M S" shows "tfr\<^sub>s\<^sub>s\<^sub>t S"
minimal_def: shows "MN.minimal \<equiv> MN.accessible = dfa.states Canon.DFA \<and> inj_on MN.right_lang (dfa.states Canon.DFA)"
fold_congs_2: fixes r :: "('b, 'c) simple_packet_scheme"   and r' :: "('b, 'c) simple_packet_scheme"   and v' :: "char list"   and f :: "char list \<Rightarrow> char list"   and f' :: "char list \<Rightarrow> char list" assumes "r = r'"   and "p_oiface r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "p_oiface_update f r = p_oiface_update f' r'"
integral_complex_of_real: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "CLINT x|M. complex_of_real (f x) = complex_of_real (integral\<^sup>L M f)"
assms_2: assumes "x \<in>\<^sub>\<circ> A" shows "P"
nonBVG_inc_rule: fixes \<alpha> :: "game"   and x :: "variable" assumes "\<And>I \<omega> X. \<omega> \<in> game_sem I \<alpha> X \<Longrightarrow> \<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> {x})" shows "x \<notin> BVG \<alpha>"
ih: fixes Ci :: "'a literal list \<times> nat"   and P0 :: "'a literal list \<times> nat" assumes "Ci = select_min_weight_clause P0 P" shows "weight (apfst mset Ci) = Min ((weight \<circ> apfst mset) ` set (P0 # P))"
jnf: shows "jordan_nf (diag_block_mat Cs) (triangular_to_jnf_vector A)"
assms: shows "diagram (\<cdot>\<^sub>J) (\<cdot>) D"
equal_eq: fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.equal equal" shows "equal x y = (x = y)"
geodesic_subsegment_1: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and s :: "real"   and t :: "real" assumes "geodesic_segment_between G x y"   and "0 \<le> s"   and "s \<le> t"   and "t \<le> dist x y" shows "geodesic_subsegment G x s t = geodesic_segment_param G x ` {s..t}"
INF_cong: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
csk1': shows "cs\<^bsup>\<rho>'\<^esup> k' = cs\<^bsup>\<pi>'\<^esup> k'"
clo3: shows "closedin (top_of_set (U - C)) H3"
HCond_Idem: fixes P :: "'\<alpha> hrel" shows "\<H> (\<H> P) = \<H> P"
HEndPhase0_HInv5_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv5_inner s q"   and "Inv1 s"   and "p \<noteq> q" shows "HInv5_inner s' q"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "'b mpoly" shows "mapping_of (liftPoly x xa xb) = higherPoly\<^sub>0 x xa (mapping_of xb)"
assms: shows "infer \<Gamma> (Var x) = Some \<tau>"
distributed_affine: fixes X :: "'a \<Rightarrow> real"   and f :: "real \<Rightarrow> ennreal"   and c :: "real"   and t :: "real" assumes "distributed M lborel X f"   and "c \<noteq> 0" shows "distributed M lborel (\<lambda>x. t + c * X x) (\<lambda>x. f ((x - t) / c) / ennreal \<bar>c\<bar>)"
bp_sol_finite: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and S :: "'b set"   and L :: "'b set"   and P :: "'b set set" assumes "BinPacking U w c S L"   and "BinPacking.bp U w c P" shows "finite P"
get_ancestors_si_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_ancestors_si ptr" shows "ptr |\<in>| object_ptr_kinds h"
weak_composition_with_homs_axioms: shows "weak_composition_with_homs (\<cdot>) (\<star>) some_src some_trg"
lg_lemma'': fixes x :: "nat"   and y :: "nat" assumes "\<not> Suc 0 < x" shows "rec_exec rec_lg [x, y] = lg x y"
ennreal_zero_less_divide: fixes a :: "ennreal"   and b :: "ennreal" shows "(0 < a / b) = (0 < a \<and> b < \<top>)"
notin_outstanding_non_volatile_takeWhile_lem': fixes a :: "nat"   and sb :: "'b memref list" assumes "a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb" shows "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
try_spmf_bind_spmf_lossless: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'e spmf"   and q :: "'e spmf" assumes "lossless_spmf p" shows "TRY p \<bind> f ELSE q = p \<bind> (\<lambda>x. TRY f x ELSE q)"
h_def: fixes i :: "nat" shows "h i = (case i of 0 \<Rightarrow> (While b c # cs, css, Normal s) | Suc j \<Rightarrow> f j)"
lin_red_imp_red: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" assumes "lin_red F p q" shows "red F p q"
Val: fixes v :: "val"   and s :: "char list \<Rightarrow> val option" shows "\<lbrakk>Val v\<rbrakk>s = Some v"
least_min_alg2: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e" assumes "alg B1 B2 s1 s2" shows "min_alg2 s1 s2 \<subseteq> B2"
Diff_insert0: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A" shows "A\<setminus>insert x B = A\<setminus>B"
lm101: fixes G :: "'c set"   and a :: "('d \<times> 'c set) set"   and N :: "'d set"   and aa :: "('d \<times> 'c set) set"   and x :: "real" assumes "finite G"   and "a \<in> allAllocations N G"   and "aa \<in> allAllocations N G"   and "x = real (sum (summedBidVector (pseudoAllocation a <| (N \<times> finestpart G)) N G) a) - real (sum (summedBidVector (pseudoAllocation a <| (N \<times> finestpart G)) N G) aa)" shows "x \<le> real (card G) \<and> 0 \<le> x \<and> (x = 0) = (a = aa) \<and> (aa \<noteq> a \<longrightarrow> sum (summedBidVector (pseudoAllocation a <| (N \<times> finestpart G)) N G) aa < sum (summedBidVector (pseudoAllocation a <| (N \<times> finestpart G)) N G) a)"
LabelFunSem_def: fixes a :: "('d, 'e, 'f) hierauto" shows "LabelFunSem a = (\<lambda>ST. if HA ST = a then let In_preds = IN ` Conf ST; En_preds = EN ` Events ST; Val_preds = {VAL P |P. P (Value ST)} in Some (In_preds \<union> En_preds \<union> Val_preds \<union> {TRUE}) else None)"
vrestr_empty: fixes f :: "'a \<Rightarrow> 'a" shows "f --` {} = {}"
induct: fixes P :: "('b, 'c) Combinators \<Rightarrow> bool"   and Combinators :: "('b, 'c) Combinators" assumes "P DenyAll"   and "\<And>x1 x2. P (DenyAllFromTo x1 x2)"   and "\<And>x1 x2 x3. P (AllowPortFromTo x1 x2 x3)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 \<oplus> x2)" shows "P Combinators"
sylvester_mat_sub_dim_1: fixes m :: "nat"   and n :: "nat"   and p :: "'b poly"   and q :: "'b poly" shows "dim_row (sylvester_mat_sub m n p q) = m + n"
add_ac_1: fixes a :: "'f"   and b :: "'f"   and c :: "'f" shows "a + b + c = a + (b + c)"
wf_dickson_less_p: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "wfP (punit.dickson_less_p d m)"
closest_point_in_set: fixes S :: "'b set"   and a :: "'b" assumes "closed S"   and "S \<noteq> {}" shows "closest_point S a \<in> S"
analz_increasing: fixes H :: "msg set" shows "H \<subseteq> analz H"
correct: fixes m :: "nat" shows "spmf (ped_commit.correct_game m) True = 1"