hcju/isabelleps huggingface dataset:
distinct_1: fixes x2 :: "t" shows "TAny \<noteq> TSome x2"
cong3_preserves_out: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet A B C"   and "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'" shows "Tarski_neutral_dimensionless.Out Bet A' B' C'"
sg6: shows "cycleLength (nC k) = cycleLength (nC j)"
true_clss_def: fixes I :: "'b set"   and CC :: "'b literal multiset set" shows "(I \<TTurnstile>s CC) = (\<forall>C\<in>CC. I \<TTurnstile> C)"
state_and_event_determine_next: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and c'' :: "('a, 'b, 'c) configuration" assumes "c \<turnstile> ev \<mapsto> c'"   and "c \<turnstile> ev \<mapsto> c''" shows "c' = c''"
create_def: shows "create = update_next \<bind> (\<lambda>_. State_Monad.get)"
negative_resolvent_is_model_res: shows "less_restrictive ordered_negative_resolvent (ordered_model_resolvent UNIV)"
prob_le_1: fixes A :: "'a set" shows "prob A \<le> 1"
removeShadowRules1_alternative_def: fixes p :: "('b, 'c) Combinators list" shows "removeShadowRules1_alternative p = rev (removeShadowRules1_alternative_rev (rev p))"
hom: fixes f :: "complex \<Rightarrow> 'a" assumes "continuous_on (sphere 0 1) f"   and "f ` sphere 0 1 \<subseteq> S" shows "\<exists>a. homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f (\<lambda>x. a)"
borel_measurable_vimage: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and x :: "'c" assumes "f \<in> borel_measurable M" shows "f -` {x} \<inter> space M \<in> sets M"
is_sg_def: shows "is_sg TYPE('b) = of_prop OFCLASS('b, semigroup_add_class)"
distinct_173: fixes x91 :: "expr"   and x92 :: "char list"   and x93 :: "char list list"   and x94 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "x91\<bullet>x92{x93} := x94 \<noteq> x121;; x122"
distinct_6: fixes x41 :: "'d regex"   and x42 :: "'d regex"   and x1 :: "nat" shows "Times x41 x42 \<noteq> Skip x1"
continuous_map_closedin: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map X Y f = ((\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>C. closedin Y C \<longrightarrow> closedin X {x \<in> topspace X. f x \<in> C}))"
mult_insert: fixes x :: "V"   and y :: "V"   and z :: "V" shows "x * vinsert y z = x * z \<squnion> lift (x * y) x"
FL_bisimilar_def: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state" shows "P \<sim>\<cdot>[F] Q \<equiv> \<exists>R. is_L_bisimulation R \<and> R F P Q"
restate_tarski2: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0" shows "changes_R_smods p (pderiv p * q) = int (card {x. poly p x = 0 \<and> 0 < poly q x}) - int (card {x. poly p x = 0 \<and> poly q x < 0})"
pre: shows "\<forall>l s. I l s \<and> \<not> bval b s \<longrightarrow> Q l s \<and> 1 \<le> E s \<and> S s = s on \<Union> (funStar SS ` (LVE \<union> LVQ))"
domA_fv_subset: fixes \<Gamma> :: "('a \<times> 'b) list" shows "domA \<Gamma> \<subseteq> fv \<Gamma>"
af_opt_unfold: shows "\<up>af\<^sub>\<UU> = (\<lambda>\<phi> \<nu>. \<up>Unf (\<up>step \<phi> \<nu>))"
C: fixes Da :: "'c pmf"   and Db :: "'d pmf" shows "set_pmf (Proj1_pmf (Sum_pmf (5 / 10) Da Db)) = set_pmf Da"
power_less_imp_less_base: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat"   and b :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less (power.power one times a n) (power.power one times b n)"   and "less_eq zero b" shows "less a b"
emb_step_at: shows "emb_step_at p dir.Left t = s"
qtable2: shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length auxlist) \<psi>) rel2"
finite2: shows "finite ((\<lambda>(y, f). f(x := y, inv f y := x)) ` ((S - {x}) \<times> derangements (S - {x})))"
wf_measure: fixes f :: "'b \<Rightarrow> nat" shows "wf (Wellfounded.measure f)"
