hcju/isabelleps huggingface dataset:
INF_inf_distrib: 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 f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "inf (Inf (f ` A)) (Inf (g ` A)) = Inf ((\<lambda>a. inf (f a) (g a)) ` A)"
gt_eq_assm: shows "dim_row (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))\<^sup>T \<le> dim_col (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))\<^sup>T"
Id_on_eqI: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" assumes "a = b"   and "a \<in> A" shows "(a, b) \<in> Id_on A"
VSLuckiest: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (VSLuckiest \<phi>) xs = eval \<phi> xs"
open_cover_from_ind_open_cover: fixes U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" assumes "is_open U"   and "open_cover_of_open_subset U it.ind_is_open U I C" shows "open_cover_of_open_subset S is_open U I C"
append_eq_first_pref_spec: fixes s :: "'c list"   and t :: "'c list"   and r :: "'c list"   and x :: "'c" assumes "s @ t = r @ [x] \<and> t \<noteq> []" shows "s \<le> r"
f1: shows "\<not> (uint a1 = 8 \<or> uint a1 = 10)"
simps: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'g) FSM"   and iseq :: "'d list"   and \<Omega> :: "('d, 'e) ATC set" shows "atc_io_reduction_on M1 M2 iseq \<Omega> = (M1 \<preceq>\<lbrakk>{iseq}\<rbrakk> M2 \<and> (\<forall>io\<in>L\<^sub>i\<^sub>n M1 {iseq}. B M1 io \<Omega> \<subseteq> B M2 io \<Omega>))"
assms_3: shows "nsqn rt dip = nsqn rt' dip"
lhopital_left: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and F :: "real filter" assumes "(f \<longlongrightarrow> 0) (at_left x)"   and "(g \<longlongrightarrow> 0) (at_left x)"   and "\<forall>\<^sub>F x in at_left x. g x \<noteq> 0"   and "\<forall>\<^sub>F x in at_left x. g' x \<noteq> 0"   and "\<forall>\<^sub>F x in at_left x. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_left x. (g has_real_derivative g' x) (at x)"   and "LIM x at_left x. f' x / g' x :> F" shows "LIM x at_left x. f x / g x :> F"
sum_SDG_call_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "n = CFG_node (sourcenode a)"   and "n' = CFG_node (targetnode a)" shows "n s-p\<rightarrow>\<^bsub>call\<^esub> n'"
ftv_ty_subst: fixes X :: "W.tvar"   and T :: "ty"   and Y :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "X \<in> set (ftv T)"   and "Y \<in> set (ftv (lookup \<theta> X))" shows "Y \<in> set (ftv (\<theta><T>))"
nonsat_potential_bound: shows "nonsat_potential f l \<le> 2 * (card V)\<^sup>2"
smc_Cod_vempty_if_Arr_vempty: assumes "\<AA>\<lparr>Arr\<rparr> = []\<^sub>\<circ>" shows "\<AA>\<lparr>Cod\<rparr> = []\<^sub>\<circ>"
ConcretenessSemantics1: fixes r :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool"   and x :: "\<omega>" assumes "Some r = d\<^sub>1 E!" shows "\<exists>w. \<omega>\<nu> x \<in> ex1 r w"
measurable_distr_prob_space2: fixes g :: "'b \<Rightarrow> 'c measure"   and L :: "'b measure"   and M :: "'c measure"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "g \<in> L \<rightarrow>\<^sub>M prob_algebra M"   and "(\<lambda>(x, y). f x y) \<in> L \<Otimes>\<^sub>M M \<rightarrow>\<^sub>M N" shows "(\<lambda>x. distr (g x) N (f x)) \<in> L \<rightarrow>\<^sub>M prob_algebra N"
STL4: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> \<box>F \<longrightarrow> \<box>G"
tetrahedral_group_def: shows "tetrahedral_group = \<lparr>carrier = complex_rotations, mult = (\<circ>), one = id\<rparr>"
