hcju/isabelleps huggingface dataset:
divisionTr3: fixes g :: "'a"   and f :: "'a" assumes "Corps S"   and "g \<in> carrier R"   and "g \<noteq> \<zero>"   and "0 < deg_n R S X g"   and "f \<in> carrier R" shows "\<exists>q\<in>carrier R. f \<plusminus> -\<^sub>a q \<cdot>\<^sub>r g = \<zero> \<or> f \<plusminus> -\<^sub>a q \<cdot>\<^sub>r g \<noteq> \<zero> \<and> deg_n R S X (f \<plusminus> -\<^sub>a q \<cdot>\<^sub>r g) < deg_n R S X g"
1: shows "local.iso (\<i>' a) \<and> \<guillemotleft>\<i>' a : a \<star> a \<Rightarrow> a\<guillemotright> \<and> (obj a \<longrightarrow> \<i>' a = \<i>[a])"
simps: fixes a1 :: "'node"   and a2 :: "'edge list"   and a3 :: "'node" shows "a1 -a2\<rightarrow>* a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> valid_node n) \<or> (\<exists>n'' as n' a n. a1 = n \<and> a2 = a # as \<and> a3 = n' \<and> n'' -as\<rightarrow>* n' \<and> valid_edge a \<and> sourcenode a = n \<and> targetnode a = n''))"
real_div_up_def: fixes p :: "nat"   and i :: "int"   and j :: "int" shows "real_div_up p i j = truncate_up (Suc p) (real_of_int i / real_of_int j)"
borel_measurable_le: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{w \<in> space M. f w \<le> g w} \<in> sets M"
submodule_subset: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "submodule R M H" shows "H \<subseteq> carrier M"
assms_4: fixes S :: "'xa set"   and \<sigma> :: "'\<sigma>" assumes "S \<subseteq> S0"   and "S \<noteq> {}"   and "\<not> c \<sigma>"   and "I S \<sigma>"   and "\<forall>x\<in>S. \<forall>y\<in>S0 - S. R y x" shows "P \<sigma>"
rel_fundef_body_nth: fixes fd1 :: "('dyn, 'd, 'e, 'f, 'opinl) Inca.instr fundef"   and fd2 :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"   and "pc < length (body fd1)" shows "body fd1 ! pc = norm_instr (body fd2 ! pc)"
pi_it_to_it: fixes I :: "('e list \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'e list \<Rightarrow> 'e list) \<Rightarrow> 'e list \<Rightarrow> 'e list" shows "proper_it (it_to_it I) (it_to_it I)"
PiE_Int: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set"   and B :: "'d \<Rightarrow> 'e set" shows "Pi\<^sub>E I A \<inter> Pi\<^sub>E I B = (\<Pi>\<^sub>E x\<in>I. A x \<inter> B x)"
base_residue: fixes s :: "complex set"   and z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex" assumes "open s"   and "z \<in> s"   and "0 < r"   and "f holomorphic_on s - {z}"   and "cball z r \<subseteq> s" shows "(f has_contour_integral complex_of_real (2 * pi) * \<i> * residue f z) (circlepath z r)"
select_convs_1: fixes aActs :: "'b"   and aTrans :: "'c"   and more :: "'d" shows "aActs \<lparr>aActs = aActs, aTrans = aTrans, \<dots> = more\<rparr> = aActs"
singular_relboundary_restrict: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "singular_relboundary p X (topspace X \<inter> S) = singular_relboundary p X S"
FW_zone_equiv_spec: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" shows "[M]\<^bsub>v,n\<^esub> = [FW M n]\<^bsub>v,n\<^esub>"
cases: fixes x :: "'g + 'h" obtains xa :: "'g" where "x = Inl xa__"   | xa :: "'h" where "x = Inr xa"
add_block_def: fixes b :: "'a set" shows "add_block b \<equiv> \<B> + {#b#}"
order_antisym: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "order r A"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> x"   and "x \<in> A"   and "y \<in> A" shows "x = y"
PartI: fixes h :: "'c \<Rightarrow> 'b"   and b :: "'c"   and A :: "'b set" assumes "h b \<in> A" shows "h b \<in> Part A h"