msteps0_msteps0_stateless: fixes w :: "((String.literal \<times> event_data list) set \<times> nat) list"   and st :: "('msaux, 'muaux) mstate" shows "fst (msteps0 w st) = msteps0_stateless w st"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> 'a"   and f5 :: "'a"   and agent :: "agent" shows "P (case agent of CA x \<Rightarrow> f1 x | Cardholder x \<Rightarrow> f2 x | Merchant x \<Rightarrow> f3 x | PG x \<Rightarrow> f4 x | Spy \<Rightarrow> f5) = ((\<forall>x1. agent = CA x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. agent = Cardholder x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. agent = Merchant x3 \<longrightarrow> P (f3 x3)) \<and> (\<forall>x4. agent = PG x4 \<longrightarrow> P (f4 x4)) \<and> (agent = Spy \<longrightarrow> P f5))"
evals_cases_1: fixes P :: "(char list list \<times> char list exp) prog"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>[],s\<rangle> [\<Rightarrow>] \<langle>e',s'\<rangle>" obtains "e' = []" and "s' = s"
fref_def: fixes P :: "'h \<Rightarrow> bool"   and R :: "('f \<times> 'h) set"   and S :: "('g \<times> 'i) set" shows "[P]\<^sub>f R \<rightarrow> S \<equiv> {(f, g). \<forall>x y. P y \<and> (x, y) \<in> R \<longrightarrow> (f x, g y) \<in> S}"
absCase_Abs_swap: fixes xs :: "'varSort"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and f :: "'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and x :: "'var" assumes "isInBar (xs, s)"   and "wls s X"   and "compatAbsSwap xs s f" shows "absCase xs s f (Abs xs x X) = f x X"
integral_map_pmf: fixes g :: "'d \<Rightarrow> 'c"   and p :: "'d pmf"   and f :: "'c \<Rightarrow> 'b" shows "measure_pmf.expectation (map_pmf g p) f = measure_pmf.expectation p (\<lambda>x. f (g x))"
pm_of_poly_minus: fixes x :: "'a"   and p :: "'b poly"   and q :: "'b poly" shows "pm_of_poly x (p - q) = pm_of_poly x p - pm_of_poly x q"
a1: shows "\<Gamma> \<turnstile> s is t : T"
ag_pOp_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus> y \<in> carrier A"
as_needed_asses_submap_exec_viii: fixes f :: "('c, 'd) fmap"   and g :: "('c, 'd) fmap" assumes "f \<subseteq>\<^sub>f g" shows "\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v"
well_founded_axioms: shows "well_founded UNIV ((\<sqsubset>) \<restriction> A)"
lvl_adjust: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree" assumes "pre_adjust \<langle>l, (a, lv), r\<rangle>" shows "lv = lvl (adjust \<langle>l, (a, lv), r\<rangle>) \<or> lv = lvl (adjust \<langle>l, (a, lv), r\<rangle>) + 1"
meas: fixes D :: "'a set" assumes "D \<in> \<D>" shows "D \<in> fmeasurable M"
itypeI: fixes c :: "'c"   and I :: "'d itself" shows "c ::\<^sub>i I"
c_def: shows "c \<equiv> (3 * real d + 1) / real (2 * (2 * d + 1))"
pivotandupdate_tableau_normalized: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "PivotAndUpdate eq_idx_for_lvar pivot_and_update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "\<triangle> (\<T> (pivot_and_update x\<^sub>i x\<^sub>j c s))"
WTrtBinOp: 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 E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e\<^sub>1 :: "expr"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "expr"   and T\<^sub>2 :: "ty"   and T :: "ty"   and bop :: "bop" assumes "P,E,h \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E,h \<turnstile> e\<^sub>2 : T\<^sub>2"   and "case bop of Eq \<Rightarrow> T = Boolean | Add \<Rightarrow> T\<^sub>1 = Integer \<and> T\<^sub>2 = Integer \<and> T = Integer" shows "P,E,h \<turnstile> e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2 : T"
measurable_embed_measure2': fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "inj_on f (space M)" shows "f \<in> M \<rightarrow>\<^sub>M embed_measure M f"
DN_UP: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "DN (UP \<mu>) = \<mu>"
ccINF_superset_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "B \<subseteq> A"   and "\<And>x. x \<in> B \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