Syntax_with_Connectives_False_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c" shows "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls \<equiv> Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi \<and> Syntax_with_Connectives_False_axioms var trm fmla Fvars subst fls"
terminationLess_def: fixes F0 :: "Literal list list"   and decisionVars :: "nat set" shows "terminationLess F0 decisionVars \<equiv> {(stateA, stateB). (getM stateA, getM stateB) \<in> lexLessRestricted (vars F0 \<union> decisionVars)}"
rt_fresher_def': fixes i :: "nat"   and rt\<^sub>1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt\<^sub>2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "(rt\<^sub>1 \<sqsubseteq>\<^bsub>i\<^esub> rt\<^sub>2) = (nsqn\<^sub>r (the (rt\<^sub>1 i)) < nsqn\<^sub>r (the (rt\<^sub>2 i)) \<or> nsqn\<^sub>r (the (rt\<^sub>1 i)) = nsqn\<^sub>r (the (rt\<^sub>2 i)) \<and> \<pi>\<^sub>5 (the (rt\<^sub>2 i)) \<le> \<pi>\<^sub>5 (the (rt\<^sub>1 i)))"
antecedent: shows "P z (x \<oplus> y) \<and> \<not> O z x"
ht_at_phl_while: fixes p :: "'a"   and s :: "'a"   and x :: "'a"   and w :: "'a" assumes "\<lbrace>t p \<cdot> t s\<rbrace>x\<lbrace>t p\<rbrace>" shows "\<lbrace>t p\<rbrace>(t s \<cdot> x)\<^sup>\<dagger> \<cdot> t w\<lbrace>t p \<cdot> t w\<rbrace>"
good_context_def: fixes s :: "'b sparc_state" shows "good_context s \<equiv> let traps = get_trap_set s; psr_val = cpu_reg_val PSR s; et_val = get_ET psr_val; rt_val = reset_trap_val s in if traps \<noteq> {} \<and> rt_val = False \<and> et_val = 0 then False else let s' = delayed_pool_write s in case fetch_instruction s' of Inl x \<Rightarrow> True | Inr v \<Rightarrow> case decode_instruction v of Inl x \<Rightarrow> False | Inr instr \<Rightarrow> let annul = annul_val s' in if annul = True then True else if supported_instruction (fst instr) then if fst instr = ctrl_type RETT then let curr_win_r = get_CWP (cpu_reg_val PSR s'); new_cwp_int_r = (uint curr_win_r + 1) mod NWINDOWS; wim_val_r = cpu_reg_val WIM s'; psr_val_r = cpu_reg_val PSR s'; et_val_r = get_ET psr_val_r; s_val_r = ucast (get_S psr_val_r); op_list_r = snd instr; addr_r = get_addr (snd instr) s' in if et_val_r = 1 then True else if s_val_r = 0 then False else if get_WIM_bit (nat new_cwp_int_r) wim_val_r \<noteq> 0 then False else if addr_r AND 3 \<noteq> 0 then False else True else True else False"
hadd_hinsert_right: fixes x :: "hf"   and y :: "hf"   and z :: "hf" shows "x @+ (z \<triangleleft> y) = x @+ z \<triangleleft> x @+ y"
r_swap: fixes f :: "recf"   and x :: "nat"   and y :: "nat" assumes "recfn 2 f" shows "eval (r_swap f) [x, y] = eval f [y, x]"
sees_field_fun: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and Cs :: "char list list"   and T' :: "ty" assumes "P \<turnstile> C has least F:T via Cs"   and "P \<turnstile> C has least F:T' via Cs" shows "T = T'"
fin_digraph_def: fixes G :: "('c, 'd) pre_digraph" shows "fin_digraph G \<equiv> wf_digraph G \<and> fin_digraph_axioms G"
cases: fixes x :: "'d \<times> 'b" obtains v :: "'d" and p :: "'b" where "x = (v, p)"
int_polyE: fixes p :: "'b poly" assumes "\<And>n. poly.coeff p n \<in> \<int>" obtains p' :: "int poly" where "p = of_int_poly p'"