complete_join_adj: fixes ts :: "'a tree23s"   and n :: "nat" assumes "\<forall>t\<in>trees ts. complete t \<and> height t = n"   and "not_T ts" shows "\<forall>t\<in>trees (join_adj ts). complete t \<and> height t = Suc n"
approximating_bigstep_fun_prepend_replicate: fixes n :: "nat"   and \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and r :: "'c rule"   and rs :: "'c rule list" assumes "0 < n" shows "approximating_bigstep_fun \<gamma> p (r # rs) Undecided = approximating_bigstep_fun \<gamma> p (replicate n r @ rs) Undecided"
greedy_path_ltl_VVpstar: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and P :: "'a llist" assumes "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v0 P"   and "v0 \<in> VV p**"   and "\<not> deadend v0" shows "\<sigma>' v0 = lhd P"
minus_nonzero: fixes x :: "'a" assumes "True"   and "x \<noteq> (0::'a)" shows "- x \<noteq> (0::'a)"
Dijkstra_Invar_axioms: shows "D_invar D S"
lcoset_join2: fixes H :: "'a set"   and g :: "'a" assumes "subgroup H (G\<lparr>carrier := H__\<rparr>)"   and "g \<in> H" shows "g <#\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> H = H"
dependent_def: fixes P :: "'d set" shows "cdependent P = (\<exists>a\<in>P. a \<in> cspan (P - {a}))"
I: shows "fr_graph_impl_loc Re node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G"
mset_link: fixes t1 :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and t2 :: "('a, 'b) BinomialHeapStruc.BinomialTree" shows "BinomialHeapStruc.tree_to_multiset (BinomialHeapStruc.link t1 t2) = BinomialHeapStruc.tree_to_multiset t1 + BinomialHeapStruc.tree_to_multiset t2"
orderE: fixes b :: "'b word"   and a :: "'b word" assumes "b \<le>s a" obtains "a = signed.max a b"
has_lock_has_locks_Suc: fixes l :: "('d \<times> nat) option"   and t :: "'d" assumes "has_lock l t" shows "\<exists>m. has_locks l t = Suc m"
primeideal_iff_prime: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R) - {\<zero>\<^bsub>poly_ring R\<^esub>}" shows "primeideal (PIdl\<^bsub>poly_ring R\<^esub> p) (poly_ring R) = pprime (carrier R) p"
InvariantCllCharacterization_def: fixes Cl :: "Literal"   and Cll :: "Literal"   and C :: "Literal list"   and M :: "(Literal \<times> bool) list" shows "InvariantCllCharacterization Cl Cll C M \<equiv> set C \<noteq> {opposite Cl} \<longrightarrow> isLastAssertedLiteral Cll (removeAll Cl (oppositeLiteralList C)) (elements M)"
G_from_list_unfold: shows "G_from_list = G.from_list"
adjust_idx_rev_def: fixes i :: "nat"   and j :: "nat" shows "adjust_idx_rev i j \<equiv> if j < i then j else j - Suc 0"
assms: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
C11_var: fixes x :: "'b"   and y :: "'b" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> (y \<cdot> x\<^sup>\<star>)\<^sup>\<star>"
reads_consistent_append: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and xs :: "'b memref list"   and ys :: "'b memref list" shows "reads_consistent pending_write \<O> m (xs @ ys) = (reads_consistent pending_write \<O> m xs \<and> reads_consistent (pending_write \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs \<noteq> {}) (acquired True xs \<O>) (flush xs m) ys \<and> (outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs \<noteq> {} \<longrightarrow> outstanding_refs is_volatile_Read\<^sub>s\<^sub>b ys = {}))"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (local.dom \<mu>) \<mu>.r.tab (tab\<^sub>0 (local.dom \<mu>)) (tab\<^sub>1 (local.dom \<mu>)) (cod \<mu>) \<mu>.s.tab (tab\<^sub>0 (cod \<mu>)) (tab\<^sub>1 (cod \<mu>)) \<mu>"