Un_commute: fixes A :: "'d set"   and B :: "'d set" shows "A \<union> B = B \<union> A"
powerset: fixes x :: "hf" shows "\<exists>z. \<forall>u. (u \<^bold>\<in> z) = (u \<le> x)"
Dg: shows "\<forall>\<^sub>F x in at_top. (g has_real_derivative g' x) (at x)"
assms_1: shows "P \<parallel> !P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
of_hypnat_le_0_iff: fixes m :: "nat star" shows "(of_hypnat m \<le> 0) = (m = 0)"
llength_simps_1: shows "LList2.llength LNil = 0"
insert_before_wf_is_l_insert_before_wf: shows "l_insert_before_wf Shadow_DOM.heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.insert_before Shadow_DOM.get_child_nodes"
exhaust_sel: fixes strand_label :: "'d strand_label" obtains "strand_label = ln the_LabelN strand_label"   | "strand_label = \<star>"
extremal: fixes P :: "indi \<Rightarrow> alt \<Rightarrow> real"   and b :: "alt" assumes "Extreme P b" shows "extreme (F P) b"
adjacent_sym: fixes x :: "'c set"   and y :: "'c set" assumes "x \<sim> y" shows "y \<sim> x"
sum_mset_replicate_mset: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum_mset plus zero (replicate_mset n a) = times (semiring_1.of_nat one plus zero n) a"
mcont_finite_chains: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and lub :: "'c set \<Rightarrow> 'c"   and lub' :: "'d set \<Rightarrow> 'd" assumes "finite_chains ord"   and "monotone ord ord' f"   and "class.ccpo lub ord (mk_less ord)"   and "class.ccpo lub' ord' (mk_less ord')" shows "mcont lub ord lub' ord' f"
less_asym: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a"   and P :: "bool" assumes "ko.lt ko x y"   and "\<not> P \<Longrightarrow> ko.lt ko y x" shows "P"
of_nat_less_imp_less: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)" shows "m < n"
subcring_axioms_def: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" shows "subcring_axioms H R \<equiv> \<forall>h1 h2. h1 \<in> H \<longrightarrow> h2 \<in> H \<longrightarrow> h1 \<otimes>\<^bsub>R\<^esub> h2 = h2 \<otimes>\<^bsub>R\<^esub> h1"
length_upto: fixes i :: "int"   and j :: "int" shows "length [i..j] = nat (j - i + 1)"
valid_Tree\<^sub>\<alpha>_Pred\<^sub>\<alpha>: fixes P :: "'state"   and \<phi> :: "'pred" shows "valid_Tree\<^sub>\<alpha> P (Pred\<^sub>\<alpha> \<phi>) = P \<turnstile> \<phi>"
Left_CosetE: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "A \<in> contained.additive.Partition"   and "\<And>a. a \<in> R \<Longrightarrow> P (additive.kernel.Left_Coset a I)" shows "P A"
quotient_axioms: shows "Quotient_Type.quotient eq Rep Abs"
add_minus_zero_rel: fixes a :: "('a \<times> 'a) set" assumes "valid_frac a" shows "add_rel a (uminus_rel a) = zero_rel"
list_split_two_elems: fixes xs :: "'c list"   and x :: "'c"   and y :: "'c" assumes "distinct xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "x \<noteq> y" shows "\<exists>pre mid suf. xs = pre @ x # mid @ y # suf \<or> xs = pre @ y # mid @ x # suf"
that: fixes c :: "'b" assumes "\<And>x. x \<in> S \<Longrightarrow> f x = c" shows "thesis"
L1: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map g (map f xs) = map (g \<circ> f) xs"
Mn_free_imp_total: fixes f :: "recf" assumes "wellf f"   and "Mn_free f" shows "Partial_Recursive.total f"
scast_ucast_4: fixes a :: "'b word" assumes "is_up UCAST('b \<rightarrow> 'c)"   and "is_down UCAST('c \<rightarrow> 'd)" shows "SCAST('c \<rightarrow> 'd) (UCAST('b \<rightarrow> 'c) a) = UCAST('b \<rightarrow> 'd) a"
inj_on_finite: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "finite A"
h: shows "h \<in> j ` S \<rightarrow> carrier G"
wt_jvm_progD: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "\<exists>wt. wf_prog wt P"
allUse_defNode_in_\<alpha>n: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and m :: "'node" assumes "v \<in> braun_ssa.allUses g m"   and "m \<in> set (\<alpha>n g)" shows "defNode v \<in> set (\<alpha>n g)"