ANDRIGHT_elim: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and C :: "ty" assumes "<a>:M \<in> ANDRIGHT (B AND C) (\<parallel><B>\<parallel>) (\<parallel><C>\<parallel>)" obtains d' :: "coname" and M' :: "trm" and e' :: "coname" and N' :: "trm" where "M = AndR <d'>.M' <e'>.N' a" and "fic (AndR <d'>.M' <e'>.N' a) a" and "<d'>:M' \<in> \<parallel><B>\<parallel>" and "<e'>:N' \<in> \<parallel><C>\<parallel>"
coeff_R_p_eq_q: shows "coeff P p (deg R r) = \<ominus> coeff P q (deg R r)"
td_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" shows "td PROB \<equiv> Sup {card (traversed_states (fst p) (snd p)) - 1 |p. fst p \<in> valid_states PROB \<and> snd p \<in> valid_plans PROB}"
simps_1: fixes committed :: "nat set"   and justifying_exec :: "('b \<times> ('a, 'b) obs_event action) llist"   and justifying_ws :: "nat \<Rightarrow> nat"   and more :: "'c" shows "committed \<lparr>committed = committed, justifying_exec = justifying_exec, justifying_ws = justifying_ws, \<dots> = more\<rparr> = committed"
is_ground_cls_union: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "is_ground_cls (C + D) = (is_ground_cls C \<and> is_ground_cls D)"
IdI: fixes a :: "'a" shows "(a, a) \<in> Id"
completeness: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<Turnstile>\<^sub>3\<^sub>' {P} c { Q}" shows "\<turnstile>\<^sub>3\<^sub>b {P} c { Q \<and>* (\<lambda>s. True)}"
length_splice: fixes xs :: "'g list"   and ys :: "'g list" shows "length (splice xs ys) = length xs + length ys"
bind_def: shows "bind = COERCE(udom\<cdot>'d \<rightarrow> (udom \<rightarrow> udom\<cdot>'d) \<rightarrow> udom\<cdot>'d, 'c\<cdot>'d \<rightarrow> ('c \<rightarrow> 'e\<cdot>'d) \<rightarrow> 'e\<cdot>'d)\<cdot>bindU"
preserves_ide: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.ide a" shows "S.ide (S.trg a)"
aargh: shows "Not \<circ> is_volatile_Write\<^sub>s\<^sub>b = (\<lambda>a. \<not> is_volatile_Write\<^sub>s\<^sub>b a)"
Method_def: fixes P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D \<equiv> \<exists>Mm. P \<turnstile> C sees_methods Mm \<and> Mm M = \<lfloor>((b, Ts, T, m), D)\<rfloor>"
int_int_eq: fixes m :: "nat"   and n :: "nat" shows "(int m = int n) = (m = n)"
epath_not_direct: fixes E :: "'b uprod set"   and u :: "'b"   and p :: "'b uprod list"   and v :: "'b" assumes "epath E u p v"   and "u \<noteq> v"   and "Upair u v \<notin> E" shows "2 \<le> length p"
Diff_idemp: fixes A :: "'c set"   and B :: "'c set" shows "A - B - B = A - B"
ginitial_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme" shows "ginitial_code node_eq_impl node_hash_impl G_impl \<equiv> \<lambda>x xa. (as_singleton (igbgi_acc G_impl x), as_singleton x, as_singleton 0, Impl_Array_Hash_Map.ahm_update node_eq_impl node_hash_impl x (int 0) (snd xa), if is_Nil (gi_E G_impl x) then as_empty () else as_singleton (0, gi_E G_impl x))"
rows_with_1_def: shows "rows_with_1 = {i. \<forall>i0\<in>set (digit_encode (nths (dims Aw) (Collect even)) i). i0 < last (butlast rs)}"
atms_empty_iff_empty: fixes C :: "'a literal multiset" shows "(atms_of C = {}) = (C = {#})"
radical_sqrt_rule_subtraction: fixes x :: "real"   and y :: "real" assumes "x \<in> radical_sqrt"   and "y \<in> radical_sqrt" shows "x - y \<in> radical_sqrt"