pt_eqvt_fun1: fixes f :: "'a \<Rightarrow> 'c" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)" shows "(supp f = {}) = (\<forall>pi. pi \<bullet> f = f)"
Above_def: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" shows "Above r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. (a, b) \<in> r}"
assms_4: shows "mix_pmf \<beta> local.some_best local.some_worst \<succeq>[\<R>] mix_pmf \<alpha> local.some_best local.some_worst"
partition_onD3: fixes A :: "'c set"   and P :: "'c set set" assumes "partition_on A P" shows "{} \<notin> P"
mset_set_def: shows "mset_set \<equiv> folding.F add_mset {#}"
avalDirekt3_correct: fixes x :: "char list"   and v :: "int"   and ps :: "char list \<Rightarrow> int option"   and m :: "nat"   and ps' :: "char list \<Rightarrow> int option"   and a :: "aexp" assumes "(x ::= N v, ps) \<Rightarrow>\<^sub>A m \<Down> ps'"   and "paval' a ps = Some v" shows "(x ::= a, ps) \<Rightarrow>\<^sub>A m \<Down> ps'"
nn_integral_conv_infsetsum: fixes f :: "'b \<Rightarrow> real"   and A :: "'b set" assumes "f abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space A = ennreal (infsetsum f A)"
tiny_dghm_is_tm_dghm: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
PO_refinesI: fixes Tc :: "('e, 'f) TS_scheme"   and R :: "('g \<times> 'e) set"   and Ta :: "('g, 'h) TS_scheme" assumes "init Tc \<subseteq> R `` init Ta"   and "{R} TS.trans Ta, TS.trans Tc {> R}" shows "PO_refines R Ta Tc"
assms_4: shows "eint (int n) < val_Zp_dist a b"
conseq_exploit_pre: fixes P :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>({s} \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
req: shows "\<forall>x\<in>S \<inter> T. r x = x"
to_bool_def: shows "to_bool \<equiv> (\<noteq>) 0"
a1: shows "as = as\<^sub>1 @ as\<^sub>2"
assms_2: shows "det (submatrix A I J) \<noteq> (0::'a)"
rec_transfer: fixes R2 :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "((R2 ===> S) ===> rel_optionT' R1 R2 ===> S) rec_optionT rec_optionT"
lt_minus_distinct_eq_max: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.lt p \<noteq> punit.lt q" shows "punit.lt (p - q) = ordered_powerprod_lin.max (punit.lt p) (punit.lt q)"
arrE: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VxVxV.arr f" obtains "arr (fst f) \<and> VxV.arr (snd f)"
order_smult: fixes c :: "'b"   and x :: "'b"   and p :: "'b poly" assumes "c \<noteq> (0::'b)" shows "order x (smult c p) = order x p"
divide_numeral_i: fixes z :: "complex"   and n :: "num" shows "z / (numeral n * \<i>) = - (\<i> * z) / numeral n"
M1_def: shows "M1 = M.summset {#lc i \<cdot>\<^sub>v us ! i. i \<in># mset_set ({0..<length us} - {i, j})#}"
abelian_groupI: fixes R :: "('c, 'd) ring_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"   and "\<zero>\<^bsub>R\<^esub> \<in> carrier R"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y\<in>carrier R. y \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>" shows "abelian_group R"
entailsD: fixes t :: "'a \<Rightarrow> 'b option"   and P :: "'b \<Rightarrow> bool"   and k :: "'a"   and x :: "'b" assumes "t entails P"   and "t k = Some x" shows "P x"
rel_compp: fixes R :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_bset (R OO S) = rel_bset R OO rel_bset S"
signed_take_bit_int_eq_self: fixes n :: "nat"   and k :: "int" assumes "- (2 ^ n) \<le> k"   and "k < 2 ^ n" shows "signed_take_bit n k = k"