full_abstraction_wrt_preorders_impl_indRelSTEQ_to_SRel_and_TRel_3: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "trans TRel"   and "refl SRel"   and "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q" shows "\<forall>SP TQ. SP \<in>S P \<and> TQ \<in>T Q \<longrightarrow> (\<lbrakk>SP\<rbrakk>, TQ) \<in> TRel"
join_commute: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
replicate_elem: fixes i :: "nat"   and n :: "nat"   and x :: "'d" assumes "i < n" shows "replicate n x ! i = x"
6: shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>g \<rightarrow> w\<^sub>g'\<guillemotright> \<and> \<beta>\<^sub>g = tab\<^sub>1 g \<star> \<gamma> \<and> \<theta>\<^sub>g = \<theta>\<^sub>g' \<cdot> (tab\<^sub>0 g \<star> \<gamma>)"
total_onI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r" shows "total_on A r"
floorlog_eq_zero_iff: fixes b :: "nat"   and x :: "nat" shows "(floorlog b x = 0) = (b \<le> 1 \<or> x \<le> 0)"
sumE: fixes s :: "'e + 'f" obtains x :: "'e" where "s = pure x"   | y :: "'f" where "s = Inr y"
connex_axioms: shows "connex X (\<sqsubseteq>)"
f22: shows "t' \<otimes> t \<otimes> (s \<otimes> s' \<otimes> (fst x \<otimes> fst x')) = t' \<otimes> (t \<otimes> snd x \<otimes> r \<otimes> snd x' \<otimes> r')"
between_inner_transitivity: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and C :: "'p" assumes "Bet A B D"   and "Bet B C D" shows "Bet A B C"
surjective: fixes r :: "('a, 'b) strips_operator_scheme" shows "r = \<lparr>precondition_of = precondition_of r, add_effects_of = add_effects_of r, delete_effects_of = delete_effects_of r, \<dots> = strips_operator.more r\<rparr>"
simps_8: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" shows "((w, I) \<Turnstile> FEXISTS \<phi>) = (\<exists>P\<subseteq>{0..length w - 1}. (w, Inr P # I) \<Turnstile> \<phi>)"
qf3_even: fixes a :: "int"   and b :: "int" assumes "even (a\<^sup>2 + 3 * b\<^sup>2)" shows "\<exists>B. a\<^sup>2 + 3 * b\<^sup>2 = 4 * B \<and> is_qfN B 3"
no_loops: fixes v :: "Vertices" shows "(v, v) \<notin> E\<^bsub>T2\<^esub>"
res_diff_zero_fact'': fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a k = b k" shows "(a \<ominus> b) k = 0"
sendT: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "((\<xi>, {l}send(s\<^sub>m\<^sub>s\<^sub>g) . p), send (s\<^sub>m\<^sub>s\<^sub>g \<xi>), \<xi>, p) \<in> seqp_sos \<Gamma>"
convex_real_interval_7: fixes a :: "real"   and b :: "real" shows "convex {a..<b}"
a2: shows "\<not> dt (Suc t) \<noteq> []"
OstrowskiTr5: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and s :: "'a"   and t :: "'a" assumes "vals_nonequiv K (Suc (Suc n)) vv"   and "s \<in> carrier K"   and "t \<in> carrier K"   and "0 \<le> vv (Suc 0) s \<and> 0 \<le> vv (Suc (Suc 0)) t"   and "Ostrowski_elem K (Suc n) (compose {j. j \<le> Suc n} vv (skip 1)) s"   and "Ostrowski_elem K (Suc n) (compose {j. j \<le> Suc n} vv (skip 2)) t" shows "Ostrowski_elem K (Suc (Suc n)) vv (s \<cdot>\<^sub>r t)"
length_zip: fixes xs :: "'b list"   and ys :: "'c list" shows "length (xs || ys) = min (length xs) (length ys)"
OPT_noStupid: fixes Strat :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and x :: "nat"   and l :: "nat" assumes "length Strat = length qs"   and "T\<^sub>p init qs Strat = T\<^sub>p_opt init qs"   and "init \<noteq> []"   and "x < length Strat"   and "l < length (snd (Strat ! x))" shows "Suc (snd (Strat ! x) ! l) < length init"