Dijkstra_Invar_def: fixes w :: "'b \<times> 'b \<Rightarrow> enat"   and s :: "'b"   and D :: "'b \<Rightarrow> enat"   and S :: "'b set" shows "Dijkstra_Invar w s D S \<equiv> ((\<forall>u. WGraph.\<delta> w s u \<le> D u) \<and> (s \<in> S \<or> D = (\<lambda>_. \<infinity>)(s := 0) \<and> S = {})) \<and> (\<forall>u. u \<in> S \<longrightarrow> D u = WGraph.\<delta> w s u) \<and> (\<forall>v u. v \<in> S \<longrightarrow> D u \<le> WGraph.\<delta> w s v + w (v, u))"
abs_eq: fixes x :: "('a \<times> 'b) set" shows "graph_pfun x = Abs_pfun (graph_map x)"
measurable_to_stream: fixes M :: "'b measure" shows "to_stream \<in> Pi\<^sub>M UNIV (\<lambda>i. M) \<rightarrow>\<^sub>M stream_space M"
image_cong: fixes M :: "'g set"   and N :: "'g set"   and f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
dom_null: shows "B.VVV.dom B.VVV.null = B.VVV.null"
simps_2: fixes k :: "nat"   and ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "order k (Node ts t) = (k \<le> length ts \<and> length ts \<le> 2 * k \<and> Ball (set (subtrees ts)) (order k) \<and> order k t)"
dickson_grading_lcs_minus: fixes d :: "'b \<Rightarrow> nat"   and s :: "'b"   and t :: "'b" assumes "dickson_grading d" shows "d (lcs s t - s) \<le> ord_class.max (d s) (d t)"
cases: fixes x :: "'c net_state" obtains i :: "nat" and s :: "'c" and R :: "nat set" where "x = NodeS i s R"   | n1 :: "'c net_state" and n2 :: "'c net_state" where "x = SubnetS n1 n2"
sup_aci_3: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
hd_rev: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "hd (rev xs) = last xs"
valid_index_append: fixes is1 :: "nat list"   and ds1 :: "nat list"   and is2 :: "nat list"   and ds2 :: "nat list" assumes "is1 \<lhd> ds1"   and "is2 \<lhd> ds2" shows "is1 @ is2 \<lhd> ds1 @ ds2"
not_dels_preserved: fixes P :: "ast_problem"   and f :: "object atom formula"   and d :: "object atom formula list"   and M :: "object atom formula set"   and a :: "object atom formula list" assumes "wf_ast_problem P"   and "f \<notin> set d"   and "f \<in> M" shows "f \<in> apply_effect (Effect a d) M"
Re_complex_div_eq_0: fixes a :: "complex"   and b :: "complex" shows "is_imag (a / b) = is_imag (a * cnj b)"
SETobj: fixes X :: "ZF" shows "X \<in> obj\<^bsub>SET\<^esub>"
wf_residual_networkD_1: fixes x :: "'v"   and y :: "'v" assumes "RES.wf_residual_network"   and "edge (residual_network f) x y" shows "\<not> edge (residual_network f) y x"
fps_carrier: fixes L :: "('a, 'b) eq_object_scheme"   and f :: "'a \<Rightarrow> 'a" shows "fps L f \<subseteq> carrier L"
invertible_fst_Gauss_Jordan_in_ij_PA: fixes P :: "(('a, 'b) vec, 'b) vec"   and i :: "'b"   and A :: "(('a, 'c) vec, 'b) vec"   and j :: "'c" assumes "invertible P"   and "\<not> (\<forall>m\<ge>i. A $ m $ j = (0::'a))" shows "invertible (fst (Gauss_Jordan_in_ij_PA (P, A) i j))"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "\<And>x y. hom (x + y) = hom x + hom y" shows "plus_hom hom"
minus_eq_not_plus_1: fixes a :: "'a" shows "- a = NOT a + (1::'a)"
dlex_pm_zero_min: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" shows "dlex_pm 0 s"
reals_Archimedean: fixes x :: "'b" assumes "(0::'b) < x" shows "\<exists>n. inverse (of_nat (Suc n)) < x"
HierAuto_def: fixes D :: "'d data"   and F :: "('e, 'f, 'd) seqauto set"   and E :: "'f set"   and G :: "'e \<Rightarrow> ('e, 'f, 'd) seqauto set option" shows "HierAuto D F E G = (\<Union> (SAEvents ` F) \<subseteq> E \<and> MutuallyDistinct F \<and> finite F \<and> IsCompFun F G)"