indefinite_integral_continuous_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "integrable (lebesgue_on {a..b}) f" shows "continuous_on {a..b} (\<lambda>x. integral\<^sup>L (lebesgue_on {a..x}) f)"
cases: fixes a :: "('Var, 'AExp, 'BExp) Stmt" assumes "no_await a" obtains x :: "'Var" and e :: "'AExp" where "a = x \<leftarrow> e"   | c1 :: "('Var, 'AExp, 'BExp) Stmt" and c2 :: "('Var, 'AExp, 'BExp) Stmt" where "a = c1__ ;; c2__" and "no_await c1__" and "no_await c2__"   | c1 :: "('Var, 'AExp, 'BExp) Stmt" and c2 :: "('Var, 'AExp, 'BExp) Stmt" and b :: "'BExp" where "a = Stmt.If b__ c1 c2" and "no_await c1" and "no_await c2"   | c :: "('Var, 'AExp, 'BExp) Stmt" and b :: "'BExp" where "a = While b ca__" and "no_await ca__"   | "a = Skip"   | "a = Stop"   | c :: "('Var, 'AExp, 'BExp) Stmt" and m :: "'Var ModeUpd" where "a = c@[m]" and "no_await c"
UP_smult_closed: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "p \<in> carrier P" shows "a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"
induct: fixes P :: "val \<Rightarrow> bool"   and val :: "val" assumes "P Unit"   and "P Null"   and "\<And>x. P (Bool x)"   and "\<And>x. P (Intg x)"   and "\<And>x. P (Addr x)" shows "P val"
mat_kernel_dim_mult_eq_right: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nc nc"   and "C \<in> carrier_mat nc nc"   and "B * C = 1\<^sub>m nc" shows "vectorspace.dim class_ring (module_vec TYPE('b) nc\<lparr>carrier := mat_kernel (A * B)\<rparr>) = vectorspace.dim class_ring (module_vec TYPE('b) nc\<lparr>carrier := mat_kernel A\<rparr>)"
a0_less_int_conv: fixes n :: "int" shows "(0 < ant n) = (0 < n)"
intros_8: fixes v :: "Syntax.term" assumes "value v" shows "value (Roll v)"
WT_callee_full: fixes callee :: "'a \<Rightarrow> ('b \<times> 'c) spmf" shows "\<I>_full \<turnstile>c callee \<surd>"
below_monicI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. f x \<sqsubseteq> f y \<Longrightarrow> x \<sqsubseteq> y" shows "below_monic f"
additive_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "Transformers.additive t \<equiv> \<forall>P Q. sound P \<and> sound Q \<longrightarrow> t (\<lambda>s. P s + Q s) = (\<lambda>s. t P s + t Q s)"
seq_assign_elim: fixes x :: "var\<^sub>C"   and e :: "aexp\<^sub>C"   and c :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "eval_abv\<^sub>C \<langle>(x \<leftarrow> e) ;; c, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C" shows "c' = Stop ;; c \<and> mds' = mds \<and> mem' = mem(x := ev\<^sub>A\<^sub>C mem e)"
relCond: shows "\<forall>(P, Q)\<in>indRelR. Pred (P, Q)"
comparator_of_def: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "ko.comparator_of ko x y = (if ko.lt ko x y then Lt else if x = y then Eq else Gt)"
insertCI: fixes a :: "'a"   and B :: "'a set"   and b :: "'a" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> b \<triangleright> B"
inject: fixes x2 :: "'c"   and y2 :: "'c" shows "(Some x2 = Some y2) = (x2 = y2)"
simps: fixes P :: "'d mat"   and A :: "'d mat"   and Q :: "'d mat"   and P' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and A' :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and Q' :: "(('d, 'f) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "HMA_M3 (P, A, Q) (P', A', Q') = (Mod_Type_Connect.HMA_M P P' \<and> Mod_Type_Connect.HMA_M A A' \<and> Mod_Type_Connect.HMA_M Q Q')"
simps: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = run \<delta> q\<^sub>0 (suffix x w) (n - x)"
cong_simp: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "A = B"   and "\<And>x. x \<in> B =simp=> g x = h x" shows "sum g A = sum h B"