mm_equiv_step: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and cms\<^sub>1 :: "'a \<times> (Mode \<Rightarrow> 'b set)"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and cms\<^sub>2 :: "'a \<times> (Mode \<Rightarrow> 'b set)"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and dma :: "'b \<Rightarrow> Sec"   and cms\<^sub>1' :: "'a \<times> (Mode \<Rightarrow> 'b set)"   and mem\<^sub>1' :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "((cms\<^sub>1, mem\<^sub>1), cms\<^sub>2, mem\<^sub>2) \<in> sifum_security.mm_equiv dma eval"   and "snd cms\<^sub>1 = snd cms\<^sub>2"   and "((cms\<^sub>1, mem\<^sub>1), cms\<^sub>1', mem\<^sub>1') \<in> eval" shows "\<exists>c\<^sub>2' mem\<^sub>2'. ((cms\<^sub>2, mem\<^sub>2), (c\<^sub>2', snd cms\<^sub>1'), mem\<^sub>2') \<in> eval \<and> ((cms\<^sub>1', mem\<^sub>1'), (c\<^sub>2', snd cms\<^sub>1'), mem\<^sub>2') \<in> sifum_security.mm_equiv dma eval"
viette2: fixes a :: "real"   and b :: "real"   and c :: "real"   and \<xi>1 :: "real"   and \<xi>2 :: "real" assumes "a \<noteq> 0"   and "0 \<le> b\<^sup>2 - 4 * a * c"   and "a * \<xi>1\<^sup>2 + b * \<xi>1 + c = 0"   and "a * \<xi>2\<^sup>2 + b * \<xi>2 + c = 0"   and "\<xi>1 \<noteq> \<xi>2" shows "\<xi>1 * \<xi>2 = c / a"
map_dual_invol: shows "\<partial>\<^sub>F \<circ> \<partial>\<^sub>F = id"
of_int_mult: fixes i :: "int"   and j :: "int" shows "\<guillemotleft>i * j\<guillemotright> = \<guillemotleft>i\<guillemotright> \<otimes> \<guillemotleft>j\<guillemotright>"
subset_UNIV: fixes A :: "'b set" shows "A \<subseteq> UNIV"
of_int_1: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 1 = one"
coprime_imp_gcd_eq_1: fixes one :: "'d"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and gcd :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and lcm :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a b" shows "gcd a b = one"
diff_cancel: fixes a :: "'q" shows "a - a = (0::'q)"
UP_subring_taylor_closed': fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "a \<in> S" shows "taylor_expansion R a g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
Rep_complex_homo_coords_cases: fixes y :: "complex \<times> complex" assumes "y \<in> {v. v \<noteq> vec_zero}" obtains x :: "complex_homo_coords" where "y = Rep_complex_homo_coords x"
Red: shows "((e\<^sub>0, s\<^sub>0), e\<^sub>1, s\<^sub>1) \<in> Red P (E(V \<mapsto> T))"
of_rat_lec_sum: fixes c :: "rat le_constraint list" shows "of_rat_lec (sum_list c) = sum_list (map of_rat_lec c)"
integer_of_nat_1: shows "integer_of_nat 1 = 1"
phl_seq: fixes p :: "'a"   and x :: "'a"   and r :: "'a"   and y :: "'a"   and q :: "'a" assumes "\<lparr>p\<rparr>x\<lparr>x \<cdot> r\<rparr>"   and "\<lparr>r\<rparr>y\<lparr>y \<cdot> q\<rparr>" shows "\<lparr>p\<rparr>x \<cdot> y\<lparr>x \<cdot> y \<cdot> q\<rparr>"
assms_3: shows "f \<cdot> \<iota> = \<iota>\<^sub>1 \<cdot> (f \<otimes>\<^sub>1 f)"
ord_of_nat_inject: fixes i :: "nat"   and j :: "nat" shows "(ord_of_nat i = ord_of_nat j) = (i = j)"
lin_depE: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc"   and "module.lin_dep class_ring (module_vec TYPE('b) n) (set (cols A))"   and "distinct (cols A)" obtains v :: "'b vec" where "v \<in> carrier_vec nc" and "v \<noteq> 0\<^sub>v nc" and "A *\<^sub>v v = 0\<^sub>v n"
viterbi_ix_viterbi: fixes os :: "'t list"   and s :: "'s"   and t :: "'s" shows "viterbi_ix os s t 0 = viterbi s t os"
Init_subset_reachable: fixes F :: "'c program" shows "Init F \<subseteq> reachable F"
e_lam_intro: fixes v :: "val"   and f :: "(val \<times> val) list"   and e :: "exp"   and x :: "nat"   and \<rho> :: "(nat \<times> val) list" assumes "v = VFun f"   and "\<forall>v1 v2. (v1, v2) \<in> set f \<longrightarrow> v2 \<in> E e ((x, v1) # \<rho>)" shows "v \<in> E (ELam x e) \<rho>"
assms: shows "a_consistent (ae, a, a' # as') (\<Gamma>, Bool b, Alts e1 e2 # S)"
fst_swap: fixes x :: "'b \<times> 'a" shows "fst (prod.swap x) = snd x"
sets_restrict_UNIV: fixes M :: "'a measure" shows "sets (restrict_space M UNIV) = sets M"
distinct_17: fixes x21 :: "POPLmarkRecord.type"   and x22 :: "trm"   and x41 :: "trm"   and x42 :: "trm" shows "(\<lambda>:x21. x22) \<noteq> x41 \<bullet> x42"
tpOf_tNN: fixes T :: "'fsym trm" shows "TE.tpOf (tNN T) = Ik.tpOf T"