list_of_oalist_OAlist_tc_update_by_fun: fixes k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and xs :: "('a, 'b) oalist_tc" shows "list_of_oalist_tc (OAlist_tc_update_by_fun k f xs) = update_by_fun_pair_tc k f (list_of_oalist_tc xs)"
intros_7: fixes s :: "'a"   and b :: "'a set"   and c2 :: "'a com"   and s' :: "'a option"   and c1 :: "'a com" assumes "s \<notin> b"   and "Sem c2 (Some s) s'" shows "Sem (Cond b c1 c2) (Some s) s'"
fmdom'_restrict_fset_precise: fixes A :: "'e fset"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmdom (fmrestrict_fset A m) = fmdom m |\<inter>| A"
le_INF_iff: fixes u :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(u \<le> \<Sqinter> (f ` A)) = (\<forall>i\<in>A. u \<le> f i)"
notSmallerTrans: fixes x :: "('b, 'c) Combinators"   and r :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators"   and z :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "x \<noteq> DenyAll"   and "r \<noteq> DenyAll"   and "singleCombinators (x # y # z)"   and "\<not> smaller y x l"   and "FWNormalisationCore.sorted (x # y # z) l"   and "r \<in> set z"   and "all_in_list (x # y # z) l" shows "\<not> smaller r x l"
identify_blocks_main: fixes i_start :: "nat"   and i_end :: "nat"   and A :: "'b mat"   and i :: "nat"   and list :: "(nat \<times> nat) list"   and k :: "nat" assumes "(i_start, i_end) \<in> set (??.Jordan_Normal_Form_Existence.identify_blocks_main A i list)"   and "\<And>i_s i_e. (i_s, i_e) \<in> set list \<Longrightarrow> i_s \<le> i_e \<and> i_e < k"   and "i \<le> k" shows "i_start \<le> i_end \<and> i_end < k"
simps: fixes p :: "nat"   and n :: "nat"   and a :: "(nat \<Rightarrow> nat) set" shows "ksimplex p n a = (\<exists>base upd s. a = s \<and> kuhn_simplex p n base upd s)"
monom_1_dvd_iff: fixes p :: "'a poly"   and n :: "nat" assumes "p \<noteq> 0" shows "(Polynomial.monom (1::'a) n dvd p) = (n \<le> Polynomial.order (0::'a) p)"
cooper: fixes p :: "fm"   and bs :: "real list" assumes "qfree p" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = Ifm bs (cooper p) \<and> qfree (cooper p)"
sup_idem: fixes x :: "'a \<times> 'b" shows "sup_prod x x = x"
homeomorphism_imp_open_map: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and U :: "'c set" assumes "homeomorphism S T f g"   and "openin (top_of_set S) U" shows "openin (top_of_set T) (f ` U)"
lt_left_comm: 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.Lt Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Lt Bet Cong B A C D"
simps: fixes a1 :: "real \<Rightarrow> real"   and a2 :: "'a"   and a3 :: "(real \<Rightarrow> real) list" shows "(a1 expands_to a2) a3 = (\<exists>F basis f. a1 = f \<and> a2 = F \<and> a3 = basis \<and> is_expansion F basis \<and> (\<forall>\<^sub>F x in at_top. eval F x = f x))"
get_child_nodes_impl: shows "get_child_nodes = a_get_child_nodes"
zoom_comp2: fixes m :: "('a, 'd) state"   and n :: "'d \<Rightarrow> ('a, 'c) state" shows "zoom x m \<bind> (\<lambda>a. zoom x (n a)) = zoom x (m \<bind> n)"
eint_ord_simps_6: fixes q :: "eint" shows "(\<infinity> < q) = False"
axioms_2: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme" assumes "g_set_xy_defs_loc ops1 ops2" shows "StdSet ops2"
card_permutations: fixes S :: "'b set"   and n :: "nat" assumes "card S = n"   and "finite S" shows "card {p. p permutes S} = fact n"