null_sets_def: fixes M :: "'c measure" shows "null_sets M = {N \<in> sets M. emeasure M N = 0}"
assms: shows "comp_tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> M"
of_nat_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (numeral n) = numeral.numeral one plus n"
assms_2: shows "(v2, w, v3) \<in> E"
hom_paired: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and G :: "('b, 'e) monoid_scheme"   and H :: "('c, 'f) monoid_scheme"   and K :: "('d, 'g) monoid_scheme" shows "((\<lambda>x. (f x, g x)) \<in> hom G (H \<times>\<times> K)) = (f \<in> hom G H \<and> g \<in> hom G K)"
le_less: fixes y :: "('d, 'e) Term.term"   and x :: "('d, 'e) Term.term" shows "y \<unrhd> x = (y \<rhd> x \<or> x = y)"
homeomorphism_compact: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and t :: "'b set" assumes "compact s"   and "continuous_on s f"   and "f ` s = t"   and "inj_on f s" shows "\<exists>g. homeomorphism s t f g"
count_mset_set_if: fixes A :: "'a set"   and a :: "'a" shows "count (mset_set A) a = (if a \<in> A \<and> finite A then 1 else 0)"
fmran_def: shows "fmran \<equiv> map_fun fmlookup Abs_fset ran"
word_upto_alt: fixes a :: "'a word"   and b :: "'a word" assumes "a \<le> b" shows "word_upto a b = map word_of_nat [unat a..<Suc (unat b)]"
right_idem: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "linordered_monoid.min (linordered_monoid.min a b) b = linordered_monoid.min a b"
chain_boundary_of: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" shows "chain_boundary p (frag_of f) = (if p = 0 then 0 else \<Sum>k\<le>p. frag_cmul ((- 1) ^ k) (frag_of (singular_face p k f)))"
count_eq_zero_iff: fixes M :: "'d multiset"   and x :: "'d" shows "(count M x = 0) = (x \<notin># M)"
axioms_3: fixes s :: "(real \<times> real) set"   and i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain_typeI :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and two_chain_typeII :: "(real \<times> real \<Rightarrow> real \<times> real) set" assumes "green_typeI_typeII_chain s i j F two_chain_typeI two_chain_typeII" shows "green_typeII_chain i j F two_chain_typeII s"
short_exact_sequence_iff: fixes A :: "'b monoid"   and B :: "'b monoid"   and C :: "'b monoid"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" shows "short_exact_sequence A B C f g = (exact_seq ([A, B, C], [f, g]) \<and> f \<in> epi B A \<and> g \<in> mon C B)"
c1: shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. p ((1 - fst x) * snd x + fst x))"
nb\<^sub>3: fixes k :: "nat"   and op\<^sub>1 :: "'a strips_operator"   and op\<^sub>2 :: "'a strips_operator" assumes "k < length \<pi>"   and "op\<^sub>1 \<in> set (\<pi> ! k)"   and "index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2"   and "are_operators_interfering op\<^sub>1 op\<^sub>2" shows "op\<^sub>2 \<notin> set (\<pi> ! k)"
inject_5: fixes x5 :: "'a MFOTL.formula"   and y5 :: "'a MFOTL.formula" shows "(formula.Exists x5 = formula.Exists y5) = (x5 = y5)"
obeh_pi_translation: fixes pi :: "'e \<Rightarrow> 'e"   and S :: "('f, 'e) spec" shows "map pi ` obeh S = obeh (S\<lparr>obs := pi \<circ> obs S\<rparr>)"
locally_connected: fixes S :: "'b set" shows "locally connected S = (\<forall>v x. openin (top_of_set S) v \<and> x \<in> v \<longrightarrow> (\<exists>u. openin (top_of_set S) u \<and> connected u \<and> x \<in> u \<and> u \<subseteq> v))"
span_image: fixes S :: "'b set" shows "l.vs2.span (f ` S) = f ` local.span S"