Suc: fixes v1 :: "'a list"   and v2 :: "'a list"   and w1 :: "'a list"   and w2 :: "'a list" shows "length v1 = length v2 \<and> length v1 = k \<and> length w1 = length w2 \<longrightarrow> scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)"
chamber_in_fold_A_im: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set"   and F :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"   and "SimplicialComplex.maxsimp X F"   and "F \<in> (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) \<turnstile> A" shows "F \<in> (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) \<turnstile> ChamberComplex.chamber_system A"
b: shows "nat_of_ipv4addr ((ipv4addr_of_nat (d + 256 * c + 65536 * b + 16777216 * a) >> 16) && mask 8) = b"
word_atLeastAtMost_Suc_greaterThanAtMost: fixes m :: "'b word"   and u :: "'b word" assumes "m \<noteq> - 1" shows "{m<..u} = {m + 1..u}"
fls_right_inverse: fixes f :: "'a fls"   and y :: "'a" assumes "fls_nth f (fls_subdegree f) * y = (1::'a)" shows "f * fls_right_inverse f y = 1"
BuildSeq3_combine: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s1 :: "hf"   and k1 :: "hf"   and y1 :: "hf"   and y1' :: "hf"   and y1'' :: "hf"   and s2 :: "hf"   and k2 :: "hf"   and y2 :: "hf"   and y2' :: "hf"   and y2'' :: "hf"   and y :: "hf"   and y' :: "hf"   and y'' :: "hf" assumes "BuildSeq3 B C s1 k1 y1 y1' y1''"   and "BuildSeq3 B C s2 k2 y2 y2' y2''"   and "C y y' y'' y1 y1' y1'' y2 y2' y2''" shows "BuildSeq3 B C (insf (seq_append (succ k1) s1 s2) (succ (succ (k1 @+ k2))) \<langle>y, y', y''\<rangle>) (succ (succ (k1 @+ k2))) y y' y''"
app_strict: fixes x :: "'c" shows "\<bottom> x = \<bottom>"
order1: fixes A :: "'b tensor" assumes "order A \<le> 1" shows "cprank_max1 A"
gpop_impl_def: fixes s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "gpop_impl s \<equiv> let (a, s) = s in pop_impl s \<bind> (\<lambda>s. ASSERT (a \<noteq> []) \<bind> (\<lambda>_. let a = butlast a in RETURN (a, s)))"
learn_cpE_1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_cp \<psi> U s" shows "environment \<psi> U s"
A6_DAcc_level1: shows "DAcc level1 sA6 = {sA71, sA72}"
refine_basic: fixes Tc :: "('a, 'b) TS_scheme"   and R :: "('c \<times> 'a) set"   and Ta :: "('c, 'd) TS_scheme" assumes "init Tc \<subseteq> R `` init Ta"   and "{R} TS.trans Ta, TS.trans Tc {> R}" shows "PO_refines R Ta Tc"
sort_eqv_sym: fixes cs :: "(String.literal \<times> String.literal) set"   and x :: "String.literal set"   and y :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_eqv cs x y" shows "sort_eqv cs y x"
compliant_stateful_ACS_no_side_effects': shows "\<forall>E\<subseteq>backflows (flows_state \<T>). \<forall>F\<in>get_offending_flows (get_ACS M) \<lparr>nodes = hosts \<T>, edges = flows_fix \<T> \<union> flows_state \<T> \<union> E\<rparr>. F \<subseteq> E"
lunit_commutes_with_L: fixes a :: "'a" assumes "ide a" shows "\<l>\<^sub>1[\<I>\<^sub>1 \<otimes>\<^sub>1 a] = \<I>\<^sub>1 \<otimes>\<^sub>1 \<l>\<^sub>1[a]"
sup_bot_left: fixes x :: "'c" shows "top \<sqinter> x = x"
finite_borel_measure'_axioms: shows "finite_borel_measure' M"
w1_def: shows "w1 = rconjby z (pair_relator_freeword s t)"
get_parent_child_dual: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr" obtains children :: "(_) node_ptr list" where "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" and "child \<in> set children"