runit_in_hom_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>PB.runit f : f \<star> src f \<Rightarrow> f\<guillemotright>"
strict_trans: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a \<prec>\<^sub>t b"   and "b \<prec>\<^sub>t c" shows "a \<prec>\<^sub>t c"
DropToIndex: fixes a :: "'a"   and l :: "nat"   and liste :: "'a list" assumes "a \<in> set (drop l liste)" shows "\<exists>i\<ge>l. i < length liste \<and> a = liste ! i"
distinct_87: shows "LessThan \<noteq> ShiftRightZeros"
some_third_chamberD_facet: fixes C :: "'a set"   and z :: "'a set"   and D :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "z \<lhd> some_third_chamber C D z"
dghm_dag_Rel_ObjMap_app: fixes x :: "V"   and \<alpha> :: "V" assumes "x \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ObjMap\<rparr>\<lparr>x\<rparr> = x"
sturm_seq_sturm_squarefree': fixes p :: "real poly" assumes "p \<noteq> 0" shows "sturm_seq (sturm_squarefree' p) (p div gcd p (pderiv p))"
ereal_of_enat_pushout_3: fixes m :: "enat"   and n :: "enat" shows "ereal_of_enat m * ereal_of_enat n = ereal_of_enat (m * n)"
prems: shows "\<forall>x\<in>fv t \<union> fv (Fun f S). (\<exists>y. \<delta> x = Var y) \<or> (\<exists>f. \<delta> x = Fun f [] \<and> P f = [])"
take_similar'_similar: fixes n :: "nat"   and x :: "Value"   and y :: "CValue" assumes "x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> y" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>x \<triangleleft>\<triangleright> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>y"
assms_10: shows "X = {(t, genericJoin J (newQuery J Q_J_pos (I, t)) (newQuery J Q_J_neg (I, t))) |t. t \<in> R_I}"
simps_2: fixes p :: "pat"   and q :: "pat" shows "pat_vars \<langle>\<langle>p, q\<rangle>\<rangle> = pat_vars q @ pat_vars p"
strict_from_inj_image': fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "inj_on f {0..<n}" shows "strict_from_inj n f ` {0..<n} = f ` {0..<n}"
grid_split1: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds' :: "nat set"   and ds :: "nat set" assumes "p \<in> grid b (ds' \<union> ds)"   and "ds \<inter> ds' = {}" shows "\<exists>!x. x \<in> grid b ds \<and> p \<in> grid x ds'"
sqrt_def: fixes x :: "int" shows "sqrt x \<equiv> SOME r'. r' * r' \<le> x \<and> x < (r' + 1) * (r' + 1)"
Greatest_is_minus_1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "\<forall>a. less_eq a (uminus one)"
wf_ordertype_eqpoll: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wf r"   and "total_on A r"   and "small A" shows "elts (ordertype A r) \<approx> A"
hcomp_isomorphic_ide: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "f \<cong>\<^sub>B g"   and "B.ide h"   and "src f = trg h" shows "f \<star> h \<cong>\<^sub>B g \<star> h"
ext_inject: fixes s_inst :: "t_context list"   and s_funcs :: "tf list"   and s_tab :: "nat list"   and s_mem :: "nat list"   and s_globs :: "tg list"   and more :: "'a"   and s_inst' :: "t_context list"   and s_funcs' :: "tf list"   and s_tab' :: "nat list"   and s_mem' :: "nat list"   and s_globs' :: "tg list"   and more' :: "'a" shows "(\<lparr>s_inst = s_inst, s_funcs = s_funcs, s_tab = s_tab, s_mem = s_mem, s_globs = s_globs, \<dots> = more\<rparr> = \<lparr>s_inst = s_inst', s_funcs = s_funcs', s_tab = s_tab', s_mem = s_mem', s_globs = s_globs', \<dots> = more'\<rparr>) = (s_inst = s_inst' \<and> s_funcs = s_funcs' \<and> s_tab = s_tab' \<and> s_mem = s_mem' \<and> s_globs = s_globs' \<and> more = more')"
l: shows "l ` Vs \<subseteq> {x. (0::'a) < x}"